earth, be inclined in a small angle to the former plane: therefore, in order to reduce the computed line B'c'to the plane of BCD, and allow A B' and the reduced line to retain the character of being two small portions of the geodetical meridian, that line B'C' may be supposed to turn on the point B' as if it moved on the surface of a cone of which B' is the vertex and DB' or B'B the axis (that is, so as not to change the angle which it makes with DB') till it falls into A B N E C/ the plane BCD. Thus let AM be part of the periphery of the terrestrial meridian passing through a in the former figure, and let AB' be the position of the chord AB'; then the first computed value of B'C' may, in the annexed figure, be represented by B'C', which is in the plane of the meridian and of the triangle A B D. And when, by the conical movement above mentioned, the line B'C' comes into the plane of the triangle BCD in the former figure, it will have nearly the position of a chord line, and may be represented by B'N which terminates at N in a line imagined to be drawn from c' to the centre of the earth.* Since the inclination of the plane ABD to that * It must be remembered that (agreeably to what is stated in art. 387.) a geodetical meridian is a curve line the plane of which is every where perpendicular to the tangent plane or the horizon, at every point on the earth's surface through which it passes; and unless the earth be considered as a solid of revolution, the geodetical meridian is a curve of double curvature: the error which arises from considering it as a plane curve is, however, not sensible. Now, if AB', B'c' (in the above figure, and in the fig. to art. 406.) be considered as two small portions of the geodetical meridian, the vertical planes passing through these lines should be, respectively, perpendicular to the horizons at the middle points of the triangles A B D, BC D. Let it be granted that the vertical plane passing through AB' is perpendicular to the plane of the triangle ABD; and let it be required to prove that while B'C' (fig. to art. 406.) in the plane B C D makes the angle D B'C' equal to the angle AB' B, in the plane A B D, the vertical plane passing through AB' and B'C' may be considered as perpendicular to the plane BCD. Imagine, in the annexed figure, a sphere to exist having its centre at B′ and any radius as B'D; and let Dm, Dn be arcs of great circles on such sphere, the former in the plane ABD and the latter as much below the plane BCD as Dm is above it let also в'm be the prolongation of AB' in the plane ABD produced. Then, by the manner in which в'm was supposed to revolve to the position B'n (keeping the angle DB'm or ABB equal to DB'n) the spherical triangle Dmn is evidently isosceles, and a great circle passing through D, bisecting mn, will cut mn at right angles. Let B'N be in the plane of this circle; it will also be in the plane BCD, and the latter will be cut perpendicularly by the plane passing through D A m B C of BCD is very small, the triangle B'NC' may be considered as right angled at N, and the angle c'B'N as that between a tangent at B' and a chord line drawn from the same point; consequently (Euc. 32. 3.) as equal to half the angle B'E N, or half the estimated difference of latitude between the points B' and N. Therefore B'N (the reduced value of B' C) can be found; and in a similar manner the reduced values of C'E', &c. may be computed. The first station A is on the surface of the earth; but the points B', c', &c., after the above reductions, are evidently below the surface: therefore the meridional arcs appertaining to the chords A B', B'N', &c. should be increased by quantities which are due to the distances of the points B', N, &c. from the said surface. When the sides of the triangles have been computed by Legendre's method (art. 402.), and the azimuthal angle between a station line and the plane of the meridian passing through one of its extremities has been observed; if perpendiculars be let fall from the stations to the meridian (the stations not being very remote from thence on the eastern or western side), the lengths of the perpendiculars and of the meridional arc intercepted between any station as A, and the foot of each perpendicular, may also be computed by the rules of plane trigonometry. For since the computed lengths of the station lines are equal to the real values of those lines on the surface of the earth, though the lines be considered as straight, the lengths of the arcs Aa, Ab, Ba, &c. (fig. to art. 406.), computed from them (one third of the spherical excess for each triangle being subtracted from each angle in the triangle), will be the true values of those arcs. Consequently the whole length of the meridional arc Ap will be correct. The following is an outline of the steps to be taken for the determination of the length of a meridional arc, as Ap, by perpendicular arcs let fall upon it from the principal stations and by arcs coinciding with the meridian, or let fall perpendicularly on the others from the several stations. Let AB (fig. to art. 406.) be the measured base; PAB the azimuthal angle observed at A; and let Ba, cb, &c. be the perpendiculars let fall from the stations B, C, &c. on the meridian AP: then, in the triangle Aa B, we have AB (supposed to be expressed in feet), the angle a AB and the right angle at A, B, m, n: or the plane n'mn, which is the plane of the geodetical meridian passing through B'c', may be considered as at right angles to the plane BCD. In like manner the plane passing through the next portion c'e' of the geodetical curve, in the plane of the triangle CDE, may be considered as at right angles to the plane of that triangle; and so on. a; to find Aa and Ba. In like manner in the triangle DAC, we have AD, the right angle at c and the angle DAC (equal to the difference between the angles DAB and PAB); to find De and AC. Again, imagining Bd to be drawn parallel to AP, in the triangle Cdв we have BC, the right angle at d and the angle CBd(=ABC-Aвa—aвd, the last being a right angle); to find cd and Bd; thus we obtain Ab (=Aa+вd) and cb (cd+aB). In the triangle DCe, we have DC, the right angle