SIMILAR PYRAMIDS.

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The volumes of similar solids, terminated by any num-ber of plane faces, are as the cubes of the correspondinglines.

We can, in fact, decompose them into any number of similar pyra-mids, all having the same relation r, to that of their correspondingsides. But two pyramids, of which the corresponding sides are toeach other, as 1 is to r, are, to one another in volume, as 1 is to thecube of r. By adding together, on the one hand, all the small pyra-mids, and on the other, all the pyramids which are r 3 times more vo-luminous, the volumes will be to each other as : : 1 : r 3 .