in Astronomy, is the difference between the places of any celestial object as seen from the surface, and from the centre of the earth at the same instant.

Let E in figure of parallax, represent the centre of the earth, O the place of an observer on its surface, whose visible horizon is OH, and true horizon EF: Now let ZDT be a portion of a great circle in the heavens, and A the place of any object in the visible horizon; join EA, and produce it to C; then C is the true place of the object, and H is its apparent place, and the angle CAH is the parallax; or, because the object is in the horizon, it is called the horizontal parallax. But OAE, the angle which the earth's radius subtends at the object, is equal to CAH: Hence the horizontal parallax of an object may be defined to be the angle which the earth's semidiameter subtends at that object. For the various methods hitherto proposed to find the quantity of the horizontal parallax of an object, see Astronomy.

The whole effect of parallax is in a vertical direction: For the parallactic angle is in the plane passing through the observer and the earth's centre; which plane is necessarily perpendicular to the horizon, the earth being considered a sphere.

The more elevated an object is above the horizon, the less is the parallax, its distance from the earth's centre continuing the same. When the object is in the zenith, it has no parallax; but when in the horizon, its parallax is greatest. The horizontal parallax being given, the parallax at any given altitude may be found by the following rule:

To the logarithmic cofine of the given altitude, add the log. fine of the horizontal parallax, the sum, rejecting 10 from the index, will be the log. fine of the parallax in altitude.

Demonstration. Let B be the place of an object; produce OB, ED to F and D; then the angle BOZ will be the apparent altitude of the object, BEZ the true altitude, and OBE the parallax in altitude. Now in the radius of the triangle AOE,

\[ R : \text{fine } OAE :: EA : EO. \]

And in the triangle OBE

\[ BE (=EA) : EO :: \text{fine } BOE : \text{fine } OBE. \]

Hence \( r : \text{cofine } BOA :: \text{fine } OAE : \text{fine } OBE. \)

As the two last terms are generally small quantities, the arch may be substituted in place of its fine without any sensible error.

Example. Let the apparent altitude of the moon's centre be \(39^\circ 25'\), and the moon's horizontal parallax \(56' 54''\). Required the parallax in altitude.

Moon's apparent alt. \(39^\circ 25'\) cofine \(9.8879260\) Moon's horizontal par. \(56' 54''\) fine \(8.2188186\)

Moon's par. in altitude \(43' 57''\) fine \(8.1067446\)

Or, to the secant of the moon's apparent altitude, add the proportional logarithms of the parallax in altitude.

As the apparent place of an object is nearer the horizon than its true place, the parallax is therefore to be added to the apparent altitude, to obtain the true altitude. Hence also an object will appear to rise later and set sooner. The fine of the parallax of an object is inversely as its distance from the earth's centre.

**Demonstration.** Let A be the place of an object and H the place of the same object at another time, or that of another object at the same instant; join EH, then in the triangles AOE, HOE,

\[ R : \text{fine } OAE :: AE : OE \] \[ \text{fine } OHE : R :: OE : EH \]

Hence fine OHE : fine OAE :: AE : EH.

The parallax of an object makes it appear more distant from the meridian than it really is.

**Demonstration.** The true and apparent places of an object are in the same vertical, the apparent place being lower than the true; and all verticals meet at the zenith: Hence the apparent place of an object is more distant from the plane of the meridian than the true place.

The longitude, latitude, right ascension, and declination of an object are affected by a parallax. The difference between the true and apparent longitudes is called the *parallax in longitude*; in like manner, the difference between the true and apparent latitudes, right ascensions, and declinations, are called the *parallax in latitude*, *right ascension*, and *declination*, respectively.

When the object is in the nonagefmal, the parallax in longitude is nothing, but that in latitude is greatest; and when the object is in the meridian, the parallax in right ascension vanishes, and that in declination is a maximum. The apparent longitude is greater than the true longitude, when the object is east of the nonagefmal, otherwise less; and when the object is in the eastern hemisphere, the apparent right ascension exceeds the true, but is less than the true right ascension when the object is in the western hemisphere. The apparent place of an object is more distant from the elevated poles of the ecliptic and equator than the true place: hence, when the latitude of the place and elevated pole of the ecliptic are of the same name, the apparent latitude is less than the true latitude, otherwise greater; and the apparent declination will be less or greater than the true declination, according as the latitude of the place, and declination of the object, are of the same or of a contrary denomination.

The parallaxes in longitude, latitude, right ascension, and declination, in the spheroidal hypothesis, may be found by the following formulae; in which L represents the latitude of the place, diminished by the angle contained between the vertical and radius of the given place; P the horizontal parallax for that place; a the altitude of the nonagefmal at the given instant; d the apparent distance of the object from the nonagefmal; \( r \) the true and apparent latitudes of the object; \( D \) the true and apparent declinations respectively; and \( m \) its apparent distance from the meridian.

