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ly ; but taken geometrically, is absurd, unless we supply the radius as a multiplier of the terms on the right hand of the fine of equality. It then becomes afin A x fin B = R(fin (A +B) + sin(A - B)); or twice the rectangle under the fines of A and B equal to the rectangle under the radius, and the sum of the lines 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


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.


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F 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 was described.

For 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 its 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. l. E. D.





NY 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 bifedt 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.


The pole of a great circle of the sphere is a point in the fu.

perficies of the fphere, from which all straight lines drawn to the circumference of the circle are equal.


A spherical angle is that which on the superficies of a sphere

is contained by two arches of great circles, and is the same with the inclination of the planes of these great circles.


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,



THE arch of a great circle, between the pole and

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, pass 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 section of the planes of these great circles, which will pass through E, the centre of the sphere : Join DA, DC 1. Because AD=DC, (Def.2.), and equal straight lines, in the same


с circle, cut off equal arches, (28. 3.) the arch AD = the

B arch DC; but ADC is a semicircle, 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 straight 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 firaight line drawn from the centre of the sphere perpendicular to the plane of any great circle, meets the superficies of the sphere in the pole of that circle.

Cor. 2. The circle ABC has two poles, one on each side 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.


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