Elements of Geometry, Conic Sections, and Plane Trigonometry

by Elias Loomis

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. 1880 ...the side AC be equal to the side AB. For if the two sides are not equal to each other, let AB be the greater; take BE equal to AC, and join EC. Then, in the triangles EBC, ACB, the two sides BE, BC are equal to the two sides CA, CB, and the included angles EBC, ACB are equal; hence the c angle ECB is equal to the angle ABC (Pr. 13). But, by hypothesis, the angle ABC is equal to ACB; hence ECB is equal to ACB, which is absurd. Therefore AB is not greater than AC; and, in the same manner, it can be proved that it is no't less; it is, consequently, equal to AC. Therefore, in an isosceles spherical triangle, etc. Cor. The angle BAD is equal to the angle CAD, and the angle ADB to the angle ADC; therefore each of the last two angles is a right angle. Hence the arc drawn from the vertex of an isosceles spherical triangle to the middle of the base is perpendicular to the tase, and also bisects the vertical angle. PROPOSITION XVI. TIIEOREM. In a spherical triangle, the greater side is opposite the greater angle, and conversely.. Let ABC be a spherical triangle having the angle A greater than the angle B; then will the side BC be greater than the side AC. Draw the arc AD, making the angle BAD equal to B. Then, in the triangle ABD, we shall have AD equal to DB (Pr. 15); that is, BC is equal to the sum of AD and DC. But AD and DC are together greater than AC (Pr. 2); hence BC is greater than AC. Conversely. If the side BC is greater than AC, then will the angle A be greater than the angle B. For if the angle A is not greater than B, it must be equal to it, or less. It is not equal; for then the side BC would be equal to AC (Pr. 15), which is contrary to the hypothesis. Neither can it be less, for then the side BC would-be less than AC by the first case, which is also c...

112 pages, Paperback

First published January 1, 1869

About the author

Elias Loomis was an American mathematician with a large interest in the sciences. He wrote about mathematics, astronomy, meteorology and natural philosophy.