The requirements of modern astronomy demand that the solar parallax shall be determined with an accuracy of -∏⅛τj∙ part of its amount—that is, within less than τ ⅛ π part of a second of arc. But measures in the neighbourhood of the sun cannot be made with any approach to this accuracy, not only on account of the effect of the sun’s heat on the various parts of the instruments employed, but also of the atmospheric currents created by heat, which tend to destroy steady atmospheric definition and to render the solar image incapable of exact observation. It is thus hopeless to look for any solution of the problem by the most direct method. Two courses remain—either to seek some method which affords a larger angle to measure, or one which permits a mode of observation affording a higher precision. There are many relations to the solar parallax which are well established.

(1) The parallax of the moon is known with very considerable precision by direct determination. The proportion of this parallax to that of the sun is an important term in the lunar theory, and the constant of this term (the ftarallactic inequality ) is a known function of the solar parallax. Hence, if the constant of the parallactic inequality is independently determined, the solar parallax becomes known. The elements of the orbits of Venus and Mars undergo secular variations which increase from year to year, from century to century, and at last acquire very large values. These secular variations (on the assumption that all the terms of the theories of the planets are mathematically accurate) have also a well-determined relation to the solar parallax, and therefore afford a means of determining that parallax with an accuracy which increases by the continuance of observation.

(2) It has been shown (Astronomy, vol. ii. p. 779 sq., and Mechanics, vol. xv. p. 708) that the proportions of the interplanetary distances can be very accurately determined and tables be computed from observations of the apparent places of the planets, without any knowledge or assumption as to absolute distances (although an accurate knowledge of the solar parallax is required for giving final perfection to the lunar and planetary tables). In astronomical ephemerides therefore the distances of planets from the earth are accurately expressed in terms of the earth’s mean distance from the sun, the latter being reckoned unity. Hence, to determine the solar parallax, it is only necessary to measure, at some favourable opportunity, the parallax of any planet, and to multiply the parallax so found by the number which expresses the relation of the distance of the planet from the earth to the earth’s mean distance from the sun.

(3) When Jupiter is in opposition he is nearer the earth by the diameter of the earth’s orbit than when in conjunction; hence, since light occupies a very sensible time to travel, eclipses of Jupiter’s satellites will seem to occur too soon in the first case and too late in the latter, the difference between the extremes of acceleration and retardation being the time occupied by light in crossing the earth’s orbit. This time is about 16⅜ minutes for the mean diameter of the earth’s orbit; hence, if the velocity of light can be independently determined, the diameter of the earth’s orbit becomes known. The determination, by employing the velocity of light, is also arrived at in another way. The constant of aberration (see Astronomy, p. 757), or the maximum apparent change of a star’s true place due to the motion of the observer, depends on the relative velocity of the motion of the observer in space compared with the velocity of light. The angular velocity of the observer is perfectly known; hence if his linear velocity is known his radius of motion is known. Thus, if the constant of aberration and the velocity of light are independently determined, the radius of motion (that is the sun’s parallax) will be found.

There are thus three distinctive typical methods:—(1) the gravitational method, depending on terms in the lunar and planetary theories, the constants of which are determined by observation; (2) the geometrical, or direct observational, method; and (3) the physical method.

1. The, Gτosυitational Method.— The moon’s parallactic inequality appears, at first sight, to furnish a very accurate method, as its constant is about 125", or fourteen times as great as the solar parallax, and the existing observations are very numerous. Unfortunately its determination is inextricably mixed up with the determination of the moon’s diameter—a diameter increased by irradiation, and therefore different for every telescope, and perhaps for every observer. But this is not all. The maximum and minimum effect of the parallactic inequality occur at first and last quarter, ι.e., when the moon is half full. One half of the observations for parallactic inequality therefore are made when the sun is above the horizon, and a great portion of the other half during twilight; whilst those on which the moon’s diameter depend are made at midnight, when the irradiation is a totally different quantity from what it is in daylight or during twilight. Newcomb has attempted to determine the correction of the diameter by the errors in right ascension, derived by comparing Hansen’s tables of the moon with observations made by daylight and at night; but he confesses that the result is so mixed up with the correction of the coefficient of the variation (and, he might have added, with the observer’s personality and the telescope employed) that it cannot be relied upon.

The following are the most important discussions :— [table]

One cannot look with confidence upon a method which thus permits discordance of more than one per cent, in the discussion of the same observations by different astronomers. The result arrived at must depend on the adopted corrections of the moon’s diameter, and, since that diameter is not capable of determination under the same circumstances of illumination as those in which the observations for parallactic inequality are made, the judgment of the theorist must step in and assign some more or less hypothetical grounds for the adoption of a particular diameter; and upon this assumption will turn the whole of the quantity of which we are in search.

It is, however, not impossible that the method of observing a spot near the centre of the moon, instead of the moon’s limb, may lead to a more reliable result. But it will have to be shown by independent methods that the position of the selected spot is not systematically affected by phase.

