326 APPENDIX CALCULATION OF THE 



OCTAHEDRON WITH A SQUARE BASE. 



The plane angles at the base are consequently 

 =. 90" -J A , , and may be called A 2 . 

 Terminal edges are all equal. 

 Edges of the base are all equal. 

 Ratio of a terminal edge : an edge of the base 

 :: sin. A 2 : sin. A,. 



Ratio of an edge d h : a perpendicular on the 

 base, d e, : : R : cos. | A l . 

 Inclination of an edge on the axis may be 

 called A^ and may be thus found. 



cos. A 3 =r cos. | /, 



The same spherical triangle from which 

 we have determined the angle A^ maybe 

 used to determine A^ which in the above 

 fig. is the angle k d h^ and is the hypothe- 

 nuse of that triangle. 



The known formula to determine A z is 



,, cot. \ I, cot. 45" 

 cos. ~ 



But as cot. 45 R, this evidently becomes 



cos. A 9 cot. | /, . 



Inclination of edge on edge over the summit 

 will consequently be 2A 3 . 

 Ratio of \ a diagonal of the base : \ the axis 

 :: tang. A 3 : R. 



The relation of/,. / 2 , and^4 2 , may be thus ex- 

 pressed, 



cos. i J 2 = cos. ^ g tang, if, 



We may observe that the angles of the 

 spherical triangle marked at the base of the 

 fig. are 7 /,, | 7 15 and 90", and that the side 

 d h c, wjiich we have called A 2 , is the hypo- 



