and vertex the point D, for they are contained by the fame Book XII plains; and the pyramid whole base is the triangle ABD, and vertex the point C, has been proved to be a third part of the prism, having the fame base, viz. the triangle ABC, and the oppofite base the triangle DEF: Which was required. COR. I. Hence every pyramid is a third part of a prism, ha. ving the fame base, and an equal altitude; for, if the base of a prism be of any other figure, it can be divided into prisms, having triangular bases. II. Prisms of the same altitude are to one another as their bases. S IMILAR pyramids, having triangular bases, are in the a 29. 11. Let the two pyramids whose bases are ABC, DEF, and vertices the points G, H, be similar and alike situate, then the pyramids ABCG, DEFH, are to one another in the triplicate ratio of BC to EF. For, compleat the parallelopipedons BGML, EHPO, then each contain two equal prisms, having triangular bases *; and a pyramid is one third of a prism, having the fame base and altitude b; but similar folid parallalelopipedons are to one another b 7. in the triplicate ratio of their homologous fides, and parts have c 33. IT. the same proportion as their like multiples d; therefore the pyra.d 15. s. mids ABCG, DEFH, are to one another in the triplicate ratio of their homologous fides. Wherefore, &c. Cor. Hence fimilar pyramids, having polygonous bases, are to one another in the triplicate ratio of their homologous lides. T HE bases and altitudes of equal pyramids, having triangu lar bases, are reciprocally proportional ; and those pyramids, having triangular bases, whose bases and altitudes are reciprocally proportional, are equal, and 7 Book XII Let the pyramids whose triangular bases 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. For, compleat the folids BGML, EHPO ; then, because the pyramids ABCG, DEFH, are equal, the solids ABGML, a 28. 11. EHPO, are equal"; but equal folid parallelopipedons have their b 34. 11 bases and altitudes reciprocally proportional b; therefore the pyramids ABCG, DEFH, have their bases and altitudes reciprocally proportional “; and, if their bases and altitudes are reci. procally proportional, they are equal. For, the fame construction remaining, the solid parallelopipedons, whose bases and altitudes are reciprocally proportional, are equal; therefore pyramids of the fame altitudes with the solids, having their bases and altitudes reci; rocally proportional, are likewise equal. Wherefore, &c. C15. S. PRO P. X. THE O R. VERY cone is the third part of a cylinder, having the same base, and an equal altitude. Let there be a cone and cylinder, having the fame 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 less than triple the cone. First , let it be greater, and let a polygon AEBFCGDH be inscribed in the circle ABCD, and let the small fegments AE, EB, BF, FC, CG, GD, DH, HA, the excefs by which the circle exceeds the polygon, be less than any assigned magnitude; and, upon the circle and polygon let a cylinder and pyramid be described, of the fame altitude with the cone; and, upon the remaining legrrents, the reinaining parts of the cylinder, which let be less than the excess 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 prism is triple the pyramid of the same base and altitude of the cone b; therefore the pyramid is greater than the cone, and likewise less, as included in it; which is absurd; therefore the cylinder is not greater than triple the cone, neither is it less ; for then, inversely, the cone would be greater than one third of the cylinder ; for, the same construction remaining, the pyramid, whole bale is the poiygon AEBFCGDH, and vertex the fame of the cone, is greater than one third of the cylinder ; but the pyramid is one third of the prism constitute on the same base, Book XII and having the same altitude ; therefore the pyramid whose base is the polygon AEBFCGDH, and altitude the same of the cone, is greater than the cone whose 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, since neither greater nor less, it must be triple the cone. Wherefore, &c. PROP. XI. THEOR. Cher, as their" bases. ONES and cylinders, of the same altitude, are to one ano- Let there be cones and cylinders of the same altitude, whose bases are the circles ABCD, EFGH, and axes KL, MN, and diameters of their bases AC, EG; then, as the circle ABCD is to the circle EFGH, so 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. First, let it be to a solid X less 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 inscribed in the circle EFGH, of which the remaining circumferences HO, OE, EP, PF, FR, RG, GS, SH, are less than any assigned magnitudes. Upon the polygon