BODE'S LAW AND THE DISCOVERY OF CERES[*]

When Copernicus's De revolutionibus appeared in 1543, it was valued by the professionals for its innovative planetary models rather than for anything it might have to say about which body is at the centre of the universe. In a volume dominated by complex geometry, and introduced by a misleading preface inserted without the author's authority to the effect that what followed was guided by the search for accuracy and convenience rather than the quest for truth, the cosmological Book I was largely overlooked. In Book I Copernicus shows in broad, qualitative terms, how so many of the hitherto-puzzling features of the observed motions of these `wandering stars' - such as their retrogressions - are readily explained if one begins from the assumption that the Earth is an ordinary planet orbiting the Sun.

Another consequence of the heliocentric hypothesis outlined in Book I, and one especially satisfying to its author, was that the planets at last formed a single, integrated system. In the accepted Ptolemaic astronomy, even the very order of the planets was uncertain. It was supposed that the planets whose movements differed least from the daily spinning of the fixed stars - Saturn, and then Jupiter and Mars - were physically closest to the stars and so furthest from the central Earth. But since Mercury and Venus appear to accompany the Sun around the sky, all three seemed to have the same period of one year, and so their order of distances was a matter of guesswork. But the heliocentric hypothesis revealed that the supposed equal periods were no more than an illusion resulting from the status of Mercury and Venus as inner planets. It also became clear that the circular movements with a period of one year that occurred in the various traditional models of the planetary orbits were no more than the reflection of the terrestrial motion; this being so, the radius of each of these circles should now be equated to the astronomical unit, giving a common scale to the models and permitting them to be seen as components in an integrated planetary system. From this it transpired that the further a planet from the central Sun, the longer it took to complete a circuit of sky - an harmonious relation that strongly appealed to Copernicus's Platonic intuition.

The planetary system is represented by Copernicus in simplified form in his famous diagram in Book I. But his figure is not to scale. A scale representation would have made it obvious that there is an astonishing gap between the orbit of the fourth planet, Mars, and that of the fifth, Jupiter.

Towards the end of the century, the young Johannes Kepler, in one of the first publications that were irrevocably heliocentric, his Mysterium cosmographicum (1596), sought to make sense of the dimensions of the planetary system. Why, he asked himself, had God been motivated mathematically to select the planetary orbits in just the way he had. ``There were'', he tells the reader in the Preface, ``three things in particular about which I persistently sought the reasons why they were such and not otherwise: the number, the size, and the motions of the circles.'' The gap between Jupiter and Mars was especially awkward to explain. After various attempts, he tried a novel and bold approach.

Eventually Kepler found suitable motivation for the divine geometer in a totally different approach, the nesting of spheres and regular solids; but it was one that commended itself to few in the generations that followed. Some found an acceptable explanation of the gap in the sheer size of the outer planets. Isaac Newton for example regarded the gap as part of the divine plan for the stable and clockwork universe: the massive planets, Jupiter and Saturn, had been located by Providence at the outside of the planetary system, well clear of the smaller planets whose orbits their gravitational force would otherwise disrupt. [2] In the middle of the eighteenth century Immanuel Kant also sees a dynamical justification for the gap in the great mass of Jupiter: ``The width between the orbit of Jupiter and Mars is so great that the space enclosed there exceeds the regions of all lower planetary orbits taken together ... that space is worthy of the greatest among all planets, namely, of that which has more mass than all the others together.'' [3] Johann Heinrich Lambert in 1761 likewise remarks on the gap. Lambert in general is as committed to an eternal, unchanging clockwork universe as was Newton, but at the level of the solar system he is prepared to accept that change has been brought about by the attractive power of Jupiter: ``And who knows whether already planets are missing which have departed from the vast space between Mars and Jupiter? Does it then hold of celestial bodies as well as of the Earth, that the stronger chafe the weaker, and are Jupiter and Saturn destined to plunder forever?'' [4]

These speculative dynamical explanations of the `gap' took place in the context of a surprising willingness on the part of professionals and informed amateurs alike to accept that there may exist planets as yet undiscovered, perhaps inside Mercury, but more plausibly beyond Saturn: surprising, because no primary planet had been discovered since history began. Significantly, just as the most interesting late seventeenth and early eighteenth century speculations on cosmology came from writers whose interests had a theological dimension, so the same is true of speculations about additional planets. So William Wall, cited in the Postscript to the second edition (1727) to Tobias Swinden's An Enquiry into the Nature and Place of Hell, [5] wrote:

William Whiston hints at the same in his Astronomical Principles of Religion, Natural and Reveal'd (London, 1717), when he says carefully that ``Mercury is the nearest to the Sun of all the known Planets'', and that Saturn is ``the highest and most remote of all the known Planets''. [6] As so often, Whiston's views in this book are reflected in the writings of that well-known maverick in both astronomy and theology, Thomas Wright of Durham. In his Clavis Coelestis (1742) he speaks of Mercury as ``the first Planet we know of in the System'' [7][italics supplied], and Venus as ``the second Planet known in the System'', while ``Saturn is the last and highest known Planet in the System''. [8] In his more famous An Original Theory or New Hypothesis of the Universe (1750) he again refers [9] to ``the known Planets''. And in his often bizarre Second Thoughts, which remained in manuscript until our own time, he is explicit: ``...I am far from supposing our present knowledge of ye solar system perfect and fully known''. [10]

