Once upon a time, there were just nine known planets in the universe - and all of those were orbiting our sun. Now the number has just zoomed past the 400 mark, following the announcement that another few dozen of the things have been discovered around other stars. Astronomers have been finding planets beyond our solar system at an ever-increasing rate since the mid-1990s. But it is a cornucopia that is now proving something of a mixed blessing for those wrestling with the mystery of how stars acquire their celestial offspring. Planets are turning up around single stars like the Sun, but also around binaries and triplets, often on their own but sometimes accompanied by a clutch of siblings.
And then there are the planets themselves: everything from colossal "gas giants" even more massive than our own Jupiter to Earth-sized balls of rock. Their orbits are no less varied - some take centuries to complete their journey around their parent star, while some whizz round in days. A few weeks ago, astronomers announced the discovery of a planet 10 times more massive than Jupiter that is hurtling round its parent star in just 23 hours.
Somehow, all of this will have to be accommodated in a single theory explaining the formation of planetary systems. Chances are it will involve a whole slew of different mechanisms, all of which can turn the raw ingredients of cold dust and gas into stars and planets. In the search for such a grand unified theory, the history of science suggests astronomers could do worse than to look for clues from apparently meaningless coincidences. Around a century ago, while devising the first-ever quantum theory of the atom, the Danish physicist Niels Bohr became intrigued by a bizarre "law" discovered by a Swiss schoolteacher. Trawling through data about the light emitted by hydrogen atoms, Johann Balmer had found a simple formula that linked the wavelength of the light to the squares of whole numbers. Some distinguished physicists dismissed the connection as a coincidence; Bohr thought otherwise, and showed that it was a natural consequence of the fact that light energy emerges from atoms in discreet "packets", or quanta.
In their quest for the structure of the genetic molecule DNA, the biologists James Watson and Francis Crick were helped by another coincidence, discovered in 1949 by the Austrian biochemist Erwin Chargaff. He noted that the four so-called chemical bases in DNA, code-named A, C, G and T, always appeared in the same proportions: with the amounts of A and T always matching each other, and similarly for C and G. Chargaff himself regarded this fact as "striking, but perhaps meaningless". A few years later, Professors Watson and Crick showed the matching proportions were an inevitable consequence of the fact that DNA is a double helix-shaped molecule.
In astronomy, no less a genius than Sir Isaac Newton was perfectly happy to take another such "coincidence" seriously. While developing his theory of planetary motions, he sought to explain the odd fact - pointed out by the German astronomer Johannes Kepler in 1619 - that if the square of the orbital period of a planet is divided by the cube of its distance from the sun, the result is always the same. Newton showed that this followed directly from a combination of the laws of motion and his famous law of gravity.
So which coincidences could help today's astronomers fathom the mysteries of planetary formation? The most striking is the curious "law" that appears to govern the layout of our own solar system. First pointed out in 1766 by the Prussian astronomer Johann Titius, it parallels Balmer's formula in relating a fact of nature - in this case, the relative distances of the planets from the sun - to whole numbers.
The formula amounts to the following: write down the numbers 0, 3, 6, 12 and so on, doubling each time. Now add 4 to each one, and divide the result by 10. The resulting sequence of numbers - 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0 - is stunningly close to the relative sizes of the orbits of all the planets known in Titius's day, from Mercury to Saturn, the only flaw being the lack of a planet at a distance of 2.8 units.
When the planet Uranus was later discovered lying at 19.2 units from the sun, within 2 per cent of its distance as predicted by Titius's "law", astronomers started to take notice. They looked for the missing planet at 2.8 units, and duly found the first of the asteroids, Ceres, in 1801. For a supposed coincidence, this was a stunning success, but it also marked the high-water mark in the reputation of the law. When Neptune was found in 1846, it turned out to be on an orbit corresponding to 30 units, rather than the 38.8 predicted by the formula. The death knell was the discovery of Pluto in 1930 at 39 units, far closer than the 77.2 given by Titius.
Ever since, most astronomers have dismissed the whole "law" as a meaningless fluke. However, recent developments suggest it may be worth a second look. To begin with, the supposedly egregious failure to account for the outermost planet has lost much of its impact following Pluto's demotion in 2006 to the lowly status of second-largest "dwarf planet". Current theories of the formation of the eight genuine planets of the solar system suggest a physical basis for the law. It is believed that the solar system formed from a vast collection of dust and gas that collapsed under its own gravity, forming a pancake-like shape. Processes taking place within this cloud - like turbulent mixing of material - are likely to generate so-called scale invariance, meaning that the cloud has features that look the same no matter how much they are magnified. This in turn could lead to concentrations of matter - and thus planets - appearing at points spaced according to rules such as that found by Titius.
Admittedly, as explanations go, this is all a bit sketchy. But as ever more planetary systems are found orbiting other stars, astronomers will soon have enough examples to put the general principle to the test. And they may well find that Titius's law is another example of a scientific "coincidence" that is actually anything but. Robert Matthews is Visiting Reader in Science at Aston University, Birmingham, England