208 which is 21 ‘547. 2.3) shows small portions of country in strictly correct forms; but the areas are — slightlytoo great at 4 " the extreme lati- " tades and too small , in the centre. At. any part of the map a degree of latitude may be used as the a true scale in any di- rection. The value ]t=:];, as suggested by Sir John Herschel, is admirably suited for a map of the world. ., The representation is fan-shaped, with remarkably little _ _ distortion (fig. 26). “-3- 29' It follows from what has been stid above that the con- dition that the scale is true at the equator is It/c=a, which (1 E (T) (l '3 .1-1 -1 A P H Y The map of .'orth Am -iic-.1 so found (fig. ' [)1 .-'l‘llE![.TIC.L. determines l‘ when It is given. The radius of the parallel whose co-latitude is '16 being p, let r be the distance of that parallel from the equator; then, keeping to the condition that the scale is true at the equator, _a. t_ hu p_]: an 2 , H '. 1' ‘ -r 1 - tan . /t( ‘.3 ) When It is very small, the angles between the meridian lilies in the representation are very small; and proceeding to the limit, when It is zero the meridians are parallel, that is, the vertex of the cone has removed to infinity. And at the limit when 1:. is zero we have ‘ll; r=a log. cot , [V which is the characteristic equation of J[crcal0r’s Pzojection. From the manner in which we have arrived at this pro- jection it is clear that it retains the characteristic property of Gauss’s projection,—namely, similarity of representation 3-) FIG. 26.——Fan-shaped Map of the Vorld. of small parts of the surface. In lIercator’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 latitudeqs from the equator is, as we have seen above, 7°=a log, tan (45°+§qS). 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 ?'=a (SiI1qS+;§ sin3¢+—}; sin 5¢~+ . . . .), 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 IIercator’s chart so valuable to seamen. For instance: join by a straight line on the chart Land’s End and Ber- mud-1, 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 we apply Mercator’s system of projection along a mericlian, as proposed by Lambert, we have the represen- tation of all possible great circles. The diagram (fig. 27 gives the projection. The two vertical bounding lines are the equator— crossed at right angles by the initial meridian passing through one of the poles. From the form of the representations of parallels round the pole it is clear that the distortion up to a distance of 30° or 40° from the initial meridian is not at all great. The representation extends to infinity upwards and downwards, and the left and righthalves are interchangeable; if interchanged the representation is on a meridian extending from pole to pole. The meridian Mercator drawn as described in the last
paragraph——with the meridians and parallels rather close——