to secure some reward for his labours. At Leverrier's request M. Rouland, the Minister of Public Instruction, communicated to Napoleon III. the result of Leverrier's visit, and on January 25 the Emperor bestowed on the village doctor the decoration of the Legion of Honour.
To return to astronomical facts.
It appears from Lescarbault's observation, that on March 26, 1859, at about four in the afternoon, a round black spot entered on the sun's disc. It had a diameter less than one-fourth that of Mercury (which he had seen in transit with the same telescope and the same magnifying power on May 8, 1845). The time occupied in the transit of this spot was about one hour seventeen minutes, and, the chord of transit being somewhat more than a quarter of the sun's diameter in length, Lescarbault calculated that the time necessary to describe the sun's diameter would have been nearly four and a half hours. The inclination of the body's path to the ecliptic seemed to be rather more than 6 degrees, and was probably comprised between 5–⅓ and 7–⅓ degrees.
From Leverrier's calculations, it appeared that the time of revolution of the new planet would be 19 days 17 hours, its distance from the sun about 147, the earth's being taken as 1,000; giving for Mars, the earth, Venus, Mercury, and Vulcan (as the new planet was named), the respective distances 1, 524, 1,000, 723, 387, and 147. Leverrier assigned 12–⅕ degrees as Vulcan's inclination, and the places where it crosses the ecliptic he considered to be in line with those occupied by the earth on or about April 3 and October 6. Judging from Lescarbault's statement respecting the apparent size of the dark spot, Leverrier concluded that the volume of the stranger must be about one-seventeenth of Mercury's, the masses being presumably in the same proportion. Hence he inferred that the new planet would be quite incompetent to produce the observed change in the orbit of Mercury.
Leverrier further found that the brilliancy of Vulcan when the planet was furthest from the sun on the sky (about eight degrees) would be less than that of Mercury when similarly placed in his orbit, and he hence inferred that Vulcan might readily remain unseen, even during total eclipse. Here, as it seems to me, Leverrier's reasoning was erroneous. If Vulcan really has a volume equal to one-seventeenth of Mercury's, the diameter of Vulcan would be rather less than two fifths of Mercury's and the disc of Vulcan at the same distance about two-thirteenths of Mercury's. But Vulcan, being nearer the sun than Mercury in the ratio of 147 to 387, or say 15 to 39, would be more brightly illuminated in the ratio of 39 times 39 to 15 times 15, or nearly as 20 to 3. Hence if we first diminish Mercury's lustre when at his greatest apparent distance from the sun in the ratio of 2 to 13, and increase the result in the ratio of 20 to 3, we get Vulcan's lustre when he is at his greatest apparent distance from the sun. The result is that his lustre should exceed Mercury's in the same degree that 40 exceeds 39. Or practically, for all the numbers used have been mere approximations, the inference is that Vulcan and Mercury, if both seen when at their greatest distance from the sun during eclipse, would probably shine with equal lustre. But in that case Vulcan would be a very conspicuous object indeed, at such a time; for Mercury when at his greatest distance from the sun, or greatest elongation, is a bright star even on a strongly illuminated twilight sky; moreover, Vulcan, when at either of his greatest elongations, ought to be visible in full daylight in a suitably adjusted telescope. For Mercury is well seen when similarly placed, and even when much nearer to the sun and on the nearer part of his path where he turns much more of his darkened than of his illuminated hemisphere towards us. Venus has been seen when so near the sun that the illuminated portion of her disc is a mere thread-like sickle of light. Nay, Professor Lyman, of Yale College, in America, has seen her when so near the sun that she appeared to be a mere circular thread of light, the completion of the circle being the best possible proof how exceedingly fine the thread must have been, and also how small its intrinsic lustre.
