312 APPENDIX CALCULATION OF THE 



REGULAR TETRAHEDRON. 



Decrements on the edges. 

 Fig. 334. 



The law of any decrement on an edge of the regu- 

 lar tetrahedron, may be ascertained by means of the 

 two lines a 6, a c, drawn from the angles b and c, 

 perpendicularly on the edge jfg, on which the decre- 

 ment is supposed to take place. 



The primary planes of this figure being equilateral 

 triangles, the lines a 6, and a c, which are perpen- 

 dicular to the edgeyg*, are equal. 



The angle b a c, is that at which the adjacent pri- 

 mary planes incline to each other, and is already 

 known and called /, . 



Let the new plane d e, be one of the planes belong- 

 ing to mod. /of the tetrahedron, and inclining on the 

 primary plane P' at an angle which we shall call J 6 . 

 And let d a, a e, be the portions of the lines a b, a c, 

 intercepted by the new plane. 



In the triangle dae, the angle aed, is =(180 7 6 ), 

 and consequently the angle a d e, is (/ 6 /,). 



Wherefore, 

 a e : a d :: p : q :: sin. (/ 6 /,) : sin. (180 >T 6 ). 



We may determine p and q, those being the lowest 

 whole numbers which will express that ratio, by the 

 means of either the natural sines, or their logarithms ; 

 and when determined, they will express the law of 

 decrement by which the new plane has been pro- 

 duced. 



