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and Barcelona described in the " Base du Système Métrique,” as also on that between Dunnose and Clifton, it being considered not only more expeditious, but also more correct. Another advantage of this method is (if all the triangles are intersected by the meridian), that by calculating the various portions of which the arc is composed from the right-angled triangle formed on each side of the meridian separately, one result serves as a check upon the other.
A modification of this method is described in Puissant's Géodesie," page 248, which consists in constructing through the vertices of the triangles parallels both to the meridian A X and the perpendicular A Y, without taking any account of the spherical excess. The intersections of these lines form, with the sides of the triangles, right-angled triangles, of which those sides are the hypothenuses; and the azimuth of each being known, all the elements can be ascertained, as is evident by reference to fig. 2. In this manner, the distances of several places from the perpendicular, and the meridian passing through the observatory of Paris, were calculated by Cassini.
The third method ("Puissant,” vol. i. page 316) of ascertaining the length of the arc AX is by determining the geographical positions of the vertices of the triangles extending along the meridian, and calculating the difference of their parallels of latitude projected on the meridian, the sum of these being the measure of the arc.
The measure of an arc of a parallel is calculated by a similar process, which is described at page 319 of the same work.
The methods of calculating, geodesically, the latitudes, longitudes, and azimuths of the different stations from one meridian, with the rigid accuracy required in such operations as the measurement of an arc of the meridian or parallel, will be found fully explained in the 12th chapter of Woodhouse's "Trigonometry;” in the 18th chapter of Puissant's "Géodesie;" and in "Francœur." Their determination by astronomical observations will be treated of hereafter.
On the supposition that the earth is a sphere, the calculations are resolved into the solution of spherical triangles.
The accurate length of the arc on the surface of the earth, between two very distant places whose latitude and longitude have
been determined, is, on account of the spheroidal figure of the globe, a problem of great difficulty, and of no real practical utility;—it is fully investigated in Puissant's "Géodesie," vol. i., page 296*. Between stations, however, within the limits of triangulation, it is often useful to calculate the distance as a check upon the geodesical operations; and in the length of an extended line of coast, or in a wild country, where triangulation may be, from local obstacles or want of means, quite impossible, the solution of this problem is of great importance for the purpose of laying down upon paper the positions of a certain number of fixed stations, between which the interior survey has to be carried on; and it is, within such bounds, one of easy application, particularly in the latter case, where the observations themselves are generally taken with portable instruments, and not with minute accuracy.
In the accompanying figure, P is the pole of the earth (considered as a sphere), and S and S' the two stations, whose latitude and longitude are determined; the angle SPS' is evidently measured by the difference of their longitude, and PS and PS' are their respective latitudes; the solution of the spherical triangle PSS' then gives the length of the arc SS'.
If it is possible, when observing at S and S', to determine the azimuths of these stations from each other, that is, the angles PSS' and
PS'S, a more accurate result will be obtained, as these angles can be determined with precision, whereas the angle P depends upon the correctness of the observations for longitude at each station, which with portable instruments is always, at best, but a close approximation; and the errors in the determination of each may lie in the same, or in different directions. In geodesical operations, if it be possible, the reciprocal azimuths of stations should
* See also Francœur's " Géodesie," p. 208.
+ In cases where the difference of longitude between the two stations can be ascertained by means of signals, or by the interchange of chronometers, as explained in the next chapter, the measure of the angle P may be obtained with great accuracy.
always be observed, as well as the angles contained between them and other trigonometrical points.
From these reciprocal azimuths, with the astronomical latitudes of each station, the difference of their longitudes, or the angle of inclination of their meridians, is found by Dalby's method of solution, which is applicable to spheroids. This mode of determining the difference of longitudes by observations of reciprocal azimuths was practised on the Ordnance Survey; and the analysis of the theorem is given at length in page 214 of Airy's "Figure of the Earth." In the course of the investigation it is proved, that the spherical excess in a spheroidal triangle is equal to that in a spherical triangle whose vertices have the same astronomical latitudes and the same difference of longitude; from whence results the following simple rule—
tan diff. longitudes = cos diff. lat.
sinsum of lat.
x cot sum of azimuthal angles.
Generally, a small error in the latitudes produces no sensible error in the determination; but in the azimuths, accuracy is of vital importance; when the latitudes are small, their correctness becomes of consequence, and the method is not therefore well adapted for stations near the equator.
The angle at the pole formed by the two meridians being thus obtained, the distance SS' between the stations can be found nearly in the triangle PSS'; this arc, however, must be converted into its corresponding value in distance on the surface of the earth; and if its spheroidal figure be taken into account, the radius of curvature must be ascertained for the middle latitude 1 (1-1).
On the other hand, to obtain geodesically the latitudes, longitudes, and azimuths of stations from others whose positions on the surface of the globe have been determined by triangulation, it is necessary to be able to convert any measured or calculated distances on the earth's surface into arcs; for which purpose also the radius of curvature of the arc in question is required, to obtain an accurate result. In a paper published by Mr. Galbraith, in the 51st number of the " Edinburgh New Philosophical Journal," tables are given to facilitate this preliminary computation, whether
the arc be in the direction of a meridian, of a perpendicular to the meridian, or forming an oblique angle with it-as also those for the azimuths, latitudes, and longitudes, and convergence of meridians.
The formula given in the "Synopsis of Practical Philosophy" for the radius of curvature at any point of the terrestrial meridian, supposing the earth to be an oblate spheroid, is as follows, a and b being the equatorial and polar semi-axes, 7 the latitude, c= (a - b) the compression :
places, and the lines drawn to the celestial pole. And if S be any star which can be observed at both places, the angle sPpsPS +SPp, and sQ q = sQS + SQq; considering, therefore, the angles SQ8, 8PS as equal, the difference of latitudes is the same as the difference of SPp, SQg; that is, it is the same as the difference of the zenith distances of the same star at the two places, and can therefore be easily found. Now, if the places P and Q be on the same meridian, their verticals will intersect in some point D; and the difference of latitudes, which is the difference of sQ q and sPp, or (Pr being parallel to Qq) the difference of s Pr and sPp, is equal to r Pp or QDP, the angles contained by the verticals. The length PQ being known from measurement, and the angle PDQ, or the difference of latitude, being found by observations of the zenith distances of a star, the length of PD or QD, or the radius of curvature, is found.
Again, if T and V be two places on different meridians, and if planes be drawn through these places, and through the axis, AC, of the earth, the angle made by these planes (or the difference of the longitudes) may be determined astronomically. Now, instead of T we have a place t, whose latitude is the same as that of V; and if we draw VW, tW perpendicular to the axis, the angle between the planes will be the same as the angle VWt. The distance Vt being measured (or otherwise obtained), and the angle VWt, or the difference of longitude being found, the length of VW, or tW, or the radius of a parallel, will be found. Either of the measures will give this line, which will materially assist in determining the earth's form and dimensions, but they cannot easily be combined: the difference of latitude can be ascertained with so much greater accuracy than the difference of longitude, that measures of the former kind have generally been relied upon."
This subject is still further pursued in the work from which the above extract has been made.
It may also be required to calculate with the greatest exactness the azimuths or true bearings of two distant stations from each other, the latitudes and difference of longitudes of these points having been determined by observation; as was the case in marking the North American boundary in 1845, when one line 64 miles. in length was cut through the dense Canadian forest upon bearings from each of the extremities computed by the following directions and formulæ furnished by Mr. Airy.
Convert the difference of longitude found in time, into arc. From the latitudes of the stations compute the following formulæ :