. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. cylinder EB to the cylinder FD, so is the axis GH to theaxis KL. Produce the axis, KL to the point N, and make LN equalto the axis GH, and let CM be a cylinder of which the base isCD, and axis LN, and because the cylinders EB, CM havethe same altitude, they are to one another as their bases 3^: but a IJ.
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. cylinder EB to the cylinder FD, so is the axis GH to theaxis KL. Produce the axis, KL to the point N, and make LN equalto the axis GH, and let CM be a cylinder of which the base isCD, and axis LN, and because the cylinders EB, CM havethe same altitude, they are to one another as their bases 3^: but a IJ. bases are equal, therefore also the cylinders EB, CM areequal. And because the cylin-der FM is cut by the plane F/ ^CD parallel to its oppositeplanes, as the cylinder CM tothe cylinder FD, so is ** theaxis LN to the axis KL. Butthe cylinder CM is equal tothe cylinder EB, and the axisLN to the axis GH : thereforeas the cylinder EB to the cylin-der FD, so is the axis GH tothe axis KL : and as the cylin-der EB to the cylinder FD, sois c the cone ABG to the cone CDK, because the cylinders are c 15. ^ of the cones: therefore also the axis GH is to the axis d , as the cone ABG to the cone CDK, and the cylinder EB tothe cylinder FD. Wherefore, cones, &c. Q. E. b 13-12, 2N 282B. XII. THE ELEMENTS PROP. XV. THEOR. See N. THE bases and altitudes of equal cones and cy-linders are reciprocally proportional, and, if the basesand altitudes be reciprocally proportional, the cones^and cylinders are equal to one another. Let the circles ABCD, EFGH, the diameters of which areAc, EG, be the bases, and KL, MN the axis, as also the alti-tudes, of equal cones and cylinders; and let ALC, ENG be thecones, and AX, EO the cylinders: the bases and altitudes ofthe cylinders AX, EO are reciprocally proportional; that is,as the base ABCD to the base EFGH, so is the altitude MN tothe altitude KL. Either the altitude MN is equal to the altitude KL, or thesealtitudes are not equal.
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
