On Napier’s Rules of the Circular Parts. 257 
Now if we apply the theorem to the given triangle and its 
two complementary ones, we shall get 
Given Rsin a@=sinhksinA 
triangle. | Rsin ) = sin Asin B 
Ist complem’ {R cos B = cos J sin A 
triangle, R cos h = cos b cos a 
2d complemY { R cos A = cos asin B 
triangle. R cos £ = cos 4 cos a. 
The solution of every case in right-angled trigonometry re- 
quires an equation, or proportion, between three parts of the 
triangle, viz. two given parts and one sought. The total num- 
ber of equations required for solving all the cases must therefore 
be 10; for 10 is the number of different ways in which five things 
can be combined, three and three. Of these 10 equations five 
have been obtained: and thus one theorem, applied to the given 
triangle and its two complementary ones, comprehends half the 
cases that occur in right-angled trigonometry. 
In a spherical triangle, the right angle being omitted, Lord 
Napier gave the name of circular parts to the two sides and the 
complements of the other three parts, namely, of the two angles 
and the hypothenuse. It is essential that the circular parts be 
taken in the natural order of their succession in going round the 
triangle: and hence it is obvious that they are susceptible of no 
more than five different arrangements. In every arrangement, 
the two parts next the middle part on the right and left are 
called adjacent parts ; and the other two, which stand first and 
last, are called opposite parts. All the possible arrangements of 
the circular parts may be thus exhibited, each part occupying 
the middle place successively, viz. 
Opposite 
Part. 
Adjacent) Middle | Adjacent Opposite 
Part. Part. Part. Part. 
| 90-—h | 90—A b 
—_—|— 
90—B | 90—A | 90U—A 
90—A a 90—B | 90—h 
_—___ 
I0—h | 9O—A b a 90—B 
a | 90—B 
b | 7 
| b 
90—B | 90—h | 90—A b a 
One of Napier’s Rules is this: 
Rule 1.—The rectangle under the radius and the sine of the 
middle part is equal to the rectangle under the cosines of the op- 
posite parts. 
Vol, 58. No, 282. Oct. 1821. K k Now 
