Con PROPOSITION XI. THEOREM X1. ONES (E À BDF & HGKIM), and cylinders (Q R BE & STKH) of the fame altitude, are to one another as their bales Hypothesis. Thesis. The cones EABDF & HGKIM, as like 1. Cone EFB : cone HMK = base EABD wise the cylinders QRBE & STKH : base HGKI. have the same altitude. II. Cylinder QRBE : cylinder STKH = base E ABD: base HGKI. DEMONSTRATION. 1. Suppofition. IN OGHIK base of cone commenti delcribe CG HIK. P. 6. B. * N ; 2. Divide the cone into pyramids (as in II. Sup. of P. 10.). 3. In the bases of the cones EFB & HMK, draw diam. EB & HK. 4. In the O E ABD base of the cone E F B, describe a polyg. v to the polyg.HbGgKLIiH, & divide it as the cone H M K. ECAUSE the cone HMK has been divided into pyramids. (Prep. 2.). If those pyramids be taken from the cone (as was done in i be fore going proposition. Arg: 13;). 1. The sum of the remaining elements will be < X. Lem B.12. Therefore, if those elements be taken from the cone H M K, & the magnitude X from 2 + X. Becau 2. The remaining pyramid HbGgKLIIM will be > Z. But those polygons inscribed in the OEABD & HGKI are r. (Prep. 4.). 3. Therefore, O AEDB: O GHIK = polyg. Cdea: polyg: SP. 2. B.12. ibg L. Cor. O AEDB: :O GHIK= cone E FB : Z. (Sup.). P. 6. B.11. 4. Consequently, pyram. Dd Ee ABCF: pyram. HbGgKLIIM = cone EFB: Z. P.II. B. 5 But the pyramid Dd Ee A a B CF is < cone E F B. Ax.8. B. 5. Therefore, the pyramid Hb Gg KLIiMis < Z. P.14. B, 5. cone HMK (the cones having the same altitude) as the cone E FB Il. Suppofition. 11. Preparation. Because P.14 4 : : P.11. B. $ ECAUSE Z is > the cone HMK. (1. Sup.). 11. The cone E F B is > X. B. 5: B. 5 But it has been demonstrated ( Arg. 10.), that the base of a cone is first cone is to a magnitude < the fecond. But X is the cone E F B. (Arg. 10.). H MK is false. HM K. (Arg. 9. & 17.). Which was to be demonstrated. i. P.10. B.12. 20. The cylin. QRBE : cyl. HSTK = base E ABD : base HGKI. P.15. B. s. Which was to be demonftrated. II. 1 B. Beeca'use } SIMILAR PROPOSITION XII. THEOR EM XII. IMILAR cones (BFE & LOM), and cylinders (Bab E & Lcd M) have to one another the triplicate ratio of that which the diameters (CD & I H) of their bases (B Y DEP & LTHMR), have. Hypothesis. Thesis. The cones BFE & LOM, likewise the I. The cone BFE is to the cone LOM in the gylinders B ab E&Lcd M, are w. triplicate ratio of C D to IH; or as CD: : IH:. the triplicate ratio of CDIH; or as DEMONSTRATION. The cone B F E is to a magnitude Z (which is < or the 1. Supposition. 1. Preparation propofition. polygon of the base of the cone LOM. CD; also the rays L N&B A. BE CAU 5 L P.11. B. Š. ECAUSE the cone L OM has been divided into pyramids. in the foregoing proposition. Arg. 1.). Lem. B.12. Therefore, if those elements be taken from the cone LOM, & the part X from the magnitude 2 + X. 2. The remainder, viz. the pyramid LTGHMSRIO will be > Z. Ax.4. B. 1. But the su cones have their axes & the diameters of their bases proportional. D.24. B.11. And the cones B FE & LOM are N. (Hyp.). 3. Consequently, CD:HI= FA: ON But, CD:HI=CA:1 N. P.15. B. 5. 4. Therefore, CA: IN = FA: ON. 5 And alternando CA:FA=IN: ON. The AFAC&ION have the VCAF = to VINO. (Prep.3). portional. (Arg. 5.). D. 1. B. 6. P. 4. B. 6. 8. Likewise, the BCA is to the ALIN. (VBAC being = VLNI). (Prep. 3.). 9. Therefore, CA: BC= IN:IL. P. 4. B. 6. But, CF:CA= 10: IN. (Arg. 7.). 10. Consequently, CF:BC=10: IL. In the ACAF & BAF, the fide C A is = to B A (D. 15. B. 1.) A F is common, & VCAF = V BA F. (Prep. 3.). u. Therefore, the base B F is = to the base CF. B. i. 12. Io like manner, L O is = to O I. But, CF:BC= 01:IL. (Arg. 10.). 13. Therefore, BF:BC=LO: IL. 14 And invertendo, BC:BF=IL: OL. P. 4. B. 5 . 15.Consequently, the three sides of the AB F C are proportional to Cor. the three sides of the ALOI. 16.From whence it follows, that those ABFC & IOL are s. P. 5. B. 6. 17.It may be demonstrated after the same manner, that all the tri angles which form the pyramid BDQF are to all the triangles which form the pyramid LHSO, each to each. P.22. B. 5. P. 7. B. 5. And as the bases of those pyramids are to polygons. (Prep. 2.). 18. The pyramid B DQF is to the pyramid LHSO. D. 9. B.11. But those pyramids being s. P. 8. B.12. 19. The pyramid B DQF: pyramid LHSO=CBS: IL.. But, CA: BC= IN:IL. (Arg. 9.). ? Cor. 20. Therefore invert. BC:CA = IL: IN. P. 4. B. S. Cor. 21. And alternando, BC:LI CA : IN. P.16. B. 5. 22. Consequently, BC: LI = CD : IH. SP.15. B. 5. 23. Therefore, three times the ratio of B C to LI is = to three times {P.11. B. 5. the ratio of C D to I H, that is, BC*: LI' =CD*:IH But CB*: IL: = pyramid BDQF : pyramid LHSO. (Arg.19). 24. Consequently, pyramid BDQF : pyramid LHSO =CD* : IÅ, P.U. B. 5: But the cone B F E:2=CD3: IH”. (Sup.). P.11. B. 25. Therefore, the pyram. BDQF : pyram. LHSO =cone BFE :2. 52 But the pyramid B D QF being < cone B E F. Ax.8. B. I, 26. The pyramid LHSO will be also < Z. P.14. B. 1. But the pyramid LHS O is > Z. (Arg. 2.). 27. Consequently, the pyram. LHSO will be < &> Z. (Arg.2. & 26). 28. Which is imposible. 29. Therefore, the supposition of 2 < the cone LOM or LTG HMSRI O is falle. |