the points where the meridians meet the planes of projec- BOOK tion. = IV. From any point whatever taken on the line AB, or its To trace prolongation, from the point F, for example, is brought the merididown perpendicularly Fk, on the line PP', making as weans ty points. know already, an angle equal to the height of the pole, and the length Fk is carried from F to k': then from this last point as centre, and with a radius Fk", or with any other radius taken at pleasure, but rather large, a circumference is described, which is likewise divided into 40 equal parts. After this, secants k' n', k' n", k n'", are brought through all those points of division; the extremities n' n", n'", of those secants, terminated in the right line SS', are on the very traces of the planes of the meridians; drawing therefore right lines, which pass through the centre of the map, such as n C'u', n" C"u". n" C"", the diameters n'', &c. will be the required traces of the meridians; and as, moreover, they must all pass through the pole p, we shall have three points of each meridian, for example, μ''', p, m''' ; the meridians therefore will be easily described according to one of the processes previously indicated. In practice, as there commonly is not space enough round the map to perform this construction, founded on the principles of descriptive geometry, Fk may be carried from F to k"; this point will then be what is called the centre divisor; in other respects, the processes are the same. Let us now examine how parallels to the equator are Tracing of described. Their planes being perpendicular to the prin- parallels. cipal meridian AB, we shall obtain the diameters of their projections, as we obtained those of the equator, that is to say, after having divided the circumference ABDE into 40 equal parts, setting out from the point P, we draw, two and two, the right lines (1) E, (1) E; and the interval v intercepted between these right lines, and taken on the meridian AB, will be the diameter of a parallel. In the present, the parallel vv belongs evidently to the 80th degree of latitude, since the arc AP measures the height IV. BOOK of the pole. But for the parallels which are very distant from the superior pole p, the construction we have just indicated can no longer be put in practice, because the point would then be too far from the centre of the map. To obviate this inconvenience, we may trace the intersections of the planes of the parallels, with the plane of projection ADBE, intersections which are necessarily parallel to the diameter DE, and distant from it by a sum equal. sin. lat. of the parallel Properties cos. height of the pole When the latitude is southern the pole p being the northern pole, this value becomes negative; and instead of bringing it on the side of AC, it is brought on the side of CB. Thence it follows that if at any distance whatever from the right line DE. (fig. 23.) the parallel line de is brought to it, the points d and e common to that parallel and to the circumference ADBE, will belong to the required parallel: but this parallel passes at the same time through a point such as v determined by the preceding method; we have therefore the three points necessary for tracing a circumference. There are other methods of explaining the three stereoof the ste- graphic projections, but we prefer indicating in a few reographic projection. words the advantages and defects of this sort of projection. It is sufficient to cast one's eyes on a map of this kind, to perceive that the quadrilaterals comprehended between two meridians and two consecutive parallels, increase in extent in going from the centre to the circumference. This increase results from the obliquity which the visual rays take, on parting from an axis perpendicular to the picture, called the optical axis. It follows thence that the regions placed towards the borders of the hemisphere have a much more considerable extent than if they were at the centre, and that we are led into error whenever we compare them with those which occupy that part. For example, the point of austral Africa, appears much broader than on a globe, and in Nova Zembla the distances, south and * R. Vaugondy, Instit. Geogr. 1. c. north are rendered by spaces much larger than the same distances are in India. This inconvenience, of no consequence to learned geographers, may lead pupils to false ideas; but this risk would be diminished, if, in teaching care were taken to explain the qualities of stereographic projections, and to place under the view of beginners the polar, equatorial, and horizontal planispheres, the defects of one always disappearing in the other. BOOK IV. distances map. The stereographic projection does not admit, in general, Measure of the employment of a rectilineal scale for comparing the re- on a stereospective distances of places, which are measured according graphic to the arc of a great circle, joining these places: but we may always, by means of the graduation itself, measure the distance between the centre of the map, and any point whatever, and consequently we may know, on the horizontal projection relative to Paris, for example, the distance of that town from all the other points of the globe. This