[ 542 ] 
Note 3. By comparing this propofition with that 
of the Lord Neper J, which makes the 39th 
of Keill’s Trigonometry, it appears, that if A C, 
AM, are two arcs, then lin. x lin. 
AC ~ -— = (f+d x b= 2 =) fin. i AC + 
fin. | AM x lin. | AC — lin. { AM. And in 
the folution of Cafe II. the firft of thefe pro- 
ducts will be the moft readily computed. 
CASE IV. 
When the part required fiands oppofite to a part , 
which is likewife unknown : Having from the data 
of Cafe I. found a fourth part, let the lines of the 
given lides be S, s ; thofe of the given angles 2 > a ; 
and the lines of half the unknown parts a and b j 
and we fhall have, as before, — b 2 = o\ 
and if the equation of the fupplements be 2 a ad -f- S z 
— /3 2 =e>; then, becaufe a 1 — 1 — b 1 — 1 — S sa z -\-d l > 
and (Z z =i — a 2 , fubftituting thefe values in the 
fecond equation, we get 
Theorem III. 
1 — 2 <r x 1 — d z — S 2 , . , . 
— 5 — = a z ; in words thus : 
1 — S 5 2 <r 
Multiply the produB of the fines of the two known 
angles by the fquare of the cofine of half the difference 
of the fides : add the fquare of the fme of half the dif 
ference of the angles ; and divide the complement of this 
% See Logarith. Canon, defer. Ediub. 1614. p. 48. 
fum 
