. 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. 272 THE ELEMENTS H B. XII. above the third part of the cyHnder. Let these be the segmentsV—V— upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the restof the cone, that is, the pyramid,of which the base is the polygonAEBFCGDH, and of which the ver-tex is the same with that of the cone,is greater than the th
. 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. 272 THE ELEMENTS H B. XII. above the third part of the cyHnder. Let these be the segmentsV—V— upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the restof the cone, that is, the pyramid,of which the base is the polygonAEBFCGDH, and of which the ver-tex is the same with that of the cone,is greater than the third part of thecylinder. But this pyramid is thethird part of the prism upon the samebase AEBFCGDH, and of the samealtitude with the cylinder. Thereforethis prism is greater than the cylinderof which the base is the circle it is also less, for it is containedwithin the cylinder ; which is impos-sible. Therefore the cylinder is not less than the triple of thecone. And it has been demonstrated that neither is it greaterthan the triple. Therefore the cylinder is triple of the cone, or,the cone is the third part of the cylinder. Wherefore, eveiy one, Sec. Q. E. PROP. XL THEOR. See N. CONES and cylinders of the same altitude are toone another as their bases. Let the cones and cylinders, of which the bases are the circlesABCD, EFGH, and the axes KL, MN, and AC, EG the diame-ters of their bases, be of the same altitude. As the circle ABCDto the circle EFGH, so is the cone AL to the cone EN. If it be not so, let the circle ABCD be to the circle EFGH,as the cone AL to some solid either less than the cone EN, orgreater than it. First, let it be to a solid less than EN, viz. tothe solid X; and let Z be the solid which is equal to the ex-cess of the cone EN above the solid X; therefore the cone ENis equal to the solids X, Z together. In the circle EFGH de-scribe tlie square EFGFI, therefore this square is greater thantlie half of the circle: upon the square EFGH erect a py
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
