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the apparent celestial sphere, would there trace the circumference of a great circle, which would be the meridian of that place. All the points of the earth's surface, which have their zenith in that circumference, will be under the same celestial meridian, and will form the corresponding terrestrial meridian. If the earth be an irregular spheroid, this meridian will be a curve of double curvature; but if the earth be a solid of revolution, the terrestrial meridian will be a plane curve.
5. If the earth were a sphere, then every point upon a terrestrial meridian would be at an equal distance from the centre, and of consequence every degree upon that meridian would be of equal length. But if the earth be an ellipsoid of revolution slightly flattened at its poles, and protuberant at the equator; then, as will be shown soon, the degrees of the terrestrial meridian, in receding from the equator towards the poles, will be increased in the duplicate ratio of the right sine of the latitude; and the ratio of the earth's axes, as well as their actual magnitude, may be ascertained by comparing the lengths of a degree on the meridian in different latitudes. Hence appears the great importance of measuring a degree.
6. Now, instead of actually tracing a meridian on the surface of the earth,-a measure which is prevented by the interposition of mountains, woods, rivers, and seas,-a construction is employed which furnishes the same result. It consists in this.
Let ABCDEF, &c, be a series of triangles, carried on, as nearly as may be, in the direction of the meridian, according
to the observations in art. 3.These triangles are really spherical or spheroidal triangles; but as their curvature is extremely small, they are treated the same as rectilinear triangles, either by reducing them to the chords of the respective terrestrial arcs AC, AB, BC, &c, or by deducting a third of the excess, of the sum of the three angles of each triangle above two right angles, from each angle of that triangle, and working with the remainders, and the three sides, as the dimensions of a plane triangle; the proper reductions to the centre of the station, to the horizon, and to the level of the sea, having been previously made. These computations being made throughout
throughout the series, the sides of the successive triangles are contemplated as arcs of the terrestrial spheroid. Suppose that we know, by observation, and the computations which will be explained in this chapter, the azimuth, or the inclination of the side AC to the first portion AM of the measured meridian, and that we find, by trigonometry, the point M where that curve will cut the side BC The points A, B, C, being in the same horizontal plane, the line AM will also be in that plane: but, because of the curvature of the earth, the prolongation MM', of that line, will be found above the plane of the second horizontal triangle BCD: if, therefore, without changing the angle CMM', the line мM' be brought down to coincide with the plane of this second triangle, by being turned about Bc as an axis, the point M will describe an arc of a circle, which will be so very small, that it may be regarded as a right line perpendicular to the plane BCD: whence it follows, that the operation is reduced to bending down the side мM' in the plane of the meridian, and calculating the distance AMM', to find the position of the point м'. By bending down thus in imagination, one after another, the parts of the meridian on the corresponding horizontal triangles, we may obtain, by the aid of the computation, the direction and the length of such meridian, from one extremity of the series of triangles, to the other.
A line traced in the manner we have now been describing, or deduced from trigonometrical measures, by the means we have indicated, is called a geodetic or geodesic line: it has the property of being the shortest which can be drawn between its two extremities on the surface of the earth; and it is therefore the proper itinerary measure of the distance between those two points. Speaking rigorously, this curve differs a little from the terrestrial meridian, when the earth is not a solid of revolution: yet, in the real state of things, the dif ference between the two curves is so extremely minute, that it may safely be disregarded.
7. If now we conceive a circle perpendicular to the celestial meridian, and passing through the vertical of the place of the observer, it will represent the prime vertical of that place. The series of all the points of the earth's surface which have their zenith in the circumference of this circle, will form the perpendicular to the meridian, which may be traced in like manner as the meridian itself.
In the sphere the perpendiculars to the meridian are great circles which all intersect mutually, on the equator, in two points diametrically opposite: but in the ellipsoid of revolu
tion, and a fortiori in the irregular spheroid, these concurring perpendiculars are curves of double curvature. Whatever be the nature of the terrestrial spheroid, the parallels to the equator are curves of which all the points are at the same latitude: on an ellipsoid of revolution, these curves are plane and circular.
8. The situation of a place is determined, when we know either the individual perpendicular to the meridian, or the individual parallel to the equator, on which it is found, and its position on such perpendicular, or on such parallel. Therefore, when all the triangles, which constitute such a series as we have spoken of, have been computed, according to the principles just sketched, the respective positions of their angular points, either by means of their longitudes and latitudes, or of their distances from the first meridian, and from the perpendicular to it. The following is the method of computing these distances.
