1819.] Haiiy on the Measuring of the Angles of Crystals. 419 
Incidence of S on S—T, 121° 45’ 24”; C, 121° 41’ 54”; 
G, 121°40’. Diff. with T, 5’ 24”; and with C, 1’ 54”. 
Incidence of P on S—T, 150° 52’ 12”; C, 150° 50° 27”; 
G, 150° 45’. Diff. with T, 7’ 12”; and with C, 5’ 27”. 
Had not Mr. Phillips imposed upon himself the law of adher- 
ing strictly to mechanical measurements, he might have deduced 
the incidence of P on s from that of 121°40’ which he had found 
between s and s; adding 90° to the half of this last, which 
would have given him 150° 50’, and would have shown him that 
his goniometer placed him in opposition with himself to the 
amount of 5’. 
Lateral Faces.—The mutual incidences of these faces are in a 
particular case, in consequence of the common base of the two 
pyramids, composing the primitive octahedron, being a square. 
They may be assimilated to those which result from the laws of 
decrement on the edges of a cube, and of which it is sufficient 
that the measure be given, to deduce geometrically the angles 
derived from them with rigid accuracy. 
A simple construction will make what I have said intelligible. 
Let ab sh (fig. 5) be the square which represents the base indi- 
cated by the same letters, and let de, dk, k x, &c. be lines 
which make with each other the same angles as the faces g, 7, / 
(fig. 4), the letters indicating which are repeated on the lines 
which we are considering. Produce kd and x z till they meet 
fi; andha, s b, till they meet fk and iv. Then draw kn 
and «i perpendicular to fi. ‘The triangles ad e, n kf will be 
similar to those which I call measuring triangles; and it is by 
resolving them that we determine the inclinations of the faces, 
such as 7, g (figs. 4 and 5), whose positions coincide with their 
exterior sides de, k d (fig. 5). But these triangles are obviously 
rectangular in the present case, and the ratios of the sides adja- 
cent to the right angle are such that ad is equal to ae, and that 
nf is triple to kn. Ladd here a table of the angles to which 
these data lead, compared with those determined by the reflect- 
ing goniometer. I shall as before denote the former by T, and 
the latter by G. 
Incidence of g on J and ’/—T, 125°.—Mr. Phillips has omitted 
this incidence. 
Incidence of J on r and r’—T, 161° 33’ 54”; G, 161° 39’. 
Diff. 1’ 6”. 
Incidence of g on r and 1’—T, 153° 26’ 6”; G, 153° 25’. 
Din. 1 6”: 
Incidence of r on 7’—T, 143° 7’ 48”; G, 143° 10’. Diff. 
2 12”. 
Incidence of r on »”—T, 126° 52’ 12”; G, 126° 45’. Diff. 
apy Sa 
I shall further remark, that the two faces 7’, r’, making equal 
angles in contrary directions with the face g, it is sufficient to 
know one of these angles to deduce from it the mutual inclination 
2pD2 
