. 101. OF THE CONNEXION OF FORMS. 99 



GM : MA = B"M : MA', 



or 



GM : MA = MG . : MA', 



V 2 



therefore 



MA' == .MfL 



In order to find the ratio of the axis of the more acute 

 pyramid, from which the more obtuse one may be derived, 

 upon the supposition of their horizontal projections being 

 equal, let FGF'G' be the base of the latter ; BCB'C' will 

 be the base of the former, if the axes of the two pyramids 

 are supposed equal. 



But we have BCB'C' = | FGF'G'. 



Therefore BC' = F-^, 



and BC' : FG = MB : MG = 1 : 1. 



Produce the line MB, and with the distance MG, de- 

 scribe from the point M the arc GG", MG" will be = MG 

 = MB. V 2. 



Now produce the axis AM, and draw G"A" parallel to 

 B A ; MA" will be half the axis of the more acute pyramid, 

 its horizontal projection being equal to FGF'G'. In the 

 similar triangles BAM, G"A"M, the following proportion 

 takes place : 



BM : MA = G"M : MA", 

 or 



BM : MA = MB. J 2 : MA", 

 therefore 



MA" = MA. V 2. 



. 101. SERIES OF ISOSCELES FOUR^SIDED PYRAMIDS. 



Every derived pyramid may again be considered 

 as a fundamental form, and the derivation maybe con- 

 tinued. This will produce a series of icosceles four- 

 sided pyramids, whose axes increase and decrease 



