Elements of Geometry : Containing the First Six Books of Euclid, With a Supplement On the Quadrature of the Circle, And the Geometry of Solids : to Which Are Added, Elements of Plane And Spherical Geometry

Cover Elements of Geometry : Containing the First Six Books of Euclid, With a Supplement On the Quadrature of the Circle, And the Geometry of Solids : to Which Are Added, Elements of Plane And Spherical Geometry
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Solid parallelepipeds which have the same altitude, are to one another as their bases.
Let AB, CD be solid parallelopipeds of the same altitude : they are to one another as their bases ; that is, as the base AE to the base CF, so is the solid AB to the solid CD.
To the straight line FG apply the parallelogram FH equal (Cor.
45. 1.) to AEj'so that the angle FGH be equal to the aui;le LCGj and complete the solid parallelopiped GK upon the base FH, one of whose insisting lines is FD, whereby the s
...olids CD. GK must be of the same altitude. Therefore the solid AB is equal (7. 3. Sup.) to the solid GK, because they are upon equal bases AE,FH, and are of the same altitude: and because the solid parallelopiped CK is cut by the plane DG which is parallel to its opposite planes, tlie base HF is (3. 3. Sup.) to the base FC, as the solid HD to the solid DC: But the base HF is equal to the base AE, and the solid GK to the solid AB : therefore, as the base AE to the base CF, so is the solid AB to the solid CD.

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