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to conclude, so far as they can be thence concluded, their real geometrical relations to each other and to the spectator. It agrees with ordinary perspective when only a small visual area is contemplated, because the concave ground of the celestial sphere, for a small extent, may be regarded as a plane surface, on which objects are seen projected or depicted as in common perspective. But when large amplitudes of the visual area are considered, or when the whole contents of space are regarded as projected on the whole interior surface of the sphere, it becomes necessary to use a different phraseology, and to resort to a different form of conception. In common perspective there is a single “point of sight,” or “ centre of the picture," the visual line from the eye to which is perpendicular to the “plane of the picture," and all straight lines are represented by straight lines. In celestial perspective, every point to which the view is for the moment directed, is equally entitled to be considered as the “ centre of the picture,” every portion of the surface of the sphere being similarly related to the eye. Moreover, every straight line (supposed to be indefinitely prolonge) is projected into a semicircle of the sphere, that, namely, in which a plane passing through the line and the eye cuts its surface. And every system of parallel straight lines, in whatever direction, is projected into a system of semicircles of the sphere, meeting in two common apexes, or vanishing points, diametrically opposite to each other, one of which corresponds to the vanishing point of parallels in ordinary perspective; the other, in such perspective has no existence. In other words, every point in the sphere to which the eye is directed may be regarded as one of the vanishing points, or one apex of a system of straight lines, parallel to that radius of the sphere which passes through it, or to the direction of the line of sight, seen in perspective from the earth, and the points diametrically opposite, or that from which he is looking, as the other. And any great circle of the sphere may similarly be regarded as the vanishing circle of a system of planes, parallel to its own.
(115.) A familiar illustration of this is often to be had by attending to the lines of light seen in the air, when the sun's rays are darted through apertures in clouds, the sun itself being at the time obscured behind them. These lines which, marking the course of rays emanating from a point almost infinitely distant, are to be considered as parallel straight lines, are thrown into great circles of the sphere, having two apexes or points of common intersection - one in the place where the sun itself (if not obscured) would be seen. The other diametrically opposite. The first only is most commonly suggested when the spectator's view is towards the sun. But in mountainous countries, the phenomenon of sunbeams converging towards a point diametrically opposite to the sun, and as much depressed below the horizon as the sun is elevated above it, is not unfrequently noticed, the back of the spectator being turned to the sun's place. Occasionally, but much more rarely, the whole course of such a system of sunbeams, stretching in semicircles across the hemisphere from horizon to horizon (the sun being near setting), may be seen.* Thus again, the streamers of the Aurora Borealis, which are doubtless electrical rays, parallel, or nearly parallel to each other, and to the dipping needle, usually appear to diverge from the point towards which the needle, freely suspended, would dip northwards (i. e. about 70° below the horizon and 23° west of north from London), and in their upward progress pursue the course of great circles till they again converge (in appearance) towards the point diametrically opposite (i. e. 70° above the horizon, and 23° to the eastward of south), forming a sort of canopy over head, having that point for its centre. So also in the phenomenon of shooting stars, the lines of direction which they appear to take on certain remarkable occasions of periodical recurrence, are observed, if
• It is in such cases only that we conceive them as circles, the ordinary conventions of plane perspective becoming untenable. The author had the good fortune to witness on one occasion the phenomenon described in the text under circumstances of more than usual grandeur. Approaching Lyons from the south on Sept. 30. 1826, about 5 h. P. M., the sun was seen nearly setting behind broken masses of stormy cloud, from whose apertures streamed forth beams of rosecoloured light, traceable all across the hemisphere almost to their opposite point of convergence behind the snowy precipices of Mont Blanc, conspicuously visible at nearly 100 miles to the eastward. The impression produced was that of another but feebler sun about to rise from behind the mountain, and darting forth precursory beams to meet those of the real one opposite.
prolonged backwards, apparently to meet nearly in one point of the sphere; a certain indication of a general near approach to parallelism in the real directions of their motions on those occasions. On which subject more hereafter.
(116.) In relation to this idea of celestial perspective, we may conceive the north and south poles of the sphere as the two vanishing points of a system of lines parallel to the axis of the earth; and the zenith and nadir of those of a system of perpendiculars to its surface at the place of observation, &c. It will be shown that the direction of a plumb-line, at every place is perpendicular to the surface of still water at that place which is the true horizon, and though mathematically speaking no two plumb-lines are exactly parallel (since they converge to the earth's centre), yet over very small tracts, such as the area of a building—in one and the same town, &c., the difference from exact parallelisnu is so small that it may be practically disregarded. * To a spectator looking upwards such a system of plumb-lines will appear to converge to his zenith; downwards, to bis nadir.
