322 On Measuring the Depths of the Moon’s Cavities. 
present the shadow projected by the side bod. Let biand gh 
represent lines drawn from the observer to the extremities J and d 
of the shadow; and ABDE the circular plane in which are found 
the straight lines gh and the point 4. Let EC be the direction 
of a line joining the centres of the earth and this plane; then 
g and bi may be considered parallel and in the same plane 
that is the plane ABDE. ‘ 
Draw the diameter AD at right angles to EC; AD then is 
the boundary of vision, First let the moon be in quadratures ; Jd 
will then be perpendicular to gf, that is to EC, and EC will be 
the boundary of illumination; therefore bd=hi, that is the 
-apparent = to the true length of the shadow. 
Hence, in the right-angled adde are given, and the 
zbde=(ZACLJ the angle made between the edge of the ca- 
-vity and the @’s limb) to find de, the depth required. 
If the moon be not in quadratures, J d is not perpendicular 
to Zi, and consequently the apparent not equal to the true 
length of the shadow. 
In this case let rt be supposed to be the direction of a ray 
touching’ the edge of the cavity r¢a. 
Let. fall the perpendiculars ¢z and ry, join rC, and draw wf 
pat? jel to DC, and from ¢ Jet fall #0 perpendicular to rq ; 
th en will wt be the apparent length of the shadow, r¢ the true 
length, and ro the depth of the cavity. : 
In the triangle wér are given the side wf and the angle 
w t r=(the elongation if the moon be in her first or last quarters, 
or to its supplement if in the second or third quarters) to find r é. 
Then iv the Av.o.t. the Zr.t.o. = the Zrtz— Zo.t.r. But 
2r.t.% = Llrxe. = £z%.C.B = the ¢’s elongation and the 
(As 0.é.r. and x.¢.q. being similar) 20.é.r. = D.C.7. the angle 
between the edge of the cavity and ¢’s limb ., 27.é.0.= Zriz 
— £o.t.z. and r ¢ are given to find r.o. the depth required. 
If (a) be put = the angle between the edge of the cavity and 
¢’s limb, (2) = the apparent length of the shadow, (e¢) = 
the elongation, and (d) = the depth of the cavity ;—the fol- 
lowing formule are deducible : 
Lx Corsi, @ 
When the moon is in quadratures d = ———- 
« 5 
sine(ccja) x! 
sine ec, 
‘The angle (a) between the edge of the cavity and the moon’s 
limb is taken by placing one wire of a micrometer so as to join 
the cusps, and moving the other from the edge of the cavity till 
it becomes a tangent to the disk: the measure thus taken ; ra- 
dius :: (’s semidiameter : to the versed sine of (4). 
The 
When not in quadratures......d= 
