PARALLAX

, is an arch of the heavens intercepted between the true place of a star, and its apparent place.

The true place of a star S, is that point of the heavens B, in which it would be seen by an eye placed in the centre of the earth at T. And the apparent place, is that point of the heavens C, where a star appears to an eye upon the surface of the earth at E.

This difference of places, is what is called absolutely the Parallax, or the Parallax of Altitude; which Copernicus calls the Commutation; and which therefore is an angle formed by two visual rays, drawn, the one from the centre, the other from the circumference of the earth, and traversing the body of the star; being measured by an arch of a great circle intercepted between the two points of true and apparent place, B and C.

The Parallax of Altitude CB is properly the difference between the true distance from the zenith AB, and the apparent distance AC. Hence the Parallax diminishes the altitude of a star, or increases its distance from the zenith; and it has therefore a contrary effect to the refraction.

The Parallax is greatest in the horizon, called the Horizontal Parallax EFT. From hence it decreases all the way to the zenith D or A, where it is nothing; the real and apparent places there coinciding.

The Horizontal Parallax is the same, whether the star be in the true or apparent horizon.

The fixed stars have no sensible Parallax, by reason of their immense distance, to which the semidiameter of the earth is but a mere point.

Hence also, the nearer a star is to the earth, the greater is its Parallax; and on the contrary, the farther it is off, the less is the Parallax, at an equal elevation above the horizon. So the star at S has a less Parallax than the star at I. Saturn is so high, that it is difficult to observe in him any Parallax at all.

Parallax increases the right and oblique ascension, and diminishes the descension; it diminishes the northern declination and latitude in the eastern part, and increases them in the western; but it increases the southern declination in the eastern and western part; it diminishes the longitude in the western part, and increases it in the eastern. Parallax therefore has just opposite effects to refraction.

The doctrine of Parallaxes is of the greatest importance, in astronomy, for determining the distances of the planets, comets, and other phenomena of the heavens; for the calculation of eclipses, and for finding the longitude.

Parallax of Right Ascension and Descension, is an arch of the Equinoctial Dd, by which the Parallax of altitude increases the ascension, and diminishes the descension.

Parallax of Declination, is an arch of a circle of declination sI, by which the Parallax of altitude increases or diminishes the declination of a star.

Parallax of Latitude, is an arch of a circle of latitude SI, by which the Parallax of altitude increases or diminishes the latitude.

Menstrual Parallax of the Sun, is an angle formed by two right lines; one drawn from the earth to the sun, and another from the sun to the moon, at either of their quadratures.

Parallax of the Annual Orbit of the Earth, is the difference between the heliocentric and geocentric place of a planet, or the angle at any planet, subtended by the distance between the earth and sun.

There are various methods for finding the Parallaxes of the celestial bodies: some of the principal and easier of which are as follow:

To Observe the Parallax of a Celestial Body.—Observe when the body is in the same vertical with a fixed star which is near it, and in that position measure its| apparent distance from the star. Observe again when the body and star are at equal altitudes from the horizon; and there measure their distance again. Then the difference of these distances will be the Parallax very nearly.

To Observe the Moon's Parallax.—Observe very accurately the moon's meridian altitude, and note the mo nent of time. To this time, equated, compute her true latitude and longitude, and from these find her declination; also from her declination, and the elevation of the equator, find her true meridian altitude. Subtract the refraction from the observed altitude: then the difference between the remainder and the true altitude, will be the Parallax sought. If the observed altitude be not meridional, reduce it to the true altitude for the time of observation.

By this means, in 1583, Oct. 12 day 5 h. 19 m. from the moon's meridian altitude observed at 13° 38′, Tycho found her Parallax to be 54 minutes.

To Observe the Moon's Parallax in an Eclipse.—In an eclipse of the moon observe when both horns are in the same vertical circle, and at that moment take the altitudes of both horns; then half their sum will be nearly the apparent altitude of the moon's centre; from which subtract the refraction, which gives the apparent altitude freed from refraction. But the true altitude is nearly equal to the altitude of the centre of the shadow at that time: now the altitude of the centre of the shadow is known, because we know the sun's place in the ecliptic, and his depression below the horizon, which is equal to the altitude of the opposite point of the ecliptic, in which the centre of the shadow is. Having thus the true and apparent altitudes, their difference is the Parallax sought.

