BOOK XII.circumference ABCD to half the circumference EFGH, fo is BK to FL; and taking their doubles, as the circumference ABCD 94.5 to the circumference EFGH, fo is BD to FH. PROP. IV. THEOR. RIANGULAR pyramids upon equal bafes, and of the fame altitude, are equal to one another. Let ABC-G, DEF-H be two pyramids of the same altitude, and having the equal triangles ABC, DEF for their bases; the pyramid ABC-G is equal to the pyramid DEF-H. If not, one of them is greater than the other: let the pyramid ABC-G be greater than the other pyramid by the folid Z: and complete the prifm, of which ABC is the bafe, and AG one II. a of its infifting lines: Then, if AG be bifected by a plane parallel a Cor. 32. to ABC, that plane will bifect the prism; for AG and the perpendicular from G to the plane ABC, are cut in the fame ratio b 17. 11. by that parallel plane ; therefore the parts are prisms on the fame bafe, and having equal altitudes, and are therefore equal a. In the fame manner, may the half of the remaining prism be cut off; and fo on: Therefore, if this be done continually, there fhall at length remain a prifm less than the folid Z: let this be the prifin ABC-LNO, and let LN, LO meet GB, GC in P, d 31. 1. Q and draw & PR, QS parallel to AG, and join PQ, RS; e 13. Def. therefore ARS LPQ is a prifm in the pyramid ABC-G: Let C 1.12. 11. the 1 n g II. 5. k 4. 5. 122.6. n Cor. 32. II. the other parts of AG equal to AL, be LK, KM, MG, and com-Book XII. plete prifms in the pyramid between the planes through L, K, M: and let the fame conftruction be made in the pyramid DEF-H: and because LP is parallel to AB, AG is to GL, as 0 16. 11. f AB to LP or AR: For the fame reason, AG is to GL, as AC £ 4. 6. to AS; therefore BA is to AR, as CA to AS; and the triangle ABC is fimilar to ARS: In the fame manner, it may be h 6. 6. proved, that the triangle DEF is fimilar to DXY: and because AG, DH are equimultiples of AL, DT; as are also GL, HT; therefore AG is to GL, as k DH to HT; wherefore BA is to AR, as ED to DX, and confequently the triangle ABC is to ARS, as DEF to DXY: and the triangle ABC is equal to DEF; therefore the triangle ARS is equal to DXY; and therefore m 14. 5. the prifm LRS is equal to the prifm TXY, for they are upon equal bafes, and have the fame part of the altitude of the pyramids for their altitude. In the fame manner, may the other prifms in the two pyramids be proved to be equal, each to each; therefore all the prifms in the pyramid ABC-G together are equal to all the prifms in the pyramid DEF-H. Produce RP, SQ to a, b, and join ab; therefore LPQ-Kab is a prism; and it is equal to the prifm ASR-LPQ, because they are upon the fame bafe, and have equal altitudes: In the fame manner, it may be proved, that the other infcribed prisms are equal to the circumfcribed prifms ftanding upon them: therefore all the infcribed prifms, together with the prifm ABC-LNO, are equal to all the circumfcribed prifms; that is, they are greater than the pyramid ABC-G: But the prifms in the pyramid ABC-G are equal to thofe in the pyramid DEF-H, and the folid Z is greater than the prism ABC-LNO; therefore the prisms in the pyramid DEF-H, together with the folid Z, are greater than the pyramid ABC-G; that is, than the pyramid DEF-H, together with the folid Z: take away the common folid Z, and the prifms in the pyramid DEF-H are greater than the pyramid itself, and they are also lefs; which is impoffible: Therefore the pyramid ABC-G is not unequal to the pyramid DEF-H, that is, it is equal to it. Q. E. D. n PROP. V. THEOR *. VERY triangular prifm may be divided into three EVE Let ABC-DEF be a triangular prifm, of which the bafe is the triangle ABC, and DEF the triangle oppofite to it: It may be divided into three equal triangular pyramids. E e 2 *This is Prop. VII. Book XII. of Euclid. Join BOOKXII. Join AF, FB, BD: and because BD is the diameter of the parallelogram AE, the triangle ABD is equal to EBD; and a 34. 1. therefore the pyramid ABD-F is equal to the b 4. 12. pyramid EBD-F, for they are upon equal D A COR. 1. From this it is manifeft, that every pyramid is the third part of a prifm which has the fame base, and is of an equal altitude with it; for if the base of the prism be any other figure than a triangle, it may be divided into prifms having triangular bafes. COR. 2. Pyramids of equal altitudes are to one another as their bases; because the prifms upon the fame bafes, and of the c 2. Cor. fame altitude, are to one another as their bases. 