130 On Crystallography. 



Thus n I : t y : : ~ : 2 : : 6 : 5. Besides, in this same 



case, the angle v t y is straight ; from which we see how 

 easy it is to find the angle y n t. 



£7- The mensuralor triangles relative to the decrements 

 on the angles may be substituted for those which we have 

 considered in the decrements on the edges, and serve 

 equally well for determining the secondary forms. Let us 

 suppose, for example, that AG (fig. 8) represents a cubical 

 nucleus, which undergoes decrements by two ranges on 

 the four edges of the base A BCD, and that we wish to 

 know the angles of the pyramid SADCB produced by this 

 decrement. Having traced the diagonals BD, AC, I draw 

 from their point o of intersection the line op perpendicular 

 upon CD, then sp. If I take upon po the part, pr equal 

 to two ridges of molecule, and from the point r I raise r u 

 perpendicular upon A BCD, and which by the hypothesis 

 will be found equal to one ridge of molecule, the triangle 

 upr will perform the function of the ordinary mensurator 

 triangle, and by means of the right angle urp, and of the 

 relation 2: 1 between the sides pr and ur, we shall easily 

 find the incidence of DSC upon the base ABCD, as well 

 as the values of the other angles. For, on account of the 

 similar triangles u pr, spo> every thing is reduced to the 

 calculation of the a?igles of a straight pyramid in which 

 the side BC of the base, which is double of po, is to the 

 axis os in the relation of 4 to 1. 



On the other hand, — If I take upon Co the part Cn 

 equal to two diagonals of molecule ; and if from the point 

 n I raise nz perpendicular upon ABCD, On will represent 

 the distance from the point C to the fust lamina of super- 

 position, taken in the direction of Co, and nz will be 

 equal to one ridge of molecule; from which it follows that 

 the triangle z C n may also perform the function of men- 

 surator triangle. 



We shall therefore have Cn : nz : : -2 v 7 2 : 1, and be- 

 cause the triangle zCn is similar to the triangle .?Co, the 

 question considered under this new point of view will be 

 reduced to seek the angles of a straight pyramid, in which 

 the demi-diagonal Co of the base is to the axis os, as 

 2 s/ e~: 1, which is sufficient for having all the rest. We 

 shall have occasion more than once thus to substitute one 

 mensurator triangle for the other, when there will result 

 from it more facility in resolving the problems. 



All the details upon which we have acted ought to be 



regarded 



