Book XII and 7. b 34. 11. C 15. 5. Let the pyramids whofe triangular bafes are ABC, DEF, and vertices the points G, H, be equal; then the base ABG is to the base DEF, as the altitude of the pyramid DEFH is to the altitude of the pyramid ABCG. b. For, compleat the folids BGML, EHPO; then, because the pyramids ABCG, DEFH, are equal, the folids ABGML, a 28. 11. EHPO, are equal; but equal folid parallelopipedons have their bafes and altitudes reciprocally proportional ; therefore the pyramids ABCG, DEFH, have their bases and altitudes reciprocally proportional ; and, if their bafes and altitudes are reciprocally proportional, they are equal. For, the fame construction remaining, the folid parallelopipedons, whose bases and altitudes are reciprocally proportional, are equal; therefore pyramids of the fame altitudes with the folids, having their bases and altitudes reciprocally proportional, are likewife equal. Wherefore, &c. a 2. b 7. E PROP. X. THE OR. VERY cone is the third part of a cylinder, having the fame bafe, and an equal altitude. Let there be a cone and cylinder, having the same base, viz. the circle ABCD, and their altitudes equal, then the cone is one third of the cylinder; that is, the cylinder is triple the cone. If not, it will be either greater or lefs than triple the cone. First, let it be greater, and let a polygon AEBFCGDH be infcribed in the circle ABCD, and let the fmall fegments AE, EB, BF, FC, CG, GD, DH, HA, the excefs by which the circle exceeds the polygon, be less than any affigned magnitude; and, upon the circle and polygon let a cylinder and pyramid be defcribed, of the fame altitude with the cone; and, upon the remaining fegments, the remaining parts of the cylinder, which let be lefs than the excefs by which the cylinder exceeds triple the cone; therefore the prism whose base is the polygon AEBFCGDH, and altitude the fame of the cone, is greater than triple the cone; but the prifm is triple the pyramid of the fame bafe and altitude of the cone ; therefore the pyramid is greater than the cone, and likewife lefs, as included in it; which is abfurd; therefore the cylinder is not greater than triple the cone, neither is it lefs; for then, inversely, the cone would be greater than one third of the cylinder; for, the fame conftruction remaining, the pyramid, whofe base is the polygon AEBFCGDH, and vertex the fame of the cone, is greater than one third of the cylinder; but the pyramid pyramid is one third of the prifm conftitute on the fame bafe, Book XII and having the fame altitude; therefore the pyramid whose base is the polygon AEBFCGDH, and altitude the fame of the cone, is greater than the cone whofe base is the circle ABCD; and likewise less, as contained in it; which cannot be; therefore the cylinder is not less than triple the cone. Therefore, fince neither greater nor less, it must be triple the cone. Wherefore, &c. PROP. XI. THEOR. ONES and cylinders, of the fame altitude, are to one ano◄ Let there be cones and cylinders of the fame altitude, whose bases are the circles ABCD, EFGH, and axes KL, MN, and diameters of their bafes AC, EG; then, as the circle ABCD is to the circle EFGH, fo is the cone AL to the cone EN. If not, the circle ABCD is to the circle EFGH, as the cone AL is to some solid greater or less than the cone EN. Firft, let it be to a folid Xlefs than the cone; and let the folid I be equal to the excess of the cone EN above the folid X; then the cone EN is equal to the folid I and X together. Let a polygon HOEPFRGS be infcribed in the circle EFGH, of which the remaining circumferences. HO, OE, EP, PF, FR, RG, GS, SH, are lefs than any affigned magnitudes. Upon the polygon HOEPFRGS let a pyramid be described, of the fame altitude with the cone, and let the remaining fegments of the cone defcribed upon the