The French mathematician Urbain Le Verrier, co-predictor with J.C. Adams of the position of Neptune before it was seen, in a lecture at 2 Jan 1860 announced that the problem of observed deviations of the motion of Mercury could be solved by assuming an intra-Mercurial planet, or possibly a second asteroid belt inside Mercury's orbit. The only possible way to observe this intra-Mercurial planet or asteroids was if/when they transited the Sun, or during total solar eclipses. Prof. Wolf at the Zurich sunspot data center, found a number of suspicious "dots" on the Sun, and another astronomer found some more. A total of two dozen spots seemed to fit the pattern of two intra-Mercurial orbits, one with a period of 26 days and the other of 38 days.

In 1859, Le Verrier received a letter from the amateur astronomer Lescarbault, who reported having seen a round black spot on the Sun on March 26 1859, looking like a planet transiting the Sun. He had seen the spot one hour and a quarter, when it moved a quarter of the solar diameter. Lescarbault estimated the orbital inclination to between 5.3 and 7.3 degrees, its longitude of node about 183 deg, its eccentricity "enormous", and its transit time across the solar disk 4 hours 30 minutes. Le Verrier investigated this observation, and computed an orbit from it: period 19 days 7 hours, mean distance from Sun 0.1427 a.u., inclination 12# 10', ascending node at 12# 59' The diameter was considerably smaller than Mercury's and its mass was estimated at 1/17 of Mercury's mass. This was too small to account for the deviations of Mercury's orbit, but perhaps this was the largest member of that intra-Mercurial asteroid belt? Le Verrier fell in love with the planet, and named it Vulcan.

In 1860 there was a total eclipse of the Sun. Le Verrier mobilized all French and some other astronomers to find Vulcan - nobody did. Wolf's suspicious 'sunspots' now revived Le Verrier's interest, and just before Le Verrier's death in 1877 some more 'evidence' found its way into print. On April 4 1875, a German astronomer, H. Weber, saw a round spot on the Sun. Le Verrier's orbit indicated a possible transit at April 3 that year, and Wolf noticed that his 38-day orbit also could have performed a transit at about that time. That 'round dot' was also photographed at Greenwich and in Madrid.

4a: Mercury's orbit

All planets orbit the Sun in an oval, as described by Kepler's Laws, so there is a point in that orbit that is closest to the Sun, called the perihelion.
We observe that as Mercury orbits the Sun, the perihelion advances by a small amount.
Newton's theory does not fully explain this precession of Mercury's perihelion.
There is a 43 arcsecond per century discrepancy between the Newtonian prediction (resulting from the gravitational tugs of the other planets) and the observed precession.

4b: How Newton's gravity explains Mercury's orbit

Newton's theory of universal gravitation has had wonderful
success in explaining the motions of the planets in our Solar System, even leading to the discovery of Neptune. However, given the known planets in our Solar System, Newton's theory does not fully explain the precession of
Mercury's orbit.
Some astronomers theorize that another planet, inside the orbit of Mercury, could account for the odd behavior of Mercury.
This position is bolstered by the success in finding Neptune using the perturbations of Uranus' orbit to calculate an orbit of the
undiscovered planet.
The proposed planet between the Sun and Mercury has been named Vulcan, but has yet to be observed.