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Let ABC be a spherical triangle: its surface is to the surface of the half of the sphere, as the excess of its three angles A, B, C, is to four right angles.

Make (Art. 211.) the isosceles quadrantal triangle PEF equal to ABC: therefore, (Art. 213.) the three

P

B

angles A, B, and C, are, together, equal to the angles PEF, EFP and FPE: but (Art. 121.) the two angles PEF, EFP are right angles: therefore the third angle EPF is equal to the excess of the three angles A, B, C, above two right angles: and (Art. 218.) the quadrantal triangle EPF is to the half of the sphere's surface, as the angle EPF is to four right angles: wherefore, (E. 7. 5:) the given triangle ABC, to which EPF was made to equal, is to the half of the sphere's surface, as the excess of the three angles A, B, C, above two right angles, is to four right angles.

(223.) COR. 1. If two great circles make, with one another, an angle equal to the half of the excess of the three angles of a spherical triangle above two right angles, the

spherical segment, contained by the semi-circumferences of the circles, shall be equal to the spherical triangle (Art. 221. and 222).

(224.) COR. 2. The surface of the hemisphere is to any spherical polygon, bounded by arches of great circles, as four right angles are to the excess of all the angles of the polygon, together with four right angles, above twice as many right angles as the polygon has sides: and, the surface of the hemisphere is to its excess above the polygon, as four right angles are to the excess of all the angles of the polygon, above twice as many right angles as the figure has sides.

The Corollary is made evident, by dividing the polygon into as many triangles as it has sides.

PROP. XVIII.

(225.) Theorem. If two spherical triangles be on different spheres, and the three angles of the one be, together, equal to the three angles of the other, the two triangles are to one another as the squares of the diameters of their respective spheres.

For (Art. 222.) each of the triangles has to the half of the surface of its sphere, the same ratio that the other triangle has to the half of the surface of its sphere: therefore, (E. 11. and 16. 5.) the triangles are to one another as the surfaces of their respective spheres; that is, (Archim. 1. 1. P. 38. and E. 2. 12.) as the squares of the diameters of the spheres.

(226.) COR. If the three angles of the one, of two spherical triangles, on different spheres, be equal to the three angles of the other, each to each, and if the sides of the triangles be also similarly posited, the triangles are proportional to the squares of the straight lines which are equal to their homologous sides.

For, it was shewn, in the demonstration of Art. 189. that any two homologous sides of two such spherical triangles, are to one another, as the circumferences of great circles, in the two spheres; that is, (Archim. de Dim. Circ. P. 1. E. 2. 12. and 22. 6.) as the diameters of the spheres: wherefore, (E. 22. 6.) the squares of the straight lines, equal to the homologous sides, are to one another, as the squares of the diameters of the spheres; that is, (Art. 224.) as the two spherical triangles.

A

TREATISE

ON

Spherics.

INTRODUCTION

ΤΟ

PART II.

ON THE

PRINCIPLES OF PLANE TRIGONOMETRY.

"De Trigonometria libri extant, fateor, non pauci. Sed, ex his, aliqui nimia copia tyronem obruunt; alii difficili brevitate discruciant."- -CASWELL.

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