. 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. s greater than the cone which it contains ; therefore the pyramidupon the square ABCD, having the same vertex with the cone,is greater than the half of the cone. Bisect the circumferencesAB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF,FC, CG, GD, DH, HA: therefore each of the triangles AEB,BF
. 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. s greater than the cone which it contains ; therefore the pyramidupon the square ABCD, having the same vertex with the cone,is greater than the half of the cone. Bisect the circumferencesAB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF,FC, CG, GD, DH, HA: therefore each of the triangles AEB,BFC, CGD, DHA is greater than half of the segment of the cir-cle in which it is: upon each of these triangles erect pyramidshaving the same vertex with the cone. Therefore each of thesepyramids is greater than the half of the segment of the cone inAvhich it is, as before was demonstrated of the prisms and seg-ments of the cylinder ; and thus dividing each of the circumfe-rences into two equal parts, and joining the points of division andtheir extremities by straight lines, and upon the triangles erect-ing pyramids having their vertices the same with that of the cone,?ind so on, there must at length remain some segments of thefone, which together shall be less than the excess of the cone. 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 demo
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
