ORTHOGRAPHIC AND STEREOGRAPHIC PROJECTIONS.

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the map. Near the centre of the map, only, will there be tolerable accuracy. Accordingly this projection is either not used at all in geography, or if it is used, small portions of the sphere only are delineated. Either the plane of à meridian, that of the equator, or of any plane parallel to the equatorial plane, may be used for this projection. If the plane of the map is that of the equator, the meridians of longitude will be straight lines radiating from the centre to the circumference of the map; the parallels of latitude will be concentric circles approaching nearer to each other as they proceed towards the circumference of the map. If a map of America were drawn in this manner, the isthmus which connects North and South America would be much diminished in length; its width would be correctly shown; so that the proportion would be lost. If the map were drawn on the plane of a meridian, the lines of longitude would each be a different figure. At the circumference, the meridian would be represented as a circle; through the middle of the map, as a straight line; and each would have a more or less oval form, according as it was nearer to, or farther from, the circumference of the map. The parallels of latitude would be straight lines at unequal distances. The inaccuracy in this case would be in both dimensions; all countries near the circumference of the map would be drawn too small, (as compared with those towards the centre,) both in direction from north to south, and from east

to west.

Stereographic. In the stereographic projection, having supposed the hemisphere placed on the map as before, we represent a given place on it, say Rome, not by a point immediately under that place on the map, but by drawing a line from Rome through the map to the middle point of the under hemisphere, and so with any other spot; all the lines from the upper hemisphere meeting at the same point beneath the map, and being marked on that point of the map where their lines pass through it. It is apparent that this method has not the inaccuracies of the orthographic method; for while the central point of the map, being projected in both cases by a perpendicular from the zenith, remains the same for both, the points on the slope of the hemisphere are projected in the stereographic mode by lines sloping more and more to the plane. In fact, this projection over-corrects the orthographic projection, for it makes the parts of the hemisphere which are projected towards the circumference of the map occupy more space than equal parts of the hemisphere projected at and near the centre. In a hemisphere stereographically projected, the length of a degree of longitude increases in passing from the centre of the map to its circumference. The stereo

graphic projection has two advantages which cause it to be much used in delineating the hemisphere. One is, that the projections of the lines of longitude and latitude are either circles, or straight lines, and therefore easily described; another, that these lines cut each other on the map at right angles, that is, as they do on the globe itself.

Globular. The globular projection is, as it were, intermediate between the two preceding. We have seen that when points on the hemisphere are projected by straight lines drawn from them to the middle point of the under-hemisphere, the degrees of longitude increase towards the edge of the map. In the orthographic projection they diminish. In the globular projection they are equal, as on the globe itself. The point to which the projecting lines must be drawn, in order to preserve this equality, is one below the middle point of the under hemisphere; the distance below it being about equal to the sine of forty-five degrees. Thus, if the diameter of the globe were two hundred feet, the distance of the point from the surface of the sphere would be about seventy feet. By this projection the circles of the sphere become in the map not quite circular, but slightly elliptical; moreover, these lines do not intersect each other at right angles. It is usual, however, for the sake of simplicity, to neglect the slight deviations from the circular form, and to project the circles of the sphere as in the stereographic projection. The globular projection is that generally used, at present, in maps of the world divided into two hemispheres.

Central. The last kind of projection which we have to notice, is the central. In looking at a map thus projected, we must suppose the entire globe of which we are projecting a portion to be suspended from the central point of the map. Straight lines are then supposed to be drawn from the centre of the globe through the surface till they meet the map. Each point on the globe through which a radius passes is projected at a point where the radius meets the map. It is evident that an entire hemisphere cannot be represented by this method, for the radii which should project the circumference of the hemisphere would be parallel to the map, and could not meet it. The incorrectness of this mode of projection is similar to that of the orthographic projection. The difference is, that in one case the projecting lines are perpendicular to the map, in the other, to the sphere. Inasmuch as the circumferential parts are represented too small in the orthographic, they are represented too large in the central projection. Notwithstanding this objection, however, maps delineated according to this method have been published by the Society for the Diffusion of Useful Knowledge. In these maps, six in number, the sphere is projected on the sides of a cube, in which it is

CONICAL DEVELOPMENT.

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supposed to be enclosed. We do not think the experiment a successful one.

We have now treated of all the modes of projection which are used in the delineation of the earth on a plane surface. Projections of the sphere are not often used, except for delineating an entire hemisphere. For drawing small portions of the sphere,single countries for instance,-developments, sometimes called projections by development, being more accurate, are almost always made use of. If a sphere be placed within a right cone, whose sides it touches, one band of the sphere contiguous to the circular parts of the cone will nearly coincide with a band of the cone. This band of the sphere may be projected on the corresponding band of the cone by means of straight lines drawn from the centre of the sphere, without much distortion of the features of the sphere. Now it is a property of the cone which may easily be verified, that it can be unrolled so as to lie flat on a plane surface. If such a band of the cone as we have described be unrolled we shall have a nearly accurate map of the corresponding band of the sphere. In a map of this description, the meridians will be straight lines proceeding from the vertex of the cone, and the parallels of latitude will be circles of which the meridians are the radii. A modification of this kind of map, invented by Flamsteed, is much used. According to Flamsteed's method, the parallels of latitude are made equal to the corresponding lines on the sphere, (which they are nearly, in the normal form of this development,) and they are drawn as straight lines perpendicular to the middle meridian. It consequently becomes necessary to draw the meridians as curves. The departure from the simple development, therefore, is considerable. Another modification of the same has been used in the maps prepared under the direction of the French government. In these, the parallels of latitude are parts of circles equal to the corresponding lines on the sphere. The meridians are also curves. An objection to both these methods (which otherwise have advantages over the simple development) is, that the lines of latitude and longitude do not cut each other at right angles. A third modification of the conical development is that invented by Murdoch. Humboldt* considers it superior to any of the other methods. The conical surface, according to Murdoch, is not supposed to be entirely without the sphere, but the cone and sphere intersect each other, and in such a manner that the conical surface is exactly equal to the portion of the sphere which it represents.

