LAWS OF DECREMENT. 305 



CUBE. 



place by 1 row of molecules on the angle afb. The 

 edges of the defect of the primary form, would, in 

 this case, be, as we have already seen, proportional 

 to the corresponding edges of the primary form, and 

 might consequently, if the secondary plane were suf- 

 ficiently enlarged, be equal to those edges. The edges 

 of the new plane might therefore coincide with the 

 lines a b, a c, b c. 



If we now draw the diagonal g /", on the terminal 

 plane, we shall observe that one half of it is inter- 

 cepted at the point /*, by the edge a b of the second- 

 ary plane. 



As a decrement by 1 row therefore intercepts the 

 half diagonal fh, at the same time that it intercepts 

 the whole of the edges f a,fb,fc, the ratio offh :fc 

 may be assumed as the unit of comparison for deter- 

 mining the law of a simple or mixed decrement on 

 the angle afb. 



For let us suppose a decrement to have taken place 

 on that angle, by 2 rows in breadth ; if this decrement 

 be conceived to be continued until the edges af, and 

 b f, are again intercepted by the new plane, it is 

 obvious from what has been already stated, that only 

 one half of the edge/c would be intercepted by the 

 same plane. 



Here then the law of decrement would be ex- 

 pressed by the ratio of fh : |/c, or 2fh : fc; and 

 if f h : fc be represented by m : n, the ratio of 

 fh : f d, should be as 2 m : n, and would thus give 

 the required law of decrement by 2 rows in breadth 

 on the angle afb. 



But the ratio of fh :fd, is that of radius : tang, of 

 the angle fhd- } yuidjh d is the supplement of the 



2Q 