Attention was first called to the method which employs the secular variations of the elements of the orbits of Venus and Mars for determining the solar parallax by a most able and comprehensive paper communicated by Leverrier to the Paris Academy of Sciences, and published in their Comptes Rendus for 1872, July 22. The most important of these variations is that of the perihelion of Mars. The earth’s attraction increases the heliocentric position of Mars at perihelion by about 50" in a century, and this change at a favourable opposition subtends an angle of 185" at the earth.

On 1672, October 1, the star ψ Aquarii was occulted by Mars. The appulse was observed by Richer at Cayenne, by Picard near Beaufort, and by Romer at Paris. The separate comparisons differ only 0" , 5, 0" ’ 8, and 0" - 3 respectively; and the star ψ Aquarii was very frequently observed by Bradley. The increase in two centuries of the geocentric longitude, corresponding to the distance of the planet Mars from the earth on 1672, October 1, is 294". Hence Μ. Leverrier concludes that (attributing an error not greater than 1" to the determination of the observed variation) the time has arrived when the solar parallax can be determined with a probable error not exceeding y⅜ 75 ∙ of its amount, or the concluded parallax will be exact to nearly ±0" 01. The value of the parallax so deduced Μ. Leverrier finds to be

8"∙866.

Similarly he finds from the latitude of Venus, determined by the transits of Venus in 1761 and 1769, combined with the latitude determined by meridian observations of the present day

8' ’ ∙853.

From the discussion of the meridional observations of Venus in an interval of one hundred and six years, he finds

8"∙859.

These values from the theories of Venus and Mars accord in a wonderful manner, and would appear at first sight to justify considerable confidence in the result. But it is impossible to forget the extraordinary intricacy of the processes through which these results have been evolved, and the liability to some systematic source of error, such, for example, as some neglected term producing a long inequality which may become mixed up with the secular variation.

In 1874 the tabular errors of Venus, as determined by the planet’s transit across the sun’s disk, amounted to more than 5" of arc both in R.A. and declination, and the tabular errors of Mars amounted to more than 8" in R.A. and to about 3" in declination at the opposition of 1877, equivalent to an error of 2'' , 45 in heliocentric longitude (Jfem. R. A. S., vol. xlvi. p. 172). Leverrier’s planetary tables do not, therefore, possess the accuracy attributed to them by their distinguished author, and the conclusions at which he arrived probably require some further modification. Tisserand (Comptes Rendus, 1881, March 21) has continued the researches of Leverrier, and finds that they require modification, and are also subject to very considerable probable error. The later researches of Tisserand appear to point to a value of the solar parallax smaller than that found by Leverrier, but his work has not yet been brought to final conclusion.

2. The Geometrical Method.— The most favourable opportunities for the application of this method are afforded, in a geometrical sense, by the planets Venus and Mars, when the former is in conjunction and the latter in opposition. Of these Venus approaches the earth within one-fourth of the sun’s mean distance, whilst Mars, in the most favourable circumstances, approaches only within one-third of the same distance.

When Venus is near conjunction she is only visible as a slender crescent in the neighbourhood of the sun, and at conjunction is only visible on the occasion of a transit across the sun’s disk. It generally happens, therefore, that the only means of determining the apparent position of Venus near conjunction is to refer that position to the sun’s limb or sun’s centre. But the sun’s place is also affected by parallax, so that when the position of Venus is referred to the sun the parallactic displacement is only the difference of the parallax of the sun and Venus. Mars, on the other hand, can be referred to stars of which the parallax is absolutely insensible; thus it happens that the advantage of Venus in point of parallactic displacement is diminished till the geometrical conditions are only 5 per cent, in favour of Venus. Transits of Venus across the sun’s disk have been observed for parallax in 1761, 1769, 1874, and 1882.

If an astronomer at each of two widely separated stations observes the absolute instant of apparent internal contact of Venus with the sun’s limb, he is sure that the centres of the sun and Venus are separated by an angular distance equal to the “semidiameter of the sun minus the semidiameter of Venus.” The difference of the absolute times at the two stations is due to parallactic displacement, and, the planet’s tabular motion being accurately known, the amount of displacement becomes known. If instead of one contact only the two observers note the instants of internal contact both at ingress and egress, then they practically find the chords described by the planet as seen from both stations. The difference of length of these chords (in time) being known, as well as the approximate diameters of the sun and Venus, and their tabular motion, we have the data for computing the difference of least distance of centres of the sun and Venus at the two stations, and this distance being due to parallax, we have the means of computing the parallax of Venus and thence the solar parallax. This latter method (originally proposed by Halley in 1716) has the advantage of not requiring a rigid determination of the absolute instant of each contact, but merely of the duration of the transit; in other words, it involves no very rigid determination of the longitude or clock error, but only an exact knowledge of the clock rate.

It was Halley’s opinion that the instants of contact could be observed with an accuracy within two or three tenths of a second of time, but experience has gone to show that the actual errors are from ten to forty times this amount, and the causes of those errors can now be assigned with considerable certainty. These causes are— (1) irradiation and diffraction; (2) disturbance of the image by irregular refraction in the earth’s atmosphere; (3) the effect of the atmosphere of Venus in complicating the phenomena at the point of contact.

