Mercator's Projection.—The line on which a ship sails, when directing her course obliquely to the meridian, is on the globe a spiral, since it cuts all the meridians through which it passes at equal angles. This circumstance, combined with others, rendered a map constructed on the principles of the spherical projections very inadequate to the wants of the navigator. Mercator considered, very justly, that mariners do not employ maps to know the true figures of countries, so much as to determine the course they shall steer, and the bearing and distance of those points or places which lie near their track ; and this projec tion is the result of his efforts to secure to the seaman these desirable ends. The merit of this most useful method is thought by many to be more justly due to Wright ; for although Mercator published his first chart in 1556, he omitted to declare the principles on which he pro ceeded, and his degrees of latitude did not preserve a just proportion in their increase towards the poles. Wright, in 1599, corrected these errors, and explained the principles of his improved construction, in which the degrees of latitude on the chart were made to increase towards the poles, in the same ratio as they decrease on the globe ; by which means the course which a ship steers by the mariner's compass becomes on the chart a straight line ; the various regions of the map, however distorted, preserve their true relative bearing, and the distances between them can be accurately measured.
The use of this projection constitutes the principal difference between the methods of travelling by land and by sea, and perhaps there is no point of navigation on which a person who is neither a seaman nor a mathematician has so little chance of gaining any informa tion from popular works.
We shall suppose the ship a mathematical point in comparison with the earth, and imagine the whole of the latter to be covered by sea. Also let the ship be always sailing before the wind, and no allowance for leeway or currents be necessary. Throw out also the variation of the compass, that is, suppose the needle always to point due north.
A ship thus circumstanced, if it should continue sailing due north, would in time reach the north pole on a meridian circle of the sphere, on which, if it still kept its course, it would proceed due south, and would at last reach the south pole : such a ship would never change its longitude, except at the moment of passing either pole, when the longitude would alter at once by 180 degrees. If however the vessel sailed continually due east or due west, it would sail upon a small circle of the sphere, being always at the same distance from the pole, and always in' the same latitude. In the first case the differences of latitude would give the distances sailed over, at the rate of 60 nautical miles to a degree ; in the second case, the differences of longitude, reduced in the same way, and the results multiplied by the cosine of the latitude, would serve the same purpose.
But suppose that the vessel took an intermediate course, say north east. It would not sail on any circle of the sphere, great or small; for by hypothesis the line of the course is always making an angle of 45 degrees with the meridian ; and there is no circle (unless it be the meridian itself, or a parallel of latitude, the equator included) which always makes the same angle with the meridian. Neither could the
vessel, keeping such a course, reach the pole; for at the moment when it touches the pole, it is sailing north, whereas by hypothesis it is always sailing north-east. The fact is, that a curve which makes equal angles with all meridians must be a spiral which approaches the pole, encircling it with an infinite number of folds, but never actually reaching it, as in the following diagram, in which the curve 1, c, 2, 3, 4, &c., is that on which a ship would sail from 1 towards the north pole on a course east-north-east, and the curve 1, 5, 6, 7, 8, &c., is that of a course west-south-west towards the south pole. The dotted part of the figure is supposed to be on the other, or the invisible, side of the sphere. A ship sailing from A to B over A 0 13, keeps one course; but were it to sail over the great circle A n B, the course must be per petually altering.
The spiral A C B is the only one on which a ship should sail directly from A 10 B. though there is an infinite number of such curves which pass through both A and B, the reason being, that in every other spiral except A B one or more complete circuits iu longitude must be made, and the ship would come again to the meridian passing through A before it reaches B. In the same manner a spiral might be found, passing through A and B, which cuts the meridian of A five hundred times before it passes through B. Of course the shortest course is always preferred ; and it is the object of Mercator's projection to lay down such a map of the world, that the straight line joining two points shall be the map of the course which must be followed in order to sail from one to the other in the most direct manner, consistently with always keeping the same point of the compass.
The spirals above described are called loxodromic spirals, or rhumb lines, and under the latter term their mathematical properties are explained. Our present object is to turn the globe into one of Mer cator's maps, in a manner which will give the unmathematical reader some idea of its construction. For this purpose suppose the map of the world to be painted on the globe, and let the globe be made of a thin and very elastic material. Let the elasticity of this material increase as we go towards tither pole, and so rapidly that it becomes as great as we please at and near the poles. Let the equator s Q he immoveably connected with the internal centre (supposed fixed) of the globe. If then the north and south poles be pulled away from the equator, the thin membrane of the sphere will be extended ; and if the be continued until the pole. are sufficiently distant, a large portion of the sphere on each side of the equator will assume a cylin drioal form, or one nearly cylindrical ; and the greater elasticity of the upper parte will cause the small fold. of the different spirals to be mob more extended than the larger ones, so as to become equal to them. Let the mathematical by thesis implied in the preceding be carried to its extreme limit, that Is, let the pulea be pulled to an intl.