It would be satisfactory to be able to say that any of the supposed planets might have been Lescarbault's Vulcan. But in reality, I fear, this cannot have been the case. In the Times, I expressed, in an article dated August 14, 1878, the opinion that the evidence obtained establishes the existence of the planet which had so long been regarded as a myth. That opinion was based on a very careful investigation of the evidence available at the time. But it does not accord with what has since been learned respecting Watson's observations.
We may dismiss planet 3 at once. If Watson is right about this body being distinct from Zeta (a point about which, I must confess, I feel grave doubts), then this must be a planet travelling in an orbit much wider than we can possibly assign to Vulcan. For even at the distance of some seven degrees from the sun it showed no sign of gibbosity. If it had then been at its greatest elongation it would have appeared only half-full. But with the power Watson was using, which enabled him to pronounce that the smaller body near Theta showed no elongation, he would at once have noticed any such peculiarity of shape. He could not have failed to observe any gibbosity approaching to that of the moon when three-quarters full. Moreover on July 29 a planet which has its points of crossing the ecliptic opposite the earth's place on April 3 and October 6, could not appear where Watson saw this body (fully two degrees from the ecliptic) unless either its orbit were far wider than that which Leverrier assigned to Vulcan, or else its inclination far greater. Neither supposition can be reconciled with Lescarbault's observation.
With regard to planets 1 and 2, the case is equally strong against the theory that Vulcan was observed. The same reasoning applies to both these bodies. When I speak therefore of planet 1, it will be understood that planet 2 also is dealt with. First, as this planet appeared with a disc appreciably round, it is clear that it must have been near the point of its orbit farthest from the earth, that is, the point directly beyond the sun. It was then nearly at its brightest. Yet it appeared as a fourth-magnitude star only. We have seen that Lescarbault's Vulcan, even when only half-full, would appear as bright as Mercury at his brightest, if Lescarbault's account can be accepted in all its details. Situated as planet 1 was, Vulcan would have shown much more brightly than an average first-magnitude star. At a very moderate computation it would have been twice as bright as such a star. But planet 1 appeared fainter than a fourth-magnitude star. Assume, however, that in reality it was shining as brightly as an average third-magnitude star. Then it shone with much less than a twentieth of the lustre Vulcan should have had, if Lescarbault's estimate were correct. Its diameter then cannot be greater than a quarter of that which Leverrier assigned to Vulcan on the strength of Lescarbault's observation. In fact, the apparent diameter of planet 1, when in transit over the sun's face, could not be more than a sixteenth of Mercury's in transit, or about two-fifths of a second,—roughly, about a 5000th part of the sun's apparent diameter. It is certain that Lescarbault could not have made so considerable a mistake as this. Nay, it is certain, that with the telescope he used he could not have seen a spot of this size at all on the sun's face.
It will be seen that Lescarbault's observation still remains unconfirmed, or rather, to speak more correctly, the doubts which have been raised respecting Lescarbault's Vulcan are now more than ever justified. If such a body as he supposed he saw really travels round the sun within the orbit of Mercury, it is certain that the observations made last July by those who were specially engaged in seeking for Vulcan must have been rewarded by a view of that planet. In July, Lescarbault's Vulcan could not have been invisible, no matter in what part of his orbit it might be, and the chances would have been greatly in favour of its appearing as a very bright star, without telescopic aid.
But on the other hand it seems extremely probable,—in fact, unless any one be disposed to question the veracity of the observers, it is certain,—that within the orbit of Mercury there are several small planets, of which certainly two, and probably three, were seen during the eclipse of July 29, 1878. All these bodies must be beyond the range of any except the most powerful telescopes, whether sought for as bright bodies outside the sun (not eclipsed) or as dark bodies in transit across the sun's face. The search for such bodies in transit would in fact be hopeless with any telescope which would not easily separate double stars one second of arc apart. It is with large telescopes, then, and under favourable conditions of atmosphere, locality, and so forth, that the search for intra-Mercurial planets in transit must in future be conducted. As the observed disturbance of Mercury's perihelion, and the absence of any corresponding disturbance of his nodes (the points where he crosses the plane of the earth's motion) show that the disturbing bodies must form a ring or disc whose central plane must nearly coincide with the plane of Mercury's path, the most favourable time for seeing these bodies in transit would be the first fortnights in May and November; for the earth crosses the plane of Mercury's orbit on or about May 8 and November 10. I believe that a search carried out in April, May, and June, and in October, November, and December, with the express object of discovering very small planets in transit, could not fail to be quickly rewarded,—unless the observations made by Watson and Swift are to be wholly rejected.
[Since this was written, Professor Swift has expressed the opinion that his planet cannot possibly have been the one seen near Theta Cancri by Professor Watson,—who it seems saw Theta in the centre of a large field of view, and must therefore have seen Swift's planet had that object been placed either as shown in fig. 2 or fig. 3. Hence Professor Swift considers that both the stars he himself saw were planets, and that he did not see Theta at all. The reasoning in the last five paragraphs of the above essay would not be in the least affected if we adopted Professor Swift's conclusion, that four and not three intra-Mercurial planets were detected during the eclipse of July last. Yet later Professor Peters of Clinton has indicated reasons for believing that while Watson simply mistook for planets the two fixed stars, Theta and Zeta Cancri, Professor Swift saw no planets at all. This interpretation would account fully, though not very satisfactorily, for all that is mysterious in the two narratives.]
FOOTNOTES:
[3] Two observations of Uranus, by Bradley, were discovered by the late Mr. Breen, and published in No. 1463 of the Astronomische Nachrichten.
[4] Let the student make the following construction if he entertains any doubt as to the statements made above. Having traced the orbits of the earth and Uranus from my chart illustrating the article 'Astronomy' in the Encyc. Brit., let him describe a circle nearly twice as large to represent the orbit of Neptune as Bode's law would give it. Let him first suppose Neptune in conjunction with Uranus in 1820, mark the place of the earth on any given day in 1842, and the place of the fictitious Neptune; a line joining these points will indicate the direction of Neptune on the assumptions made. Let him next make a similar construction on the assumption that conjunction took place in 1825. (From the way in which the perturbation of Uranus reached a maximum between 1820 and 1825, it was practically certain that the disturber was in conjunction with Uranus between those years.) These two constructions will give limiting directions for Neptune as viewed from the earth, on the assumption that his orbit has the dimensions named. He will find that the lines include an angle of a few degrees only, and that the direction line of the true Neptune is included between them.
[5] The problem is in reality, at least in the form in which Lescarbault attacked it, an exceedingly simple one. A solution of the general problem is given at p. 181 of my treatise on the Geometry of Cycloids. It is, in fact, almost identical with the problem of determining the distance of a planet from observations made during a single night.