Jules Verne wrote a novel called From the Earth to the Moon, explaining that Earth may have two moons. I went web-surfing, and found this:
In 1846, Frederic Petit, director of the observatory of Toulouse, stated that a second moon of the Earth had been discovered. It had been seen by two observers, Lebon and Dassier, at Toulouse and by a third, Lariviere, at Artenac, during the early evening of March 21 1846. Petit found that the orbit was elliptical, with a period of 2 hours 44 minutes 59 seconds, an apogee at 3570 km above the Earth's surface and perigee at just 11.4 km (!) above the Earth's surface. Le Verrier, who was in the audience, grumbled that one needed to take air resistance into account, something nobody could do at that time. Petit became obsessed with this idea of a second moon, and 15 years later announced that he had made calculations about a small moon of Earth which caused some then-unexplained peculiarities in the motion of our main Moon. Astronomers generally ignored this, and the idea would have been forgotten if not a young French writer, Jules Verne, had not read an abstract. In Verne's novel "From the Earth to the Moon", Verne lets a small object pass close to the traveller's space capsule, causing it to travel around the Moon instead of smashing into it:
"It is", said Barbicane, "a simple meteorite but an enormous one, retained as a satellite by the attraction of the Earth."
"Is that possible?", exclaimed Michel Ardan, "the earth has two moons?"
"Yes, my friend, it has two moons, although it is usually believed to have only one. But this second moon is so small and its velocity is so great that the inhabitants of Earth cannot see it. It was by noticing disturbances that a French astronomer, Monsieur Petit, could determine the existence of this second moon and calculated its orbit. According to him a complete revolution around the Earth takes three hours and twenty minutes. . . . "
"Do all astronomers admit the the existence of this satellite?", asked Nicholl
"No", replied Barbicane, "but if, like us, they had met it they could no longer doubt it. . . . But this gives us a means of determining our position in space . . . its distance is known and we were, therefore, 7480 km above the surface of the globe where we met it."
Jules Verne was read by millions of people, but not until 1942 did anybody notice the discrepancies in Verne's text:
A satellite 7480 km above the Earth's surface would have a period of 4 hours 48 minutes, not 3 hours 20 minutes.
Since it was seen from the window from which the Moon was invisible, while both were approaching, it must be in retrograde motion, which would be worth remarking. Verne doesn't mention this.
In any case the satellite would be in eclipse and thus be invisible. The projectile doesn't leave the Earth's shadow until much later.
Dr. R.S. Richardson, Mount Wilson Observatory, tried in 1952 to make the figures fit by assuming an eccentric orbit of this moon: perigee 5010 km and apogee 7480 km above Earth's surface, eccentricity 0.1784.
Nevertheless, Jules Verne made Petit's second moon known all over the world. Amateur astronomers jumped to the conclusion that here was opportunity for fame -- anybody discovering this second moon would have his name inscribed in the annals of science. No major observatory ever checked the problem of the Earth's second moon, or if they did they kept quiet. German amateurs were chasing what they called Kleinchen ("little bit") -- of course they never found Kleinchen.
W. H. Pickering devoted his attention to the theory of the subject: if the satellite orbited 320 km above the surface and if its diameter was 0.3 meters, with the same reflecting power as the Moon, it should be visible in a 3-inch telescope. A 3 meter satellite would be a unaided-eye object of magnitude 5. Though Pickering did not look for the Petit object, he did carry on a search for a secondary moon -- a satellite of our Moon ("On a photographic search for a satellite of the Moon", Popular Astronomy, 1903). The result was negative and Pickering concluded that any satellite of our Moon must be smaller than about 3 meters.
Pickering's article on the possibility of a tiny second moon of Earth, "A Meteoritic Satellite", appeared in Popular Astronomy in 1922 and caused another short flurry among amateur astronomers, since it contained a virtual request: "A 3-5-inch telescope with a low-power eyepiece would be the likeliest mean to find it. It is an opportunity for the amateur." But again, all searches remained fruitless.
The original idea was that the gravitational field of the second moon should account for the then inexplicable minor deviations of the motion of our big Moon. That meant an object at least several miles large -- but if such a large second moon really existed, it would have been seen by the Babylonians. Even if it was too small to show a disk, its comparative nearness would have made it move fast and therefore be conspicuous, as today's watchers of artificial satellites and even airplanes know. On the other hand, nobody was much interested in moonlets too small to be seen.
If Earth did have a second moon, I think they would have found it by now.