Elements of geometry and trigonometry . )roved tobe greater than that oi the two pyramids ; which latter diffe-rence we su[)pos(d to be e<jual to the [)rism r/-ABC : hence thef)rism ABC-D, must be greater than the prism «-ABC. But inreality it is less ; for they have the same base ABC, and thoaltitude Ax of the first is less than Aa the altitude of the the supposed ine(|uality b(;tween the two [)yrami(is can-not <*xist ; h«*nce the two pyramids S-ABC, S-^/Ar, having ecjualaltitudes and e(|uivalent bases, are themselves e(juivalent. !60 GEOMETRY. PROPOSITION XVI. THEOREM. Eve
Elements of geometry and trigonometry . )roved tobe greater than that oi the two pyramids ; which latter diffe-rence we su[)pos(d to be e<jual to the [)rism r/-ABC : hence thef)rism ABC-D, must be greater than the prism «-ABC. But inreality it is less ; for they have the same base ABC, and thoaltitude Ax of the first is less than Aa the altitude of the the supposed ine(|uality b(;tween the two [)yrami(is can-not <*xist ; h«*nce the two pyramids S-ABC, S-^/Ar, having ecjualaltitudes and e(|uivalent bases, are themselves e(juivalent. !60 GEOMETRY. PROPOSITION XVI. THEOREM. Every triangular pyramid is a third part of the triangular prismhaving the same base and the same Let F-ABC be a triangularpyramid, ABC-DEF a triangularprism of the same base and thesame altitude ; the pyramid willbe equal to a third of the prism. Cut off the pyramid F-ABCfrom the prism, by the planeFAC ; there will remain the solidF-ACDE, which may be consi-dered as a quadrangular pyramid,whose vertex is F, and whose baseis the parallelogram the diagonal CE ; and passthe plane FCE, which will cut thequadrangular pyramid into two triangular ones F-ACE, two triangular pyramids have for their common altitudethe perpendicular let fall from F on the plane ACDE ; theyhave equal bases, the triangles ACE, CDE being halves of thesame parallelogram ; hence the two pyramids F-ACE, F-CDE,are equivalent (Prop. XV.). But the pyramid F-CDE and thepyramid F-ABC have equal bases A BC,DEF; they have also thesame altitude, namely, the distance between the parallel planesABC, DEF ; hence the two pyramids are equivalent. Now thepyramid F-CDE has already been proved equivalent to
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
