V. line perpendicular and equal to Ca. If from the point a BOOK we let down on Co the perpendicular ae, it will be the radius of the parallel at the lat. of 50 deg. (new measure), taking P for the pole, and Q for a point of the equator. This being done, we may consider aO as the side of a cone-tangent to the sphere; and then the surface near the circle of contact will coincide sensibly with the spherical surface. But, since, on one hand, we have to develope only the quarter of the circumference of which ae is the radius, or which comes to the same, the quarter of the curve surface of the right cone which has O a for side, and that, on the other hand, ae is the sine of 50 degrees, when the radius. ac is taken for the total sine, we shall have the logarithm of the sine of 50° 9,8494850, and the sine of 50° = 0,70,711. Then of the circumference which has for radius ae, is 1,1101627; finally, since the arc a M b. (Fig. 39.) described with a radius a0 = 1, should have for length 1,1101627, we shall find the number of degrees of this arc by the following proportion: 3,14: 200°:: 1, 1101627: x = 70°, 71. Such is the value of the angle a O b, or the amplitude of the arc ab, Fig. 39. Now, if we wish to have the degrees of longitude from 5 to 5, we must divide the arc ab into 20 equal parts, and the middle M of that arc will be on the axis OM of the map. But as it is not possible to determine the position of the other parallels, as well as the length of their respective degrees, without having a scale of equal parts, constructed according to the number of metres. contained in the mean radius a of the earth; a radius which, as we know, is 6,366198 metres, we shall proceed previously to the construction of this scale. For this purpose, we must bring on an indefinite line mC', Fig. 40, 636 parts, and from C' to m, and must take a' C' equal to the radius aC, Fig. 38; then through all the points of division of the line mC, are brought parallel to a' m, the right the map. lines xx', yy', &c. The line a' C' being by this means divided into parts proportional to mC', we form on this module the scale of the Fig. 39. 10 Scale of BOOK V. Remark on the map. Having thus constructed the scale of the map, we take in it a length of 50 parts, or myriametres, for the value of the degrees of the meridian, taken from 5 to 5, and we carry that length on the axis of the map, ten times above and ten times below the mean parallel a b, Fig. 39. We then describe from the point 0, as centre, indefinite arcs, passing through all the points of division of the axis O M; we as then have the parallels from 5 to 5 degrees. Finally, on each parallel, we take distances equal each to five times the value of the degree of longitude, known by the geographical tables. Thus, on the parallel of 55 degrees, the length of the degree of longitude is 6 myriametres 49; consequently, we must, setting out from the axis of the map, and on both sides of that axis, carry ten times the interval 6,49 × 5 = 32 myriametres 45, taken on the scale. When all the points through which the meridians must pass have been determined in this manner, it is easy to trace those curves. It must be confessed, that the amplitude of the arc of any parallel whatever determined by this method, will be a little greater than it ought to be, since we give to the cord of an arc of 5 degrees, the length itself of that arc; but the error which results from it is the less, according as the curvature of the parallel is smaller. Moreover, to obtain a rigorous exactness, we may determine the amplitude of all the parallels like that of the mean parallel, by the angle of which the two radii brought to the extremities of that parallel form. Instead of taking arbitrarily, as in the above example, the scale of the radius of the sphere, the length is most commonly fixed by means of a scale constructed beforehand, the parts of which also are in a determined relation with the metre. For example, in the Depot de la Guerre, the scale for the drawing and engraving of the map of each of the four parts of the world is that is to say, that, 2,000,000 metres taken on the ground, will be represented on the map by the real length of a metre. According to this, the radius of 1 2000000' the earth, which is 6,366198 metres, will be only on the BOOK 6m, S66198 2 map =S, 18. Hence, that the scale of this V. The various modifications of the conical projection having been sufficiently explained, we shall now consider the cylin- Cylindrical drical developments of the surface of the globe, and the ma- develop rine charts deduced from them.