10.0276497 deduct the constant log. 9.3267737 0.7008760, the number answering to which is 5."022=5" nearly. Hence the error in the three observed angles above, is 5"-1".25=3".75. This spherical excess will enable the observer to examine the accuracy of his observations. Having corrected the angles, the sides of the triangles may be calculated by the rules of spherical trigonometry. 5. Given two sides of a spherical triangle, and the angle contained between them, to find the angle contained between the chords of these sides, supposing the chords not to differ materially from the arcs which they subtend. Let the three angles of the spherical triangle be represented by A, B, C; and their opposite sides by a, b, c; and let a', b', c', represent the chords of these sides, which chords are supposed not to differ materially from the arcs; r=radius of the sphere, and A' rectilenear angle: then cos. A'=r3 [(cos. A. cos. b. cos. c) +(r sin. b. sin. c)]. The spherical excess was constantly calculated on the trigonometrical survey of England, not altogether for the purpose of diminishing the observed angles by the amount, but to correct the observations. In one of the great triangles in Dorsetshire, the sum of the three angles was less than 180° by 0".5, while the spherical excess amounted to 1".29: their difference, 1".79, is the amount of error in the observation. Now if one-third of the error thus found be added to each of the angles, they are corrected as angles of a spherical triangle; and if we deduct one-third of the spherical excess from each of the corrected spherical angles, they are converted into angles of a plane triangle, ready for calculation. The following table shews, at one view, what we have said respecting the correction of the angles. Here one-third of the spherical excess is deducted from each angle, which might have been otherwise done by reducing the angles of the spherical triangles to those formed by the chords. By the assistance of a trigonometrical survey accurately executed, the length of an arc of the meridian may be measured. This was executed with great precision in the trigonometrical survey of France, from the measure of nineteen bases. Let us suppose NS to be the meridian of any place A, (see figure, next page,) and let o represent the point of the horizon where the sun sets; measure the angle BAO. Now, having the latitude of the place, the amplitude NAO can be found; therefore the angle BAF =NAO-BAO) is given, as also the angle BFA, being right. Then in the tri- S A With a proper instrument, find the true zenith distance of a star at A and D, and the difference of these zenith distances will give the difference of latitude between A and D. Now if the distance AL, before determined, be divided by the number of degrees thus obtained for the difference of latitude, the quotient will give the length of one degree of the meridian at that particular place, allowing the earth to be a perfect sphere.* As the earth is not a perfect sphere, being flattened at the poles, the above requires correction, which is easily effected. The annexed figure exhibits the most interesting portion of this survey. In the following register, each angle in the successive triangles is marked by the single letter affixed to it, and the computed length of its opposite side in feet, ranges in the same line. Some * For the meaning of the terms amplitude, zenith distance, latitude, &c. see the author's Epitome of Astronomy. |