(1) Irradiation increases the diameter of the sun and diminishes that of Venus. Its extent depends on the aperture of the telescope, the perfection of its optical quality, and the perfection of the focal adjustment. Its amount is also changed by the brilliancy of the sun, i.e., is affected by the transparency of the sky and the density of the sun shade employed. Also, when the space between the limbs of the sun and Venus becomes smaller than the diffraction disk of the object-glass employed, a greyness or shadow is perceived at the point of past or approaching contact; therefore, within a minute angle equal to the separating power (the diameter of the diffraction disk) of the object-glass, the actual instant of contact can only be estimated by changes in the diffraction phenomena. (2) When the images are thrown into rapid vibration by irregular refraction in the earth’s atmosphere, it becomes impossible to distinguish between the vibration of the image of the dark body of Venus across the sun’s limb near the point of contact and the regular phenomena of irradiation, provided that the atmospheric vibrations are sufficiently rapid to produce a persistent image on the retina of the observer’s eye. Thus at the transit of Venus in 1882 observers were instructed to note at ingress the time when there was “a well-marked and persistent discontinuity in the illumination of the apparent limb of the sun.” Now it so happened that at the Royal Observatory, Cape of Good Hope, the definition was very bad—a south-easter was blowing, the effect of which was, as is almost invariably the case, to create a rapid minute vibration in the images of celestial objects (see Sir John Herschel’s Results of Observations at the Cape of Good Hope, p. xiv.). Thus “a well-marked and persistent discontinuity in the illumination of the apparent limb of the sun” was seen by all of five observers at the Royal Observatory from 10 to 20 seconds of time longer than at the adjoining stations in the Cape Colony, where the images were seen comparatively steady and well-defined. The instant of occurrence of the above-described phase is therefore a function of the state of the atmospheric definition, and no accurate means exist of estimating such influence. (3) The observation is besides complicated by the illuminated atmosphere of Venus, which forms an arc of light round the planet near the point of contact. In many cases this light has been confounded with the light of the sun, and has thus caused very considerable errors of observation.

From these various causes the apparent phenomena are different at different stations; and probably also the same phenomena are described by different observers in very different language. The real difficulty of the discussion of the results arises when these different and differently described phenomena have to be combined. It is of no consequence whether a real or seeming contact has been observed; it is only necessary to be certain that those observations are combined which represent the same phenomenon. The same phenomenon would correspond with the same apparent angular distance of centres of the sun and Venus, if all the telescopes were alike, if all the telescopes were in perfect focal adjustment, and if the atmospheric conditions of definition at all the stations were perfect or identical. But if these conditions are not realized (and they cannot be realized in practice) the same apparent phenomena will not represent corresponding phases; and, further, the observers at different stations use such different language to express what they saw that it becomes impossible to select even apparent corresponding phases with any certainty.

The value of the solar parallax deduced from a series of observations of the contacts of Venus with the sun’s limb will therefore entirely depend upon the interpretation put .upon the language of the various observers. The result will besides be systematically affected if the state of atmospheric definition is systematically different in the opposite stations.

It is thus not surprising that very different results have been arrived at by different astronomers from different transits, and even from different discussions of the same observations of the same transit. [table]

Besides observing the contacts, another method was employed by the Germans, the Russians, the Dutch, by Lord Lindsay’s expedition at the transit of Venus in 1874, and by the Germans in 1882, viz., the heliometric method. This consists in observing with the heliometer (see Micrometer, vol. xvi. pp. 252-254) the distance of V en us from opposite limbs of the sun along known position-angles nearly in the line of greatest and least distance of Venus from the sun’s limb. The method possesses many apparent advantages, because it affords the opportunity of multiplying the observations and of eliminating many sources of error.

At first sight it seems as if the method is free from the necessity for any accurate determination of the scale-value of the instrument, because, if measures are made from opposite limbs of the sun, the sun’s diameter may be taken as the standard for all observers, and the place of the planet may be interpolated relatively to the opposite limbs. Unfortunately it happens that there is à very marked difference in observing the sun’s diameter due both to instrument and observer. Thus two observers with different instruments, who have compared scale-value by a number of pairs of stars, or zones of stars, will measure sun-diameters with a marked constant difference. If the sun’s diameter is assumed to be constant, it, in fact, determines the scale-value; hence the distance of centres measured by the two observers will differ by the proportionate part × ∆d, where s is the distance of centres, d the true diameter, and ∆d the difference of diameter as measured by the two observers. Thus it is only when s = 0 (that is, when the planetis near the centre of the sun) that this method can be used,—a condition that does not exist in practice.

In the case of the transit of Venus fully one-third of the whole of this personality would enter into the result by this method of reduction. For rigid reduction therefore it is absolutely necessary to have a rigid determination of scale-value in seconds of arc. Unfortunately this value, when determined for any uniform instrumental condition of temperature, is liable to change, because, in observations of the sun, difference of temperature between the tube, the object-glass, and the scale of the instrument is produced, and the focal adjustment is also disturbed. The scalevalue depends on the relation of the focal length of the objectglass to the length of a part of the scale, and is besides...