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direction, from which we then set off a series of angles, AO B, A OC, &c. equal to the observed longitudes of the sun throughout the year, and in these respective directions measure off from the distances O A, OB, O C, &c. representing a the distances deduced from the observed diameter, and then connect all the extremities A, B, C, &c. of these lines by a continuous curve, it is evident this will be a correct representation of the relative orbit of the sun about the earth. Now, when this is done, a deviation from the circular figure in the resulting curve becomes apparent; it is found to be evidently longer than it is broad - that is to say, elliptic, and the point O to occupy, not the centre, but one of the foci of the ellipse. The graphical process here described is sufficient to point out the general figure of the curve in question; but for the purposes of exact verification, it is necessary to recur to the properties of the ellipse *, and to express the distance of any one of its points in terms of the angular situation of that point with respect to the longer axis, or diameter of the ellipse. This, however, is readily done; and when numerically calculated, on the supposition of the excentricity being such as above stated, a perfect coincidence is found to subsist between the distances thus computed, and those derived from the measurement of the apparent diameter.
(350.) The mean distance of the earth and sun being taken for unity, the extremes are 1•01679 and 0.98321. But if we compare, in like manner, the mean or average angular velocity with the extremes, greatest and least, we shall find these to be in the proportions of 1.03386, 1.00000, and 0.96670. The variation of the sun's angular velocity, then, is much greater in proportion than that of its distance - fully twice as great; and if we examine its numerical expressions at different periods, comparing them with the mean value, and also with the corresponding distances, it will be found, that, by
• See Conic Sections, by the Rev. H. P. Hamilton, or any other of the very numerous works on this subject.
whatever fraction of its mean value the distance exceeds the mean, the angular velocity will fall short of its mean or average quantity by very nearly twice as great a fraction of the latter, and vice versa. Hence we are led to conclude that the angular velocity is in the inverse proportion, not of the distance simply, but of its square; so that, to compare the daily motion in longitude of the sun, at one point, A, of its path, with that at B, we must state the proportion thus:
O BP : 0 A? :: daily motion at A : daily motion at B. And this is found to be exactly verified in every part of the orbit.
(351.) Hence we deduce another remarkable conclusion viz. that if the sun be supposed really to move around the circumference of this ellipse, its actual speed cannot be uniform, but must be greatest at its least distance and less at its greatest. For, were it uniform, the apparent angular velocity would be, of course, inversely proportional to the distance; simply because the same linear change of place, being produced in the same time at different distances from the eye, must, by the laws of perspective, correspond to apparent angular displacements inversely as those distances. Since, then, observation indicates a more rapid law of variation in the angular velocities, it is evident that mere change of distance, unaccompanied with a change of actual speed, is insufficient to account for it; and that the increased proximity of the sun to the earth must be accompanied with an actual increase of its real velocity of motion along its path.
(352.) This elliptic form of the sun's path, the excentric position of the earth within it, and the unequal speed with which it is actually traversed by the sun itself, all tend to render the calculation of its longitude from theory (i.e. from a knowledge of the causes and nature of its motion) difficult; and indeed impossible, so long as the law of its actual velocity continues unknown. This law, however, is not immediately apparent. It does not come forward, as it were, and present itself at once, like the elliptic form of the orbit, by a direct comparison of angles and distances, but requires an attentive consideration of the whole series of observations registered
during an entire period. It was not, therefore, without much painful and laborious calculation, that it was discovered by Kepler (who was also the first to ascertain the elliptic form of the orbit), and announced in the following terms:- Let a line be always supposed to connect the sun, supposed in motion, with the earth, supposed at rest; then, as the sun moves along its ellipse, this line (which is called in astronomy the radius vector) will describe or sweep over that portion of the whole area or surface of the ellipse which is included between its consecutive positions: and the motion of the sun will be such that equal areas are thus swept over by the revolving radius vector in equal times, in whatever part of the circumference of the ellipse the sun may be moving.
(353.) From this it necessarily follows, that in unequal times, the areas described must be proportional to the times. Thus, in the figure of art. 349. the time in which the sun moves from A to B, is to the time in which it moves from C to D, as the area of the elliptic sector A O B is to the area of the sector DOC.
