+log. versed sine 2x log, sine arc, or, 9.6989700+ log. versed sine 2 x log. sine arc. Therefore, log. versed sine= 2x log. sine arc-9'6989700.

(Q) The following logarithmical formulæ for the sine, cosine, &c. of any arc, may in some cases be useful, viz. Sinecos+tang-10=tang+10-sec=cos+10-cot=

20-cosec.

Cos=sine+10-tang=20-sec=sine+cot-10= cot+

10-cosec.

Tang=sine+10-cos=20-cot.

Cot cos+10-sine-20-tang.

Sec-tang+10-sine 30-sine-cot-cosec+10—cot=

20-cos.

Cosec cot+10-cos 30-cos-tang- sec+10-tang= 20-sine.

THE END OF PLANE TRIGONOMETRY.

BOOK III.

SPHERICAL TRIGONOMETRY.

CHAPTER I.

DEFINITIONS, &c. OF SPHERICAL ANGLES, ARCS, AND TRI

ANGLES.

(A) SPHERICAL TRIGONOMETRY treats on the properties of spherical triangles, or the position and magnitudes of arcs of circles described upon the surface of a sphere or globe. (B) A sphere, or globe, is a round solid body, each part of its surface being equidistant from its centre. This centre is common to every circle described on the surface of the sphere, wherein spherical trigonometry is concerned, and such circles are called great circles.

(C) All great circles divide the globe into two equal parts, and consequently bisect each other at the distance of a semicircle or 180°. Hence two arcs cannot enclose a space, unless they are both semi-circles.

(D) A spherical angle is the inclination of the planes of two great circles to each other, which circles intersect or meet each other on the surface of the sphere, in a point called the angular point. The inclination of these planes must always be ineasured on the arc of a great circle, 90° from the angular point.

(E) A spherical triangle is formed on the surface of the sphere by the intersection of three great circles, and consists of three sides and three angles; any three of which parts being given the rest may be found.

(F) All the sides of a spherical triangle are arcs of equal circles, and the angles of a spherical triangle are measured by arcs of circles, having the same radii as the sides. Hence the sides and angles of spherical triangles are always expressed in degrees, and parts of degrees.

(G) If one angle of a triangle be 90°, it is called a rightangled triangle.—If one side be 90°, a quadrantal triangle. -If no angle or side be 90°, it is called an oblique-angled triangle.

(H) The pole of a circle is a point on the surface of the

sphere equidistant from every part of that circle of which it is the pole. Consequently every circle has two poles diametrically opposite to each other, and the arc of a circle comprehended between each of these poles, and the circumference of such a circle, is a quadrant. No two circles can have the same, or a common pole. If a straight line be drawn from the pole of any circle to the centre of the sphere, it will cut the diameter of that circle at right angles.

(I) All great circles passing through the pole of another great circle, cut that circle at right-angles; and if two circles cut each other at right-angles, in the poles of a third circle, the four points of intersection with that third circle, will be the four poles of the cutting circles; viz. the two opposite points will be the poles of that circle which is described between them.

(K) Sides and angles of spherical triangles are said to be of the same species, kind, or affection, when by comparing any two sides, any two angles, or an angle and a side together, you discover each to be greater or less than a right-angle, or equal to a right-angle.

But when by comparing a side with a side, an angle with an angle, or a side with an angle, you discover one to be less and another greater than a right-angle; such sides and angles are said to be of different species.

(L) Spherical triangles are equilateral, isosceles, or scalene, according as they have three equal sides, two equal sides, or three unequal sides.

N.B. In any of the following propositions, wherever the word circle, or arc of a circle occurs, it must always be understood to be a great circle, or the arc of a great circle. And, that all circles concerned in spherical trigonometry are equal to each other.

PROPOSITION I.

(M) If one great circle intersect another great circle in any point a, all the angles deA, scribed from the point a, on the same side of any arc CAD, or EAB, are equal to two rightangles; and all the angles made about the L point a, are equal to four right-angles.

D

DEMONSTRATION. Let CA be perpendicular to EB, then will the angles EAC and CAB be each of them a right-angle. If AL be drawn, then LAB and EAL are equal to two right-angles, for the angle CAB is increased by the angle LAC, and CAE is diminished by the same, therefore LAB+EAL=EAC+ CAB.

