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ly; but taken geometrically, is abfurd, unless we supply the radius as a multiplier of the terms on the right hand of the fine of equality. It then becomes 2fin A x fin B= R (fin (A + B) + fin (A — B)); or twice the rectangle under the fines of A and B equal to the rectangle under the radius, and the fum of the fines 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 fame in every term, otherwise we will compare unlike magnitudes with one another.

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

ELE.

ELEMENTS

OF

SPHERICAL TRIGONOMETRY.

IF

.PRO P. I.

F a fphere be cut by a plane through the centre, the fection is a circle, having the fame centre with the sphere, and equal to the circle by the revolution of which the fphere was described.

For all the ftraight lines drawn from the centre to the fuperficies of the fphere are equal to the radius of the generating femicircle, (Def. 7. 3. Sup.). Therefore the common fection of the spherical fuperficies, and of a plane paffing through its centre, is a line, lying in one plane, and having all its points equally diftant 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 fphere, that is of the femicircle by which the sphere has been defcribed. It is equal, therefore, to the circle, of which that femicircle was a part. Q. E. D.

A

DEFINITIONS.

I.

NY circle, which is a fection 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 bifect one another.

They are all equal, having all the fame radii, as has juft been fhewn; and any two of them bifect one another, for as they have the fame centre, their common fection is a diameter of both, and therefore bifects both.

II.

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

III.

A spherical angle is that which on the fuperficies of a sphere is contained by two arches of great circles, and is the fame with the inclination of the planes of these great circles.

IV,

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

PROP.

PROP. II.

HE arch of a great circle, between the pole and

TH the circumference of another great circle, is a

quadrant.

Let ABC be a great circle, and D its pole; if DC, an arch of a great circle, país through D, and meet ABC in C, the arch DC is a quadrant.

Let the circle, of which CD is an arch, meet ABC again in A, and let AC be the common fection of the planes of thefe great circles, which will pafs through E, the centre of the fphere: Join DA, DC 1. Becaufe ADDC, (Def. 2.), and equal ftraight lines, in the fame

E

A

circle, cut off equal arches,

C

(28. 3.) the arch AD the

B

arch DC; but ADC is a femicircle, therefore the arches AD, DC are each of them quadrants. Q. E. D.

COR. 1. If DE be drawn, the angle AED is a right angle; and DE being therefore at right angles to every line it meets with in the plane of the circle ABC is at right angles to that plane, (4.2.Sup.). Therefore the ftraight line drawn from the pole of any great circle to the centre of the sphere is at right angles to the plane of that circle; and, conversely, a flraight line drawn from the centre of the phere perpendicular to the plane of any great circle, meets the fuperficies of the sphere in the pole of that circle.

COR. 2. The circle ABC has two poles, one on each fide of its plane, which are the extremities of a diameter of the sphere perpendicular to the plane ABC; and no other point but these two can be a pole of the circle ABC.

PROP.

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