14. 5. the cone ABCDL, fo is the cone EFGHN to fome folid, Book XII. which must be less than the cone ABCDL, because the folid Ź is greater than the cone EFGHN: Therefore the cone EFGHN has to a folid which is lefs than the cone ABCDL, the triplicate ratio of that which EG has to AC, which was demonftrated to be impoffible: Therefore the cone ABCDL has not to any folid greater than the cone EFGHN, the triplicate ratio of that which AC has to EG; and it was demonftrated that it could not have that ratio to any folid lefs than the cone EFGHN: Therefore the cone ABCDL has to the cone EFGHN, the triplicate ratio of that which AC has to EG: But as the cone is to the cone, fob the cylinder to the cylinder; for every b 15. 5. cone is the third part of the cylinder upon the fame base, and of the fame altitude: Therefore alfo the cylinder has to the cylinder, the triplicate ratio of that which AC has to EG. Wherefore fimilar cones, &c. Q. E. D, I F a cylinder be cut by a plane parallel to its opposite See N. planes, or bafes; it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other. Book XII. drawn from the point K to the line GH may be proved to be equal to those which are drawn from the centre of the circle AB to its circumference, and are therefore all equal to one ana 15. def. I. other. Therefore the line GH is the circumference of a circle' of which the centre is the point K: Therefore the plane GH divides the cylinder AD into the cylinders AH, GD; for they are the fame which would be described by the revolution of the parallelograms AK, GF about the straight lines EK, KF: And it is to be fhown that the cylinder AH is to the cylinder HC, as the axis EK to the axis KF L P N S A E B Produce the axis EF both ways; and take any number of ftraight lines EN, NL, each equal to EK; and any number FX, XM, each equal to FK; and let planes parallel to AB, CD pafs through the points L, N, X, M: Therefore the common fections of thefe planes with the cylinder produced are circles the centres of which are the points R L, N, X, M, as was proved of the plane GH; and thefe planes cut off the cylinders, PR, RB, DT, TQ : And because the axes LN, NE, EK are all equal; therefore the cylinders b 11. 12. PR, RB, BG are to one another as their bates; but their bafes are equal, G and therefore the cylinders PR, RB, BG are equal: And because the axes LN, NE, EK are equal to one another, as alfo the cylinders PR, RB, BG, and that there are as many axes as cylinders; therefore, whatever multiple the axis KL is of the axis KE, the fame multiple is the cylinder PG of the cylinder GB: For the fame reafon, whatever multiple the axis MK is of the axis KF, the fame multiple is the cy linder QG of the cylinder GD: And if the axis KL be equal to the axis KM, the cylinder PG is equal to the cylinder GQ; and if the axis KL be greater than the axis KM, the cylinder PG is greater than the cylinder GQ; and if lefs, lefs: Since therefore there are four magnitudes, viz. the axes EK, KF, and the cylinders BG, GD, and that of the axis EK and cylinder BG there has been taken any equimultiples whatever, viz. the T K H F D x Y M Q axis axis KL and cylinder PG; and of the axis KF and cylinder Book XII. GD, any equimultiples whatever, viz. the axis KM and cylin- W der GQ; and it has been demonftrated, if the axis KL be greater than the axis KM, the cylinder PG is greater than the cylinder GQ; and if equal, equal; and if lefs, lefs: Therefore the axis EK is to the axis KF, as the cylinder BG to the cylin- d 5. def. der GD. Wherefore, if a cylinder, &c. Q. E. D.. PRO P. XIV. THE OR. ONES and cylinders upon equal bases are to one an- CONE Let the cylinders EB, FD be upon the equal bafes AB, CD: As the cylinder EB to the cylinder FD, fo is the axis GH to the axis KL. Produce the axis KL to the point N, and make LN equal to the axis GH, and let CM be a cylinder of which the bafe is CD, and axis LN; and because the cylinders EB, CM have the fame altitude, they are to one another as their bafes: But a 11. 12. their bafes are equal, therefore alfo the cylinders EB, CM are equal. And because the cylin- A F K C LD b 13. 