If any parallel of co-latitude z is true to scale hk(tan 1⁄2z1)h = sin z, if this parallel is the equator, so that z1 = 90°, kh = 1, then equation (i.) becomes ρ = (tan 1⁄2z)h/h, and the radius of the equator = 1/h. The distance r of any parallel from the equator is 1/h − (tan 1⁄2z)h/h = (1/h){1 − (tan 1⁄2z)h}.
If, instead of taking the radius of the earth as unity we call it a, r = (a/h){1 − (tan 1⁄2z)h}. When h is very small, the angles between the meridian lines in the representation are very small; and proceeding to the limit, when h is zero the meridians are parallel—that is, the vertex of the cone has removed to infinity. And at the limit when h is zero we have r = a loge cot 1⁄2z, which is the characteristic equation of Mercator’s projection.
| Fig. 25.—Elliptical equal-area Projection, showing the whole surface of the globe. |
Mercator’s Projection.—From the manner in which we have arrived at this projection it is clear that it retains the characteristic property of orthomorphic projections—namely, similarity of representation of small parts of the surface. In Mercator’s chart the equator is represented by a straight line, which is crossed at right angles by a system of parallel and equidistant straight lines representing the meridians. The parallels are straight lines parallel to the equator, and the distance of the parallel of latitude φ from the equator is, as we have seen above, r = a loge tan (45° + 1⁄2φ). In the vicinity of the equator, or indeed within 30° of latitude of the equator, the representation is very accurate, but as we proceed northwards or southwards the exaggeration of area becomes larger, and eventually excessive—the poles being at infinity. This distance of the parallels may be expressed in the form r = a (sin φ + 1⁄3 sin3 φ + 1⁄5 sin5 φ + ...), showing that near the equator r is nearly proportional to the latitude. As a consequence of the similar representation of small parts, a curve drawn on the sphere cutting all meridians at the same angle—the loxodromic curve—is projected into a straight line, and it is this property which renders Mercator’s chart so valuable to seamen. For instance: join by a straight line on the chart Land’s End and Bermuda, and measure the angle of intersection of this line with the meridian. We get thus the bearing which a ship has to retain during its course between these ports. This is not great-circle sailing, and the ship so navigated does not take the shortest path. The projection of a great circle (being neither a meridian nor the equator) is a curve which cannot be represented by a simple algebraic equation.
If the true spheroidal shape of the earth is considered, the semiaxes being a and b, putting e = √(a2 − b2) / a, and using common logarithms, the distance of any parallel from the equator can be shown to be
(a / M) {log tan (45° + 1⁄2φ) − e2 sin φ − 1⁄3 e4 sin3 φ ...}
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 adφ is represented in this projection by a sec φ dφ, and the length of the meridian in the projection between the equator and latitude φ,
√φ0 a sec φ dφ = a loge tan (45° + 1⁄2φ),
which is the direct way of arriving at the law of the construction of this very important projection.