circumferences HO, OE, EP, PF, FR, RG, GS, SH, and vertex the fame as the pyramid be lefs than the folid!; therefore the pyramid HOEPFRGS, and altitude the fame of the cone, will be greater than the solid X. Upon the circle ABCD let the polygon DTAYBQCV be described fimilar and alike fituate to HOFPFRGS, and let a pyramid EN be erected, of the fame altitude as the cone AL; but the polygons DTAYBQCV, HOEPFRGS, are to one another as the fquares of their diameters AC, EG, and the circlesa I. ABCD, EFGH, are to one another as the fquares of their diameters AC, EG ; therefore, as the circle ABCD is to theb 2. circle EFHG, fo is the polygon DTAYBQCV to the polygon HOEPFRGS; but, as the circle ABCD is to the circle LFGH, fo is the cone AL to the folid X; therefore the polygon. DTAYBQCV is to the polygon HOEPFRCS as the conec 13. §. AL is to the folid Xd; but the pyramid DTAYBQCVL is tod hyp. the Book XII the pyramid HOEPFRGSN as their bafes ; therefore the pyramid DTAYBQCVL is to the pyramid HOEPFRGSN as the e 5. and 6. cone AL is to the folid X; but the pyramid is greater than the folid X, and the cone AL graeater than the pyramid in it; therefore, likewife the cone EN is greater than the pyramid in it; but the pyramid in the cone EN is greater than X; therefore the cone EN is much greater than X; but it was put lefs; which is abfurd; therefore the circle ABCD, to the circle EFGH, is not as the cone AL to a folid lefs than the cone EN; and it is proved, in the same manner, that the circle EFGH is not to the circle ABCD, as the cone EN is to a folid lefs than the cone AL. Again, the circle ABCD to the circle EFGH, is not as the cone AL to a folid Z greater than the cone EN; then, inversely, as the circle EFGH is to the circle ABCD, fo is the folid Z to the cone AL; but the folid Z is greater than the cone EN. Then, as the folid Z is to the cone AL, fo is the cone EN to fome folid less than the cone AL; therefore, as the circle EFGH is to the circle ABCD, fo is the cone EN to fome folid lefs than the cone AL; which is impoffible; therefore the circle ABCD to the circle EFGH is not as the cone AL to fome folid greater or less than EN, therefore, to the cone EN; but, as cone is to cone, fo is cylinder to cylinder f. Wherefore, &c. PRO P. XII. THE OR. SIMILAR cones and cylinders are to one another, in the triplicate ratio of the diameters of their bafes. Let there be fimilar cones and cylinders, whofe bafes are the circles ABCD, EFGH, their diameters BD, FH, and axes of the cones and cylinders KL, MN; then the cone whofe base is the circle ABCD, and vertex the point L, to the cone whofe base is the circle EFGH, and vertex the point N, hath a triplicate ratio of BD to FH. For, if the cone ABCDL be not to the cone EFGHN, in the triplicate ratio of BD to FH, let it be in the triplicate ratio to fome folid greater or less than the cone EFGHN. First, let it be to a folid X, lefs than the cone EFGHN, and let the polygon EOFFGRHS be the greatest polygon poffible infcribed in the circle EFGH; that is, that the excess of the circle above the infcribed polygon be lefs than any affigned magnitude; upon the polygon LOFPGRHS let a pyramid be defcribed, of the fame altitude altitude of the cone, and the fegments of the cone defcribed upon Book XII the segment of the circle, greater than the polygon, be lefs than the excess by which the cone EFGHN exceeds the folid X; then the pyramid defcribed on the polygon EOFPGRHS, of the fame altitude as the cone, is greater than the folid X. Let the polygons ATBYCVDQ be infcribed in the circle ABCD, fimilar to the polygon EOFPGRHS, and upon it defcribe a pyramid a 18. 6, of the fame altitude of the cone. For, upon the polygon EOFPGRHS, fuppofe prifms erected, of the fame altitude of c 8. the cone; then thefe prifms are to one another as their bafes b. b 32. 