one and the same Pyramid. с mids ACBF, ACDF, DFEC, into which the Prifm is divided, are equal to each other. Q. E. D. B с D COROL Hence, any Pyramid is the third part of a Prism F of the fame Base and Altitude or any Prism is the triple of a Pyramid of the fame Base and Altitude. E For fuppofe the Polygoneous Prifm ABCDEG HIKF to be divided into Triangular Prifms, and the Pyramid ABCDEH into Triangular Pyramids; then each part of the Prifm fhall be the Triple of each part of a the Pyramid and confequently the whole. Prifm ABCDEGHIKF fhall be the Triple e 1. 5. of the whole Pyramid ABCDEH. Q. E. D. PROP. 7.12. PROP. VIII. Similar Pyramids ABCD, EFGH, having Triangular Bafes ABC, EFG, are in the Triplicate proportion of the Homologous Sides AC, EG. 27. 11. 9 def.II. 28. 11. ກາ b Compleat the Parallelepipedons ABICDM KL, EFNGHQOP; which are Similar and Sextuple of the Pyramids ABCD, EFGH; and fo are in the fame Ratio to each other with them. That is, in the Triplicate Ratio 33.1 of the Homologous Sides. Q. E. D. 7.12. d 15.5. d CORO L. Hence, likewife fimilar Polygoneous Pyramids are to one another in the triplicate Proportion of their Homologous Sides, as may be cafily demonftrated by dividing these Polygoneous Pyramids into Triangular ones. PROP. IX. The Bafes and Altitudes of equal Pyramids ABCD, EFGH having Triangular Bases ABC, EFG are reciprocally Proportional; and thofe Pyramids, whofe Bafes and Altitudes are reciprocally Proportional, are equal. Hyp. a Hyp. 1. Compleat the Parallelepipedons ABI CDMKL, EFNGHQOP, which are each fix times their refpective equal Pyramids ABCD, 28. 11. EFGH, and fo equal to each other. There- & 7. 12. fore the Alt. (H) : Alt. (D)'::ABIC: EFNG ↳ :: ABC: EFG. Q. E. D. e d C f с e 34. II. hyp. 15. 5. 34. 11. Hyp. 2. Alt. (H): Alt. (D) :: ABC: g This is applicable likewife to Polygoneous Pyramids for these may be divided into Triangular one's. COROL What has been demonftrated in Prop. 6, 8, 9, of Pyramids, likewife extends to any kinds of Prifms, fince thefe are the triples of Pyramids having the fame Bafes and Altitudes. Therefore, (1.) Prifms that have the fame Altitude, are to one another as their Bases. (2.) The Proportion of Similar Prifms is the triplicate of the Proportion of the Homologous Sides. (3.) The 7 a Cor. 1. of this, Schol. 40. 11. 7. 12. (3.) The Bafes and Altitudes of equal Prifms, are reciprocally Proportional; and thofe Prifms, whofe Bafes and Altitudes are reciprocally Proportional, are equal. SCHOL. From what has been demonftrated, we may have the Dimenfion of any Prifms and Pyramids. The Solidity of a Prifm is gotten a by multiplying the Altitude into the Bafe and fo that of a Pyramid by of the Altitude into the Bafe. Schol. 7. 4, & cor. 9.12. b ing upon the Square circumfcribing the fame Circle, and the Cylinder of the fame Altitude. Therefore, a Prifm ftanding upon the Square ABCD does exceed the half of the Cylinder. In like manner, the Prifm ftanding upon the Bafe AFB, being of the fame Altitude as the a fchol. 27. 3. & cor. Cylinder, is greater than the half of the Seg9. 12... ment of the Cylinder AFB. Now continue on the Bifection of the Arches, and take away the Prifms till at length there be left the Segments whofe Bales are AF, FB, &c. of the Cylinder 1 PROP. XII. lind. ilar Cones and Cylinders ABCDK, PFGHM on the Baf the triplicate Proportion of the Diameters Cylind.H, of their Bafes ABCD, EFGH. the Pyran faid Prifm ving the i of the fan. for its Bafe until at leng Bases are AF, i E. Therefore i Pyr. AFBGCHL e. A be to fome folid N in the &c.) Whence th Pyramid (of the ortion of TX to FHI fay GHM. for if poffible, let N greater than a Cyd let the Excefs be O. Then the Part than the whence the Cylind LM be the Axes of the Propofitions, NPyr. SPFQ triple of the Cone. he right Lines VK, CK, VI, PR, QL, RL. Because the herefore VI: IK :: GL: Cylinders and Cones es VIK, GLM are Same Altitude, are to he Triangles VIK, Whence VC: VI Let the Circ. ABCI: GL: GM. ABCDK: Solid N. IK:: GR: GM ABCD, EFGH. f right 24 def. II. b18 def. GLM II. GR: 6. 6. And fod 4. 6. But also For if poffible, let M. Therefor again by 7.5. and let the Excefs be thGR: RM Whence the pofe the fame preparat2 are fimilar and by the f 5. 6. ing as in the last Prop. the remaining Triangles than the Conic Segme remaining Triangles of and fo the Solid Nramids hemfelves are 9 def.11. |