If it be possible, let some magnitude E be the common measure. Then because AB taken from CD, as often as it can be, leaves a magnitude FD less than it self, and FD taken froin AB leaves GB, and fo forward (a) therefore at length fome magnitude GBE fffall be left, therefore E (6) measuring AB, (c) and fo CF, (b) and the whole CD, (d) shall alfo measure the residue FD, (c) consequently also AG; (d) wherefore it shall likewise mcafure the remainder GB, less than it self. Which is abfurd. PROP. III. Plate IV. Fig. 15. Two commensurable magnitudes being given AB, CD, to find out their greatest common measure FB. Take AB from CD, and the refidue ED from AB, and FB from ED, till FB meafsure ED (which will come to pass at length, (a) because by the Hyp. AB LCD) FB shall be the magnitude required. ai. 1o. b byp. C 2. ax. 10. d 3 ax. 10: 22.10. For FB (6) measures ED, (c) and so also AF; but it measures it felf too, (d) therefore likewise AB, (c) and confequently CE, (d) and so the whole CD. Wherefore FB is the common measure of AB, CD. If you affirm G to be a common measure greater than that, then G measuring AB and CD, (e) measures also CE and (f) the remainder ED, (e) and so AF; and (f) f 3. ax. 10. confequently the remainder FB, the greater the lefs. Which is abfurd. b const. C2. ax. 10. d 1. ax. 10. e 2. ax. 10. Coroll. Hence, a magnitude that measures two magnitudes, does alfo measure their greatest common meafure. PROP. IV. Fig. 16. Three commenfurable magnitudes being given, A, B, C, to find out their greatest common measure. (a) Find out D the greatest common meafure of any two A, B; (a) alfo E the greatest common measure of D and C, therefore E is the magnitude fought. (a) For it is clear, E measuring Dand C, (6) meafares the three, A, B, C. Conceive another magnitude F greater than that to measure them; (c) then F meafures D, (c) and consequently E the greatest common meafure of D, and C, the greater the less. Which is abfurd. Coroll a 3. 10. 2 ax. 10. b conftr. & C cor. 3.10. Coroll. Hence also it appears, that if a magnitude measures three magnitudes, it shall likewise measure their greatest common measure. a 3. 10. b 20. def. 7. C 22.5. (a) C being found the greatest common measure of A, B; as often as C is contained in A and B, so often is I contained in the numbers ID and E; (6) therefore C: A ::1:D; wherefore inversely A:C:: D: 1, (6) but likewife C: B: : 1. E; (c) therefore by equality A: B ::D:E::N: N. Note, The letter N only signifies number in general, and refers not to any particular space or magnitude as the other letters do, and is to be read, as A to B, so D to E, and so number to number. Which was to be demonstrated. a fch. 10. 6. b constr. c hyp. d 22. 5. c 5.ax. 7. f 20. def. 7. g const. h 1. def. 10. 26.10. D, those magnitudes A, B, shall be commenfurable. :I: What part I is of the number C, (a) that let E be of A. Therefore because E: A (6) :: 1 : C, and A: B (c) :: C: D, (d) therefore by equality shall E: B :: 1 : D. Wherefore seeing I (e) measures the number D, (f) likewife Emeasures B; but it (g) also measures A, (b) therefore AB. Which was to be demonstrated. PROP. VII. Plate IV. Fig. 17. Incommensurable magnitudes A, B, have not that proportion one to another, which number hath to number. If you affirm A: B:: N: N, (a) then AB, against the Hypothesis, PROP. VIII. Plate IV. Fig. 17. If two magnitudes A, B, have not that proportion one to another, which number hath to number, those magnitudes are incommensurable. Conceive ALB. (a) then A: B:: N: N, contrary to the Hypothesis. PROP. IX. Ả B E, 4. The Squares described upon right 1. Hyp. AB. I say that Aq: Bq :: Q:Q Aq A 1 E Eq a 5. 10. a 5. 10. b 20. 6. Bq (6) twice (c)= twice, (d) =Fq, (e) therefore Aq: cfch. 23.5. B Bq: : Eq: Fq:: Q: Q. Which was to be demonftrated. 