at e and the angle DCe (the complement of DCb); to find De and Ce: hence we obtain Ac and Dc a second time. The values may be compared with those which were determined before; and if any difference should exist, a mean may be taken. In the like manner the computation may be carried on to the end of the survey; and the whole extent of the meridional arc from A to p as well as the lengths of the several perpendiculars may be found. But at intervals in the course of the survey other azimuthal angles as PMF must be obtained by observation: then, since the angle FMP will have been found from the preceding computations, and the angle PMP by the solution of the right angled spherical triangle PMP; the sum of these two may be compared with the observed azimuth, and the accuracy of the preceding observations may thus be proved. The angle PHM being computed in the spherical triangle PHM, that azimuthal angle may be employed to obtain the meridional arcs and the perpendiculars beyond the point H. The process above described is particularly advantageous when it is intended to make a trigonometrical survey of a country as well as to determine the length of an extensive meridional arc; for the spherical latitudes and longitudes of the stations A, B, C, &c. might be found from the above computations, and thus the situations of the principal objects in the country might be fixed. For this purpose it is convenient to imagine several meridian lines to be traced at intervals from each other of 30 or 40 miles; and to refer to each, by perpendiculars, the several stations in the neighbourhood. The lengths of these perpendiculars will not, then, be so great as to render of any importance the errors arising from a neglect of the spherical excess in employing the rules of plane trigonometry for the purpose of making the reductions to the several meridian lines. 408. If a chain of triangles be carried out nearly in the direction of an arc perpendicular to any meridian, the situations of the stations may, in like manner, be referred to that arc by perpendiculars imagined to be let fall on the latter; and the lengths of the arcs and of the perpendiculars may be computed as before. The difficulty of obtaining the longitudes of places with precision is an objection to the employment of this method in the survey of a country; and the same objection exists to the measurement of an arc on a parallel of terrestrial latitude. The measured length of an arc on a perpendicular to a meridian, and on a parallel of latitude, have, however, been used in conjunction with the measured arc of the meridian at the same place, as means of determining the figure of the earth. In a triangulation carried out from east to west, or in the contrary direction, the sides of the triangles may be computed as arcs of great circles of the sphere: then with these sides and the included angles, the distances AB, AC, AD, &c. of the several stations may be obtained by spherical trigonometry; and from the last of these as M, letting fall Mp perpendicularly on the meridian of A, the arcs Ap, Mp may be computed in the right angled spherical triangle Арм. There must subsequently be obtained the distance from M to q on the arc of a parallel circle, as Mq, drawn through M, and the distance from p to q on the meridian. It has been shown in art. 71. that pq in seconds is approximately equal to p2 sin. 2 PM sin. 1" (fig. to that article), the radius of the sphere being unity. Now if the arc Mp, computed as above mentioned, were in feet, and MC the semidiameter of the earth be also expressed in feet; since мр is = MC equal to the measure of the angle мCp at the centre, and that MC sin. MCPMC' sin. MC'q, each member being equal to MN; also, since MC' MC sin. PM, we have, considering Mp as an arc of small extent, and putting Mp in feet for MC sin. MCP, also for sin. Mc'q putting its equivalent sin. P or P sin. 1", putting 2 sin. PM cos. PM for sin. 2 PM (Pl. Trigon., art. 35.), and 7 for the latitude of M, pq (in arc, rad. = 1) = Mp2 2 MC2 tan 7; In the same article it has been shown that the difference between Mq and Mp (in arc, rad. 1) is approximatively equal to Mp3 sin.3 1" tan.2 l. Now if the arc Mp were in feet, and MC the semidiameter of the earth be also expressed in feet, Mp sin. 1" in the last expression, in which мp is sup мр мр posed to be in seconds, would be equivalent to and that Mp3 MC tan.27; therefore the differ expression would become ence between Mp and мq in feet is, when Mp and MC are in feet, equal to мр. 1 Mp3 tan.2 l, by which quantity Mq exceeds If, after the several distances AB, BC, CD, &c. have been computed in the triangulation, the latitudes of the stations and the bearings of the station lines from the terrestrial meridian passing through one extremity of each be observed or computed; the lengths of the several arcs of parallel circles, as Bb, Cc, Dd, &c., drawn from each station to the meridian passing through the next may be calculated and subsequently reduced to the corresponding arcs qh, hk, &c. on the parallel of terrestrial latitude мq, which passes through any one, as M, of the stations. The sum of all such arcs will be the value of that whose length it may have been proposed to obtain. Whether the chain of triangles extend in length eastward and westward, or in the direction of the meridian, the value of pq must be subtracted from the computed value of Ap in order to obtain the length of the meridional arc comprehended between A and the parallel of latitude passing through м then the latitudes of A and M being determined by computation or found by celestial observations, the difference between the latitudes of A and M will become known, and such difference compared with the measured length of aq will, by proportion, give the length of a degree of latitude at or near A. In like manner the difference between the longitudes of A and M, obtained by celestial observations, by chronometers or otherwise, if compared with the measured lengths of мp and мq, will, by proportion, give the lengths near A of мр an arc of one degree on a great circle perpendicular to the meridian and on a parallel of terrestrial latitude. |