416 Haiiy on the Measuring of the Angles of Crystals. (Juxx, 
or:: 419: /31. This gives us crs = 51° 56’ andesr= 
38° 4’, the first of which is too great, and the second too small, 
in consequence of the quantities neglected. I see that if I add 
unity to each of the terms of the ratio, c 7 will be more aug- 
mented in proportion than cs, which has a tendency to make 
the two angles approach to the result given by observation. I 
shall then have cr: cs :: ¥ 20: ¥ 32, or:: /5: “8, Thus 
the ratio has all the requisite simplicity to give it the character 
of a limit. This ratio gives 51° 40’ for the measure of the angle 
crs, and 141° 40’ 16” for the incidence / s g ord gd xn, results 
which approach very nearly to the mechanical measurement. 
On the same hypothesis, the ratio of the two demi-diagonals 
g and_p of the faces of the primitive rhomboid is that of “ 15 
to / 13; and the cosine of the angle which measures the small- 
est incidence of the faces of the rhomboid is ,,th of radius, 
which gives for that incidence 85° 36’, and for the greatest. 
94° 24’. Setting out from the same ratio, we have 133° 48’ 46” 
for the angle which two adjacent faces on the same pyramid 
make with each other. 
We find in the beautiful work published by Malus on double 
refraction a determination of the mutual incidences of the faces 
of a rhomboid of quartz which this celebrated philosopher ascer- 
tained by reflection making use of the repeating circle. He 
gives 94° 16’ for the greatest, and 85° 44’ for the smallest.* 
I was curious to know how far the differences between the two 
measurements would go relatively to the other incidences, and 
what would be the ratios between the principal dimensions of 
the rhomboid of quartz which would result from Malus’s mea- 
surement. I found, by following a method similar to that which 
led me to the ratio / 5: 8, that in the present hypothesis, 
we should have g: p:: / 718: / 625; that the cosine of the 
smallest incidence of the faces would be +23, of radius, and that 
the ratio between c r and c s would be / 1157 to / 718. We 
should have for the incidence of /gs or g t s 133° 44 46” 
instead of 133° 48’ 46”, making a difference of 4’. 
We might substitute for the ratio / 1157 to / 718 between 
cr and cs that of / 14y to / 240, which is more simple, and 
which gives only half a minute of difference in the angles that 
depend upon it from those obtained by the first ratio. This 
suggests a reflection which, I think, I ought not to omit. 
If I were to show to a mathematician the ratio of / 149 to 
/ 240, informing him that it is the ratio which exists in the 
* Theorie de 1a Double Refraction, p. 242. Mr. Phillips gives 94° 15! and 
85° 45’, which differs only 1’ from the result of Malus. It was occasioned by Mr. 
Phillips’ goniometer being only divided into five and five minutes. } 
+ M. Malus appears to have neglected the seconds in measuring the angles 
quoted in the text, 
