110 TERMINOLOGY. . 109. 



Let AX, Fig. 45., be the axis, and ACXB the principal 

 section (. 37.) of the given rhombohedron ; AC will be 

 one of its terminal edges. 



According to the method of derivation applied, the ter- 

 minal point B' of the inclined diagonal of the derived rhom- 

 bohedron will be situated in the prolongation of the termi- 

 nal edge AC of the fundamental form. But this point lies 

 also in the prolongation of Q.S', a line perpendicular to the 

 axis in Q, AQ, being = AX (. 50. 11.). Therefore the 

 terminal point B' of the inclined diagonal of the derived 

 rhombohedron coincides with the intersection of the two 

 produced lines AC and QS', and AB' is this diagonal 

 itself. 



If in the same way, the lines XB and PS are pro- 

 duced till they intersect each other in C' ; XC' is the other 

 inclined diagonal of the derived rhombohedron parallel to 

 the former ; and the lines joining the point X with B', and 

 A with C', represent the terminal edges, AB'XC' accord- 

 ingly the principal section of the derived rhombohedron. 



The line AX, or the axis, is common to both principal 

 sections ACXB, and AB'XC' ; but the side of the horizon- 

 tal projection of the given rhombohedron is PC, while 

 that of the derived rhombohedron is QB'. 



In the similar triangles APC, AQB' we have 

 PC : QB 7 = AP : AQ = ^ AX : AX = 1 : 2 (. 50. 11.). 



Hence the side of the horizontal projection of the deriv- 

 ed rhombohedron is double the same line of the given 

 rhombohedron, their axes being equal ; and the inclined 

 diagonal AB' = XC' of the derived rhombohedron, is 

 double the terminal edge AC = XB of the given rhombo- 

 hedron. 



Draw now the line BA' parallel to C'A, and B"A' paral- 

 lel to B'A ; BA'B"X will be the principal section, and 

 A'X the axis of the derived rhombohedron ; provided the 

 two rhombohedrons have the same horizontal projection, 

 whose side is = BQ.. 



But in the similar triangles AC'X and A'BX the follow- 

 ing proportion takes place : 



