98 TERMINOLOGY. . 100. 



of the more obtuse one, and vice versa. This Is termed 

 the diagonal position. If from the more obtuse pyramid ano- 

 ther still more obtuse is derived, and from the more acute one 

 another, still more acute ; these new pyramids are diagonal- 

 ly situated to that one from which they are derived ; but to 

 the fundamental form they are in such positions as to have 

 their sides and diagonals parallel to the analogous lines of 

 the other, and this is termed the parallel position. 



. 100. EATIO BETWEEN THE DERIVED AND THE 

 FUNDAMENTAL FOEM. 



The axis of an isosceles four-sided pyramid, 

 whose faces touch the edges of another, is to the 

 axis of this latter pyramid, in the ratio of */i : 1 ; 

 the axis of that pyramid, whose edges are touched 

 by the planes of another, is to the axis of the latter, 

 in the ratio of *J2 : I. In both cases, the sides of 

 the horizontal projection are supposed equal. 



Let AM, Fig. 45., be half of the axis, BCB'C' the base 

 of the fundamental form ; FAG will be a plane, laid on 

 the terminal edge AB, GAF' another, kid on the termi- 

 nal edge AC, &c. : therefore FGF'G' will be the base, and 

 FA, GA, &c. the terminal edges of the derived pyramid. 

 The axis AM is common to both. 



The square FGF'G' = 2. BCB'C', 

 hence FG = GF' = BCV2, 

 and FG : BC' = MG : MB = V 2 : 1- 



Describe from M with the distance MB, the arc BB"; it 

 will follow, that 



MB" = MB = . 



V2 



Draw the line B"A' parallel to GA ; MA' will be half of 

 the axis of the derived pyramid, its horizontal projection 

 being equal to BCB'C'. In the similar triangles AGM 

 and A'B"M, is 



