& dent'with the lines AB, AC, whose inclination is the 
real angle of the planes ; and , if we measure 
. bya instrum i ent, the it angle contained 
the oblique lines A 6, Ac, we obtain a measure of 
dle of tha'chystal ure Eo rr 
dangles is con r plane angles are first 
red with great accuracy by the goniometrical mi: 
croscope, or the angular micrometer adapted to a mi- 
croscope, and the inclination of the planes is deduced 
a trigonometrical formula. Whatever be the num- 
ber of plane angles which contain the solid ‘angle, we 
_ ean always reduce the’solid angle to one which is form- 
ed by three plane angles, and determine by the formu+ 
‘inclination’ of any two of them. Thus, if the 
‘angle at A, Fig. 8. is contained by five plane 
; and if it is required to find the inclination of 
planes’ ABC, ACD, we first measure the plane 
les CAB, CAD, and also the angle contained by 
lines AB; AD ; so that we have now reduced the 
solid angle contained by five angles, into one con+ 
_ tained by three e angles, CAB, CAD, BAD. 
-~ Legendre, in his Elements of Geometry, has given 
tet Pi egant solution of this problem by a plain con- 
struction ; and it is easy, from his ‘solution, to form an 
instrument for shewing the angles of the planes ‘with- 
out the trouble of ealculation. Thus let the angles 
. ‘BAC,;CAD, DAE, Fig. 9. be made equal to the three 
plane angles by which the solid angle is contained. 
_ Make AB”equal to AE, and from the points B,'E let 
fall the’ diculars BC, ED on the lines AC, AD, 
and let them meet at O. From the point C, as a centre 
‘with the radius CB, describe the semicircle BFG. From 
___ the point O draw OF at right angles to CO, and from 
___F, where it meets the semicircle, draw FC. The angle 
_ » GCF is the inclination of the two planes, CAD, CAB. 
In order to construct an instrument on this principle, 
to save the trouble of projection or calculation, we have 
only to form a ted circle BHEG, with three 
moveable radii, AC, AD, AE, and a fixed radius AB. 
The moveable ‘radii must have vernier scales at their 
extremities, that they may be set so as to contain the 
three plane angles which form the solid angle. Two 
moyéuble arms BG, EO, the former of which is divided 
into any viumber of equal parts, turn round the extre- 
mities’ B, E ; ‘and, <4 means of a reflecting mirror on 
their exterior sides, they can be: set in such a position 
as to be perpendicular to the radii AC, AD. \When 
this is done, the number of equal parts between € 
and O, divided by the number between B and C, is the 
fatural co-sine of the angle GCF; and therefore, by 
entering’a table of sines with'this number, the inclina- 
tion of the two planes will be found. 
“In order to obtain a more accurate result, however, 
‘we must have recourse to a trigonometrica!l formula. 
Let A, ‘Fig. 10. be the solid'angle, and let it be required 
to determine, by means of the three plane angles, the 
inclination of the surfaces ACB; ACD. Draw AM, 
AN in the planes ACB, ACD, and perpendicular to the 
common section AC ; join BM, DN. Then it is ob- 
vious, that the angle MAN is the inclination of the 
planes required, and that the angle BAD, which is an 
oblique section of the prism BM, will be equal to MAN 
when it is reduced to the plané AMN. By considering 
that the inclinations of the bounding lines of the oblique 
section of the prism, to the bounding lines of the per- 
icular section, are measured’ by the angles DAN, 
BAM, the complements of the two given lane angles 
SO MOL, x. PARTI. 
"ye eter Se veer 
GONIOMETER © 
837 
CAD, CAB, we shall obtain, by spherical trigonome- 
try, the following formula : ae ot 
Sin. =_ = 
- BAD+C€CAD—CAB  «, BAD+4+CAB—CAD' 
Ry | Sin. “Bi . Sin. 7 
Sin. CAB) Sin. CAD 
Or, calling 2 the angles of the surfaces of the crystal, 
B, C the plane angles at the vertex of these surfaces, 
and A the other plane angle, then we shall have. 
Sin. ? ga Rad(Sin. sate . Sin. “+ 
2 A i 2 g 
* ' Sin. By Sin. C. 
a formula from which thé value of @ may be obtained 
by a very simple calculation. 
Let the angle BAD, for example, be 62° 56’, the 
angle CAD = 100° 2’, and the angle CAB = 106° 10’, 
then we shall haye, by the preceding formula, 
«Op 45 Sin. 28° 24 Sin. 34/ 39” 
Sin.? 9= Rad Sin. 106" 10” Sin. 100° 2” * 
- Now we have, 
Log. sin. 28° 24! - 9.6772640 
Log, sin, 34° 32’... :9.7584954 
19.4307594 
Add 2 Log. of Rad... 20.0000000 
39.4307594 
Log. sin. 106° 10’ . 9.9824774 
Log. sin. 100° 2’ . . 9.9933068 
19.9757842 
Krom Hse io 39.43807594- 
Subtract. . 199757842 
2 Log. sin. © . os 19.4549759 
Log. sino + ee) 967274876 
Q 
Hence>= $2° 16’ 18" 
.. and ? = 64° 32! 36” 
the angle of the surfaces:of the crystal. 
A goniometer, upon another principle, has recently 
Goniometer, 
Brewster's 
PLATE ~ 
CCLXXVLL. 
Burrow’s 
been invented by the Rev. E. J. Burrow, fellow of goniometer. 
Magdalene College, Cambridge. The following is the 
description of it given by himself: 
« BG (Fig am) is'a steel bar, of about ,*, of aninch 
square, chamfered off to the point B. On BG is taken 
exactly an inch; BA, and A’ is made the centre of motion 
of the legs DE and de, of which DE is also brought to 
a point at D tocorrespond with B. To the other end of 
these legs is attached by the pin F, a moveable qua- 
drant, passing through the bar BG at C, and graduated 
to read towards the side of the shorter leg de: The 
handle GH is made’to project on the opposite: side to 
that on which the legs move, that it may not interfere 
with the use of a brass degree divided into minutes, to 
be attached to the centre A upon longer radii. The 
leg AB is divided accurately into tenths, and the two 
nearest the point B into: twentieths, and these again 
into fortieths. The whole instrument is about four 
inches and a half long. 
Now if the crystal, the angle of whieh is to be mea- 
20 
Fig. 11. 
