256 On Napier’s Rules of the Circular Parts. 
the oblique angles of a right-angled spherical triangle, as a pole, 
so as to meet the side opposite that angle and the hypothenuse, 
both produced if necessary ; another right-angled triangle will be 
formed by the intersections of the three circles, which is said to 
be complementary of the proposed triangle. 
Every triangle has two complementary ones, according as the 
pole of the great circle is made to coincide with one or other of 
the two oblique angles. 
The relations between a triangle and its complementary, are 
reciprocal. They have a common angle, namely, that which is 
adjacent to the produced side. The other four parts are the 
complements of one another; the hypothenuse of one triangle 
being the complement of the side adjacent to the common angle 
of the other; and the third side of one, the complement of the 
remaining angle of the other. It is sufficient kere to mention 
these properties, as the complementary triangle is treated of in 
all the elementary books *. 
Let 4 denote the hypothenuse of a right-angled' triangle; a 
and J, the two sides; A and B, the angles opposite to a and J, 
respectively: then the following Table will exhibit at one view, 
all the parts of the proposed triangle and its two complementary 
ones. 
; J. Hypo- . 53 
Side. Angle. shane Angle. Side. 
Given triangle. b A h B a 
1 90—h A 90—b | 90—a | 90—B 
——— |§ ———_ SS | | | 
triangles 
In this Table the hypothenuses of the three triangles occupy 
the middle column; and each angle is placed between the hypo- 
thenuse and the other containing side. 
2. Theorem 1.—In every right-angled spherical triangle, the 
rectangle under the radius and the’ sine of either side is equal to 
the rectangle under the sine of the angle opposite to that side, 
and the sine of the hypothenuse. 
This proposition is demonstrated in all the elementary trea- 
tisest. It is no more than an application to right-angled tri- 
angles of a general property of all triangles, namely, that the sines 
of the sides are proportional to the sines of the angles opposite 
to them. 
* See in Playfair’s Geometry, Spher. Trig. Prop. 20; or in Simson’s 
Euclid, Spher. Trig. Prop. 19. 
+ See in Playfair’s Geometry, Spher. Trig. Prop. 19; or in Simson’s 
Euclid, Spher. Trig. Prop. 18, 
Now 