De la Hire makes the greatest horizontal Parallax 1° 1′ 25″, and the least 54′ 5″. M. le Monnier determined the mean Parallax of the Moon to be 57′ 12″. Others have made it 57′ 18″.

From the Moon's Parallax EST, and alitude SF (last fig. but one); to find her distance from the Earth. —From her apparent altitude given, there is given her apparent zenith distance, i. e. the angle AES; or by her true altitude, the complement angle ATS. Wherefore, since at the same time, the Parallactic angle S is known, the 3d or supplemental angle TES is also known. Then, considering the earth's semidiameter TE as 1, in the triangle TES are given all the angles and the side TE, to find ES the moon's distance from the surface of the earth, or TS her distance from the centre.

Thus Tycho, by the observation above mentioned, found the moon's distance at that time from the earth, was 62 of the earth's semidiameters. According to De la Hire's determination, her distance when in the perigee is near 56 semidiameters, but in her apogee near 63 1/2; and therefore the mean nearly 59³, or in round numbers 60 semidiameters.

Hence also, since, from the moon's theory, there is given the ratio of her distances from the earth in the several degrees of her anomaly; those distances being found, by the rule of three, in semidiameters of the earth, the Parallax is thence determined to the several degrees of the true anomaly.

To Observe the Parallax of Mars.—1. Suppose Mars in the meridian and equator at H; and that the observer, under the equator in A, observes him culminating with some fixed star. 2. If now the observer were in the centre of the earth, he would see Mars constantly in the same point of the heavens with the star; and therefore, together with it, in the plane of the horizon, or of the 6th horary: but since Mars here has some sensible Parallax, and the fixed star has none, Mars will be seen in the horizon, when in P, the plane of the sensible horizon; and the star, when in R, the plane of the true horizon: therefore observe the time between the transit of Mars and of the star through the plane of the 6th hour.—3. Convert this time into minutes of the equator, at the rate of 15 degrees to the hour; by which means there will be obtained the arch PM, to which the angle PAM, and consequently the angle AMD, is nearly equal; which is the horizontal Parallax of Mars.

If the observer be not under the equator, but in a parallel IQ, that difference will be a less arch QM: wherefore, since the small arches QM and PM are nearly as their sines AD and ID; and since ADG is equal to the distance of the place from the equator, i. e. to the elevation of the pole, or the latitude; therefore AD to ID, as radius to the cosine of the latitude; say, as the cosine of the latitude ID is to radius, so is the Parallax observed in I, to the Parallax under the equator.

Since Mars and the fixed star cannot be commodiously observed in the horizon; let them be observed in the circle of the 3d hour: and since the Parallax observed there TO, is to the horizontal one PM, as IS to ID: say, as the sine of the angle IDS, or 45° (since the plane DO is in the middle between the meridian DH and the true horizon DM), is to radius, so is the Parallax TO to the horizontal Parallax PM.

If Mars be likewise out of the plane of the equator, the Parallax found will be an arch of a parallel; which must therefore be reduced, as above, to an arch of the equator.

Lastly, if Mars be not stationary, but either direct or retrograde, by observations for several days find out what his motion is every hour, that his true place from the centre may be assigned for any given time.

By this method Cassini, who was the author of it, observed the greatest horizontal Parallax of Mars to be 25″; but Mr. Flamsteed found it near 30″. Cassini observed also the Parallax of Venus by the same method.

