_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 [phi] from the equator is, as we have seen above, r = a log_e tan (45° + ½[phi]). 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 [phi] + 1/3 sin ^3[phi] + 1/5 sin ^5[phi] + ...), 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 = [root] (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° + ½[phi]) - e² sin [phi] - 1/3 e^4 sin ^3[phi] ...}