A second mode of developing the sphere is that in which a cylinder is substituted for a cone. In this case, the meridians and parallels of latitude are straight lines, cutting each other at

* Introduction to La Nouvelle Espagne.

right angles. It is partly according to this method that the very useful nautical maps known as Mercator's projection, or charts, are drawn. Kauffman, (generally known by the Latin synonyme of Mercator) a native of East Flanders, published the first maps according to cylindrical development, about the middle of the sixteenth century. It is said, however, that he did not discover the principle on which charts (which are not simple developments of the cylinder) are constructed, but that Wright, an English mathematician, has the merit of the discovery. The chart is the only part of the invention deserving of much notice. A chart is a map intended for a different purpose from that of other maps.

Its object is, not to show the form and size of the different parts of the globe, but to inform the mariner what course he must steer in order to reach a given place. The principle on which it is constructed, and by which it serves this purpose, is such, that a path on the globe, cutting all the meridians at equal angles,-in other words, the path which a ship would sail which never altered her course, is represented on the chart by a straight line. Now, as this path, as a general rule, is the nearest course between any two places, so the nearest route, according to the chart, is generally that indicated by a straight line drawn between the same places. In order to draw a map on this principle, the degrees of longitude are made equal in all parts of the chart, while those of latitude increase rapidly north and south of the central line. It was some time before the chart so universally used at present gained its present position. William Burrough, a celebrated navigator, who rose to be comptroller of the navy in the reign of Queen Elizabeth, objected to the use of Mercator's chart, because, by Mercator's augmenting his degrees of latitude towards the poles, the same is more fit for such to behold as study in cosmegraphy, by reading authors upon the land, than to be used in navigation at the sea,' showing thus his entire ignorance of its peculiar merits. But indispensable as the chart is to modern navigation, it must not be followed blindfold. Old mariners, indeed, have become so used to chart-ist associations (if we may be excused a pun) that they verily believe their distorted image of the world to be a true one; and nothing will persuade them that the shortest way between two points is not always the straightest. As a general rule, indeed, the shortest path between two places on a globe is that which cuts all the meridians at equal angles, but it is not always so. Attention was first drawn to this fact about three years ago, by Mr. John Towson, and directions for following his suggestions were published by the Admiralty. The mode of sailing thus recommended was called composite great circle sailing.' Increased attention has beca

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given to the new method by the success of Captain Godfrey, who, by following the directions of the Admiralty, performed the voyage to Adelaide in the unusually short time of seventy-seven days.

The name by which this kind of sailing goes, correctly indicates its nature, which may be thus illustrated. If a vessel situated on the equator, wishes to pass to the opposite point of the same parallel, there are three paths of equal length, which (supposing no practical difficulties) she might take. One would be along the equator, the other two along the meridian, either north or south, and through one or other of the poles. It is the first of these paths which will be indicated by the chart. In the case we put, the three paths are equal, and each describes a great circle of the sphere. But if we suppose a vessel nearer one or other of the poles, the case is different. In that case the path through the pole which is along a great circle of the sphere, may be shorter than that along a corresponding parallel of latitude, which is a small circle, and represented as a straight line on the chart. In this example, the line on the chart corresponds with a small circle on the sphere, but it is an exceptional case. Straight lines on the chart represent any curves making equal angles with the meridians. Of course, a small circle is one of these; but, generally, such curves are spirals. And generally, as we have said, the mariner who wishes to pass in the shortest way from one spot on the earth to another, cannot do better than sail on a spiral. In certain cases, however, there is a shorter path, and when a mariner should sail on a great circle, and when on a spiral, is a question for the mathematician, from whose results, general rules, such as those of the Admiralty, are prepared for the guidance of the mariner. In the case of the Constance,' alluded to above, instead of doubling the Cape of Good Hope in the usual hanner, Captain Godfrey began from the latitude of 24° S. to shape his course on the arc of a great circle of the sphere; he did not, therefore, keep on one course, but varied his course with his latitude. He followed this method till he had attained the parallel of 50° of S. latitude, having then made about 68° of longitude. He then sailed due east along a small circle, by which he added 72° 40′ to his longitude. Finally, he repeated the new method at the other end of his voyage, by a northerly course along the arc of a great circle to the place of his destination. The distance sailed by Captain Godfrey from the point where he commenced the new method of sailing (in S. lat. 24°) was 8145 miles. The voyage from the same latitude by the Cape, and thence, by Mercator's sailing, to Adelaide, is 9080 miles; so that the saving in this route is 935 miles.