To Find the Sun's Parallax.—The great distance of the sun renders his Parallax too small to fall under even the nicest immediate observation. Many attempts have indeed been made, both by the ancients and moderns, and many methods invented for that purpose. The first was that of Hipparchus, which was followed by Ptolomy, &c, and was founded on the observation of lunar| eclipses. The second was that of Aristarchus, in which the angle subtended by the semidiameter of the moon's orbit, seen from the sun, was sought from the lunar phases. But these both proving desicient, astronomers are now forced to have recourse to the Parallaxes of the nearer planets, Mars and Venus. Now from the theory of the motions of the earth and planets, there is known at any time the proportion of the distances of the sun and planets from us; and the horizontal Parallaxes being reciprocally proportional to those distances; by knowing the Parallax of a planet, that of the sun may be thence found.

Thus Mars, when opposite to the sun, is twice as near as the sun is, and therefore his Parallax will be twice as great as that of the sun. And Venus, when in her inferior conjunction with the sun, is sometimes nearer us than he is; and therefore her Parallax is greater in the same proportion. Thus, from the Parallaxes of Mars and Venus, Cassini found the sun's Parallax to be 10″; from whence his distance comes out 22000 semidiameters of the earth.

But the most accurate method of determining the Parallaxes of these planets, and thence the Parallax of the sun, is that of observing their transit. However, Mercury, though frequently to be seen on the sun, is not fit for this purpose; because he is so near the sun, that the difference of their Parallaxes is always less than the solar Parallax required. But the Parallax of Venus, being almost 4 times as great as the solar Parallax, will cause very sensible differences between the times in which she will seem to be passing over the sun at different parts of the earth. With the view of engaging the attention of astronomers to this method of determining the sun's Parallax, Dr. Halley communicated to the Royal Society, in 1691, a paper, containing an account of the several years in which such a transit may happen, computed from the tables which were then in use: those at the ascending node occur in the month of November O. S. in the years 918, 1161, 1396, 1631, 1639, 1874, 2109, 2117; and at the descending node in May O. S. in the years 1048, 1283, 1291, 1518, 1526, 1761, 1769, 1996, 2004. Philos. Trans. Abr. vol. 1, p. 435 &c.

Dr. Halley even then concluded, that if the interval of time between the two interior contacts of Venus with the sun, could be measured to the exactness of a second, in two places properly situated, the sun's Parallax might be determined within its 500dth part. And this conclusion was more fully explained in a subsequent paper, concerning the transit of Venus in the year 1761, in the Philos. Trans. numb. 348, or Abr. vol. 4, p. 213.

It does not appear that any of the preceding transits had been observed; except that of 1639, by our ingenious countryman Mr. Horrox, and his friend Mr. Crabtree, of Manchester. But Mr. Horrox died on the 3d of January, 1641, at the age of 25, just after he had finished his treatise, Venus in Sole visa, in which he discovers a more accurate knowledge of the dimensions of the solar system, than his learned commentator Hevelius.

To give a general idea of this method of determining the horizontal Parallax of Venus, and from thence, by analogy, the Parallax and distance of the sun, and of all the planets from him; let DBA be the earth, V Venus, and TSR the eastern limb of the sun. To an observer at B, the point t of that limb will be on the meridian, its place referred to the heavens will be at E, and Venus will appear just within it at S. But to an observer at A, at the same instant, Venus is east of the sun, in the right line AVF; the point t of the sun's limb appears at e in the heavens, and if Venus were then visible she would appear at F. The angle CVA is the horizontal Parallax of Venus; which is equal to the opposite angle FVE, measured by the arc FE. ASC is the sun's horizontal Parallax, equal to the opposite angle eSE, measured by the arc eE; and FAe or VAe is Venus's horizontal Parallax from the sun, which may be found by observing how much later in absolute time her total ingress on the sun is, as seen from A, than as seen from B, which is the time she takes to move from V to v, in her orbit OVv.

If Venus were nearer the earth, as at U, her horizontal Parallax from the sun would be the arch fe, which measures the angle fAe; and this angle is greater than the angle FAe, by the difference of their measures Ff. So that as the distance of the celestial object from the earth is less, its Parallax is the greater.

