. 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. 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 pyra-mid of ihe same altitude with tlie cone ; this pyramid is greaterthan half of the cone. Fo
. 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. 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 pyra-mid of ihe same altitude with tlie cone ; this pyramid is greaterthan half of the cone. For, if a square be described about th^circle, and a pyramid be erected upon it, having the same ver- OF EUCLID. 27a. tex with the cone*, the pyramid inscribed in the cone is half B. the pyramid circums,cribed about it, because they are to one ^ —^mjanother as their bases »: but the cone is less than the circum- a pyramid ; therefore the pyramid of which the base is thesquare EFGH, and its vertex the same with that of the cone, isgreater than half of the cone: divide the circumferences EF,FG, GH, HE, each into two equal parts in the points O, P, R,S, and join EO, OF, FP, PG, GR, RH, HS, SE: therefore eachof the triangles EOF, FPG, GRH, HSE is greater than half of. the segment of the circle in which it is: upon each of thesetriangles erect a pyramid having the same vertex with the cone ;each of these pyramids is greater than the half of the segmentof the cone in which it is: and thus, dividing each of these cir-cumferences into two equal parts, and from the points of divisiondrawing straight lines to the extremities of the circumferences,and upon each of the triangles thus made erecting pyran)idshaving the same vertex with the cone, and so on, there must atlength remain some segments of the cone which are togt;therless than the solid Z : let these be the segments upon EO, OF, * Vertex is put in place of ahitude which is in the Greek, because the pyra-
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
