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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1772 Excerpt: ... lest than the great circle Bcde. Theorem Theorem 7. 185. If the angular points of one spherical triangle, are poles to the fides of another the measures of the angles in the one will be supplemental to the fides of the other. Let Abc be a spherical triangle, whose angular points are poles of the great circles Forming the triangle Def; then the measures of the angles A, B, C, are supplements to the sides De, Ef, Fd; and the measures of the angles D, E, F, are supplements to the sides AC, AB, BC. Produce the sides of Aec both ways, to meet the sides of Def; at G, H, J, &c. Since A is the pole of Ed (con.) the pole of Ab will fall in the circle Ed (175), and because B is the pole of Ef (con.) the pole of Ba will fall in the circle Ef (175)1 therefore the point E is the pole of Ab; for the fame reason F is the pole of Bc, and D is the pole of Ac. Secondly, Eo, Nd are quadrants (168); therefore a semicircle == (eg 4-Nd = En + No + Ng+ Gd =) Ed + Ng; therefore Ng is the supplement of Ed (29), and consequently Ih, Ml, are supplements to Ef, Fd. Thirdly, Ng is the measure of A (171), and Ng is the supplement of Ed (above), therefore the measure of the angle A is the supplement of the side Ed, and Ih, Ml, the measures of the angles B and c, are the supplements of Ef, Fd; and after the fame manner it is proved, that the measures of the angles D, E, F, are supplements of the sides Ac, Ab, Bc. Theorem Theorem 8. 18 6. The three angles of a spherical triangle taken together, are greater than two right angles, and less thanstx right angles. Let Abc be the triangle, produce its sides both ways, and let De, Ef, Fd, be arches of great circles described about its angular points as poles. Then the measures of the angles A, B, C, together with the three sides of the suppl...

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