. 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. gle ABC, and DEFis the opposite triangle. Q. E. D. CoR. 1. From this it is manifest, that every pyramid is thethird part of a prism which has tUe same base, and is of an equalaltitude with it; for if the base of the prism be any other figurethan a triangle, it may be divided into prisms having triangularb
. 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. gle ABC, and DEFis the opposite triangle. Q. E. D. CoR. 1. From this it is manifest, that every pyramid is thethird part of a prism which has tUe same base, and is of an equalaltitude with it; for if the base of the prism be any other figurethan a triangle, it may be divided into prisms having triangularbases. Cor. 2. Prisms of equal altitudes are to one another as theirbases ; because the pyramids upon the same bases, and of thet , same altitude, are ^ to one another as their bases, PROP. VIII. THEOR, SIMILAR pyramids having triangular bases areone to another in the triplicate ratio of that of theirhonmologous sides. Let the pyramids having the triangles ABC, DEF for theirbases, and the points G, H for their vertices, be similar, and si-milarly situated; the pyramid ABCG has to the pyramid DEFH,the triplicate ratio of that which the side BC has to the homolo-gous side EF. Complete the parallelograms ABCM, GBCN, ABGK, andthe solid parallelepiped BGML contained by these planes an(i. B C E F those opposite to them ; and, in like manner, complete the solidparallelepiped EHPO contained by the three parallelogramsDEFPj IIEFR, DEHX, and those opposite tq them; and, be? OF EUCLID. cause the pyramid ABCG is similar to the pyramid DEFH, the B. ABC is equal ^ to the angle DEF, and the angle GBC to the v.^-y-^1*angle HEF, and ABG to DEH: and AB is b to BC, as DE to EF ; a. 11. is, the sides about the equal angles are proportionals ; vvhei-e- !-•fore the parallelogram BM is similar to EP : for the same reason, b parallelogram BN is similar to ER, and BK to EX: there-fore the three parallelograms BM, BN, BK are similar to thethree EP, ER, EX : but t
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Keywords: ., bookauthoreuclid, bookcentury1800, booksubje, booksubjectgeometry
