where M, the modulus of common logarithms, = 0.434294. Of course Mercator's projection was not originally arrived at in the manner above described; the description has been given to show that Mercator's projection is a particular case of the conical orthomorphic group. The introduction of the projection is due to the fact that for navigation it is very desirable to possess charts which shall give correct local outlines (i.e. in modern phraseology shall be orthomorphic) and shall at the same time show as a straight line any line which cuts the meridians at a constant angle. The latter condition clearly necessitates parallel meridians, and the former a continuous increase of scale as the equator is departed from, i.e. the scale at any point must be equal to the scale at the equator × sec. latitude. In early days the calculations were made by assuming that for a small increase of latitude, say 1´, the scale was constant, then summing up the small lengths so obtained. Nowadays (for simplicity the earth will be taken as a sphere) we should say that a small length of meridian ad[phi] is represented in this projection by a sec [phi] d[phi], and the length of the meridian in the projection between the equator and latitude [phi], /[phi] / a sec [phi] d[phi] = a log_e tan (45° + ½[phi]), \/ 0 which is the direct way of arriving at the law of the construction of this very important projection. Mercator's projection, although indispensable at sea, is of little value for land maps. For topographical sheets it is obviously unsuitable; and in cases in which it is required to show large areas on small scales on an orthomorphic projection, that form should be chosen which gives two standard parallels (Lambert's conical orthomorphic). Mercator's projection is often used in atlases for maps of the world. It is not a good projection to select for this purpose on account of the great exaggeration of scale near the poles. The misconceptions arising from this exaggeration of scale may, however, be corrected by the juxtaposition of a map of the world on an equal-area projection. It is now necessary to revert to the general consideration of conical projections. It has been shown that the scales of the projection (fig. 23) as compared with the sphere are p´q´/pq = dp/dz = [sigma] along a meridian, and p´r´/pr´ = [rho]h / sin z = [sigma]´ at right angles to a meridian. Now if [sigma][sigma]´ = 1 the areas are correctly represented, then h[rho] d[rho] = sin z dz, and integrating ½h[rho]² = C - cos z; (i.) this gives the whole group of _equal-area conical projections_. As a special case let the pole be the centre of the projected parallels, then when z = 0, [rho] = 0, and const = 1, we have p = 2 sin ½z/[delta]h (ii.)