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 = σ along a meridian, and p′r′ / pr′ = ρh / sin z = σ′ at right angles to a meridian.
Now if σσ′ = 1 the areas are correctly represented, then
hρ dρ = sin z dz, and integrating 1⁄2hρ2 = 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, ρ = 0, and const = 1, we have p = 2 sin 1⁄2z / δh
(ii.)