2. Hyp. Aq: Bq:: EqFq::Q:Q. I fay ATLB. For Bq Eq B::E:F:: N: N, (k) wherefore AB. Which was to be demonstrated. 3. Hyp. AB I deny that Aq: Bq::Q:Q. For suppose Aq: Bq:: Q: Q, then AB, as is shewn before, against the Hypothesis. 4. Hyp. Not Aq: Bq:Q:Q I say that AB. For conceive A τι B, then Aq: Bq :: Q: Q, as above, against the Hypothesis. d 11.8. e 11.5. f 20.6. g byp. h11.8. isch. 23.5. k6.10. PROP. X. Plate IV. Fig. 18. C If four magnitudes are proportional (C:A::B: D) and the first C be commensurable to the second A, the third B shall be commensurable to the fourth D. And if the first C be incommensurable to the second A, also the third B shall be incommensurable to the fourth D. If CA, (a) then C: A:: N: N, (b) : : B : D, (b) therefore BD. But if CA, (c) then shall not C: A:: N: N:: 3 :D, (e) wherefore B Which was to be demonftrated. Lemma 1. D. To find out two plane numbers, not having the proportion which a square number hath to a square. Any two plane numbers not like, will fatisfy this Lemma, as those numbers which have super-particular, superbipartient, or double proportion; or any two prime numbers, See Schol. 27.8. Lemma 2. Fig. 19. To find out a line HR, to which a right-line given KM hath the proportion of two numbers given B, C. (a) Divide KM into as many equal parts as there are units in the number B, and let as many of these, as there are units in the number C, (b) make the right-line HR, it is manifest that KM: HR::B:C. Lemma 3. To find out a line D, to the square of which the square of a right-line given KM hath the proportion of two numbers given B, C. Make B: C (a): : KM: HR, and between KM and HR, (6) find a mean proportional D. Therefore KMq: Dq (c): : KM: HR():: B: C. PROP. XI. Fig. 20. To find two right-lines incommensurable to a right-line given A, one Din length only, the other E in power also. 1. Take the numbers B, C, (a) so that it be not B: C :: Q: Q, (b) and let B: C:: Aq: Dq, (c) it is plain that AD. But Aq (d). Dq. Which was to be done. 10. b 3. lem.. 10. 10. C 9. 10. d 6. 10. 2.(d) 2. (d) Make Ą: E:: E: D. I say AqEq. For A: Die):: Aq: Eq, therefore since AD, as before; (f) therefore Aq Eq. Which was to be done. PROP. XII. Plate IV. Fig. 20. Magnitudes (A,B) commensurable to the fame magnitude C, are also commensurable one to the other. Because ATLC, and CLB, (a) let A: C::M:N :: D: E, and C: B:: M:N:: F: G, (b) take three numbers H, I, K, the least in the proportion of D to E, and F to G. Now because d 13. 6. е 20. 6. f10.10. a5.10. D, 8. E, 8. b 4. 8. H, 5.1, 4. K, 6. A:C(c) :: D: E(c): : H: I, and C: B (c) ::F:G ::I: K, (d) therefore by equality, A: B::H:K:: M: N, (e) therefore AB. Which was to be demonstrated. c confir. d 22. 5. c 6.10. Schol. 12. 10. & def. 9. Hence, every right-line commensurable to a rational line is also it felf rational. And all rational right-lines def. 6. are commenfurable to one another, at least in power. Also every space commenfurable to a rational rational space space is is rational too: And all rational spaces are commenfurable one to another. But magnitudes whereof one is rational, the other irrational, are incommensurable amongst themselves. PROP. XIII. Fig. 21. If there are two magnitudes A, B, and one of them A, commensurable to a third C, but the other B incommenfurable, those magnitudos AB, are incommenfurable. Conceive BL A, then since C (a) LA, (b) therefore CB, against the Hyyothesis. ! PROP. XIV. Fig. 20. If there are two magnitudes commensurable A, B, and one of them A incommensurable to any other magnitude C, the other alfo B shall be incommensurable to the same C. Imagine BC, then for that A (a) LB, (b) there fore AC, against the Hyp. def. 7. and a byp. a hyp. b12 10. |