(354.) The circumstances of the sun's apparent annual motion may, therefore, be summed up as follows: — It is performed in an orbit lying in one plane passing through the earth's centre, called the plane of the ecliptic, and whose projection on the heavens is the great circle so called. In this plane, however, the actual path is not circular, but elliptical; having the earth, not in its centre, but in one focus. The excentricity of this ellipse is 0.01679, in parts of a unit equal to the mean distance, or half the longer diameter of the ellipse; i.e. about one sixtieth part of that semi-diameter; and the motion of the sun in its circumference is so regulated, that equal areas of the ellipse are passed over by the radius vector in equal times.
(355.) What we have here stated supposes no knowledge of the sun's actual distance from the earth, nor, consequently, of the actual dimensions of its orbit, nor of the body of the sun itself. To come to any conclusions on these points, we must first consider by what means we can arrive at any knowledge of the distance of an object to which we have no
access. Now, it is obvious, that its parallax alone can afford us any information on this subject. Suppose P A B Q to represent the earth, C its centre, and S the sun, and A, B
two situations of a spectator, or, which comes to the same thing, the stations of two spectators, both observing the sun S at the same instant. The spectator A will see it in the direction A Sa, and will refer it to a point a in the infinitely distant sphere of the fixed stars, while the spectator B will see it in the direction B S b, and refer it to b. The angle included between these directions, or the measure of the celestial arc a b, by which it is displaced, is equal to the angle A SB; and if this angle be known, and the local situations of A and B, with the part of the earth's surface A B included between them, it is evident that the distance C S may be calculated. Now, since A S C (art. 339.) is the parallax of the sun as seen from A, and BSC as seen from B, the angle A S B, or the total apparent displacement is the sum of the two parallaxes. Suppose, then, two observers — one in the northern, the other in the southern hemisphere-at stations on the same meridian, to observe on the same day the meridian altitudes of the sun's centre. Having thence derived the apparent zenith distances, and cleared them of the effects of refraction, if the distance of the sun were equal to that of the fixed stars, the sum of the zenith distances thus found would be precisely equal to the sum of the latitudes north and south of the places of observation. For the sum in question would then be equal to the angle Z C X, which is the meridional distance of the stations across the equator. But the effect of parallax being in both cases to increase the apparent zenith distances, their observed sum will be greater than the sum of the latitudes, by the sum of the two parallaxes, or by the angle A SB. This angle, then, is obtained by subducting the sum of the north and south latitudes from that of the zenith distances; and this once determined, the horizontal parallax is easily found, by dividing the angle so determined by the sum of the sines of the two latitudes.
(356.) If the two stations be not exactly on the same meridian (a condition very difficult to fulfil), the same process will apply, if we take care to allow for the change of the sun's actual zenith distance in the interval of time elapsing between its arrival on the meridians of the stations. This change is readily ascertained, either from tables of the sun's motion, grounded on the experience of a long course of observations, or by actual observation of its meridional altitude on several days before and after that on which the observations for parallax are taken. Of course, the nearer the stations are to each other in longitude, the less is this interval of time, and, consequently, the smaller the amount of this correction; and, therefore, the less injurious to the accuracy of the final result is any uncertainty in the daily change of zenith distance which may arise from imperfection in the solar tables, or in the observations made to determine it.
(357.) The horizontal parallax of the sun has been concluded from observations of the nature above described, performed in stations the most remote from each other in latitude, at which observatories have been instituted. It has also been deduced from other methods of a more refined nature, and susceptible of much greater exactness, to be hereafter described. Its amount so obtained, is about 8":6. Minute as this quantity is, there can be no doubt that it is a tolerably correct approximation to the truth; and in conformity with it, we must admit the sun to be situated at a mean distance from us, of no less than 23984 times the length of the earth's radius, or about 95000000 miles.
(358.) That at so vast a distance the sun should appear to us of the size it does, and should so powerfully influence our condition by its heat and light, requires us to form a very grand conception of its actual magnitude, and of the scale on which those important processes are carried on within it, by