. 89- OF THE CONNEXION OF FORMS. 73 



described above. For this purpose inscribe that intermediate 

 form which belongs to the pyramid sought, in that which 

 is given. This is effected by bisecting each of the lateral 

 edges, and joining the points thus determined by straight 

 lines ; after which, lines must be drawn from the angles of 

 the oblong figure, which has thus been inscribed in the 

 base of the pyramid, to the terminal points of the axis of" 

 the fundamental form. A rhomb is now inscribed in the 

 rectangular base of the intermediate form ; the angles of 

 the rhomb coinciding with the centres of the sides of the 

 oblong. This rhomb will be similar and parallel to the 

 base of the fundamental form. But it likewise represents 

 the base of the derived form, which will be completed, if we 

 draw straight lines from the angles of this rhombic figure, 

 towards the terminal points of the axis of the fundamental 

 form, and lay planes on every two of those lines, which 

 are adjacent or contiguous to each other. 



. 89. BATIO BETWEEN THK DERIVED AND THE 

 FUNDAMENTAL FO11M. 



The axes of two scalene four-sided pyramids, of 

 which the one is derived from the other, according 

 to . 88., are towards each other in the ratio of 

 J: 1, if the derived pyramid is more obtuse; in 

 the ratio of 2 : 1, if the derived pyramid is more 

 acute than the given one. The horizontal projec- 

 tions of all these forms are supposed equal ; and 

 the axis of the fundamental pyramid =: I. 



Let BCB'C', Fig. 37., represent the base of the fundamen- 

 tal form, which bisects the axis AX in M, the centre of 

 the pyramid ; GAF will be a plane tangent to the termi- 

 nal edge AC, HAF another plane tangent to the terminal 

 edge AB, and consequently AFGIHX the intermediate 

 form, whose base is the rectangle FGIH. 



Circumscribe about this rectangle, a rhomb 10C1B'C' ? sirai- 



