TERMINOLOGY. . 88. 



1. DERIVATIONS FROM THE SCALENE FOUR-SIDED PYRAMID. 



. 88. DERIVATION OF MORE ACUTE AND MORE OB- 

 TUSE PYRAMIDS, OF SIMILAR BASES WITH THE 

 FUNDAMENTAL FOKM. 



From every scalene four-sided pyramid may be 

 derived a more obtuse pyramid of the same kind, 

 possessing a similar base with the fundamental form. 



First of all, one of the prismatic axes of the given py- 

 ramid is fixed upon as its principal axis (. 41.) ; according 

 to this the pyramid itself is brought into its upright po- 

 sition (. 42.). Apply, after the first process (. 80.), tan- 

 gent planes to the terminal edges AB, AC, &c. of this form 

 ABCB'CX, Figs. 37., and enlarge them, till they intersect 

 each other from all sides. There will arise a form AFGIHX, 

 contained under eight isosceles triangles, which, by four 

 and four, are equal and similar to each other. If the form 

 be considered as a four-sided pyramid, its base FGIH has 

 the form of an oblong or rectangle. This is the intermediate 

 form (. 80.). It will be proper to observe here, that these 

 intermediate forms are not mere geometrical conceptions, 

 but that they are very frequently found in nature, and 

 will be farther explained in . 97 



Place now, after the third process (. 82.), one plane 

 on each of the terminal edges AF, AG, &c. of this inter- 

 mediate form, the inclination of this plane to the faces of 

 the form, and to each other, being such as to enable them 

 to intersect each other, after the necessary enlargement, in 

 the plane of the base, and thus to produce a plane figure 

 QSra'C', similar and parallel to BCB'C', the base of the 

 fundamental form. These planes will contain the required 

 form, namely, a scalene four-sided pyramid. 



This process may also be applied inversely, that is to 

 say, for any given scalene four-sided pyramid, we may find 

 the one from which it is derivable, according to the method 