Then par. in long. \( = P \cdot \text{fine } a \cdot \text{fine } d \cdot \text{secant } l \), to radius unity; and par. in lat. \( = P \cdot \text{cofine } a \cdot \text{cofine } \lambda \)

\[ \lambda = p \cdot \text{cofine } d \cdot \text{fine } a \cdot \text{fine } \lambda \]

The sign \( - \) is used when the apparent distance of the object from the nonagefmal and from the elevated pole of the ecliptic are of the same affection, and the sign \( + \) if of different affection. If the greatest precision be required, the following quantity \( 0.0000121216 \) is to be applied to the parallax in latitude found as above, by addition or subtraction, according as the true distance of the object from the elevated pole of the ecliptic is greater or less than \( 90^\circ \).

Again, par. in right ascen. \( = P \cdot \text{cofine } L \cdot \text{fine } m \) secant \( D \), to radius unity; and par. in declination \( = P \cdot \text{fine } L \cdot \text{cofine } 2 \cdot \text{cofine } L \cdot \text{cofine } \lambda \), cofine \( m \).

The upper or lower sign is to be used, according as the distance of the object from the meridian and from the elevated pole of the equator are of the same or different affection. Part 2d of par. in declination \( = 0.0000121216 \) par. in right ascen. \( ^2 \), fine \( 2D \); which is additive to, or subtractive from, part first of parallax in declination, according as the true distance of the object from the elevated pole of the equator is greater or less than \( 90^\circ \). For the moon's parallax, see Astronomy.

There is also a curious paper in the first volume of Asiatic Researches, p. 372, &c., on the same subject, to which we refer our readers.

**Parallax of the Earth's annual Orbit**, is the difference between the places of a planet as seen from the sun and earth at the same instant. The difference between the longitudes of the planet as seen from the sun and earth is called the *parallax in longitude*; and the difference between its latitudes is the *parallax in latitude*.

**Parallax of the Fixed Stars**, see Astronomy, art. 268, which contains an account of the method used by Dr Herschel, to ascertain the parallax of a star which appears to be double, from observations made at opposite points of the orbit of the earth. M. Piazzi, the discoverer of the planet Ceres, has made many observations of the zenith distances of \( \alpha \) Lyrae, Arcturus, Procyon, and Aquila, &c., at those times when the effects of parallax ought to be the greatest. His observations are published in the 10th volume of the Italian Society. Let \( p \) be the absolute parallax, consequently,

\[ \text{fine } p = \frac{\text{distance of the star from the earth}}{I} \]

then the parallax of Arcturus in declination will be \( 0.595p \); and that in right ascension, \( 1.005p \); hence, he observes, that observations of the right ascension of this star are preferable to those of the declination, for determining the parallax of this star.

M. Calendrelli, in a work printed at Rome in 1836, has given the result of his observations of the zenith distances of \( \alpha \) Lyrae, made with the sector of Meffl. Maire and Bolcovich. By comparing five observations in June, with four in December 1835, and five in March with the same number in June, he deduced the parallax of \( \alpha \) Lyrae in declination to be \( 4'' \), and that in right ascension \( 6.8'' \). According to M. Piazzi, the parallax is less than half of these quantities; and, hence, the required quantity not exceeding the unavoidable differences attending observations, it appears difficult to determine it, so as to be free from doubt.

Meffl. Delambre and Mechain have made many observations of the pole star, and \( \beta \) Ursae minoris, being those stars which ought to have the greatest parallax in declination, at the times most proper to discover their parallax; but from the comparison, which M. Delambre made, of the zenith distances of these stars, he discovered nothing that could give the least suspicion of a parallax; and the small anomalies which he observed are often in a contrary direction. M. Delambre adds, Parallax that these stars being of the second magnitude, may be too far distant from us to have a parallax; however, although this may be the case, yet it appears to him that the fixed stars have no parallax.

The parallax of Venus affords the most correct method, hitherto proposed, of finding the distance of the earth from the sun; and, hence, the distances of the other planets, and also their magnitudes. For this discovery we are indebted to the celebrated Dr Halley. From observations of the transits of this planet, in 1761 and 1769, the parallax of the sun has been more accurately determined than previous thereto. The parallax of Mars has also been employed for the same purpose.

Parallax is also used to denote the change of place in any object arising from viewing it obliquely with respect to another object. Thus the minute hand of a watch is said to have a parallax when it is viewed obliquely; and the difference between the images shown by it, when viewed directly and obliquely, is the quantity of parallax in time.