17. For the fame reafon, the prifms defcribed on the polygon and 15. 5. ATBYCVDQ, equiangular to thofe on the polygon EOPGRHS, and of the fame altitude of the cone, are to one another as their base; but the bafes are fimilar to one another; therefore the equiangular prifms are likewife fimilar, and likewife the pyramids; therefore the pyramids are to one another, in the triplicate ratio of their homologous fides; that is, of BD to FH but the cone ABCDL is to the folid X, in the triplicate ratio of BD to FH; therefore, as the cone ABCDL is to the folid X, fo is the pyramid ATBYCVDQL to the pyramid EOFPGRHSN; but the cone is greater than the pyramid EOFPGRHSN 4; but d 14. s it is proved less; which is abfurd; therefore the cone ABCDL has not to a folid less than the cone EFGHN, a triplicate ratio of BD to FH. For the fame reafon, the cone EFGHN has not to fome folid lefs than the cone ABCDL a triplicate ratio of FH to BD. Again, the cone ABCDL has not to a folid Z, greater than EFGHN, a triplicate ratio of BD to FH; for, then, inversely, the folid Z has to the cone ABCDL a triplicate ratio of FH to BD ; but the solid Z is greater than EFGHN ; therefore the folid Z, to the cone ABCDL, is as the cone EFGHN to fome folid less than the cone ABCDL; therefore the cone EFGHN, to fome folid less than the cone ABCDL, has a triplicate ratio of FH to BD; but it is proved that it has not; therefore the cone ABCDL, to a folid greater or lefs than the cone EFGHN, has not a triplicate ratio of BD to FH; therefore the cones ABCDL, EFGHN, have to one another the triplicate ratio of their bafes BD to FH; and, as cone is to cone, fo is cylinder to cylinder. Wherefore, &c. e 15. S: Book XII a 11. a cylinder be divided by a plain parallel to the oppofite plains, then, as one cylinder is to the other cylinder, fo is the axis of the one to the axis of the other. Let the cylinder AD be divided by the plain GH, parallel to the oppofite plains AB, CD, and meeting the axis EF in the point K; then, as the cylinder BG is to the cylinder GD, fo is the axis EK to KF. a For, let the axis EF be produced both ways to L and M; let EL be taken any multiple of EK; and FM any multiple of FK; through the points L, N, X, M, draw plains parallel to AB, CD; and with the centers L, N, X, M, draw the circles OP, RS, TY, VQ, each equal to AB; and compleat the cylinders PR, RB, DT, TQ; then, because the axis LN, NE, EK, are equal, the cylinders PR, RB, BG, are equal . For the fame reafon, the cylinders HC, DT, TQ, are equal; therefore the cylinder PG is the fame multiple of the cylinder BG, that the axis LK is of EK. For the fame reafon, the cylinder GQ is the fame multiple of GD that KM is of KF; therefore, if KL is equal to KM, PG will be equal to GQ; if greater, greater, and, if lefs, lefs. Therefore, AH is b def. 5. 5. to GD as EK is to HF. Wherefore, &c. a ir. b 13. C PRO P. XIV. THE O R. ONES and cylinders, conflituted upon equal bases, are to one another as their altitudes. Let the cylinders EB, FD, ftand upon equal bases AB, CD, then the cylinder EB is to the cylinder FD, as the altitude GH is to the altitude KL. For, produce the axis KL to the point N, and put LN equal to GH, and let the cylinder CM be drawn about the axis LN; then the cylinders EB, CM, are to each other as their bafes "; but their bafes are equal; therefore the cylinders EB, CM, are equal; but the cylinders CM, FD, are as their axes LN, KL^; but the cylinders CM, EB, are equal; and their axes GH, LN, likewife equal; therefore the cylinder EB is to the cylinder FD as the axis GH to the axis KL; but, as the cylinder EB is to 15. 5, and the cylinder FD, fo is the cone ABG to the cone CDK ; therefore, as the axis GH is to KL, fo is the cone ABG to CDK and fo the cylinder EB to FD. Wherefore, &c. 1 C PROP. |