398 APPENDIX ON THE 



if we draw n ,r, parallel d /*, the triangles xjn, dfh 

 are similar, and x n = d o is of d h. 



But dh m dg- 3 fig. 364. The plane o' is produced 

 therefore by a decrement, consisting of 3 rows in 

 breadth, on the plane d b af. 



This demonstration, it may be remarked, is purely 

 geometrical, and limited in its application, to this 

 particular case. The same method might however 

 be adapted to other cases; but the problem would 

 frequently become extremely complicated, and diffi- 

 cult of solution by the aid of geometry alone. 



Perceiving this difficulty, and the limited nature of 

 the method itself, Mr. Levy has generalised the pro- 

 blem by giving it an algebraical form, and has pub- 

 lished an interesting paper on the subject in the 6th 

 vol. of the Edinburgh Philosophical Journal, p. 227. 

 In this paper, Mr. L. has given formulae for deter- 

 mining the law of decrement, by which any secondary 

 plane, modifying any parallelopiped, is produced, 

 whenever two of the edges of that plane, not being 

 parallel to each other, are parallel to two known 

 edges of the crystal. 



The following brief abstract of Mr. Levy's paper 

 is inserted here, for the purpose of affording the 

 reader a more immediate reference to the formulae it 

 supplies ; and as an additional example of a method of 

 investigation, which may be advantageously applied 

 to other points of crystallographical research. 



To derive these formulae, Mr. Levy has first sup- 

 posed the edges of the primary form to be represented 

 by three co-ordinate axes, and the primary planes, 

 consequently, to correspond to the three co-ordinate 

 planes. He has then found the equations of all the 

 planes concerned in the solution of the problem; 

 and by combining these equations, has obtained the 



