Again, adding the second and third, cos A x cos B + sin A x sin B = cos (A — B) ; And, lastly, subtracting the third from the second, cos A x cos B-sin A x sin B = cos (A + B).

7. Again, since by the first of the above theorems, sin A x cos B sin (A+B) + sin (A→B), if A+B=S,

and A-B-D, then (Lem. 2.) A= -,and B=

S+D
2

S-D 2

=sin S+ sin D.

But as S and D may be any arches whatever, to preserve the former notation, they may be called A and B, which also express any arches whatever: thus,

In the same manner, from Theor. 2. is derived,

A B

2

4th, 2 cos

A+B
2

X sin

A-B
2

sin A-sin B.

In all these theorems, the arch B is supposed less than A.

Theorems of the same kind with respect to the tangents of arches may be deduced from the preceding. Because the tangent of any arch is equal to the sine of the sin (A+B)

arch divided by its cosine, tan (A+B)

cos (A+B)`

But it has just been shewn, that sin (A+B) = sin A. x cos B + cos A x sin B, and that cos (A+B) cos A x cos B-sin A x sin B; therefore

9. If the theorem demonstrated in Prop. 3. be expressed in the same manner with those above, it gives

10. In all the preceding theorems, R the radius is supposed = 1, because in this way the propositions are most concisely expressed, and are also most readily applied to trigonometrical calculation. But if it be required to enunciate any of them geometrically, the multiplier R, which has disappeared, by being made 1, must be restored, and it will always be evident from inspection in what terms this multiplier is wanting. Thus, Theor. 1, 2 sin A x cos B = sin (A + B) + sin (A — B), is a true proposition, taken arithmetically; but taken geometrically, is absurd, unless we supply the radius as a multiplier of the terms on the right hand of the sign of equality. It then becomes 2 sin A x cos BR (sin (A + B + sin (A—B)); or twice the rectangle under the sine of A, and the cosine of B equal to the rectangle under the radius, and the sum of the sines of A+B and A-B.

In general the number of linear multipliers, that is, of lines whose numerical values are multiplied together, must be the same in every term, otherwise we will compare unlike magnitudes with one another.

The propositions in this section are useful in many of the higher branches of the Mathematics, and are the foundation of what is called the Arithmetic of Sines.

ELEMENTS

OF

SPHERICAL

TRIGONOMETRY.

PROP. I.

If a sphere be cut by a plane through the centre, the section is a circle, having the same centre with the sphere, and equal to the circle by the revolution of which the sphere has been described.

FO

NOR all the straight lines drawn from the centre to the superficies of the sphere are equal to the radius of the generating semicircle; (Def. 7. 3. Sup.) Therefore the common section of the spherical superficies, and of a plane passing through the centre, is a line, lying in one plane, and having all its points equally distant from the centre of the sphere; therefore it is the circumference of a circle, (Def. 11. 1.), having for its centre the centre of the sphere, and for its radius the radius of the sphere; that is, of the semicircle by which the sphere has been described. It is equal, therefore, to the circle, of which that semicircle was a part. Q. E. D.

DEFINITIONS.

I.

ANY circle, which is a section of a sphere by a plane through its centre, is called a great circle of the sphere.

COR. All great circles of a sphere are equal; and any two of them bisect one another.

They are all equal, having all the same radii, as has just been shewn; and any two of them bisect one another : for, as they have the same centre, their common section is a diameter of both, and therefore bisects both.

II.

The pole of a great circle of a sphere is a point in the superficies of the sphere, from which all straight lines drawn to the circumference of the circle are equal.

III.

A spherical angle is an angle on the superficies of a sphere, contained by the arches of two great circles which intersect one another; and is the same with the inclination of the planes of these great circles.

IV.

A spherical triangle is a figure, upon the superficies of a sphere, comprehended by three arches of three great circles, each of which is less than a semicircle.