property results from this, that all the great circles which pass through the centre of the map, cutting each other according to the optical axis, have for perspective, right lines drawn through that centre, and admit of a graduation similar to that which is observed on the equator of maps of the world, constructed on the plane of the meridian. If we wish to measure the distance from two points on a stereographic map, we may (Fig. 24.) make use of the following construction. Let Z be the zenith of a place, C the centre of the horizon, or the projection of Z and ZMB, XMB' the respective verticals of the two points MM' given on the globe by their longitudes and latitudes. These points will have evidently for perspectives or traces mm', supposing the eye in E. But if we prolong the right lines MM', mm', they will meet in a point R, and the right line COR will mark on the plane of projection CBB, the trace of the plane MCM' of the great circle to be pro IV. BOOK jected. Therefore the four points m, m', O, O', are on the projection of the great circle which passes through MM'; thus this projection, which is itself a circle, will be entirely determined. This being laid down, we may trace the shortest distance on the map, in the following manner. Carry Cm (Fig. 25.) from C to , and Cm' from C to; draw the right lines Eun, Fu'n': then on mm' construct the triangle m' E' m, so that m E' be equal to E, and m' E' to ' E; next, on the prolongations of E' m' and Em, draw un from m to n', and μ'n' from m' to n"; finally, let the common section R of the two right lines m' m and n" n", be determined, and draw the right line RCO, which will be the trace required. We may now trace the arc of a circle O' m' vm 0, of which the portion m'vm is the shortest distance. The number of degrees contained in the shortest distance, will be ascertained by considering the right line n'" n', which is equal to MM' (Fig. 24. and 25.) as the cord of the circumference ADB. Origin of The stereographic projection was not known to the anthe stereo- cients. The first map of the world of this kind is found in projection, a work of the beginning of the 16th century, by the same graphic Orthogra. Werner of Nuremberg, who gave the first indication of the method of lunar distances * He was indebted for the idea to his master, Stabius the astronomer.† The use of this projection appears to have been general 150 years later. Varenius marks its three modifications. Hesius, a German geographer, who lived in the beginning of the 18th century, applied stereographic projections to special maps. This method, laborious, but favourable to the exactness of the details of position, is little followed in France, where the stereographic projection is reserved for maps of the World. We shall now proceed to the explanation of orthographic phic pro- projections, which might also be called planetary, since their jections. * J. Werner, de quatuor orbis terrarum figurationibus, ad calcem. Ptolomæi geograph. lib. i. vers. ab eodem. + Comp. Weidler, Hist. Astron. cap. xiv. nos. 3 and 4. IV. principal object is to show the direct image of the half of BOOK the globe, the eye being supposed at an infinite distance, that is to say, great enough for all the visual rays to be reckoned parallel. As these rays are perpendicular to the plane of projection, while the lateral parts of the sphere present themselves more and more obliquely to this same plane, it is easy to perceive, even without demonstration, that this projection, offering the contrary defect of the stereographic, makes the space diminish from the centre to the circumference. This diminution, which is infinitely greater than that remarked in the preceding projection, gives to the extremities of a planisphere orthographically projected, an aspect too much disfigured to fulfil, in general, any of the objects proposed by geography. This is a sufficient reason for indicating here only very briefly what regards orthographic constructions. Figure 26, indicates the polar projection. The lines AB Polar proand CD, are two meridians which cut each other at rightjection. angles in E, which is the projection of the pole, and the centre of the map. The circumference ABCD is the equator, on the plane of which the map is projected. This circumference is divided into equal parts, from 10 degrees to 10, or from 5 to 5; the diameters which pass through the points a' a", b' b", &c. and by the centre E will be the meridians. Let fall from the points a' b', &c. perpendiculars on the diameter CD; they will determine the radii E 1, E 2, &c. with which you will describe the circles parallel to the equator. In the projection on a meridian, the process is constructed Equatorial in the following manner. Draw the lines AB and CD (Fig. 27.) cutting each other at right angles; one will be the meridian of the middle, the other the equator. Their intersection E is the centre of the plane of projection, circumscribed by the meridian ABCD. This circumference must be divided into equal parts, then unite the points of division, the diameters a' a", b' b", &c. will be the common sections of the meridians, with the plane of the equator. The angles a' ED, &c. will mark the inclination of these meridians on |