PROP. XX. THEOR. THE surface of a spherical triangle is equal to a rectangle contained by a semi-diameter of the sphere, and a straight line equal to the excess of the arc (of a great circle) which measures the sum of its three angles, above the semicircumference of such circle. Let ABC be a spherical triangle formed by the arcs of three great circles of the sphere which intersect one another at A, B, and C on one side of the sphere, and at E, D, and F on the opposite side. Since great circles intersect one another in a diameter of the sphere, ABE, BED, BCD, CDF, ACE, and CEF are semicircumferences, consequently equal to one another; therefore, taking away parts which are common, there remain AB equal to ED, BC equal to DF, and AC equal to EF: that is, in the two triangles ABC, DEF the sides of the one are equal to the sides of the other; therefore (4. Sph. Geom.) the triangles are equal to one another. Let P represent the spherical surface BCE, Q represent the space CED, and R the space ACD: also, B A E let AE be the diameter of the sphere, and let arc A, arc B, arc C, represent the arcs which measure the spherical angles at A, B, and c respectively. Then (19. Sph. Geo.) the spherical surface between the semicircumferences ABE and ACE; that is, ABC + P is equal to the rectangle AE. arc A; the surface between CEF and CDF; that is, DEF+Q, or ABCQ is equal to AE. arc c; and the surface between BCD and BAD; that is, ABC + R is equal to AE. arc B: therefore 3 ABC + P+Q+ R is equal to the rectangle contained by AE and a line equal to the sum of the arcs A, B, and c. But ABC + P+Q+ R is equal to half the surface of the sphere; that is, to the rectangle A E. circumf. AE; therefore, AE. 2 ABC AE. circumf. AEAE (arc A+ arc в + arc c). Consequently, taking equals from equals, 2 ABC: = AE (arc A+ arc B+ arc C) or ABC is equal to the rectangle contained by AE, and by a line equal to the difference between the half circumf. AE, circumference and the sum of the arcs which measure the angles at A, B, and c. Q. E. D. COR. The surface of a spherical polygon may be obtained on dividing it into triangles by drawing great circles through the angular points, and adding together the expressions for the surfaces of all the triangles. It will be found that the surface of the polygon is equal to a rectangle contained by a line equal to an arc which measures the sum of all the angles of the polygon and a line equal to that multiple of the half circumference of a great circle which is expressed by the number of sides of the polygon diminished by 2. THE END. LONDON: Printed by A. SPOTTISWOODE, A CATALOGUE OF PATERNOSTER Row, BOOKS ON EDUCATION, IN ALL BRANCHES OF KNOWLEDGE; NEW COURSE OF TEXT-BOOKS FOR THE USE OF SANDHURST ROYAL MILITARY COLLEGE, But applicable to Mathematical Education in general. §‡ The Works in this Catalogue may be had of all Booksellers. CAUTION.-Early and immature Editions of some of Messrs. Longman and Co.'s School Books having been reprinted by other Booksellers, it is necessary to caution purchasers that the only correct and genuine Editions, with the Authors' latest Additions and Improvements - all of which are Copyright-bear the imprint of Messrs. Longman, Brown, and Co. 2 WORKS ON EDUCATION PUBLISHED BY 1. Elementary Greek Works, Lexicons, Grammars, &c. BRASSE'S GREEK GRADUS.-NEW EDIT. BY REV. F. E. J. VALPY. 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