422 Haiiy on the Measuring of the Angles of Crystals. (June, 
of crystals sent me by M. Selb, First Counsellor of Mines to the 
Great Duke of Baden, and Director of the Mines of the Prince 
of Furstemberg. These crystals are diaphanous, of a consider- 
able size, a regular figure, and all their faces are smooth. And 
the satisfaction of owing them to a philosopher so justly cele- 
brated has doubled the value which they derive in my eyes from 
their perfection, and from the happy influence which they have 
had upon the results of my investigations. 
I have continued to adopt as the type of the species, the 
rectangular octahedron obtained by mechanical division. But 
* [have changed its dimensions in conformity with new measures, 
taken with all possible care by means of the common gonio- 
meter. Let s s’ (fig. 7) be the octahedron in question; if I draw 
cs, the axis of the pyramid, then c r and c ¢, the one perpendi- 
cular to k x, the other tos¢, then rs and¢ s, the angle s 7c will 
measure half the incidence of P on P” (fig. 8), and the angle 
s tc (fig. 7) half the incidence of P” on P’. But if we make 
erpes: 713: 83andct:cs: VS 2: / 3%, I find 76° 12’ 
for the first incidence, and 101° 32’ for the second. In the same 
hypothesis, the incidence of P on P” (fig. 8) = 119° 51’. 
‘The cosine of the angle which measures the incidence of the 
face ns 2, (fig. 7) on the face ks h, situated on the opposite 
side of the same pyramid is + radius, and that of the angle which 
measures the incidence of ks x on k s w is , of it; so that if 
we represent the first cosine by .4,, it will be sufficient to add 
unity to each of the terms of the fraction to have the expression 
for the other cosine. 
Mr. Phillips has found for the inclination of P on P” (fig. 8) 
the same angle, 76° 12’, as that which results from my determi- 
nation. But he gives 101° 20’ instead of 101° 32’ for the inci- 
dence of P” on P’, which makes a difference of 12’. That of P 
on P”, deduced from the preceding would be equal to 119° 54’ ; 
the difference of which from that which I have obtained is 
only 3’, The ratio of » ¢to ¢ s, which leads to the incidence of 
P” on P, such as Mr. Phillips gives, itis / 100 to / 149. This 
report is similar to those in the two preceding articles: a slight 
modification of one of the two terms would be sufficient to reduce 
it to my ratio. If we add unity to the last figure of the second 
term, we make it / 100: / 150, or VW 2: 3, as in my 
determination. 
According to the ratio / 100 to / 149, the cosine of the 
angle which measures the incidence of » s a (fig. 7) on ks h is 
the 42, of the radius, instead of the 1, which presents a kind of 
discordance between the two ratios, so well conneeted together 
mmy determination. If this last is not the true ratio, it must be 
admitted at least that it is the most satisfactory to the mind, 
* If we wish to put the two ratios under a form in which they will have the line 
es for a common term, we will makecs = V 24, ct = V l6,cr = v 39. 
