258 On Napier’s Rules of the Circular Parts. 
Now the truth of this rule will be established by applying it 
successively to all the arrangements of the circular parts; for, 
when this is done, the very same results will be found that have 
already been obtained by applying the foregoing theorem to the 
given triangle and its two complementary ones. The two pro- 
cesses are equivalent. Napier’s Rule is not only true, but it is 
sufiicient for solving half the cases of right-angled triangles. 
3. Theorem I1.—In any right-angled triangle, the rectangle 
under the radius and the sine of one side is equal to the rectan- 
gle under the co-tangent of the angle adjacent to that side, and 
the tangent of the other side. 
This theorem, in the form of a proportion, is likewise demon- 
strated in all the elementary treatises of trigonometry *. 
If we apply it to the given triangle and the two complementary 
triangles, we shall get 
Given {R sina = cot Btan J 
triangle. | R sin ) = cot A tana 
Ist complemY { R cos B = tan a cot h 
triangle. R cosh = cot A cot B 
2d complemY f R cos h = cot A cot B 
triangle. ReosA = tan 4 cot h. 
These five equations, together with the other five already ob- 
tained by means of the first theorem, embrace the whole com- 
pass of right-angled trigonometry. 
The remaining Rule of Napier is this, viz. 
Rule 11.—The rectangle under the radius and the sine of the 
middle part is equal to the rectangle under the tangents of the 
adjacent parts. 
This rule is true ; because, when it is applied to all the possi- 
ble arrangements of the circular parts, it brings out the same 
five results that have just been found by applying the second 
theorem to the given triangle and its two complementary trian- 
gles. The two Rules of Napier, taken together, are therefore suf- 
ficient for solving all the cases of right-angled triangles. 
It appears therefore that the whole doctrine of right-angled 
trigonometry may be brought within the compass of two theorems 
or rules in two different ways. First, we may employ the two 
complementary triangles; and then no more is necessary than 
two of the theorems found in every elementary treatise, with- 
out any artificial arrangement or new denominations. The two 
theorems, applied either immediately to the data in the given 
triangle, or to the same data transferred to one or other of the 
two complementary triangles, will solve every case. Or, se- 
* See in Playfair’s Geometry, Spher. Trig. Prop. 18, cor; or, in Sim- 
son's Euclid, Spher. Trig. Prop. 17, cor. 2. 
condly, 
