82 TERMINOLOGY. . 94?. 



far as they belong to one and the same diagonal ; because 

 the axes of the fundamental pyramids have no influence 

 upon the dimensions of the bases. It will be exactly the 

 same, if the axes are in the ratio of the powers of the 

 number 2, that is to say, if the fundamental pyramids are 

 members of one series; as, for instance, P, P + 1, 

 P + 2, &c. 



. 94. EATIO OF THE DERIVED AND THE FUNDA- 

 MENTAL FORM. 



If in the pyramid P, the ratio of the axis, the 

 longer, and the shorter diagonal is expressed by 



a : b : c, 

 or in P -f n, by 



2 n . a : b : c; 



the ratio of the analogous lines in the same succes- 

 sion will be, 



for (?) m = m. a : b : m. c, or 

 for(f + n) m =2 n .m.a : b : m. c; 

 for (P) m = m. a : m. b : c, or 

 for(P-f n) m = 2 n .m.a :m.b : c. 

 Since, Fig. 40., in the ratio of 



3M : MB : MC' 



SIM and MC' are known quantities, being expressed by a, 

 c and m (for 3M is = m. a, or = 2 n . m. a, and MC' = c) ; 

 the only thing still to be effected, is to express MB in the 

 same manner, or only by b and m. 



Draw the line SN parallel to BM ; the triangles SC^N, 

 3BC'B', will be similar to each other, and 

 C'NcSN^C'B'rBB', 



= C'B' : MB -f MB'. 

 But 



C'N = C'B' NB' = CB' MS 



m -f 1 