This is indeed the chief difficulty in Lescarbault's supposed observation. If he really saw a body in transit across the sun, moving at the observed rate, and having anything like the observed diameter, that body ought to have been seen repeatedly during total eclipses of the sun, and ought not to have escaped the search which has been made over and over again near the sun for intra-Mercurial planets. Either we must reject Lescarbault's narrative absolutely, or we must suppose that he greatly over-estimated the size of the body he observed.
Another difficulty almost equally important is found to exist when we consider the circumstances of Lescarbault's supposed discovery. Suppose the path of Vulcan to be inclined about twelve degrees or thereabouts to the ecliptic or to the plane in which the earth travels. Then, as seen from the earth on April 3, and October 6, this path, if it were a material ring, would appear as a straight line across the sun's centre, and extending on either side of the sun to a distance of about 16 sun-breadths. As seen on January 3 and July 5, when it would have its greatest opening, Vulcan's path would appear as an oval whose longest axis would be about 32 sun-breadths, while its shortest would be little more than 6 sun-breadths, the sun of course occupying the centre of the ellipse, which, where closest to him, would lie but about 2½ sun-breadths only from the outline of his disc. Now it is easily seen that the path of Vulcan, changing in this way from apparent straightness to a long oval (whose breadth is about one-fifth its length), back to straightness but differently inclined, then to the same oval as before but opened out the other way, and so back to its original straightness and inclination, must, for no inconsiderable portion of the year on either side of April 3 and October 6, intersect the outline of the sun's disc. From a rough but sufficiently accurate calculation which I have made, I find that the interval would last about 36 days at each season, that is, from about March 16 to April 21 in spring, and from about September 18 to about October 24 in autumn. But during a period of 36 days there would generally be two passages of Vulcan between the earth and sun, and there would always be one (in any long period of time two such passages would be five times as common an event during one of these intervals as a single passage). Consequently there would be at least two transits of Vulcan every year, and there would generally be four transits; the average number of transits would be about eleven in three years. With a wider orbit and a greater inclination transits would be fewer; but even with the widest orbit and the greatest inclination that can possibly be allowed, there would be at least one transit a year on the average.
Now when we remember that, so far as the northern hemisphere is concerned, the sun is observed on every fine day in almost every country in Europe and in half the States of the American Union, to say nothing of observations in Asia, where England and Russia have several observatories, while in the southern hemisphere there are many observatories, in Australia, South Africa, and South America (on both side of the Andes), we see how exceedingly small must be the chance that Vulcan could escape detection even for a single year. Far less could Vulcan have escaped all the years which have elapsed since Lescarbault announced his discovery, to say nothing of all the observations made by Carrington, Schwabe, and many others, before the year 1860. If Vulcan really exists, and really has the dimensions and motions described by Lescarbault, the planet must long ere this have been repeatedly seen upon the sun's disc by experienced observers.
As a matter of fact, Wolf has collected nineteen observations of dark bodies unlike spots on the sun, during the interval between 1761 and 1865. But as Professor Newcomb justly points out, with two or three exceptions, the observers are almost unknown as astronomers. In one case at least the object seen was certainly not a planet, since it was described as a cloud-like appearance. 'On the other hand,' says Newcomb, 'for fifty years past the sun has been constantly and assiduously observed by such men as Schwabe, Carrington, Secchi, and Spörer, none of whom have ever recorded anything of the sort. That planets in such numbers should pass over the solar disc, and be seen by amateur astronomers, and yet escape all these skilled astronomers, is beyond all moral probability.'
It must be remembered that an inexperienced observer of the sun might readily mistake a spot of unusual roundness and darkness for a planet's disc. The practised observer would perceive peculiarities at once indicating the object as a spot on the sun; but these peculiarities would escape the notice of a beginner, or of one using a telescope of small power. Again, an inexperienced observer is apt to mistake the change of position which a spot on the sun undergoes on account of the diurnal motion, for a change of place on the sun's disc. At noon, for instance, the uppermost point of the sun's disc is the north point; but in the afternoon the uppermost point is east of the true north point. Thus a spot which at noon was a short distance