10.0983695 log. tan. 51°.26′.2" = 10.0983695

log. tan. 51° 26' 2" log. tan. 25 43 1

9.6827151

20.6123143

... log. x = .6123143

and x, or a = 4.09557

Hence the two roots are 4.09557 and - 17.6556, and the sum of these two roots is -13.56, the coefficient of the second term of the equation, as it ought to be.

The equation that has been solved is x2 + px - q = 0; if it had been x2-px - - q = 0, the 2 roots would have been

that is, the 2 roots of the former equation taken negatively.

If the equation to be solved be x2 - px + q = 0, then

x=1

±

49

49

* = {1 + √(1-1)}; assume = sin.20

2

θ

and x = (1 + cos. 6) = p. cos., or = p sin..

2

2

:

Hence the rule of computation, logarithmically expressed, is log. sin. 0 = (20 + 2 log. 2 + log. q

2 log.p) and p+2 log. sin. 2-20.

20, or = log. p

If the equation be x2 + px + q = 0, its roots are the negatives of the roots of the preceding equation.

The preceding solutions are, in fact, the same as those given by Dr. Maskelyne, page 56 in the Introduction to Taylor's Logarithms.

If, in mathematical researches, equations, like those that have been given of the second and third degree, presented themselves to be solved, their solution would be conveniently effected by the preceding methods, and by the aid of the Trigonometrical Tables; but, the truth is, in the application of Mathematics to Physics, the solution of equations is an operation that very rarely is requisite, and consequently the preceding application of Trigonometrical formulæ is to be considered as a matter rather of curiosity than of utility.

SPHERICAL TRIGONOMETRY.

CHAP. VIII.

Definitions.

1. A SPHERE is a solid terminated by a curve surface, of which all the points are equally distant from an interior point, called the centre of the sphere.

The surface of a sphere may be conceived to be generated by the revolution of a semicircle round its diameter.

2. Every section of a sphere, made by a plane, (so it will be demonstrated) is a circle. A great circle is that, the plane of which passes through the centre ; a small circle, that, the plane of which does not pass through the centre.

3. The pole of a circle of a sphere, is a point in the surface, equally distant from every point of the circumference of the circle.

4. A plane is said to be a tangent of a sphere, when it has one point only common with the sphere.

5. A spherical triangle is a portion of the surface of a sphere included within three arcs* of three great circles, which arcs are called sides of the triangle.

* Each arc is supposed to be less than a semicircle, for the properties of a triangle that has its sides a, b, and c = ᅲ + x are always known from those of a triangle that has its sides a, b, and a third side

6. The angles of a spherical triangle, are the angles contained between the planes in which the arcs or sides lie. (See Euclid, Book XI. Def. 6.)

A spherical triangle is called rectangular, isosceles, equilateral, in the same cases that a plane triangle is.

PROPOSITION I.

Every section of a sphere made by a plane is a circle.

Let An Bm be the plane, draw CO* perpendicular to it, which consequently, (by Euclid, Book XI. Def. 3.) is perpen

dicular to every straight line meeting it in that plane; hence, since

Cn=Cm, and, ∠ Con = ∠ Com = 90°.

CO2, Om2 = Cm2 — CO2; .. On = Om,

and similarly, On = Op; .. Om, On, Op are equal; ... Ap Bm is a circle, and O is its centre.

COR. 1. When CO=0, or when the plane passes through the

* CO is not drawn in the diagram.

centre C of the sphere, On = 0m=Cm=CB, the radius of the sphere.

COR. 2. Hence two great circles always bisect each other; for, their common intersection passing through the centre is a diameter.

COR. 3. Through two points on the surface of a sphere, such as A, p, a great circle, (part of which is represented in the Figure by the dotted curve from A to p) may be made to pass ; for, the two points A, p, with C the centre as a third point, determine the position of a plane, the intersection of which with the surface of a sphere is a great circle passing through the points A, p. In like manner a great circle may pass through the points

If the two points, instead of being A, p, be A, a, the two ends of a diameter, then the three points A, C, a lying in the same line, do not determine the position of a plane; but, through these three points, innumerable planes may pass.

PROP. II. In every spherical triangle as Apq, (Fig. p. 126.) any one side is less than the sum of the two others.

For the arcs Ap, Aq, pq, measure the angles ACP, ACq, pCq; but by Euclid, Book XI. Prop. 20, any angle, as ACp*, is less than the two others ACq, pCq;∴∴ Ap is less than Aq+pq.

PROP. III. The sum of the three sides of a spherical triangle is less than the circumference of a great circle (2π).

Let ACB be the spherical triangle; then, CB < CD + BD

by the former Proposition, and AC + AB = AC + AB; •·.

* Conceive, in Fig. p. 126, a straight line to be drawn from A to the centre C.