* ments. The points of the compass which navigators follow, having the property of cutting under the same angle all the meridians which they traverse, and which, for that reason, form on the globe the spiral named loxodromic, are necessarily projected by curve lines, of the same kind in all maps where the meridians are not parallel. This is demonstrated of loxodroby Fig. 41, in which we see a half of a hemisphere projected mic lines. on the plane of the equator. Let P be the north pole, AMB the equator; the right lines drawn from the centre to the circumference are meridians, and the concentric circles represent the parallels. Supposing the navigator wishes to go from C, a point of the equator right northwest, the course of his vessel must constantly make with the meridian of the place, or with the line north and south, an angle of 45 deg. * Neptune Français, Disc. prelim. Bezout, Cours de Mathém. Marine. Dubourguet, Traité des Navig. &c. V. BOOK (an. meas.) Supposing him to be arrived at G, the meridian line GNP, preserves no longer its parallelism with the meridian CG; if he continues his route north-west, keeping always the angle of 45°, he will reach the point H, to the point from J, and will thus describe the loxodromic curve CG H I, which constantly approaches the pole, without, however, ever reaching it. The greater the constant angle under which the route cuts the meridians, the longer the loxodromic curve becomes, as is seen in Fig. 41, by the line CRS. It is plain that mariners, who must direct their courses on these lines, cannot conveniently trace on these maps, neither the road they have made, nor that they have still to run, on account of the difficulty of measuring with the compass the arc of a curve. To obviate this inconvenience, they have endeavoured to contrive a projection of maps, in which the meridians should be right parallel lines. The development of a cylinder immediately presents itself to the mind, as the means of obtaining such a projection. When we merely wish to trace a zone of very little extent in latitude, it is evident that the spherical zone may, without any sensible error, be represented by the development of a cylinder, either inscribed, or circumscribed on that zone, and the axis of which coincides with that of the globe. The meridians which will result from the sections of the cylinder, by planes passing through its axis, are rcpresented by right lines parallel to that axis; the planes of the parallels cut the cylinder according to circles parallel to its base, and which become right lines in the development. Construc- Such is the construction of flat maps, the invention of which tion of flat is falsely attributed to Don Henry, Infanta of Portugal, naps. since Marinus of Tyre, anterior to Ptolemy, condemns their use, and has attempted to modify them.* Their defects are analogous to those of the conic projection; they are even more considerable; for, in the latter, one may Marin. Tyr. Ap. Ptolem. i. 20. Comp. Gosselin, Recherches sur la Geogr. des Grecs, li. 33. sqq. i. 46-50, &c. &c. V. give two parallels their true length with respect to the de- BOOK grees of latitude, while on the flat map this proportion can be observed with respect to one only; namely, for the inferior, in the development of the circumscribed cylinder, and for the superior in the development of that which is inscribed. It is true that one may avoid this inconvenience, by employing a cylinder constructed on one of the intermediate parallels, which would be partly interior, and partly exterior to the sphere; in this manner, the extent in longitude would be exact towards the middle, but the error would be divided between the two extremities. Cylindrical projections have even been attempted, in which the basis of the cylinder would be any vertical circle whatever;* but we shall not mention them, and shall merely observe, that the parallel which serves as base to the cylinder, may be placed so as that the area of the development shall be equal to that of the spherical zone. The tracing of flat maps is effected without difficulty, as soon as the position of the terrestrial parallel to be developed is fixed; it only remains to give to the degrees of longitude on this parallel, the size they ought to have with respect to that which is assigned to the degree of latitude. fat maps. The line HG, Fig. 42. being supposed parallel to the Lefects of axis CP, and equal to the development of the arc BF, will be the meridian of the map destined to represent the zone comprehended between the parallels of the points B and F. The development of the mean parallel, the radius of which is E e, will give the degrees of longitude. This figure shows the defect of the map on the extreme parallels, sinco the radius G g is smaller than B b, and the radius H larger than F f. These maps can only serve for very small parts of the globe; the least defective are those which represent the rogions near the equator, because at a little distance from this circle the cosines of the latitude do not vary much. D'Anville made use of them in such a case, but it was almost unique. *Textor, dans Zach, Corresp. xviii. 190. Carte de Guinée, 1776. D'Anville, Consid, sur la Geogr. p. 30. |