: side b be supposed equal to (90° - 0); then, by Naper's first Rule, 1 x cos. b = cot. (90° - 0). cot. A, and tan. 0 = cos. b. tan. A, which agrees with the assumption [c], p. 182. Again, by Naper's first Rule, sin. { 90° - (90° - 0) } or, sin. 0 = tan.p.cot. b, and ... tan p = sin. 0. tan. b. And finally, by Naper's first Rule, cos. {C- (90° -0) } = tan.p.cot. a, or ... sin. (C + 0) = sin. 0. tan. b.cot. a, which agrees with the result [d] in the preceding page. A great variety of instances to the preceding methods might easily be collected from Plane Astronomy. It is not, however, necessary to give any; since, amongst other purposes, the present Treatise is meant to be merely preparatory and subservient to the study of the latter science, and to be intelligible to the Student who may happen to be unacquainted with its technical terms and language. Astronomical Examples, stated and numerically resolved, would, indeed, be useful to the Student. One part of their utility would be, to communicate the art of translating Astronomical conditions into bare Mathematical conditions; it is not, however, the special business of a Trigonometrical Treatise to teach such art. Another part of their utility would consist in teaching the method of transforming general symbolical results and formulæ into numerical values; but, of this method sufficient specimens, it is hoped, have been given in the preceding pages. Still, however, it is desirable to apply and illustrate the preceding formulæ ; and, it happens fortunately, we can effect this without introducing either the principles or the terms of a new science. The accounts of those Trigonometrical Surveys, by which the figure and dimensions of the Earth have been attempted to be determined, will furnish us with very interesting instances of exemplification. In the next Chapter we will turn our attention to this point. We shall there perceive how results may be obtained by the direct application of the preceding methods of solution; and, besides, for what reasons and by what means, those methods, in certain circumstances, are either modified, or are completely superseded by methods of approximation. CHAP. XII. Application of Formula to certain Cases that occur in the Trigonometrical Surveys. Reduction of the Oblique to an Horizontal Angle, by the direct, by Delambre's approximate, Method.-Excess of the Sum of the Three Angles of a Spherical Triangle above Two Right Angles.Roy's Theorem. - Legendre's Reduction of a small Spherical to a Rectilinear Triangle. --Delambre's and Mudge's Method of computing the Length of the Meridian. In the Trigonometrical Survey, the two extremities of the line to be measured, are connected by a series of triangles, the angular points of which are the stations, at which, by proper instruments, the angles are observed. The sides of the triangles are determined by computation. But, since the sides cannot be determined solely from the angles, it is necessary to measure, at least, one side of one of the series of triangles. It is usual to measure the length of a side of the first triangle, near one extremity of the line to be measured; which side is called the Base; and also another line, the side of one of the triangles, near the other extremity; which latter line, serving as a test of the accuracy of the observations and computations, is called a Base of Verification. In the Trigonometrical Survey conducted in 1784, &c. under General Roy, the difference between the base of verification on Romney Marsh, measured and computed from the original base measured on Hounslow Heath was found to be about 2 feet in 28533 feet. The connecting chain of triangles by which the computation was made, extended over a space of eighty miles. After the first important operation of the measurement of a base, and which, (the side of the first triangle,) will be a spherical arc, if the measurement be conducted by spherical triangles; or, A a straight line, if by plane triangles of the chords of the spherical arcs, the angular distances between certain objects that mark the several stations must be observed. But as the objects, probably, are not situated on the same horizontal plane, nor on the same spherical surface, the angles observed are oblique angles. Since, however, the lines, &c. to be computed, are supposed to lie in the surface of the earth, they are horizontal angles which are required. The oblique angle observed then must be reduced to an horizontal angle: for instance, if the two objects be A, B; O the observer, and Z his zenith, and a, b, two points on the surface of the earth (supposed to be spherical), the observed or oblique angle, is AOB, measured by AB, and the horizontal angle, which is required, is equal to AZB. Now, by the third method of solution, Case 1, page 159, (AB+ZB − ZA) . sin. (AB+ sin.2 AZB 2 = sin. sin. ZA. sin. ZB sin. (a+H-h) sin. (a+h-H) and in logarithms, log. sin. = ZA – ZB) if AB=a, Aa=H, Bb=h {20+log. sin. (a + H - h)+log. sin. (a + h - H) - log. cos. H - log. cos. h } From this formula, a, h, H, being determined by observation, the horizontal angle may be computed. For example: in the Trigonometrical Survey, of 1787, made for the purpose of joining the Observatories of Greenwich and Paris, the angular distance of Watten and Fiennes, at Calais, was observed to be 66° 30′ 38′′.9: moreover, H was = -1" (a depression) and h = 25′ 47′′.2: hence, if O represent Calais, A and B Watten and Fiennes, AB = 66° 30′ 38′′.9. log. r2 20 (a+H-h) = 33°2′25′′.35........sin. = 9.7365802 39.4781321 log. cos. - 1" + log. cos. 25′ 47′′.2........=19.9999878 Here, the reduced or horizontal angle differs from the oblique or observed angle by a quantity a little more than 2"; and although the preceding process of reduction is not a long process, yet, for practical purposes, it is not sufficiently short, when several hundred reductions are to be made; and for this cause, M. Delambre * has investigated a formula, and constructed Tables of easy use, for finding a small correction; which, applied to the oblique angle, should give the horizontal angle. A formula similar to Delambre's, will be investigated in the Appendix. But, whether we use the preceding direct method, or Delambre's approximate method, all observed angles, if it is proposed to conduct the operation with nicety, must be reduced to horizontal angles ; * See Connoissance des Temps, 1793; Maskelyne, Phil. Trans. 1797, page 451; Legendre, Mem. Acad. 1787, page 313; Delambre, Mem. Inst. 1806, p. 112; Suanberg, Exposition des Operations faites, &c., p. 38. |