Suppose that the triangles ABC, BCD, &c, (see the fig. to art. 6) make part of a chain of triangles, of which the sides are arcs of great circles of a sphere, whose radius is the distance from the level or surface of the sea to the centre of the earth; and that we know by observation the angle CAX, which measures the azimuth of the side AC, or its inclination to the meridian Ax. Then, having found the excess E, of the three angles of the triangle ACC (Ce being perpendicular to the meridian) above two right angles, by reason of a theorem which will be demonstrated in prob. 8 of this chapter, subtract a third of this excess from each angle of the triangle, and thus by means of the following proportions find AC, and cc. sin (90° - E): cos (CAC-E): : AC: AC; sin (90°-E): sin (CAC-E) :: AC CC. The azimuth of AB is known immediately, because BAX = CAB-CAX; and if the spherical excess proper to the triangle ABM' be computed, we shall have
To determine the sides AM', BM', a third of E must be deducted from each of the angles of the triangle ABM'; and then these proportions will obtain : viz,
sin (180°-M'AB-ABM+E): sin (ABM'-E): AB AM', in (180°- M'AB- ABM'+E): sin (M'ABE) :: AB: BM'. In each of the right-angled triangles Abв, м'dD, are known two angles and the hypothenuse, which is all that is necessary to determine the sides ab, bв, and м'd, dD. Therefore the distances of the points B, D, from the meridian and from the perpendicular, are known.
9. Proceeding in the same manner with the triangle ACN, or M'DN, to obtain AN and DN, the prolongation of CD; and then with the triangle DNF to find the side NF and the angles DNF, dfn, it will be easy to calculate the rectangular coordinates of the point F.
The distance fr and the angles DEN, NFf, being thus known, we shall have (th. 6 cor. 3 Geom.)
fFP = 180°
So that, in the right-angled triangle ƒFP, two angles and one side are known; and therefore the appropriate spherical excess may be computed, and thence the angle FPf and the sides fP, FP. Resolving next the right-angled triangle eEP, we shall in like manner obtain the position of the point E, with respect to the meridian Ax, and to its perpendicular AY; that is to say, the distances Ee, and Ae=AP-ep. And thus may the computist proceed through the whole of the series. It is requisite however, previous to these calculations, to draw, by any suitable scale, the chain of triangles observed, in order to see whether any of the subsidiary triangles ACN, NFP, &c, formed to facilitate the computation of the distances from the meridian, and from the perpendicular to it, are too obtuse or too acute.
Such, in few words, is the method to be followed, when we have principally in view the finding the length of the portion of the meridian comprised between any two points, as A and x. It is obvious that, in the course of the computations, the azimuths of a great number of the sides of triangles in the series is determined; it will be easy therefore to check and verify the work in its process, by comparing the azimuths found by observation, with those resulting from the calculations. The amplitude of the whole arc of the meridian measured, is found by ascertaining the latitude at each of its extremities; that is, commonly by finding the differences of the zenith distances of some known fixed star, at both those extremities.
10. Some mathematicians, employed in this kind of operations, have adopted different means from the above. They draw through the summits of all the triangles, parallels to the meridian and to its perpendicular; by these means, the sides of the triangles become the hypothenuses of right-angled triangles, which they compute in order, proceeding from some known azimuth, and without regarding the spherical excess, considering all the triangles of the chain as described on a plane surface. This method, however, is manifestly defective in point of accuracy.
Others have computed the sides and angles of all the triangles, by the rules of spherical trigonometry. Others again,
reduce the observed angles to angles of the chords of the respective arches; and calculate by plane trigonometry, from such reduced angles and their chords. Either of these two methods is equally correct as that by means of the spherical excess so that the principal reason for preferring one of these to the other, must be derived from its relative facility. As to the methods in which the several triangles are contemplated as spheroidal, they are abstruse and difficult, and may, happily, be safely disregarded: for M. Legendre has demonstrated, in Mémoires de la Classe des Sciences Physiques et Mathématiques de l'Institut, 1806, pa. 130, that the difference between spherical and spheroidal angles, is less than one sixtieth of a second, in the greatest of the triangles which occurred in the late measurement of an arc of a meridian, between the parallels of Dunkirk and Barcelona.
11. Trigonometrical surveys for the purpose of measuring a degree of a meridian in different latitudes, and thence inferring the figure of the earth, have been undertaken by different philosophers, under the patronage of different governments. As by M. Maupertius, Clairaut, &c, in Lapland, 1736; by M. Bouguer and Condamine, at the equator, 1736— 1743; by Cassini, in lat. 45o, 1739-40; by Boscovich and Lemaire, lat. 43°, 1752; by Beccaria, lat. 44° 44′, 1768; by Mason and Dixon in America, 1764-8; by Major Lambton, in the East Indies, 1803; by Mechain, Delambre, &c, France, &c, 1790-1805; by Swanberg, Ofverbom, &c, in Lapland, 1802; and by General Roy, Colonel Williams, Mr. Dalby, and Colonel Mudge, in England, from 1784 to the present time. The three last mentioned of these surveys are doubtless the most accurate and important.
The trigonometric survey in England was first commenced, in conjunction with similar operations in France, in order to determine the difference of longitude between the meridians of the Greenwich and Paris observatories: for this purpose, three of the French Academicians, M. M. Cassini, Mechain, and Legendre, met General Roy and Dr. (now Sir Charles) Blagden, at Dover, to adjust their plans of operation. In the course of the survey, however, the English philosophers, selected from the Royal Artillery officers, expanded their views, and pursued their operations, under the patronage, and at the expence of the Honourable Board of Ordnance, in order to perfect the geography of England, and to determine the lengths of as many degrees on the meridian as fell within the compass of their labours.
12. It is not our province to enter into the history of these surveys a