Elements of geometry and trigonometry . -don AL equivalent toAG, and having its late-ral faces AK, BL, & Hence if thebase ABCD is a rectan-gle, AL will be a rectan-gular parallelopipedon equivalent to AG, and consequently,the parallelopipedon required. But if ABCD is not a rectangle,draw AO and BN perpendicular to CI), and mQ IiP OQ and NP perpendicular to the base ; youwill then have the solid ABNO-IKPQ, w hichwill be a rectangular parallelopipedon : forby construction, the bases ABNO, and IKPQare rectangles ; so also are the lateral faces,the edges AI, OQ, &:c. being perpendicul
Elements of geometry and trigonometry . -don AL equivalent toAG, and having its late-ral faces AK, BL, & Hence if thebase ABCD is a rectan-gle, AL will be a rectan-gular parallelopipedon equivalent to AG, and consequently,the parallelopipedon required. But if ABCD is not a rectangle,draw AO and BN perpendicular to CI), and mQ IiP OQ and NP perpendicular to the base ; youwill then have the solid ABNO-IKPQ, w hichwill be a rectangular parallelopipedon : forby construction, the bases ABNO, and IKPQare rectangles ; so also are the lateral faces,the edges AI, OQ, &:c. being perpendicularto the plane of the base ; hence the solid APis a rectangular parallelopipedon. But thetwo parallelopipedons AP, AL may be con-ceived as having the same base ABKI andthe same altitude AO : hence the parallelopipedon AG, whichwas at first changed into an equivalent parallelopipedon AL,Ls again changed into an equivalent rectangular parallelopipe-don AP, having the same altitude AI, and a base ABNO equi-valent to the base PROPOSITION XI. THEOREM. Two rectangular parallelopipedons, which have the same basCfare to each other as their altitudes. 154 GEOMETRY. O-f ?m A vF \ H K A Let the parallelopipedons AG, AL, have the same base BD,then will they be to each other as their altitudes AE, AT. Firsts suppose the altitudes AE, AI, to be 33 ]j to each other as two whole numbers, as 15 isto 8j for example. Divide AE into 1Ô equalparts \ whereof AI will contain 8 \ and , y, 2, &Cs the points of division, draw planesparallel to the base» These planes will cutthe solid AG into 15 partial parallelopipedons,all equal to each other, because they haveequal bases and equal altitudes—equal bases,since every section MIKL, made parallel tothe base ABCD of a prism, is equal to that base (Prop* II.), equal altitudes, because thealtitudes are the equal divisions Aa:, xy, yx,&c. But of those 15 equal parallelopipedons, 8 are con*tained in AL 5 hence the solid AG is to the solid AL
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