32. II. * This is the 5th and 6th Propofition of Book XII. of Euclid. PROP. VI. THEOR. F a cylinder and a parallelopiped be upon equal I bafes, and of the fame altitude, they are equal to one another. Let ABCD be a cylinder, and EF a parallelopiped of the same altitude; and let the circle ABG, which is the base of the cylinder, be equal to the base EH of the parallelopiped; the cylinder ABCD is equal to the parallelopiped EF. : If not, it is either greater or less than the parallelopiped : First, let it be lefs, and let it be equal to the parallelopiped EK of the fame altitude with FF, but having the base EL lefs than EH and in the circle AGB let rectangles be made as in the fecond propofition, which together shall be greater than EL; and let the fame be done in the oppofite bafe CTD; and let NOPQ, RSTV be correfponding rectangles in them; therefore, because the circles are equal, the rectangle PN is equal to RT, as was fhown in Propofition III; join NR, OS, PT, QV: and because NQ is equal and parallel to RV, NR is equal and parallel to a 33. 1. QV: For the fame reason, PT, OS are equal and parallel to b-24. 11. NR or QV; therefore NT is a parallelopiped: conftruct, in the fame manner, parallelopipeds upon the other rectangles in the the circle AGB and because the parallelopipeds NT, EK are of Book XII. the fame altitude, NT is to EK, as the base NP to the bafe EL: For the fame reason, the parallelopipeds upon the other C 32. 11. rectangles in the circle AGB are each of them to EK as the rect 24. 5. angle upon which it stands to the base EL; therefore all the parallelopipeds together are to EK as all the rectangles in the d 2. Cor. circle AGB to the bafe EL: But the rectangles in the circle AGB are greater than EL; therefore the parallelopipeds upon them are together greater than EK : and EK is equal to the e A. 5. cylinder ABCD; therefore the parallelopipeds upon the rectangles are greater than the cylinder; and they are also less, because they are contained in it; which is impoffible: Therefore the cylinder ABCD is not lefs than the parallelopiped EF. : Neither is it greater; for, if poffible, let the cylinder ABCD be equal to the parallelopiped EW of the fame altitude with EF, but upon a base EX greater than EH; that is, than the circle AGB and because EX is greater than the circle AGB, re&tangles can be made about the circle, which together are less than EX: and, if upon these rectangles be made parallelopipeds in g 1. Cor. the way that NT was made, it may be proved, as before, that they are together less than the parallelopiped EW; that is, than the cylinder ABCD: and they are alfo greater, because they contain the cylinder; which is impoffible; therefore the cylinder is not greater than the parallelopiped EF: and it has been shown, that it is not lefs than it; therefore the cylinder is equal to the parallelopiped. Wherefore, &c. Q. E. D. COR. I. *Hence cylinders of the fame altitude are to one another as their bafes, for the parallelopipeds upon bases equal to them, and of the fame altitude, are to one another as their bafes c. COR. 2. Cylinders have to one another the ratio which is compounded of the ratios of their bafes, and of their altitudes. For this is the ratio of the parallelopipeds, which are upon equal bases with them, and have the fame altitudes . *This is the 11th Propofition of Book XII, of Euclid, 2. 12. f 2. Cor. D. 11. PROP. BOOK XII. 2. T2. PROP. VII. THEOR. IF a cone and a pyramid be upon equal bases, and of the fame altitude; they are equal to one another. Let the cone of which the bafe is the circle ABC, and the vertex D, be of the fame altitude with the pyramid of which the base is the figure EFG, and the vertex H; and let the circle ABC be equal to the figure EFG: the cone ABC-D is equal to the pyramid EFG.H. If not, let them be unequal; and, first, let the cone be lefs than the pyramid; that is, let it be equal to fome pyramid FGK-H of which the bafe FGK is less than FGE, or than the circle ABC; therefore, in the circle there can rectangles be a 1. Cor. made, which together are greater than FGK : let LMNO be one of them, and join DL, DM, DN, DO: and because the pyramids LMNO-D and FGK-H are of the fame altitude, the b 2. Cor. pyramid LMNO-D is to the pyramid FGK-H, as the base LMNO to the bafe FGK: In the fame manner, if pyramids be erected on all the other rectangles in the circle ABC, it may be proved, that each of them is to the pyramid FGK-H, as the 5. 12. 24. 5. rectangle on which it ftands to the bafe FGK: wherefore all the c 2. Cor. pyramids upon the rectangles are to the pyramid FGK-H, as © all the rectangles to the base FGK: But the rectangles together are greater than the bafe FGK; therefore the pyramids upon them are together greater than the pyramid FGK-H; that is, than the cone ABC-D: and they are alfo lefs, for they are contained in it; which is impoffible; therefore the cone ABC-D is not lefs than |