_Measurement of Distance._--The shortest distance between two places on the surface of a globe is represented by the arc of a great circle. If the two places are upon the same meridian or upon the equator the exact distance separating them is to be found by reference to a table giving the lengths of arcs of a meridian and of the equator. In all other cases recourse must be had to a map, a globe or mathematical formula. Measurements made on a topographical map yield the most satisfactory results. Even a general map may be trusted, as long as we keep within ten degrees of its centre. In the case of more considerable distances, however, a globe of suitable size should be consulted, or--and this seems preferable--they should be calculated by the rules of spherical trigonometry. The problem then resolves itself in the solution of a spherical triangle. In the formulae which follow we suppose l and l´ to represent the latitudes, a and b the co-latitudes (90-l or 90° - l´), and t the difference in longitude between them or the meridian distance, whilst D is the distance required. If both places have the same latitude we have to deal with an isosceles triangle, of which two sides and the included angle are given. This triangle, for the convenience of calculation, we divide into two right-angled triangles. Then we have sin ½ D = sin a sin ½ t, and since sin a = sin (90° - l) = cos t, it follows that sin ½ D = cos l sin ½ t. If the latitudes differ, we have to solve an oblique-angled spherical triangle, of which two sides and the included angle are given. Thus, cos D - cos a cos b cos t = ------------------- sin a sin b cos D = cos a cos b + sin a sin b cos t = sin l sin l´ + cos l cos l´ cos t. In order to adapt this formula to logarithms, we introduce a subsidiary angle p, such that cot p = cot l cos t; we then have cos D = sin l cos(l´ - p) / sin p. In the above formulae our earth is assumed to be a sphere, but when calculating and reducing to the sea-level, a base-line, or the side of a primary triangulation, account must be taken of the spheroidal shape of the earth and of the elevation above the sea-level. The error due to the neglect of the former would at most amount to 1%, while a reduction to the mean level of the sea necessitates but a trifling reduction, amounting, in the case of a base-line 100,000 metres in length, measured on a plateau of 3700 metres (12,000 ft.) in height, to 57 metres only. These orthodromic distances are of course shorter than those measured along a loxodromic line, which intersects all parallels at the same angle. Thus the distance between New York and Oporto, following the former (great circle sailing), amounts to 3000 m., while following the rhumb, as in Mercator sailing, it would amount to 3120 m.