"It is", said Barbicane, "a simple meteorite but an enormous one, retained as a satellite by the attraction of the Earth."Jules Verne was read by millions of people, but not until 1942 did anybody notice the discrepancies in Verne's text:
"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."
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 naked-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.
There have been other proposals for additional natural satellites of the Earth. In 1898 Dr Georg Waltemath from Hamburg claimed to have discovered not only a second moon but a whole system of midget moons. Waltemath gave orbital elements for one of these moons: distance from Earth 1.03 million km, diameter 700 km, orbital period 119 days, synodic period 177 days. "Sometimes", says Waltemath, "it shines at night like the Sun" and he thinks this moon was seen in Greenland on 24 October 1881 by Lieut Greely, ten days after the Sun had set for the winter. Public interest was aroused when Waltemath predicted his second moon would pass in front of the Sun on the 2nd, 3rd or 4th of February 1898. On the 4th February, 12 persons at the post office of Greifswald (Herr Postdirektor Ziegel, members of his family, and postal employees) observed the Sun with their naked eye, without protection of the glare. It is easy to imagine a faintly preposterous scene: an imposing-looking Prussian civil servant pointing skyward through his office window, while he reads Waltemath's prediction aloud to a know of respectful subordinates. On being interviewed, these witnesses spoke of a dark object having one fifth the Sun's apparent diameter, and which took from 1:10 to 2:10 Berlin time to traverse the solar disk. It was soon proven to be a mistake, because during that very hour the Sun was being scrutinized by two experienced astronomers, W. Winkler in Jena and Baron Ivo von Benko from Pola, Austria. They both reported that only a few ordinary sunspots were on the disk. The failure of this and later forecasts did not discourage Waltemath, who continued to issue predicitons and ask for verifications. Contemporary astronomers were pretty irritated over and over again having to answer questions from the public like "Oh, by the way, what about all these new moons?". But astrologers caught on -- in 1918 the astrologer Sepharial named this moon Lilith. He considered it to be black enough to be invisible most of the time, being visible only close to opposition or when in transit across the solar disk. Sepharial constructed an ephemeris of Lilith, based on several of Waltemath's claimed observations. He considered Lilith to have about the same mass as the Moon, apparently happily unaware that any such satellite would, even if invisible, show its existence by perturbing the motion of the Earth. And even to this day, "the dark moon" Lilith is used by some astrologers in their horoscopes.
From time to time other "additional moons" were reported from observers. The German astronomical magazine "Die Sterne" reported that a German amateur astronomer named W. Spill had observed a second moon cross our first moon's disc on May 24, 1926.
Around 1950, when artificial satellites began to be discussed in earnest, everybody expected them to be just burned-out upper stages of multistage rockets, carrying no radio transmitters but being tracked by radar from the Earth. In such cases a bunch of small nearby natural satellites would have been most annoying, reflecting radar beams meant for the artificial satellites. The method to search for such natural satellites was developed by Clyde Tombaugh: the motion of a satellite at e.g. 5000 km height is computed. Then a camera platform is constructed that scans the sky at precisely that rate. Stars, planets etc will then appear as lines on the photographs taken by this camera, while any satellite at the correct altitude will appear as a dot. If the satellite was at a somewhat different altitude, it would produce a short line.
Observations began in 1953 at the Lowell Observatory and actually invaded virgin territory: with the exception of the Germans searching for "Kleinchen" nobody had ever paid attention to the space between the Moon and the Earth! By the fall of 1954, weekly journals and daily newspapers of high reputation stated that the search had brought its first results: one small natural satellite at 700 km altitude, another one 1000 km out. One general is said to have asked: "Is he sure they're natural?". Nobody seems to know how these reports originated -- the searches were completely negative. When the first artificial satellites were launched in 1957 and 1958, the cameras tracked those satellites instead.
But strangely enough, this does not mean that the Earth only has one natural satellite. The Earth can have a very near satellite for a short time. Meteoroids passing the Earth and skimming through the upper atmosphere can lose enough velocity to go into a satellite orbit around the Earth. But since it passes the upper atmosphere at each perigee, they will not last long, maybe only one or two, possibly a hundred revolutions (about 150 hours). There are some indications that such "ephemeral satellites" have been seen; it is even possible that Petit's observers did see one.
In addition to ephemeral satellites there are two more possibilities. One is that the Moon had a satellite of its own -- but despite several searches none has been found (in addition it's now known that the gravity field of the Moon is uneven or "lumpy" enough for any lunar satellite orbit to be unstable -- any lunar satellite will therefore crash into the Moon after a fairly short time, a few years or possibly a decade). The other possibility is that there might be Trojan satellites, i.e. secondary satellites in the lunar orbit, travelling 60 degrees ahead of or behind the Moon.
Such "Trojan satellites" were first reported by the Polish astronomer Kordylewski of Krakow observatory. He started his search in 1951, visually with a good telescope. He was hoping for reasonably large bodies in the lunar orbit, 60 degrees away from the Moon. The search was negative, but in 1956 his compatriot and colleague, Wilkowski, suggested that there may be many tiny bodies, too small to be seen individually but many enough to appear as a cloud of dust particles. In such a case, they would be best visible without a telescope i.e. with the naked eye! Using a telescope would "magnify it out of existence". Dr Kordylewski was willing to try. A dark night with clear skies, and the Moon being below the horizon, was required.
In October 1956, Kordylewski saw, for the first time, a fairly bright patch in one of the two positions. It was not small, subtending an angle of 2 degrees (i.e. about 4 times larger than the Moon itself), and was very faint, only about half as bright as the notoriously difficult Gegenschein (counterglow -- a bright patch in the zodiacal light, directly opposite to the Sun). In March and April 1961, Kordylewski succeeded in photographing two clouds near the expected positions. They seem to vary in extent, but that may be due to changing illumination. J. Roach detected these cloud satellites in 1975 with the OSO (Orbiting Solar Observatory) 6 spacecraft. In 1990 they were again photographed, this time by the polish astronomer Winiarski, who found that they were a few degrees in apparent diameter, that they "wandered" up to ten degrees away from the "trojan" point, and that they were somewhat redder than the zodiacal light.
So the century-long search for a second moon of the Earth seems to have succeeded, after all, even though this 'second moon' turned out to be entirely different from anything anybody had ever expected. They are very hard to detect and to distinguish from the zodiacal light, in particular the Gegenschein.
But people are still proposing additional natural satellites of the Earth. Between 1966 and 1969 John Bargby, an American scientist, claimed to have observed at least ten small natural satellites of the Earth, visible only in a telescope. Bargby found elliptical orbits for all the objects: eccentricity 0.498, semimajor axis 14065 km, which yields perigee and apogee heights of 680 and 14700 km. Bargby considered them to be fragments of a larger body which broke up in December 1955. He based much of his suggested satellites on supposed perturbations of artificial satellites. Bargby used artificial satellite data from Goddard Satellite Situation Report, unaware that the values in this publication are only approximate and sometimes grossly in error and can therefore not be used for any precise scientific analysis. In addition, from Bargby's own claimed observations it can be deduced that when at perigee Bargby's satellites ought to be visible at first magnitude and thus be easily visible to the naked eye, yet no-one has seen them as such.
In 1997, Paul Wiegert (et al) discovered that Asteroid 3753 has a very strange orbit and can be considered a companion to Earth, though it certainly does not orbit the Earth directly.