Now it has been already observed, that the horizontal Parallaxes of the planets are inversely as their distances from the earth's centre, therefore as the sun's distance at the time of the transit is to Venus's distance, so is the Parallax of Venus to that of the sun: and as the sun's mean distance from the earth's centre, is to his distance on the day of the transit, so is his horizontal Parallax on that day, to his horizontal Parallax at the time of his mean distance from the earth's centre. Hence his true distance in semidiameters of the earth may be obtained by the following analogy, viz, as the sine of the sun's Parallax is to radius, so is unity or the earth's semidiameter, to the number of semidiameters of the earth in the sun's distance from the centre; which number multiplied by the number of| miles in the earth's semidiameter, will give the number of miles in the sun's distance. Then from the proportional distances of the planets, determined by the theory of gravity, their true distances may be found. And from their apparent diameters at these known distances, their real diameters and bulks may be found.

Mr. Short, with great labour, deduced the quantity of the sun's Parallax from the best observations that were made of the transit of Venus, on the 6th of June, 1761, (for which see Philos. Trans. vol. 51 and 52) both in Britain and in foreign parts, and found it to have been 8″.52 on the day of the transit, when the sun was very nearly at his greatest distance from the earth; and consequently 8″.65 when the sun is at his mean distance from the earth. See Philos. Trans. vol. 52, p. 611 &c. Whence,

As sin. 8″.65	log.  5.6219140
to radius	10.0000000
So is 1 semidiameter	0.0000000
to 23882.84 semidiameters	4.3780860

that is, 23882 84/100 is the number of the earth's semidiameters contained in its distance from the sun; and this number of semidiameters being multiplied by 3985, the number of English miles contained in the earth's semidiameter, (though later observations make this semidiameter only 3956 1/2 miles), there is obtained 95,173,127 miles for the earth's mean distance from the sun. And hence, from the analogies under the article Distance, the mean distances of all the rest of the planets from the sun, in miles, are found as follow, viz,

Mercury's distance	36,841,468
Venus's distance	68,891,486
Mars's distance	145,014,148
Jupiter's distance	494,990,976
Saturn's distance	907,956,130.

In another paper (Philos. Trans. vol. 53, p. 169) Mr. Short states the mean horizontal Parallax of the sun at 8″.69. And Mr. Hornsby, from several observations of the transit of June 3d, 1769 (for which see the Philos. Trans. vol. 59) deduces the sun's Parallax for that day equal to 8.65, and the mean Parallax 8″.78; whence he makes the mean distance of the earth from the sun to be 93,726,900 English miles, and the distances of the other planets thus:

Mercury's distance	36,281,700
Venus's distance	67,795,500
Mars's distance	142,818,000
Jupiter's distance	487,472,000
Saturn's distance	894,162,000

See the Philos. Trans. vol. 61, p. 572.

But others, by taking the results of those observations that are most to be depended on, have made the sun's Parallax at his mean distance from the earth to be 8.6045; and some make it only 8.54. According to the former of these, the sun's mean distance from the earth is 95,109,736 miles; and according to the latter it is 95,834,742 miles. Upon the whole there seems reason to conclude that the sun's horizontal Parallax may-be stated at 8″.6, and his distance near 95 millions of miles. Hence, the following horizontal Parallaxes:

Mean Parallax of the sun	0′	8″.6
Moon's greatest	61	32
Moon's least	54	4
Moon's mean	57	48
Mars's	0	25

Of the Parallax of the Fixed Stars. As to the fixed stars, their distance is so great, that it has never been found that they have any sensible Parallax, neither with respect to the earth's diameter, nor even with regard to the diameter of the earth's annual orbit round the sun, although this diameter be about 190 millions of miles. For, any of those stars being observed from opposite ends of this diameter, or at the interval of half a year between the observations, when the earth is in opposite points of her orbit, yet still the star appears in the same place and situation in the heavens, without any change that is sensible, or measurable with the very best instruments, not amounting to a single se<*> cond of a degree. That is, the diameter of the earth's annual orbit, at the nearest of the fixed stars, does not subtend an angle of a single second; or, in comparison of the distance of the fixed stars, the extent of 190 millions of miles is but as a point!

Parallax is also used, in Levelling, for the angle contained between the line of true level, and that of apparent level. And, in other branches of science, for the difference between the true and apparent places.