Chap. VIII.

GEOMETRY.

855

down, they would occupy a greater area than they ought: the triangles must, therefore,be considered as spherical, and treated as such, when their sides are to be calculated ; whenangular distances are measured by a theodolite, the plane of the instrument is so adjustedto the horizon that the angle observed is the horizontal angle of the station. The threesummits of the triangle so measured are equally distant from the earth’s centre, which isnot the case when the sextant and repeating circle are used for the purpose of measuringangles. The centre of the theodolite being placed directly under the centre of the signaldestroys the necessity of any calculation for reducing the observed angle to the centre ofthe station.

In all spherical angles the three angles together exceed 180°, by what is called thespherical excess , which is proportional to the area of the triangle; ABC, being the anglesof a spherical triangle, r the radius of the sphere expressed in feet, x = 3T4159, the pro-portion the circumference has to the diameter, and S the number of square feet containedin the area of the triangle ; then by trigonometry,

g_A + B + C—180° 0

180°

Let E denote the spherical excess, = A + B + C — 180° ; then E =

S x 180°

degrees,

E =

S x 648000,

■ as expressed in seconds: in any triangle which can be measured on the

surface of the earth, S is very small in comparison with r‘ 2 , and therefore E is a very smallquantity : in practice it seldom exceeds four or five seconds ; but in the triangle of a side of100 miles in length, it would amount to more than thirty or forty : an approximate valueof S is sufficient then to compute the value of E with precision; and for this purpose, wemust consider the triangle a plane-one : let abc be the number of feet in the respectivesides opposite to ABC: we shall have for the area S = £a6sin.C. Substituting this in the

formula for the spherical excess, we obtain in seconds, E = -—- * ^ 1 ^ . 299 ; this formula

2 r 2 x

is deduced on the hypothesis that the surface of the triangle is spherical; but it equallyapplies to triangles on the surface of a spheroid; for the spherical excess is the same fortriangles on a spheroid and sphere, when the latitude of the stations and their differencesof longitude are the same. To compute the spherical excess of any triangle, the value of r,the radius, should be ascertained; the curvature of the arc joining any two points orstations on a spheroid varies with the latitude, and also with the direction of the arc inrespect of the meridian : the value of r may then be considered sufficiently near the truth,by assuming its value that which corresponds to the curvature of the meridian at themean latitude of the stations, and even to suppose it constant for all the triangles withinsimilar parallels of latitudes. To calculate the two remaining sides of the triangles afterthe three spherical angles have been determined, one side being always known either bymeasurement or computation, three different methods have been adopted: the first is totransform the side whose length is already known in feet into an arc of a circle, which isdone by comparing it with the radius of the earth, and solving the triangle by spherical trigo-nometry. But this process is not generally adopted ; that which was followed in calculatingthe triangles of the Ordnance survey was to consider, as the distance of any two stationsmutually visible from each other is very small in comparison of the whole circumferenceof the earth, the chord of the intercepted arc will differ from the arc itself, by a quantitywhich may be computed from the known ratio of the chord to the radius of the earth, butwhich in general is so small as to be insensible. If, therefore, from the observed sphericalangles we deduce in each case the corresponding angles formed by the chords, and withthese compute the sides by plane trigonometry, we shall obtain the chords of the arcsintercepted between the stations, and thence the arcs themselves. The third method wasafter the demonstration, that a triangle on a sphere or spheroid, which is small in comparisonwith the whole spherical surface, differs insensibly from a plane triangle, of which the sidesare respectively equal in length to the sides of the triangle on the sphere, and whose anglesare respectively equal to those of the spherical triangle, each diminished by one third ofthe spherical excess.

Tables for the reduction of spherical angles to the plane of the chords, and likewise forthe computation of the spherical excess, are given by Delambre in his Base Metrique.

To determine the Meridian Line the theodolite is fixed at one of the stations; and somehours before mid-day the telescope is directed, in such a manner that the cross-wires shalltouch the upper or lower limb of the sun in the east; and then the horizontal and verticalreadings of the arc are to be set down; this operation is to be repealed several times: inthe afternoon, the vertical arc being clamped to the last reading off, the horizontal angleat the time of the sun’s limb touching the intersection of the cross-wires is to be noted ;the vertical arc being clamped in succession in the descending series of the vertical angles,all the horizontal readings at the time of each successive intersection are entered : the

3 i 4