[ 74i ] 
Crons with the planes of the Tides, as QT, and R/, 
will make right angles with PC; that is, QT and 
R/ are the lines (S, 5 .) of the Tides PQJPR, and 
MC NC are whole fines. Now the ifofceles tri- 
angles MCN, QTy, rt R, being manifeftly limilar; 
as alfo MN, the fubtenfe of the arc which meafures 
the angle QP R, being equal to (2a) twice the 
line of half that angle ; we (hall have MN : MC : : 
Qy : QT :: rR : RQ or, in the notation of the 
theorem, Qy = 2Stf, rR = isa. And further, 
the chords Q r_ qR being equal, and equally diftant 
from the center of the fphere, as alfo equally inclined 
to the axis PC, will, if produced, meet the axis 
produced, in one point Z. Whence the points Q, 
y, R, r, are in one plane (2. el. 1 1.), and in the cir- 
cumference in which that plane cuts the furface of 
the fphere : the quadrilateral Q jq R r is alfo a feg- 
ment of the. ifofceles triangle ZQ q, cut off by a line 
parallel to its bafe, making the diagonals QR, yr, 
equal. And therefore, by a known property of the 
circle, Q q x r R -f jR 2 = QR 2 ; which, fubflitut- 
ing for Q q and Rr the values found above, 2 d for 
Or. ** for QR, and taking the fourth part of the 
whole, becomes S s a 2 + d 2 = the propofition 
that was to be demonftrated. 
Note 1. If this, or the preceding, is applied to a 
plane triangle, the lines of the Tides become the 
Tides themfelves ; the triangle being conceived 
to lie in the furface of a fphere greater than 
any that can be afligned. 
Mote 2. If the two lides are equal, d vanilhing, 
the operation is Ihorter : as it likewife is when 
#ne or both Tides are quadrants. 
Note 