In the same manner it may be proved that the angles EAD and DAB are together equal to two right-angles, for if ac be

perpendicular to EB, AD will likewise be perpendicular to it, therefore all the angles about the point A are equal to four right-angles. Q. E. D.

(N) COROLLARY I. If two arcs of circles intersect each other, the vertical or opposite angles will be equal,

For the angles EAC and CAB are equal to two right-angles, also CAB and BAD are equal to two-right angles; from each of these equals take the angle CAB, then EAC is equal to BAD. In the same manner it may be shown that the angle Cab is equal to the angle EAD.

(0) COROLLARY II. All the exterior and interior angles of a spherical triangle are together equal to six right-angles.

For the interior angle CAB and exterior angle BAD are equal to two right-angles; likewise ACB and BCG are equal to two right-angles; and ABC and CBG are also equal to two rightangles,

PROPOSITION II.

(P) If a great circle BC be described meeting two great circles ABG and ACG, which pass through the pole a of the circle BC; the angle CDB at the centre of the sphere, upon the circumference BC, is

D T.

H

A

the same with the spherical angle BAC, and the arc BC is called the measure of the spherical angle BAC.

DEMONSTRATION. Since A is the pole of BC, AB and AC are quadrants (H. 133.) and the angles ABC and ACB are rightangles (I. 134.)

Let D be the centre of the sphere, and join DC and DB; then because the arcs AC and AB are each of them quadrants, the angles ADC and ADB are right-angles (H. 133.): therefore the angle CDB is the inclination of the planes of the circles ABG and ACG to each other, and consequently, (D. 133.) it is the measure of the angle CAB. 2. E. D.

(Q) COROLLARY. If two circles cut each other in two points, the angles at these points are equal to each other, and to the distance between the poles of these circles. Viz. the angle BAC is equal to the angle BGC, for the arc BC is the measure of them both; and the distance between the point в and the pole of ABG is a quadrant, also the distance between c and the pole of ACG is a quadrant, (H. 133.) therefore the difference of their distances will be BC.

PROPOSITION III.

(R) If to the point of intersection A of two great circles, two tangents se and af be drawn; the angle enf will be the measure of the spherical angle BAC.

For AG is the diameter of the sphere, and ADC and ADB are right-angles, but fAD and eAD are also right-angles, being tangents to the circles ABG and ACG; therefore fa and ea are parallel to DB and DC, hence the angle eaf is equal to the angle CDB; but CDB is the measure of the spherical angle BAC, therefore eaf is the measure thereof.

In the same manner it may be shewn that the angle ILH is equal to CDB, for it is likewise the inclination of the planes ABG and ACG; hence all spherical angles must be measured on the arc of a great circle 90° distant from the angular point, for CB, and not IH, is the measure of the angle BAC.

PROPOSITION IV.

(S) If from the angular points A, B, C, of a spherical triangle ABC as poles, there be described three arcs of circles EF, DE, and DF, forming a new spherical triangle DFE; each side of this new triangle is the supplement of the measure of the angle at its pole, viz. the side FE is the supplement of the measure of the angle ; DE of the angle B; and DF of the angle c.-Likewise each angle of this new triangle is the supplement of the measure of that side of the original triangle ABC, to which it is opposite; viz. the angle D is the supplement of the measure of BC, the angle F of AC, and the angle E is the supplement of the measure of the side AB.

DEMONSTRATION. Since B is the pole of the circle DGHE, every part of this circle is 90° distant from B (H. 133.), therefore the distance between B and E is a quadrant, and вн and BG are quadrants.

Since A is the pole of the circle ELMF, every part of this circle is 90° distant from a (H. 133.), therefore the distance between A and E is a quadrant, and AM and AL are quadrants.

Since A and B have each been proved to be 90° distant from E, and AM and BG quadrants, being the side AB produced to M and G; E is the pole of the circle GABM.

In the same manner it may be proved that F is the pole of KALC, and that D is the pole of NBCH

Therefore EM, EG; FL, FK; DN and DH are quadrants, likewise AM, AL; BH, BG; CN and CK are quadrants; consequently (D. 133.) LM is the measure of the angle A, and NH the measure of the angle D.

Now EM and FL are together equal D to a semi-circle, or FL, LE, and LM are together equal to a semi-circle; but FL and LE are equal to FE, therefore FE and LM are together equal to a semicircle; that is they are supplements of ch other; hn LM is the measure of

K

F

Mad

AL

C

E

G

H