i MAP. 



into circular instead of elliptical arcs, the deviation from the strict law 

 of the projection being too slight to affect the practical utility of the 



MAP. 



4S2 



map. 



Of Projection by Development. 



The developments to be mentioned are two the Conical and 

 Cylindrical. 



Conical Projection. In this projection the sphere is supposed to be 

 circumscribed by a cone, which touches the sphere at the circle 



intended to represent the middle parallel of the map. If the points on 

 the sphere be now projected on the cone by lines drawn from the 

 centre, it is clear that in a zone extending but a short distance on 

 each side the middle parallel, as the zone a a' b b', the points on the 

 cone would very nearly coincide in position with the corresponding 

 points on the sphere. All the delineations having been thus made, the 

 cone is then conceived to be unrolled, or developed on a plane surface. 



Should the map be made to extend much above or below the 

 middle parallel, the distant parts will be very much distorted. To 

 remedy the defects of this projection, various modifications have been 

 suggested, among which those of Flamsteed are generally held in the 

 highest estimation. [Cojnc PROJECTION.] 



Cylindrical Projection. From what has been said of the cone, it 

 will be easily understood that a cylinder may be applied to the sphere 

 in a similar manner, and that a zone of very limited extent in latitude 

 may, without very material error, be developed on a cylinder. The 

 peculiarity of this method is, that the meridians, as well as the latitude 

 circles, are projected in parallel straight lines ; a condition of the map 

 which makes it very applicable to nautical purposes, and on which 

 (partly) is founded the very ingenious method called Afercator's Pro- 

 jection, which is now so universally adopted in our charts, and which, 

 in conclusion, we will briefly describe. 



ifercator'i Projection. The line on which a ship sails, when 

 directing her course obliquely to the meridian, is on the globe a spiral, 

 since it cute 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 ia 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 



ABTS ASD SCI. I>1V. VOL. V. 



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 B, keeps one course ; 

 but were it to sail over the great circle A D B, the course must be per- 

 petually altering. 



North Pole. 



The spiral A c B is the only one on which a ship should sail directly 

 from A to 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 in 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 either pole, and so rapidly that it becomes 

 as great as we please at and near the poles. Let the equator K Q be 



1 1 



