giren ; and because the ratio of AB to CD is greater than the ratio of (AE to CF, that is, than A GB the ratio of) AG to CD; AB is C 10. S. greater c than AG: And AB, С F D AG are given ; therefore the remainder BG is given : And because as AE O CF, fo is AG to CD, and so is a EG to FD; a 19. s. the ratio of EG to FD 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 cale is shewn in the same way. IF there be three magnitudes, the first of which has see N. a 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. a 2. dat b 19. S. 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 magritude 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 Al AG to CF; therefore the ratio of AG to CF is given ; and CF is given, wherefore a AG is given : And because as AB to CD, so is AG с to CF, and to is b GB to FD: the ratio of GB to FD is given. And the ratio of FD to E is F given, wherefore c the ratio of GB to ,E is given, and AG is given; therefore GB the ex: cess of AB above a given magnitude AG has a given ratio to E. B D 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 mag. nitude 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 fame with the propofition. Cor. COR. 2. Also if the first has a given ratio to the second, and the excess of the third above a given magnitude has also a given ratio to the second, the fame excefs fhall have a given ratio to the first; as is evident from the gth dat. IF 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 thall 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. 1 Let AB, C, DE be three magnitudes, and let the exceffes of cach 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 ex A cess of DE above the given magnitude DC D have a given ratio to C; and because FB, GEF have each of them a given ratio to C, they 2. 9. dat. have a given ratio a to one another. But to FB, GH GE the given magnitudes AF, DG are add 6. 18. dat. ed; therefore b the whole magnitudes AB, DE have either a given ratio to one another, o. B'C' E PRO P. XXVI. IF there be three magnitudes, the exceffes of one of which above given magnitudes have given ratios to the other two magnitudes; these two thall either have a given ratio to one another, or the excess of one of them above a given magnitudie ihall have a giyen ratio to the other. Let AB, CD, EF be three magnitudes, and let GD the ex. cess 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 excels 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 10 AB, so make CG to HA; therefore the ratio of CG 10 HA is given; and CG is given, wherefore a HA is given : And because as a 2 dat. GD to AB, fo is CG to HA, and so is b CD 10 HB; the ra. b 12. s. tio of CD to HB is given : Also because KD has a given ratio to EF, as KD to EF, so makę CK LE ; H therefore the ratio of CK to LE is given ; and CK is given, wherefore LE a is given : And C because as KD to EF, fo is CK to LE, and A fo b is CD to LF; the ratio of CD to LF is given : But the ratio of CD to HB is given, K wherefore c the ratio of HB to LF is given : and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF fhall BI DI F 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. d 19. dat. " Another demonstration. Let AB, C, DE be three magnitudes, and let the excefies 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. Becaufe the excess of C above a given magnitude has a given ratio to AB; therefore a AB together with a given mag. a 14. dat: nitude has a given ratio to C: Let this given F magnitude be AF, wherefore FB has a given ratio to C: Also, because the excefs of Carovi A a given magnitude has a given ratio to DE; therefore a DE together with a given magnitude has a given ratio to C: Let, this given magnitude be DG, wherefore GE has a given B C ratio io C: And FB has a given :atio to C, therefore by the ratio 1 9. dar. of FB O GE is given : And from IB GE the given magnitudes AF DG being taken, the remainders AB. DE erher have a given crio to one another, or the excess of one of them above a given magnitude has a given ratio to the other." Bb PROP. C 19. date 19. PRO P. XXVII. IF there be three magnitudes : the excess of the first of which above a given magnitude has a given ra. tio 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 firít 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, fo makc GH to CF; therefore the ratio of GHA a 2. dat. to CF is given ; and CF is given, wherefore a GH is given ; and AG is given, wherefore G the whole AH is given : And because as GB b 19.5. to CD), fo is GH to CF, and fo is b the re-H+ F mainder HB to the remainder FD; the ratio of HB to FD is given: And the ratio of FD E is given : And AH is given ; therefore HB • Otherwise. Let AB, C, D be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a givea raco to C, and the excess of C above a given magnitude has a given ratio to D: The excess of AB above a given magniiude has a given ratic to D. Because EB has a given ratio to C, and the excets of C above a given magnitude has a gi. F d 24. dat. ven ratio to D; therefored the excess of EB above a given nagnitude has a given ratio to are given : Therefore FB the excess of AB above a given magnitude AF has a given ratio to D.” IF two two lines given in position cut one another, the See N. point or points in which they cut one another are given. -B a 4. def. 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 A always the same situation a, and therefore the point, or points in which they cut one another D have always the same situation : E And because the lines AB, CD A -B can be found a, the point, or points, in which they cut one C another, are likewise found; and therefore are given in position 4. PROP. XXIX. IF the extremities of a straight line be given in posi tion; the straight line is given in position and may nitude. Because the extremities of the straight line are given, they can be found a : let these be the points A, B, between which a 4 def. a straight line AB can be drawn b; b 1. Pofta this has an invariable position, be A B late. cause between two given points there can be drawn but one straight line: And when the straight line AB is drawn, its magnitude is at the fame time exhibited, or given : Therefore the straight line AB is given in pofition and magnitude. B b 2 PROP. |