THE PROPOSITION IX. THEOREM IX. HE bafes (A B C & E F G), and altitudes (BD & F H), of equal pyramids, (A B C D & E F G H), having triangular bases, are reciprocally proportional, (that is, the base A BC: bafe E F G altitude F H : altitude B D), and triangular pyramids (A B C D & E F G H), of which the bafes (A B C & E FG), and altitudes (B D & F H), are reciprocally proportional: are equal to one another. Hypothefis. I. The pyrams. ABCD & EFGH are triangular. Preparation. Thefis. Bafe ABC: bafe EFG altitude Complete the EBO & FK having the fame altitude with 1. DEMONSTRATION. BECAUSE the prifms PNB & LIF, have the fame base & altitude with the given pyramids A B C D & E F G H. (Prep). 1. Each prifm will be triple of its pyramid, (that is, the prifm P N B triple of the pyramid A B C D, & the prifin L IF triple of the SP. pyramid E F G H).. 2. Confequently, the prifm PNB is to the prifm LIF. But the BO is double of the prifm P N B, & the double of the prifm LIF. 3. Therefore, the BO is to the FK. FK But the equal (BO & FK) have their bafes and altitudes re- And thofe are each fextuple of their pyramids, (that is, the 7. B.12, Cor. 1. Ax.6. B. 1, P.28. B.11. Moreover, the base of the pyramid ABCD is the half of the base of the B O. And the bafe of the pyramid EFGH is the half of the bafe P.41. B. 1. of the F K. Confequently, base A B C : base EFG alt. F H : alt. B D. Hypothefis. SP.15. B. 5. Which was to be demonftrated. Thefis. 1. The pyramids ABCD & EFGH are triangular. The triangular pyramid ABCD is = 11. Bafe ABC bafe EFG alt. FH: alt. BD. to the triangular pyramid EFGH. BECAUS II. DEMONSTRATION. ECAUSE the AABC: AEFG=FH: BD. (Hyp. 2). And the pgr. BQ is double of the AABC, the pgr. F M double of the AEF G. But P.41. B. 1. (Prep.). P.34. B.11. P.18. B.11. i. It follows, that the pgr. BQ: pgr. F MFH: BD. And thofe prifms PNB & LIF are each triple of their pyramids SP. 3. Therefore, the triangular pyramid A B C D is to the triangular 7. B.12. Cor. 1. Ax.7. B. 1. Which was to be demonftrated. COROLLAR Y EQUAL polygon pyramids have their bases and altitudes reciprocally propor tional; polygon pyramids whofe bafes & altitudes are reciprocally proportional: are equal. PROPOSITION X. THEOREM X. EVERY cone (BRC) is the third part of the cylinder (HG FE ABDC) which has the fame base, (BD CA) and the fame altitude (BH) with it. Hypothefis. The cone BRC, & the cylinder HFADC, bave the fame base B DCA, & the fame altitude BH. I. If not, Thefis. The cone BRC is equal to the third part of the cylinder HFCABD. DEMONSTRATION, The cone will be <or > the third part of the cylinder, by I. Suppofition. Let the third part of the cylinder HC be cone BRC 1. Preparation. IN the bafe A BDC of the cone & cylinder, defcribe the A B D C. 2. About the fame base describe the □POQS. Sf. P. 6. B. 4 P. 7. B. 4. * We have omitted a part of the preparation in the figure to avoid confufion. 4. Bifect the arches A TC, Cd D, D b B, & B a A, in T,d,b, & a. 8. Do the fame with respect to the other fegments A a B, B b B, &c. BECAUSE the PO QS is defcribed about, & the □ POQS BDCA defcribed in the O. (Frep. 1. & 2). 1. The PO QS is double of the BDC A. P.30. B. 3. Pol.1. B. 1. P.17. B. 3. the upon POQS is > the given cylinder. And the defcribed upon thofe fquares having the fame altitude, 2. Therefore, the upon POQS is double of the P.47. B. 1. (Prep. 3). upon BDCA. P.32. B.11. But Ax.8. B. 1. P.19. B. 5. P.41. B. i. (P 28. B.11. P.34. B.11. Rem.1.Cor.3. Ax.8. B. 1. 3. Therefore, the upon BDCA is > the half of the fame cylinder. The Ed defcribed upon the pgt. A K is > the element of the P.19. B. 5. 7. Until there remains feveral elements of the cylinder which together will be <Z.. Lem. B.12. But the cylinder is to three times the cone B R C + Z. (Sup.). Therefore, if from the whole cylinder be taken thofe elements (Arg. 7.). And from three times the cone B RC+Z, the magnitude Z. 8. The remaining prifm (viz. that which has for base the polygon Aa Bb Dd CT) will be > the triple of the cone. But this prifm is the triple of the pyramid of the fame base & altitude (viz. of the pyramid TA a B b D dC TR). 9. Confequently, the pyramid A BDCR is the given cone. IF 13. Let the cone be > the the third part of the cylinder. II. Suppofition. third part of the cylinder by the mgn. the third part of the cylinder + Z. II. Preparation. Ax.4. B. 1. P. 7. B.12. {Cor. 2. Divide the given cone into pyramids, in the fame manner. F from the given cone be taken the pyramid which has for base the There will remain feveral elements of the cone which together Therefore, if from the cone thofe elements be taken which are < to the third part of the prifm, 14. The remainder, viz. the pyramid Aa Bb DdCTR is to the the bafe of the prifm fince 18. Therefore, the third part of the cylinder is not the cone. 19. Therefore, the cone is the third part of the cylinder of the fame bafe & altitude. Ax.7. B. 1. Ax.8, B. 1. Lem. B.12 Ax.5. B. 1. SP. 7. B.12. Cor. 2. Ax.6. B. 1. Ax.8. B. 1. Which was to be demonftrated. |