Elements of geometry and trigonometry . ar 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 F-ACE ;hence the
Elements of geometry and trigonometry . ar 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 F-ACE ;hence the three pyramids F-ABC, F-CDE, F-ACE, whichcompose the prism ABC-DEF are all equivalent. Hence thepyramid F-ABC is the third part of the prism ABC-DEF> whichhas the same base and the same altitude. Cor. The solidity of a triangular pyramid is equal to a thirdpart of the product of its base by its altitude. PROPOSITION XVII. THEOREM. The solidity of every pyramid is equal to the base multiplied bya. third of the altitude. BOOK VII. IGl. I-et S-ABCDE be a pyramid. Iass the planes SEB*, SEC, tliroiigh tliediagonal EB, EC ; the polygonal pyramidS-ABCDE will be divided into several trian-gular pyramids all having the same altitudeiSO. But each ot these pyramids is measuredby multiplying its base ABE, BCE, or CDE,by the third j)artof its altitude SO (Prop. ) ; hence the sum of these triangular pyra-mids, or the polygonal pyramid S-ABCDEwill be measured by the sum of the trianglesABE, BCE, CDE, or the polygon ABCDE,multiplied by one third of SO ; hence every pyramid is mea-sured by a tliird part of the product of its base by its altitude. Cor. 1. Every pyramid is the third part of the prism whichlias the same base and the same altitude. Cor. 2. Two pyramids having the same altitude are to eachother as their bases. Cor. 3. Two pyramids havmg equivalent bases are to each(•ther as their altitudes. Cor. 4. Pyramids arc to each other as the products of theirbases by their altitu
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
