CHAPTER V.

On Parallax, Refraction, and the Equation of Time.

ART. 78. HAVING in the foregoing chapter explained some of the methods which are made use of in determining the relative positions of the fixed stars, we might now proceed to enquire into the distances, sizes, apparent and real motions, of the sun, moon, and other heavenly bodies; but previous to this it is necessary to shew the nature of some corrections which must be applied to our observations, before they will be of use in our enquiries. Of these corrections the first we shall notice is that arising from

PARALLAX.

79. HITHERTO we have considered the various phænomena of the celestial bodies as though they were viewed from the centre of the earth, whereas in reality they were observed from the surface; and this we have done, by conceiving the earth to be, as it were, a point, or that the distance between the centre and surface is insensible when compared with the vast distance of the heavens. This is so nearly true with respect to the fixed stars, as to produce no assignable error; but with respect to the sun, moon, and planets, the admission of this idea of the earth's exceeding minuteness would involve some mistakes which may be prevented by duly attending to what follows.

80. In fig. 12, Pl. I. let A be the place of an observer on the earth's surface, A BD; H, G, F, and E, the real places of four heavenly bodies; then the apparent places of these bodies in the supposed concave sphere of heaven, as seen from A, would be O, M, K, and Z, and their places with respect to the centre of the earth would be N, L, I, and Z: now the lines which mark out the apparent places as seen from the surface and from the centre respectively, intersect at the real places of the body, and form angles, as AHC, AGC, AFC; one of these angles, as AG C, is called the parallax of the body G in that situation, AFC and AHC the parallaxes of the bodies F and H: that is, the parallax of a heavenly body relative to the observer, is the inclination of two visual rays passing from the body, one to the earth's centre, the other to the point where the observer is placed on its surface.

81. The parallax of a heavenly body depends upon two circumstances, the distance of that body from the centre of the earth, and its position with respect to the zenith. The separate and joint effects of these two circumstances we shall now exhibit.

82. The sines of the parallaxes of two bodies at unequal distances from the centre of the earth, but at equal apparent distances from the zenith, are reciprocally as the distances from the earth's centre. For, iff and h are the places of the bodies, it is manifest from the principles of plane trigonometry, that the sine of the angle Ah C is to the sine of the angle hf C or of its supplement AfC; that is, that the sine of the parallax of h, is to the sine of the parallax at f, as fC to h C; that is, reciprocally as the distances of h and ƒ from the centre C.

83. Of bodies at equal distances from the centre of the earth, the sines of the parallares are always as the sines of the apparent distances from the zenith, er as the co-sines of the apparent altitudes. For, if

EFH be a vertical circle, in the triangle AFC, AC: CF: sine AFC: sine FAC sine EAF: and, in the triangle AG C, AC: CG or CF:: sine AGC sine GAC sine EAG. Therefore, the antecedents of both proportions being equal, the consequents are proportional, and sine AFC: sine EAF sine AG C: sine EAG.

=

From this it is obvious that the parallax of a body is greatest when it is in the horizon; this is called the horizontal parallax: it is also manifest, that an object in the zenith of an observer has no parallax, for the two lines which form the parallactic angle then coincide.

84. The sine of the parallax of one body, is to the sine of the parallax of another body, in a ratio compounded of the inverse ratio of the distances from the earth's centre, and the direct ratio of the sines of the apparent distances from the zenith. For, when the distance from the zenith is given, the sine of the parallax is reciprocally as the distance from the earth's centre (Art. 82.); and when the distance from the earth's centre is given, the sine of the parallax is as the sine of the apparent zenith distance (Art. 83.); consequently, when neither is given, the sine of the parallax is conjunctly, as the distance from the earth's centre reciprocally, and as the sine of the zenith distance directly.

85. The general effect of the parallax of a star is to make it appear nearer the horizon than it really is and as the position of the stars is determined by their right ascension and declination, that is, by the period of their passage over the meridian and their meridian altitude; it follows that, in the meridian, which is a vertical circle, the parallax of a star is all in declination, and nothing in right ascension.

86. The parallax of the stars makes them appear farther from the meridian than they really are. For, since all the verticals concur at the zenith, and

recede gradually from one another as they approach the horizon, and since the parallax of stars does not remove them from their verticals, but causes them to appear nearer the horizon than they really are, they consequently appear farther from the meridian than they are in reality. Hence the stars appear to rise later and set sooner, on account of parallax.

87. Parallax causing the stars to appear in other points of the heavens than those they are actually in, must also change the latitudes, longitudes, declina. tions, and right ascensions. The difference of the longitude observed from the centre, called the true longitude, and that seen from the observer's place, called the apparent longitude, is called the parallax of longitude. The same might be remarked of the parallax of latitude, of right ascension, of declination, and of altitude.

When

88. When any three of the five following things are given, namely, the observer's distance from the star, the star's distance from the centre of the earth, the earth's semi-diameter, the true or apparent altitude of the star, and the parallax ;-the other two may be determined: for then in the right-lined parallactic triangle, three things are known, whence the other two may be found. Hence it appears that an observer situated on the earth, may know the real distance of any heavenly body from the earth's centre, if he can find its parallax, and particularly its horizontal parallax, which being largest is most fit for the purpose. It may be proper therefore to explain a few of the methods by which the parallax of a celestial object is discovered,

89. The parallax of a body has been sometimes found, by measuring the apparent distance of the body from a fixed star when they are both on the same vertical; then observing when the body and star are at equal altitudes above the horizon, at which time the distance between the body and star must be

again measured: the difference of these distances will be very nearly the parallax required.

90. But the following method, when practicable, is both more certain in practice and more accurate in theory. Two observers must be placed on the same terrestrial meridian, each about 50 or 60 degrees from the equator, one towards the north, the other towards the south: each person must determine, by a very accurate instrument, the distance of the body from his zenith at the instant it transits the meridian, and the zenith distance of some fixed star whose parallel is nearly known, and differing but little from that of the body. Then on each side comparing the body's zenith distance with that of the fixed star, the distance of the body's apparent parallel from that of the star would be known. If this comparison gave the body the same parallel relatively to each observer, it would have no parallax: but if the comparison shewed that the body appeared in different parallels to the two observers, the distance of these parallels would shew the body's horizontal parallax by this analogy: as the sum of the sines of the two distances of the body from the zenith is to radius, so is the distance of the two apparent parallels of the body to its horizontal parallar. In the two last terms of this analogy, the distance of the parallels and the parallax are put instead of their sines, because, on account of their minuteness, they may safely be substituted the one for the other.

When the two observers are on the same side of the equator, use the difference of the sines of the zenith distances instead of their sum in the above analogy. If the two observers were not under the same meridian, but distant from one another by a known quantity, regard must be paid to the motion of the body in declination between the times of its passage over the two meridians.