PRO P. XIX. CDE, 30. DEF, 60. FGE, 120. FGH, 240. C, 2. D,3. E, 5. F, 3. G, 6. H, 10. Between two like folid numbers CDE, FGH, there are two mean proportional numbers DFE, FGE. And the folid CDE is to the folid FGH,in treble proportion of that which the homologous fide C has to the homologous fide F. * Whereas by the hyp. C.D:: F.G, & D.E::* 21.def.7. CDE. FGH, fall two mean proportionals DFE, Coroll. Hereby it is manifeft, that between two like folid numbers there fall two mean proportionals in the proportion of the homologous fides. PROP. XX. A, 12. C, 18. B, 27. If between two numbers D,2. E,3. F,6. G,9. A, B, there fall one mean proportional number C; thefe numbers A, B, are like plane numbers. a Take D and E the leaft in the proportion of A a 35. 7. to C, or C to B. then D measures A equally as E does C,viz.by the fame number F; balfo Dequally b 21.7. measures C, as E does B, viz. by the fame number G. Therefore DF-A, and EG B. dand con- c 9.4x.7. fequently A and B are plane numbers. But because d 16.def.7. EFc=Cc=DG, e fhall D. E :: F. G. and alter- e 19. 7. nately f21. def.7.nately D.F: E.G. f Therefore the plane numbers A and B are also like. Which was to be dem. PROP. XXI. H,2. A, 16. C, 24. D,36. B, 54. If between two numbers A, B there fall two mean proportional numbers C,D; á 2. 8. bro. 8. thofe numbers A, B are like folid numbers. in the proportion of A to C. b then E and G are like plane numbers: let the fides of this be H and P, and of that e21.def.7 K and L. c therefore H.KP.LdE. F. But E, d cor.18.8. F, G, do e equally meafure A, C, D, viz. by the fame number M. and likewife the laid numbers e 21.7. E,F,G, do equally measure the numbers C,B, D, viz. by the lame number N. f Therefore A EM f 9. ax. 7.=HPM, f and B-GN KLN; g and io A and g17.def.7.B are folid numbers. But for that C f-FM, and h17.7. DfFN, therefore fhall M. Nb: FM. FN k k 7. 5. :: C.DI:: E.F.: H.K:: P. L. m wherefore A and 1 conftr. B are like solid numbers. Which was to be dem. m21.def.7. Lemma. a 19.7. AE, BF, CG, DH, numbers (E, F, G, H,) If proportional numbers A, B, C, D, measure proportional numbers AE, BF, CG, DH, by the numbers E, F, G, H, thefe fhall be proportional. For being AEDH a BFCG, a and AD=BC, b 1.ax. 7.bthence will = AD € 9. ax. 7. BC c that is, EH-FG. a Therefore E.F:: G.H. Which was to be dem. Bq B B Coroll. d15.def.7. Hence- = xd For 1.B::B.Bq d and 1. A:: Aq A "A elem.prec. A.Aq.e theref. In like manner X d theref. Ac Ac Acc' Bq B B Aq A A -=-X and fo of the reft." PROP. Aq, B, C. PROP XXII. If three numbers Aq, B, C be 4. 8. 16. continually proportional, and the first Aq a Square, the third C Shall also be a Square.. For because Aq Ca-Bq, b thence is C= B Bq Aq a 20. 7. b ax.7. Q; But it is plain that is a number,d be- c cor.of the B Α Bq A caufe or C is a number. Theref. if three,&c. Aq PRO P. XXIII. Ac, B, C, D. 8, 12, 18, 27. If four numbers Ac, B, C, D, be continually proportional, and the first of them Ac a cube, the fourth alfo D fhall be a cube. For because Ac Da BC, b therefore D B CC; that is (because Ac Cd Bq, BC a 19. 7. Acb 7.ax. 7° and thenceC Bq B Bq Bc C = CX cor.of the prec. lem Bd 20.7. Ac Bc e 15.8. Acc D= X- B Ac But it is evident c that is a number,because Ac. or D is fuppofed a number. Therefore if four numbers, &c. PRO P. XXIV. A, 16. 24. B, 36. C,4. 6. D,9. If two numbers A,B, be in the fame proportion one to another, that a fquare number C is to a Square number D, and the firft Abe a fquare number, the fecond alfo B fhall be a fquare number. Between C and D being quare numbers, * and * 8.8. so between A and B having the fame proportion, a 11. 8. a falls one mean proportional. Therefore b being b byp c2z. 8. A is a fquare number, c B also shall be a square number. Which was to be dem. Coroll. 1. Hence, If there be two like numbers AB, CD (A.B.: C.D) and the firft AB be a fquare, *12.& 18 the fecond alfo CD fhall be a square. For AB. CD:: Aq. Cq. 8. a 12.8. b 8.8. 2. From hence it appears, that the proportion of any fquare number to any other not square, cannot poffibly be declared in two fquare numbers. Whence it cannot be Q. Q :: 1. 2, nor 1. 5: Q. Q, &c. PRO P. XXV, C,64.96. 144. D, 216. A, 8. 12. 18. B, 27. If two numbers A,B, be in the fame proportion one to another, that a cube number Cis to a cube number D, the first of them A being a cube number; the fecond В shall likewife be a cube number. a Between the cube numbers C and D, b and fo between A and B having the fame proportion, fall two mean proportionals; therefore c because A is a cube, d fhall B be a cube alfo. Which was to be dem. Coroll. 1. Hence, If there be two numbers ABC, DEF (A. B:: D. E, and B. C:: E. F;) and the firft ABC be a cube, the second DEF fhall be a cube also. *For ABC. DEF:: Ac. Dc. *12,19, 1. It is perfpicuous from hence, that the 8. proportion of any cube number to any other number not a cube cannot be found in two cube numbers. PROP. PROP. XXVI. A,20. C,30. B,45. D.4. E,6. F,9. Like plane numbers A, B, are in the Jame proportion one to another, that a square num ber is in to a fquare number. Between A and B a falls one mean propor-a 18.8. tional number C; b take three numbers D,E,F,b 2.8. the leaft in the proportion of A to C. the extremes D, F, b fhall be fquare numbers. But of equality A. Bc: D.F. therefore A, B :: Q.Q·c 14. 7. Which was to be dem. PROP. XXVII. A,16. C,24. D,36. B,54. Like folid numbers A, B, are in the fame proportion one to ano ther, that a cube number is in to a cube number. a Between A and B fall two mean proportioa 19.8. nal numbers, namely C and D: take four b 2.8. numbers E, F, G, H the leaft in the proportion of A to C; b the extremes E, H, are cube numbers. But A. B c :: E. H:: C. C. Which was to be dem. Schol. € 14. 7. 1. From hence is inferred, that no numbers in See Cla proportion fuperparticular, fuperbipartient, or vius. double, or any other manifold proportion not denominated from a fquare. number, are like plane numbers. 2. Likewise, that neither any two prime numbers, nor any two numbers prime one to another, not being fquares, can be like plane numbers. The End of the eighth Book. M THE |