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 2-1/2 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