PROP. IX. THEOR. Any two sides (as AB and BC, see figure to the seventh propo sition), of a spherical triangle (ABC), are together greuter than the third (AC.) Let D be the centre of the sphere, and join DA, DB and DC. Of the three plain angles which contain the solid angle D, the two ADB and BDC are together greater than the third ADC (17. 2 Sup.), and the sides AB, BC and AC of the spherical triangle ABC are the measures of the angles ADB, BDC and ADC (Def. 2 Pl. T'r ), therefore the sides AB and BC of the spherical triangle ABC are together greater than the third AC. PROP. X. THEOR. In any spherical triangle (ACB) the greater (ACB) of two une qual angles ( ACB and CAB) is subtended by the greater side (3B); and the greater (AB) of two unequal sides ( AB and CB) is subtended by the greater angle (ACB). Part 1. From the greater angle ACB take a part ACD equal to the less A, and the side CD is equal to AD (8. Sph. Tr.); A a Iding to cach DB, the whole AB is equal to CD and DB together, but CD and DB together are greater than CB(9. Sph. Tr.), therefore AB is greater than CB. Part 2. If AB be greater than BC, the angle ACB is greater than the angle A ; for the angle ACB is not equal to A, for if it were, the side A B would be equal to BC (8. Sph. Tr.), contrary to the supposition; and the angle ACB is not less than A, for if it were AB would be less than BC (by part 1), which is also contrary to the supposition; therefore the angle ACB being neither equal to nor less than A, is greater han it. PROP. XI. THEOR. The three sides ( AB, BC and AC), of a spherical triangle (ABC), are together less than a whole circle. Produce AB and AC to meet each other in D; the arches ABD and ACD are each of them A 1 semicircles (1 Sph. Tr.); but in the triangle BDC, the side BC is less than BD and DC together (9. Sph. Tr.). adding to each the arches AB and AC, the three sides AB, BC and AC of the triangle A30, are together less than the two arches ABD and ACD (Ax. 4. 1 Eu., or, which is equal, than a whole circle. Otherwise, see fig. to prop. 7. Let D be the centre of the sphere, on which the triangle ABC is formed ; join DA, DB and DC; and, because the three plain angles ADB, BDC and ADC, which contain the solid angle D, are together less than four right angles (18.2 Sup.), the three sides AB, BC and AC, which are the measures of them (Def. 2 | Pl. Tr.), are together less than a whole circle. PROP. XII. THEOR. In any spherical triangle (ABC, see fig. to prec. prop.), accord ing as the sum of the legs (AC and CB), are equal to, or greater or less than, a semicircle, the sum of the angles (A and ABC) at the base, are equal to, or greater or less than, two right angles. Let AB and AC be produced to meet in D. Part 1. If AC and CB be together equal to a semicircle or to ACD, taking from each the common arch AC, the arches CB and CD are equal (Ax 3. 1 Eu), therefore the angle CBD is equal to the angle CDB (7. Sph. Tr.), or, which is equal (Cor. 2. 1 Sph. Tr.), CAB ; adding to each the angle CBA, the two angles at the base CAB and CBA together are equal to the angles CBD and CBA together, or which is equal (4. Sph. Tr.), to two right angles. Part 2. If AC and CB together, be greater than a semicircle, or than ACD, CB is greater than ČD; and therefore the angle D or A is greater than the angle CBD (10. Sph. Tr.); adding to each CBA, the angles CAB and CBA together, are greater than CBD and CBA together, or than two right angles. Part 3. In like manner, if AC and CB together, be less than a semicircle, it may be shewn, that the angle D or A is less than CBD, and of course, the angles A and CBA together, less than CBD and CBA together, or than two right angles. Cor. It follows, that, according as the sum of the sides (AC and CB] is equal to, or greater or less than, a semicircle, either angle at the base [as A] is equal to, or greater or less than, the external angle (CBP) at the other extreme of the base. 52 PROP. XIII. THEOR. If a spherical triangle ( DEF), be formed by great circles, joining the poles (F, D and E), of the sides (AB, BC and AC), of another spherical triangle (XBC); the sides of the former triangle (DEF), are complements of the measures of the opposite angles of the latter, to semicircles; and the measures of the angles of the former, are complements of the opposite sides of the latter, to seinicircles. B Let AB, produced as necessary, meet DF and EF in G and H; BC meet DF and DE in M and N; and AC meet ED M and EF in K and L. Because F is the pole of the arch GAH (Hyp.), an arch drawn from F to B is a A quadrant (2 Sph. Tr.), and, because D is K the pole of the arch MCN (Hyp.), an arch dra vn from D to B is a quadrant (2 Sph.IS H Tr.) ; whence, DF being less than a semicircle (6 Sph. Tr ), B is the pole of the arch DGMF (Cor. 3. Sph. Tr.), therefore EN BG and BM are quadrants (2 Sph. Tr.), and of course GM is the measure of the angle CBA (3. Sph. Tr). In like manner it may be shewn, that LH is the measure of the angle CAB: and KN of the angle ACB. And, because F is the pole of the arch GAH (Hyp.), the arch FG is 'a quadrant (2 Sph. Tr.), and because D is the pole of the circle MCN (Hyp.), the arch DM is a quadrant (2 Sph. Tr.), therefore FG and DM together, or DF and GM together, are equal to a semimircle, and so the side FD, which is opposite the angle CBA, is the complement of GM, which is above shewn to be the measure of the angle CBA, to a simicircle. In like manner it may be shewn, that FE is the complement of the measure of the angle CAB, and DE, of the measure of the angle ACB, to seinicircles. And, because BM and CN are quadrants, these arches together, or MN and CB together, are equal to a semicircle, and therefore MN, the measure of the angle FDE, is the complement of the opposite side CB of the triangle ACB, to a semicircle. In like manner it may be demonstrated, that KL the measure of the angle DEF, is the complement of the side AC, and GH the measure of the angle F, of the side AB, to a semicircle. Scholium. Although, in this proposition, the word complement is used in its customary meaning ; yet the complement of an arch or angle, to a semicircle or two right angles, is also called its supplement, as mentioned in Def. 5. PI. Tr.; and therefore one of the triangles mentioned in this proposition, is said to be supplemental of the other. PROP. XIV. THEOR. The three angles of a spherical triangle (ABC, see fig. to the preced ing proposition ), are greater than two right angles, and less than six. For the three measures of the angles of the triangle ARC, together with the three sides of its supplemental triangle DEF; are equal to three semicircles (13. Sph. Tr.); but the three sides of the triangle DEF are less than two semicircles (11. Sph. Tr.); therefore the three measures of the angles of the triangle ABC, are greater than a semicircle, and of course these three angles are together greater than two right angles. And these three angles are less than six right angles, for the internal and external angles together of the triangle ABC, are equal to six right angles (4. Sph. Tr.), therefore the three interwal angles are less than six right angles. |