44 ROUGH WA YS MADE SMOOTH. 



over and over again near the sun forintra-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 z\ sun-breadths only from the outline of his 

 disc. Now it is easily seen that the path of Vulcan, chang- 

 ing in this way from apparent straightness to a long oval 

 (whose breadth is about one-fifth its length), back to straight- 

 ness 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 



