termined, the method employed in the English surveys for obtaining the angle which the measured base, or any side of a triangle, made with the meridian, was to compute from the elements in the Nautical Almanac the moment when the star a Polaris was on the meridian, or when it was at the greatest eastern or western elongation; then, at the place where the angle was to be observed, having first directed the telescope of the theodolite to the star, at the moment, the telescope was subsequently turned till the intersection of the wires fell on the object which marked the other extremity of the base or side. With a telescope capable of showing the star during the day, should it come in the plane of the meridian while the sun is above the horizon, this angle may be thus taken; or it may be obtained at night, when the star culminates after sun-set, if a luminous disk be used to indicate the place of the station whose bearing from the meridian is required. When the observation was made at the time of either elongation, the azimuthal deviation of the star, computed as above, was either added to, or subtracted from the observed azimuth according as the star was on the same, or on the opposite side of the meridian with respect to the station; and the sum, or difference, was of course the required azimuth of the latter: but if the bearings of the station were observed at both elongations of the star, half their sum expressed immediately the required bearing. Another method of obtaining the azimuth of a terrestrial object is given in art. 341. By some of the continental geodists a well-defined mark or, by night, a fire-signal was set up very near the meridian of the station whose azimuth was to be obtained; then, by means of circumpolar stars or otherwise, they obtained the correct azimuthal deviation of that mark from the meridian; and the sum or difference of the deviation, and the observed angle between the mark and the station, was consequently, equal to the required azimuth.

406. Different processes have been employed for reducing the sides of the triangles to the direction of a meridian passing through a station at one extremity of the series. Among the most simple is that which was adopted by M. Struve in measuring an arc of the meridian from Jacobstadt on the Dwina to Hochland in the Gulf of Finland. It consisted in computing all the sides of the triangles as if they were arcs of great circles of the sphere, either by the rules of spherical trigonometry or by the theorem of Legendre above mentioned (arts. 398, 402.), one of them as AB being the measured base. Then, C, D, E, F, &c. being stations, imagining A and F, A and M, &c. to be joined by great circles; in the

B B

P

triangles ADF, AFM, &c. each of the arcs AF, AM, &c. were found by means of the two sides previously determined and the included angles ADF, AFM. Finally, the azimuthal angle PAD having been obtained by observations, the angle PAM was computed and imagining a great circle Mp to be let fall perpendicularly on the meridian AP, the meridional arc Ap was calculated in the right-angled spherical triangle AMP.

F

M

H

E/

E

h

C

D

IC

b

B

B

A

A second process is that of computing by spherical trigonometry, or by the method of Legendre (art. 402.), the sides of the triangles; and then, in like manner, the lengths of the meridional arcs AB', B'C', &c. between the points where the sides of the triangles, produced if necessary, would cut the meridian. For the latter purpose an azimuthal angle, as PA B, must be observed: then, in the spherical triangle ABB', there would be given AB and the two angles at A and B; to find AB', BB' and the angle B'. Again, in the triangle B'DC', there would be given DB' (the difference between the computed sides BD and BB') the observed angle at D and the computed angle at B'; to find C'B', DC' and the angle c'. In like manner, in the triangle DC'E', may be found DE', C'E' and the angle E'. In the triangle E'F'D, formed by producing DF till it cuts the meridian in F'; with DE' and the angles at D and E' may be computed E'F', DF' and the angle F'. Lastly, letting fall Fk perpendicularly on the meridian, in the right angled triangle FF', with FF' (=DF'-DF) and the angles at F and F', the arc kF' may be determined; and thus with E'k (E'F'-F'k) and AB', B'C', C'E', before computed, the length of the meridional arc from a to k is obtained.

407. When the sides of the triangles are the chords of the spherical arcs, and are computed by the rules of plane trigonometry, the following process is used for reducing those sides to the meridian. After calculating the chords AB, AD, BD, &c., the distance AB' and the angle AB'B are determined by means of the angle PAB. Subsequently, with the angle DB'C' (ABB), the side B'C' is computed in the triangle B'DC', and this is to be considered as a continuation of the former line AB', which is supposed to have nearly the position of a chord within the earth's surface; but the plane of the triangle ABD being considered as horizontal, that of the next triangle BCD will, on account of the curvature of the

C/

B/

N

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 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 ABD. 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

B

M

* 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 ABD, 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 в'n (keeping the angle DB'm or A B B 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'EN, 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 вa, 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 в'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.

(=

C

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+aв). 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