But though we may consider 80 millions of miles as a fair average distance at which a few of the most closely approaching asteroids may be observed, and though this distance seems very great by comparison with Mars’s occasional opposition distance of 35 million miles, yet there are two points in which asteroids have the advantage over Mars. First, they are many, and several among them can be observed under favourable circumstances; and in the multitude of observations there is safety. In the second place, which is the great and characteristic good quality of this method of determining the sun’s distance, they do not present a disc, like the planet Mars, but a small star-like point. When we consider the qualities of the heliometric method of measuring the apparent distance between celestial objects, the advantage of points of light over discs will be obvious. If we are measuring the apparent distance between Mars and a star, we must, by shifting the movable object-glass, bring the star’s image into apparent contact with the disc-image of Mars, first on one side and then on the other, taking the mean for the distance between the centres. Whereas, when we determine the distance between a star and an asteroid, we have to bring two star-like points (one a star, the other the asteroid) into apparent coincidence. We can do this in two ways, making the result so much the more accurate. For consider what we have in the field of view when the two halves of the object-glass coincide. There is the asteroid and close by there is the star whose distance we seek to determine in order to ascertain the position of the asteroid on the celestial sphere. When the movable half is shifted, the two images of star and asteroid separate; and by an adjustment they can be made to separate along the line connecting them. Suppose, then, we first make the movable image of the asteroid travel away from the fixed image (meaning by movable and fixed images, respectively, those given by the movable and fixed halves of the object-glass), towards the fixed image of the star—the two points, like images, being brought into coincidence, we have the measure of the distance between star and asteroid. Now reverse the movement, carrying back the movable images of the asteroid and star till they coincide again with their fixed images. This movement gives us a second measure of the distance, which, however, may be regarded as only a reversed repetition of the preceding. But now, carrying on the reverse motion, the moving images of star and asteroid separate from their respective fixed images, the moving image of the star drawing near to the fixed image of the asteroid and eventually coinciding with it. Here we have a third measure of the distance, which is independent of the two former. Reversing the motion, and carrying the moving images to coincidence with the fixed images, we have a fourth measure, which is simply the third reversed. These four measures will give a far more satisfactory determination of the true apparent distance between the star and the asteroid than can, under any circumstances, be obtained in the case of Mars and a star. Of course, a much more exact determination is required to give satisfactory measures of the asteroid’s real distance from the earth in miles, for a much smaller error would vitiate the estimate of the asteroid’s distance than would vitiate to the same degree the estimate of Mars’s distance: for the apparent displacements of the asteroid as seen either from Northern and Southern stations, or from stations east and west of the meridian, are very much less than in the case of Mars, owing to his great proximity. But, on the whole, there are reasons for believing that the advantage derived from the nearness of Mars is almost entirely counterbalanced by the advantage derived from the neatness of the asteroid’s image. And the number of asteroids, with the consequent power of repeating such measurements many times for each occasion on which Mars has been thus observed, seem to make the asteroids—so long regarded as very unimportant members of the solar system—the bodies from which, after all, we shall gain our best estimate of the sun’s distance; that is, of the scale of the solar system.
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Since the above pages were written, the results deduced from the observations made by the British expeditions for observing the transit of December 9, 1874, have been announced by the Astronomer Royal. It should be premised that they are not the results deducible from the entire series of British observations, for many of them can only be used effectively in combination with observations made by other nations. For instance, the British observations of the duration of the transit as seen from Southern stations are only useful when compared with observations of the duration of the transit as seen from Northern stations, and no British observations of this kind were taken at Northern stations, or could be taken at any of the British Northern stations except one, where chief reliance was placed on photographic methods. The only British results as yet “worked up” are those which are of themselves sufficient, theoretically, to indicate the sun’s distance, viz., those which indicated the epochs of the commencement of transit as seen from Northern and Southern stations, and those which indicated the epochs of the end of transit as seen from such stations. The Northern and Southern epochs of commencement compared together suffice of themselves to indicate the sun’s distance; so also do the epochs of the end of transit suffice of themselves for that purpose. Such observations belong to the Delislean method, which was the subject of so much controversy for two or three years before the transit took place. Originally it had been supposed that only observations by that method were available, and the British plans were formed upon that assumption. When it was shown that this assumption was altogether erroneous, there was scarcely time to modify the British plans so that of themselves they might provide for the other or Halleyan method. But the Southern stations which were suitable for that method were strengthened; and as other nations, especially America and Russia, occupied large numbers of Northern stations, the Halleyan method was, in point of fact, effectually provided for—a fortunate circumstance, as will presently be seen.
The British operations, then, thus far dealt with, were based on Delisle’s method; and as they were carried out with great zeal and completeness, we may consider that the result affords an excellent test of the qualities of this method, and may supply a satisfactory answer to the questions which were under discussion in 1872–74. Sir George Airy, indeed, considers that the zeal and completeness with which the British operations were carried out suffice to set the result obtained from them above all others. But this opinion is based rather on personal than on strictly scientific grounds. It appears to me that the questions to be primarily decided are whether the results are in satisfactory agreement (i) inter se and (ii) with the general tenor of former researches. In other words, while the Astronomer Royal considers that the method and the manner of its application must be considered so satisfactory that the results are to be accepted unquestionably, it appears to me that the results must be carefully questioned (as it were) to see whether the method, and the observations by it, are satisfactory. In the first place, the result obtained from Northern and Southern observations of the commencement ought to agree closely with the result obtained from Northern and Southern observations of the end of transit. Unfortunately, they differ rather widely. The sun’s distance by the former observations comes out about one million miles greater than the distance determined by the latter observations.
This should be decisive, one would suppose. But it is not all. The mean of the entire series of observations by Delisle’s method comes out nearly one million miles greater than the mean deduced by Professor Newcomb from many entire series of observations by six different methods, all of which may fairly be regarded as equal in value to Delisle’s, while three are regarded by most astronomers as unquestionably superior to it. Newcomb considers the probable limits of error in his evaluation from so many combined series of observations to be about 100,000 miles. Sir G. Airy will allow no wider limits of error for the result of the one series his observers have obtained than 200,000 miles. Thus the greatest value admitted by Newcomb falls short of the least value admitted by Sir G. Airy, by nearly 700,000 miles.
The obvious significance of this result should be, one would suppose, that Delisle’s method is not quite so effective as Sir G. Airy supposed; and the wide discordance between the several results, of which the result thus deduced is the mean, should prove this, one would imagine, beyond all possibility of question. The Astronomer Royal thinks differently, however. In his opinion, the wide difference between his result and the mean of all the most valued results by other astronomers, indicates the superiority of Delisle’s method, not its inadequacy to the purpose for which it has been employed.
Time will shortly decide which of these views is correct; but, for my own part, I do not hesitate to express my own conviction that the sun’s distance lies very near the limits indicated by Newcomb, and therefore is several hundred thousand miles less than the minimum distance allowed by the recently announced results.