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The six cases there considered are, 1. When the station is out of the triangle made by the objects, but in one of its sides produced. 2. When the station is in one of the sides of the triangle. 3. When the three objects lie in a right line. 4. When the station is not within the triangle formed by the objects. 5. When the station is within that triangle. 6. When, by reason of the relation of the sides and angles, the points c and D (see the preceding diagram) fall so near together as to make the continuation to P of doubtful accuracy. To these, later writers on trigonometry have added another case, viz, 7. When the point c falls between the line AB and the station P.
It will be advantageous to the student, to modify the construction and computation to suit all these cases.
- Spherical Trigonometry.
SECTIon I. Fundamental Principles, General Properties, and Formulae. 1.THREE planes, Aoc, Aob, Boc, all of which pass through the centre o of a sphere, in- C tersect the surface of that sphere in & portions of great circles which form a S$ spherical triangle ABC. Thus also is 2× Fo
constituted the spherical pyramid or tetraedron which has for its base the triangle ABG, and for its vertex the centre o of the sphere. The angle A
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 mamifestly the measures of dependent plane angles, viz. a the measure of the angle cob, b the measure of coA, c the measure of AoE. 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 hypothen use. * 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 F.
sphere having its centre either at the centre of the earth, or at the eye of the observer.
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 coB, 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 perpendicular to the three edges; they will form another tetraedron oA'B'C'; their faces will be respectively perpendicular, two and two. But, in the quadrilateral AoEA' since the angles A and B are right angles, the plane angle o is the supplement of AA'B which measures the diedral angle AA'o'B. The same may be shown with respect to the other plane angles that meet at of 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 ame sphere, their bases will be spherical triangles respactively 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 A, B, 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 % a spherical triangle is less than 3 times 2, or than 6 right. angles. At the same time, it is #. than 2 right angles: for the sum a' + b + c of the sides of the supplemental triangle is less than 360° (art. 15 above) : . trking the supplements, we have 3 × 180° — (a' + b + c ) > 180°, or A + B + c > 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 os, and Ac to oc. It is evident, therefore, that the angles AcD, AHD, measure : the angles between the planes Aoc, cob, and AoE, cob, that is, the angles C and B of the spherical triangle ABc. It is also evident that the plane angles in o, are AoE = c, Aoc = b, Boc = a. This being premised, the tri-angles Aco, AcD, the former right angled in c, the latter" in D, give