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.

(296.) From this it necessarily follows, that in unequal times, the areas described must be proportional to the times. Thus, in the figure of art. 292. the time in which the sun moves from A to B, is the time in which it moves from C to D, as the area of the elliptic sector AO B is to the area of the sector D O C.

(297.) 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; 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.

(298.) What we have here stated supposes no knowledge of the sun's actual distance from the earth, nor,

CHAP. V. DIURNAL OR GEOCENTRIC PARALLAX.

189

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. Parallax may be generally defined to be the change of apparent situation of an object arising from a change of real situation of the observer. Suppose, then, PA BQ 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 Sb, 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 S B; 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.

(299.) Parallax, however, in the astronomical acceptation of the word, has a more technical meaning. It is restricted to the difference of apparent positions of any celestial object when viewed from a station on the surface of the earth, and from its centre. The centre of the earth is the general station to which all astronomical observations are referred: but, as we observe from the surface, a reduction to the centre is needed; and the

amount of this reduction is called parallax. Thus, the sun being seen from the earth's centre, in the direction CS, and from A on the surface in the direction AS, the angle ASC, included between these two directions, is the parallax at A, and similarly BSC is that at B.

Parallax, in this sense, may be distinguished by the epithet diurnal, or geocentric, to discriminate it from the annual, or heliocentric; of which more hereafter.

(300.) The reduction for parallax, then, in any proposed case, is obtained from the consideration of the triangle ACS, formed by the spectator, the centre of the earth, and the object observed; and since the side CA prolonged passes through the observer's zenith, it is evident that the effect of parallax, in this its technical acceptation, is always to depress the object observed in a vertical circle. To estimate the amount of this depression, we have, by plane trigonometry,

CS: CA: sine of CAS= sine of ZAS: sine of ASC.

(301.) The parallax, then, for objects equidistant from the earth, is proportional to the sines of their zenith distances. It is, therefore, at its maximum when the body observed is in the horizon. In this situation it is called the horizontal parallax; and when this is known, since small arcs are proportional to their sines, the parallax at any given altitude is easily had by the following rule:Parallax= (horizontal parallax) x sine of zenith distance.

The horizontal parallax is given by this proportion :— Distance of object : earth's radius :: rad. : sine of horizontal parallax.

It is, therefore, known, when the proportion of the object's distance to the radius of the earth is known: and vice versâ—if by any method of observation we can come at a knowledge of the horizontal parallax of an object, its distance, expressed in units equal to the earth's radius, becomes known.

(302.) To apply this general reasoning to the case of the sun. Suppose two observers - one in the northern, the other in the southern hemisphere at stations on the

CHAP. V.

PARALLAX OF THE SUN.

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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 whole amount of the two parallaxes, or by the angle A S B. This angle, then, is obtained by subducting the sum of the 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.

(303.) 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.

(304.) The horizontal parallax of the sun has been concluded from observations of the nature above de

scribed, 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 23,984 times the length of the earth's radius, or about 95,000,000 miles.

(305.) Tha 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 which it is enabled to keep up its liberal and unceasing supply of these elements. As to its actual magnitude we can be at no loss, knowing its distance, and the angles under which its diameter appears to us. An object, placed at the distance of 95,000,000 miles, and subtending an angle of 32′ 3′′, must have a real diameter of 882,000 miles. Such, then, is the diameter of this stupendous globe. If we compare it with what we have already ascertained of the dimensions of our own, we shall find that in linear magnitude it exceeds the earth in the proportion of 1111 to 1, and in bulk in that of 1,384,472 to 1.

(306.) It is hardly possible to avoid associating our conception of an object of definite globular figure, and of such enormous dimensions, with some corresponding attribute of massiveness and material solidity. That the sun is not a mere phantom, but a body having its own peculiar structure and economy, our telescopes distinctly inform us. They show us dark spots on its surface, which slowly change their places and forms, and by attending to whose situation, at different times, astronomers have ascertained that the sun revolves about an axis inclined at a constant angle of 82° 40′ to the plane