HQEPFRGS let a pyramid be described, of the same altitude with the cone, and let the remaining segments of the cone described upon the circumferences HO, OE, EP, PF, FR, RG, GS, SH, and vertex the same as the pyramid be less than the folid l; therefore the pyramid HOEPFRGS, and altitude the fame of the cone, will be greater than the folid X. Upon the circle ABCD let the polygon DTAYBQCV be described similar and alike situate to HOEPFRGS, 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 squares of their diameters AC, LG, and the circlesa I. ABCD, EFGH, are to one another as the squares 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 EFGH, fo is the cone AL to the folid X; therefore the polygon DTAYBQCV is to the polygon HOEPFRGSC as the concc 13. 5, AL is to the solid Xd; but the pyramid DTAYBQCVL is tod hyp. the Book XII the pyramid HOEPFRGSN as their bases "; 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 solid X, and the cone AL graeater than the pyramid in it; therefore, likewise 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 lets ; which is abfurd ; therefore the circle ABCD, to the circle EFGH, is not as the cone AL to a solid less 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 less than the cone AL Again, the circle ABCD to the circle EFGH, is not as the cone AL to a solid Z greater than the core IN; then, inversely, as the circle EFGH is to the circle ABCD, fo is the solid Z to the cone AL; but the folid Z is greater than the cone EN. Then, as the solid Z is to the cone AL, so is the cone EN to some folid less than the cone AL; therefore, as the circle EFGH is to the circle BCD, so is the cone EN to some solid less than the cone AL; which is impossible; 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, so is cylinder to cylinderf. Wherefore, &c. f 15. 5. S IMIL AR cones and cylinders are to one another, in the triplicate ratio of the diameters of their bases. Let there be similar cones and cylinders, whose bases are the circles ABCD, EFGH, their diameters BD, FH, and axes of the cones and cylinders KL, MN; then the cone whose base is the circle ABCD, and vertex the point L, to the cone whose 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 some solid greater or less than the cone EFGHN. First, let it be to a solid x, less than the cone EFGHN, and let the polygon EOFPGRHS be the greatest polygon possible inscribed in the circle EFGH; that is, that the excess of the circle above the infcribed polygon be less than any assigned magnitude ; upon the polygon EOFPGRHS let a pyramid be described, of the same altitude of the cone, and the segments of the cone described upon Book XII the segment of the circle, greater than the polygon, be less than the excess by which the cone EFGHN exceeds the solid X; then the pyramid described on the polygon EOFPGRHS, of the fame altitude as the cone, is greater than the folid X. Let the polygons ATBYCVDO be inscribed in the circle ABCD, similar to the polygon EOFPGRHS “, and upon it describe a pyramid a 18. 6. of the fame altitude of the cone. For, upon the polygon EOFPGRHS, suppose prisms erected, of the fame altitude of the cone; then these prisms are to one another as their bafes b. b 32. 11. For the same reason, the prisms described on the polygon and 15. 5. ATBYCVDQ, equiangular to those on the polygon EOPGRAS, and of the same altitude of the cone, are to one another as their base ; but the bases are similar to one another; therefore the equiangular prisms are likewise similar, and likewise the pyramids; therefore the pyramids are to one another, in the triplicate ratio of their homologous fides; that is, of BD to FH ; C8. but the cone ABCDL is to the solid X, in the triplicate ratio of BD to FH ; therefore, as the cone ABCDL is to the folid X, fo is the pyramid ATBYCVDOL to the pyramid EOFPGRHSN; but the cone is greater than the pyramid EOFPGRHSN d; but d 14. 5. it is proved less; which is absurd ; therefore the cone ABCDL has not to a solid less than the cone EFGHN, a triplicate ratio of BD to FH. For the same reason, the cone EFGHN has not to some folid less than the cone ABCDL a triplicate ratio of FH to BD. Again, the cone ABCDL has not to a solid Z, greater than EFGHN, a triplicate ratio of BD to FH ; for, then, inversely, ļ the solid Z has to the cone ABCDL a triplicate ratio of FH to BD; but the folid Z is greater than EFGHN; therefore the folid Z, to the cone ABCDL, is as the cone EFGHN to some solid 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 solid greater less 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 bases BD to FH; and, as cone is to cone, so is cylinder to cylinder e. Wherefore, &c. e 15. S to a or PROP. |