(117.) So also the celestial equator, or the equinoctial, must be conceived as the vanishing circle of a system of planes parallel to the earth's equator, or perpendicular to its axis. The celestial horizon of any spectator is in like manner the vanishing circle of all planes parallel to his true horizon, of which planes his rational horizon (passing through the earth's centre) is one, and his sensible horizon (the tangent plane of his station) another.
(118.) Owing, however, to the absence of all the ordinary indications of distance which influence our judgment in respect of terrestrial objects, owing to the want of determinate figure and magnitude in the stars and planets as commonly seen— the projection of the celestial bodies on the ground of the heavenly concave is not usually regarded in this its true light of a perspective representation or picture, and it even requires an effort of imagination to conceive them in their true relations, as at vastly different distances, one behind the other, and forming with one another lines of junction violently foreshortened, and including angles altogether differing from those which their projected representations appear to make. To do so at all with effect presupposes a knowledge of their actual situations in space, which it is the business of astronomy to arrive at by appropriate considerations. But the connections which subsist among the several parts of the picture, the purely geometrical relations among the angles and sides of the spherical triangles of which it consists, constitute, under the name of Uranometry *, a preliminary and subordinate branch of the general science, with which it is necessary to be familiar before any further progress can be made. Some of the most elementary and frequently occurring of these relations we proceed to explain. And first, as immediate consequences of the above definitions, the following propositions will be borne in mind.
* An interval of a mile corresponds to a convergence of plumb-lines amount. ing to somewhat less space than a minute.
(119.) The altitude of the elevated pole is equal to the latitude of the spectator's geographical station.
For it appears, see fig. art. 112., that the angle PA Z between the pole and the zenith is equal to NCA, and the angles Z An and NCE being right angles, we have P An=ACE. Now the former of these is the elevation of the pole as seen from E, the latter is the angle at the earth's centre subtended by the arc E A, or the latitude of the place.
(120.) Hence to a spectator at the north pole of the earth, the north pole of the heavens is in his zenith. As he travels southward it becomes less and less elevated till he reaches the equator, when both poles are in his horizon—south of the equator the north pole becomes depressed below, while the south rises above his horizon, and continues to do so till the south pole of the globe is reached, when that of the heavens will be in the zenith.
(121.) The same stars, in their diurnal revolution, come to the meridian, successively, of every place on the globe once in twenty-four sidereal hours. And, since the diurnal rotation is uniform, the interval, in sidereal time, which elapses between the same star coming upon the meridians of two different places is measured by the difference of longitudes of the places.
* Ovpavos, the heavens ; MetpeLv, to measure : the measurement of the heavens.
(122.) Vice versa — the interval elapsing between two different stars coming on the meridian of one and the same place, expressed in sidereal time, is the measure of the difference of right ascensions of the stars.
(123.) The equinoctial intersects the horizon in the east and west points, and the meridian in a point whose altitude is equal to the co-latitude of the place. Thus, at Greenwich, of which the latitude is 51° 28' 40", the altitude of the intersection of the equinoctial and meridian is 38° 31' 20". The north and south poles of the heavens are the poles of the equinoctial. The east and west points of the horizon of a spectator are the poles of his celestial meridian. The north and south points of his horizon are the poles of his prime vertical, and his zenith and nadir are the poles of his horizon.
(124.) All the heavenly bodies culminate (i. e. come to their greatest altitudes) on the meridian; which is, therefore, the best situation to observe them, being least confused by the inequalities and vapours of the atmosphere, as well as least displaced by refraction.
(125.) All celestial objects within the circle of perpetual apparition come twice on the meridian, above the horizon, in every diurnal revolution; once above and once below the pole. These are called their upper and lower culminations.
(126.) The problems of uranometry, as we have described it, consist in the solution of a variety of spherical triangles, both right and oblique angled, according to the rules, and by the formulæ of spherical trigonometry, which we suppose known to the reader, or for which he will consult appropriate treatises. We shall only here observe generally, that in all problems in which spherical geometry is concerned, the student will find it a useful practical maxim rather to consider the poles of the great circles which the question before him refers to than the circles themselves. To use, for example, in the relations he has to consider, polar distances rather than declinations, zenith distances rather than altitudes, &c. Bear