12. N M axis LN to the axis KL. But the cylinder CM is equal to the cylinder EB, and the axis LN to the axis GH: Therefore as the cylinder EB to the cylinder FD, fo is the axis. GH to the axis KL: And as the cylinder EB to the cylinder FD, fo is the cone ABG to the 15. 5. cone CDK, because the cylinders are triple of the cones: Therefore alfo the axis GH is to the axis KL, as the cone ABG to the cone CDK, and the cylinder EB to the cylinder FD. Wherefore cones, &c. Q. E. D. d d 10. 12. PROP. Book XII. See N. THE HE bafes and altitudes of equal cones and cylinders, are reciprocally proportional; and if the bafes and altitudes be reciprocally proportional, the cones and cy linders are equal to one another. Let the circles ABCD, EFGH, the diameters of which are AC, EG, be the bafes, and KL, MN the axes, as alfo the altitudes, of equal cones and cylinders; and let ALC, ENG be the cones, and AX, EO the cylinders: The bafes and altitudes of the cylinders AX, EO are reciprocally proportional; that is, as the bafe ABCD to the base EFGH, fo is the altitude MN to the altitude KL. Either the altitude MN is equal to the altitude KL, or thefe altitudes are not equal. First, let them be equal; and the cylinders AX, EO being alfo equal, and cones and cylinders a 11. 12. of the fame altitude being to one another as their bases, thereb A 5. fore the bafe ABCD is equal to the bafe EFGH; and as the bafe ABCD is to the bafe EFGH, fo is the altitude MN ta parallel to the op B F pofite planes of the circles EFGH, RO; therefore the common fection of the plane TYS and the cylinder EO is a circle, and confequently ES is a cylinder, the bafe of which is the circle EFGH, and altitude MP: And because the cylinder AX is equal to the cylinder EO, as AX is to the cylinder ES, fo is the cylinder EO to the fame ES. But as the cylinder AX to the cylinder ES, fo is the base ABCD to the base EFGH; for the cylinders AX, ES are of the fame altitude; and as the d 13. 12. cylinder EO to the cylinder ES, fod is the altitude MN to the altitude MP, because the cylinder EO is cut by the plane € 7.5. TYS TYS parallel to its oppofite planes. Therefore as the bafe Book XII. ABCD to the bafe EFGH, so is the altitude MN to the altitude MP: But MP is equal to the altitude KL; wherefore as the bafe ABCD to the base EFGH, fo is the altitude MN to the altitude KL; that is, the bafes and altitudes of the equal cylinders AX, EQ are reciprocally proportional. But let, the bafes and altitudes of the cylinders AX, EO, be reciprocally proportional, viz. the bafe ABCD to the bafe EFGH, as the altitude MN to the altitude KL: The cylinder AX is equal to the cylinder EO. First, let the bafe ABCD be equal to the bafe EFGH; then, because as the bafe ABCD is to the bafe EFGH, fo is the altitude MN to the altitude KL; MN is equal,b to KL, and b A. 5: therefore the cylinder AX is equal to the cylinder EO. But let the bafes ABCD, EFGH be unequal, and let ABCD . be the greater; and becaufe, as ABCD is to the bafe EFGH, fo is the altitude MN to the altitude KL; therefore MN is greater than KL. Then, the fame conftruction being made as before, because as the base ABCD to the bafe EFGH, fo is the altitude MN to the altitude KL; and because the altitude KL is equal to the altitude MP; therefore the base ABCD is to the base EFGH, as the cylinder AX to the cylinder ES; and as the altitude MN to the altitude MP or KL, fo is the cylinder EO to the cylinder ES: Therefore the cylinder AX is to the cylinder ES, as the cylinder EO is to the fame ES: Whence the cylinder AX is equal to the cylinder EO: And the fame reafoning holds in cones. Q. E. P. T defcribe in the greater of two circles that have the fame centre, a polygon of an even number of equal fides, that fhall not meet the leffer circle. Let ABCD, EFGH be two given circles having the fame centre K: It is required to infcribe in the greater circle ABCD a polygon of an even number of equal fides, that shall not meet. the leffer circle. Through the centre K draw the ftraight line BD, and from the point G, where it meets the circumference of the leffer circle |