130
SIMILAR PYRAMIDS.
rallel to itself, in such a manner, that the point a, is placed on A, abwill fall on AB, ac on AC, and ad on AD; whence the planes abcand ABC, abd and ABD, acd and ACD, will correspond, one withanother ; the two solid angles, therefore, a and A, of the two pyra-mids, will also be equal. In the same manner, it may be demonstrat-ed, that the solid angles, B and b, C and c, D and d, are equal ; andthus, all the conditions required in order that these two figures maybe similar, are fulfilled by the single condition of the two pyramidshaving their corresponding sides parallel.
If two pyramids, not having their sides parallel, havetheir edges proportionals, they will also be similar.
The three sides of each of their corresponding faces being propor-tionals, these faces will be similar; the plane angles, and consequent-ly, also the solid angles which they form, three and three, will beequal. Thus, all the conditions of proportion will be fulfilled.
Two solids, terminated by plane faces, are similar whentheir corresponding edges are proportionals, and their cor-responding angles, whether plane or solid, equal to oneanother.
We can always, in fact, decompose these solids into pyramids, thesides of which will be proportionals, and, consequently, the corres-ponding angles will be equal.
The volumes of the similar pyramids ABCDE abode ,fig. 26, pi. 7, are proportionals to the cubes of the corres-ponding edges.
In fact, the volume of each pyramid equals its base, multiplied bythe third part of its height ; or the bases, BCDEF, bcdef, beingsimilar figures, are proportionals to the square constructed on oneof their sides. Of these bases, we have, therefore, fig. 26, the sur-faces,
BCDEF : bcdef : : BCMN : bcmn.
Let us now, on BCMN, and on bcmn, as bases, construct cubes,and we shall have, for the volumes of the two cubes,
BC 3 = BC 2 x BC and be 3 = be 2 x be.
But BC : 6e : : 1 AH : 1 ah.
Therefore BC 3 : 6c 3 : : BC 2 x AH : bi a x J ah.
In this last proportion, the two latter terms represent the volumeof the two pyramids, and the two former terms represent the volumeof the two cubes