of the triangle, is the same as the diedral angle, between the two planes BAO, CAO; it is, also, the angle formed by the tangents to the two arcs AC, AB. The like may be said of the other angles. The sides are manifestly the measures of dependent plane angles, viz. a the measure of the angle сов, 6 the measure of con, c the measure of AOB. 2. A right angled spherical triangle has one right angle; the sides about the right angle are called legs; the side opposite to the right angle is called the hypo thenase. 3. A quadrantal spherical triangle has one side equal to 90°, or is a quarter of a great circle. 4. An isosceles, or an equilateral spherical triangle, has respectively two sides or three sides equal. 5. When the sides of a triangle are each 90°, it is not only an equilateral, but a quadrantal, and a right angled triangle. All its angles as well as its sides are equal; and these sides may any of them be regarded as an hypothenuse, any of them as legs. Such is the case with the triangle that would be formed on a celestial or terrestrial globe, by the horizon, the brazen meridian, and a quadrant of altitude, fixed at the zenith, and passing through the east or west point. 6. Two arcs or angles, when compared together, are said to be alike, or of the same affection, when both are less, or both greater than 90°. They are said to be unlike, or of different affections, when one is greater and the other less than 90°. 7. Every spherical triangle has three sides and three angles; of which, if any three be given, the remaining three may be found. 8. In plane trigonometry, the knowledge of the three angles is not sufficient for ascertaining the sides (chap. i. 6): but, in spherical trigonometry, the sides may always be determined when the angles are known. In plane triangles, again, two angles always determine the third; in spherical triangles they never do. So, farther, the surface of a plane triangle cannot be determined from its angles merely; that of a spherical triangle always can. 9. A line perpendicular to the plane of a great circle, passing through the centre of the sphere, and terminated by two points diametrically opposite, at its surface, is called the axis of such circle; and the extremities of the axis, or the points where it meets the surface, are the poles of that circle. If we conceive any number of less circles, each parallel to the said great circle, this axis will be perpendicular to them likewise; and the poles of the great circle will be their poles. 10. Hence, each pole of a great circle is 90° distant from every point in its circumference; and all the arcs drawn from either pole of a little circle to its circumference, are equal to each other. 11. It likewise follows that all the arcs of great circles, drawn through the poles of another great circle, are perpendicular to it; for, since they are great circles by the supposition, they all pass through the centre of the sphere, and consequently through the axis of the said circle. The same thing may be affirmed in reference to small circles. 12. Hence, in order to find the poles of any circle, it is merely necessary to describe, upon the surface of the sphere, two great circles perpendicular to the plane of the former, the points where these circles intersect each other will be the poles required. 13. All great circles bisect each other. For, as they have a common centre, their common section will be a diameter; and that manifestly bisects them. 14. The small circles of the sphere do not fall under consideration in spherical trigonometry; but such only as have the same centre with the sphere itself. Hence appears the reason why spherical trigonometry is of such great use in practical astronomy, the apparent heavens being regarded as in the shape of a concave sphere having its centre either at the centre of the earth, or at the eye of the observer. PROBLEM. 15. To investigate properties and equations from which the solution of the several cases of spherical trigonometry may be deduced. In order to this let us recur to the spherical tetraedron OABC, where the angles A, B, C, of the spherical triangle are the diedral angles between each two of the three planes AOC, AOB, &c. and the sides a, b, c, are the measures of the plane angles сов, COA, &c. Here it is 1st, evident that the three sides of a spherical triangle are together less than a circle, or, a + b + c < 360°. For the solid angle at o is contained by three plane angles, which (Euc. xi. 21) are together less than four right angles; therefore, the sides a, b, c, which measure those plane angles are together less than a circle. 16. Let the tetraedron OABC be cut by planes per pendicular to the three quadrilateral AOBA' since B C A the angles A and B are right angles, the plane angle o is the supplement of AA'B which measures the diedral angle ΑΑ'Θ ́Β. The same may be shown with respect to the other plane angles that meet at o; as well as of the plane angles at o', in reference to the diedral angles of the tetraedron OABC. Therefore, either of these tetraedrons, has each of its plane angles supplement to a diedral angle in the other: it is hence called the supplementary tetraedron. And if they become spherical tetraedrons referred to equal spheres, or to different parts of the same sphere, their bases will be spherical triangles respectively supplemental to each other. 17. It is obvious from this that the problems in spherical trigonometry become susceptible of reduction to half their number; since, if there are given, for example, the three angles A, B, C, and the three sides a, b, c, are required; let the triangle which has for its sides a', b', c', the supplements of the measures of A, B, and c, have its angles л', в', c', determined; their measures will be the supplements of the required sides a, b, and c. 18. On the surface of the sphere, the supplemental triangle is formed by the intersections of three great circles described from the angles of the primitive triangle as poles. Besides the supplemental triangle, three others are formed in each hemisphere by the mutual intersections of these three great circles; but it is the central triangle (of each hemisphere) that is supplementary. 19. Every angle between two planes being less than two right angles, it follows, that the sum of the angles of a spherical triangle is less than 3 times 2, or than 6 right angles. At the same time, it is greater than 2 right angles: for the sum a + b + c' of the sides of the supplemental triangle is less than 360° (art. 15 above): taking the supplements, we have 3 x 180° - (a + b + c') > 180°, or A + B + с > 180°. 20. To deduce the fundamental theorems, we may proceed thus. From any point a of the edge AO of the tetraedron, let fall on the plane or face Boc the perpendicular AD: draw, also, in that plane, the lines DH, DC, perpendicular to ob, oc, respectively; and join AH, AC; then will AH be perpendicular to on, and AC to oc. It is evident, therefore, that the angles ACD, AHD, measure the angles between the planes AOC, COB, and AOB, сов, that is, the angles cand B of the spherical triangle ABC. It is also evident that the plane angles in o, are AOB = c, AOC = b, вос = а. This being premised, the triangles ACO, ACD, the former right angled in c, the latter in D, give AC = AO sin AOC = AO sin b AD = AC sin ACD = AO sin b sin c. In like manner, the triangles AOH, ADH, right angled in H and D, give Hence, the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. 21. Draw ce and DF, respectively perpendicular and parallel to ob; then will the angle DCF = EOC = a. But the right angled triangles Aoc, ACD, FCD, give AC = AO sin b, DC = AC Cos c = AO sin b cos c and FD = DC sin a = AO sin a sin b cos c. Now OH = OE + EH = OE + FD, or ao cos c = OC cos a + FD = AO cos a cos b + FD = AO cos a cos b + Ao sin a sin b cos c. Therefore, dividing by Ao, we have cos c = cos a cos b + sin a sin b cos c. Similar relations are deducible for the other sides a and 6: hence, generally cos a = cos b cos c + sin b sin c cos A cos b cos a cos c + sin a sin c cos в cos c = cos a cos b + sin a sin b cos c }(2.) 22. These equations apply equally to the supplemental triangle. Thus, putting for the sides a, b, c, |