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THE measurement of the distance between the meridians of Paris and of Greenwich in 1787, formed a new era in the art of Trigonometrical Surveying. The instruments employed in that operation were of such a superior construction, as to afford a measure of many quantities which were before only known from theory to exist. Though it was perfectly understood that the three angles of a triangle on the surface of a spherical body like the earth, must necessarily exceed two right angles, yet a quantity so minute as to bear the same proportion to four right angles which the area of the triangle bore to half the superficies of the globe, had eluded the best instruments yet applied to the purposes of practical geometry. It was not till the survey just mentioned, that the new theodolite of Ramsden, in

* From the Edinburgh Review, Vol. XXI. (1813.)—En.

the hands of General Roy, and the repeating circle of Borda, in those of the French mathematicians, were able to measure a quantity, where even fractions of a second must be accurately ascertained. The exquisite division of the former of these instruments, and the power possessed by the latter, of not only measuring any angle, but any multiple of it, and any number of multiples, rendered them perfectly equal to such delicate observations. The advantage of this was quickly perceived; for the spherical excess, or the excess of the three angles of the triangle above two right angles, depending entirely on the area of the triangle, could be estimated with sufficient accuracy before the angles were correctly determined, and therefore might serve for a check on the observations, as effectual as that which is furnished by the well known property of plane triangles, that the three angles are always equal to 180 degrees. This was remarked by General Roy, and applied to the purpose of estimating the accuracy and correcting the errors of his observations. The French geometers carried their views farther; and in seeking to turn the knowledge of this limit to the greatest advantage, Legendre discovered, that if each of the angles of a small spherical triangle be diminished by onethird of the spherical excess, their sines become proportional to the lengths of the opposite sides of

the triangle, so that the ratios of the sides may be found by the rules of plane trigonometry.

In a science where all the parts are necessarily connected with one another, one improvement can seldom fail of leading to many more. It now became evident, that to carry, through the whole process of a trigonometrical survey, the same accuracy that was employed in measuring the angles of the triangles, methods of calculation must be introduced to which it was before quite unnecessary to resort. Thus, if the object was the measurement of an arch of the meridian, the reduction of the sides of the triangles to the direction of that line, by the usual method of letting fall perpendiculars on it, from the extremities of those sides, and finding the lengths of the parts intercepted, by the rules of plane trigonometry, did not possess a degree of accuracy equal to that which belonged to other parts of the process. The perpendiculars drawn to the meridian from any two points are not in strictness to be regarded as straight lines, but as arches of two great circles perpendicular to it, which would meet if produced in the pole of the meridian, or in the point of the horizon which is due east or west from the place of observation. It is therefore by the solution of a spherical triangle, of which the sides are nearly quadrants and the base very small, that the reduction required is to be made. This is the method followed by Delambre in the measureB b


ment of the great arch of the meridian carried across France, for the purpose of determining the length of the mètre. It is a refinement which was not thought of by General Roy; and we are not sure that it has been followed by any of the geometers who succeeded him in the conduct of the British survey. It is one however which, when the utmost accuracy is aimed at, ought not to be neglected, especially in high latitudes, where the convergency of the meridians is considerable.

Another refinement, which one should suppose might be even more easily dispensed with than the former, applies to the measurement of the base from which the sides of the triangles are determined. That line is usually measured by placing rods of equal lengths, or chains stretched with great care, at the ends of one another, for a distance of five or six miles. It has been usual to consider the base, thus measured, as a straight line, the length of which is just equal to the sum of the lengths of all the rods or chains which have been consecutively placed at the ends of one another. The truth however is, that these rods have not been placed exactly in the same straight line, and that they constitute the sides of a polygon inscribed in a circle, the radius of which is the radius of curvature of the earth at the point, and in the direction in which the base is extended. The line measured is therefore, in fact, an arch, passing

through the angles of this polygon; and this arch, which is the real base, is longer than the sum of the rods or chains. It is, however, easy to see that the deduction of the real length from the apparent, is not, in this case, a matter of much difficulty.

There is another way of including these corrections, which has been thought preferable by some geometers, and is recommended by the authority of Delambre. According to it, the spherical angles, or those actually measured, are reduced to the angles of the chords; and thus the lengths of the chords are calculated by plane trigonometry, and thence the lengths of the arches themselves are afterwards deduced. The base, measured as above, is also reduced on the chord. This method, though less direct than the former, has considerable advantages in calculation. It was followed by Major Lambton in the survey of which we are now to


A third source of inaccuracy, which had never before been thought of, drew the attention both of the French and English mathematicians engaged in the survey. Triangles, as we have seen, on the surface of the earth, cannot be regarded as plane triangles, because the plummets at the three angular points are not parallel to one another, and of course, the theodolites at these three stations can neither be in the same, nor in parallel planes. But

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