a7id Mineralogy. 167 
to the axis-edges ofy*; a is likewise a scalene six-sided pyramid 
depending onj^ or r -|- 1. Conceive the faces of P and y to be 
enlarged till they intersect ; the lines of their intersection are pa- 
rallel to the axis-edges of P, Sec. ; and y is therefore a scalene 
six-sided pyramid depending on P, To find the proportions 
which these pyramids bear to each other, and to the rhomboids, 
the following considerations are sufficient. The edges formed 
between a and y are parallel to those between y and m. The 
edges between y aadi m are parallel to the oblique diagonals of 
and consequently to each other. Hence the edges between 
a and y are parallel to the obtuse axis-edges of «/, and to the 
acute ones of a. The edges between x and y are likewise pa- 
rallel to each other. (It is necessary to consider two of such a 
kind as have the horizontal edge formed by y* and h lying between 
them). Therefore /r, y and a are three consecutive members 
in a series of scalene six-sided pyramids (35). If in the preced- 
ing general expression, we put w rrl, 7i = 0, n — and so 
on, we shall obtain (;? — 1)'", {y)% (p + l'), (p)'", (p + 1)", 
(j9-|-2y, (pq-1)'", (^o-b^)", (^ + 3)'. Out of these, three 
consecutive terms must be chosen, such that jo + 1 may appear 
twice, and p once ; in other words, that two pyramids may re- 
sult depending on r -{- 1 ', and one on r. The proper terms are 
{P + ly, {pT and (p -b I)-'''. Hence x {p + 1)' \ y= (p)"'; 
and a t=z {p -^iy*. The pyramid t remains undetermined. 
But if the combination included {p)'% whose more acute pyra- 
midal edges have the same position as the oblique diagonals of 
f% this particular hgure would produce horizontal edges of com- 
bination with t ; t therefore must either be {p — 2)' or 
{p — 4)" — (35), because in that case the parallel edges of 
combination appear for the first time. Now', could w'e discover 
the magnitude of any edge belonging to this figure, — of the ho- 
rizontal edge, for instance, between h and/J which easily admits 
of measurement by the French goniometer ; or, knowing the mi- 
neral to be calcareous-spar, could we remark that P is parallel 
to the form of the cleavage, — wo should then know every thing 
necessary for computing any required dimension of this cry- 
stal. 
43. Designation of Simple Forms and Co^nhmations. — Our 
second application is an easy and clear mode of designating 