Another of the mid-eighteenth century speculators to anticipate an undiscovered planet was Immanuel Kant. In his Universal Natural History ...he writes:

These suggestions concerning planets as yet undiscovered relate mostly to the regions outside Saturn. Only a few were concerned with the gap between Mars and Jupiter. One who did `surmise' the presence of one or more planets in the gap was apparently the Scottish mathematician Colin Maclaurin. [13] Another to focus on the gap was Thomas Wright. In one of those unexpectedly insightful speculations that make him so fascinating a figure, Wright actually suggests in his unpublished manuscript that the gap between Mars and Jupiter results from a planet having broken up following collision with a comet:

Yet the gap was readily apparent to anyone who glanced at the data for the planetary orbits. Near the beginning of the eighteenth century, for example, William Whiston, Newton's successor in Cambridge, gives the actual distances of the planets in millions of miles as 32, 59, 81, 123, 424, 777. [15] We note that there are four planets within 123 million miles of the Sun, but the gap before the next planet, Jupiter, is over 300 million miles.

Whiston's contemporary, David Gregory, in his widely-read The Elements of Astronomy [16] puts the planetary distances into proportional numbers: ``...supposing the distance of the Earth from the Sun to be divided into ten equal Parts, of these the distance of Mercury will be about four, of Venus seven, of Mars fifteen, of Jupiter fifty two, and that of Saturn ninety five.'' Gregory's work was published in Latin in 1702 and again in 1726, and an English translation appeared in 1715 with a second edition in 1726. The words quoted appear at the very beginning of the work, in Proposition 1 of Section 1 of Book I, and are therefore in a very prominent position. But they have been overlooked by historians, who have found exactly the same numbers - indeed, a paraphrase of the same sentence - in a work published in 1724 by Christian Wolff: Vernünfftige Gedanken von den Absichten der natürlichen Dinge, which was to go through several editions. [17] In 1764, the French natural philosopher Charles Bonnet published his Contemplation de la Nature, a successful work that was quickly translated into other European languages. The German translation was undertaken by Johann Daniel Titius of Wittenberg. It had long been common for translators to supplement the text they were translating, usually to bring it up to date, for in those days when book publishing was even slower than it is today, many years often elapsed between first publication and translation. Translators, that is, took a greater initiative than is now thought proper; indeed, it was not unknown for a translator to conduct a running battle with his author through the medium of footnotes. Titius, probably because he was by nature self-effacing, not only left his additions unsigned but actually incorporated them in the text itself, with no hint that they were not the original work of the author. He chose to make such an addition to the paragraph where Bonnet remarks that ``We know seventeen planets that enter into the composition of our solar system [that is, major planets and their satellites]; but we are not sure that there are no more'', going on to anticipate more discoveries as telescopes improve. Titius then inserts what we now know as Bode's Law:

It is interesting to note that these numbers are not exactly the ones listed by Gregory and Wolff; nor do they follow from the actual distances published by Whiston, which would give 96 for Saturn in place of the 95 of Gregory and the 100 of Titius. But it seems that Wolff was indeed the immediate source for Titius, for in the fourth edition of his translation, by which time he was clearly identifying his own contributions as such, he adds the comment: ``This relationship and the related considerations which Herr Bonnet thought had first been observed by Herr Lambert had already been recited by Freyherr von Wolf in his German Physics more than forty years earlier.'' [19] How Titius could declare that Bonnet had drawn his ideas of unknown planets from Lambert is not clear, though perhaps Titius and Bonnet may have corresponded over the translation; but this reference of Titius to Wolff suggests that Wolff had indeed been Titius's original source.

As it happened, Titius published a second edition of his translation - with the law now properly located in a footnote - just as the promising young astronomer Johann Elert Bode was putting the finishing touches to the second edition of his introduction to astronomy, Anleitung zur Kenntniss des gestirnten Himmels, which he had published in 1768 when he was only nineteen. Bode came across the relationship proposed by Titius, was convinced by it, and inserted it as a footnote in his text:

It is clear from the wording that Bode is following Titius, although he of course realized that the suggestion that the missing planet was a moon of Mars was preposterous, a fact he emphasized in the third edition of his book. But he makes no acknowledgement to Titius; indeed, it is only in later editions that Bode identifies his source (possibly because Titius had pressed him to do so). In the hands of Bode the relationship assumed a new importance, for Bode was a professional astronomer soon to take on international stature, and he was well-placed to act as apostle of the new law.

Given the willingness on the part of many astronomers to believe that there were planets as yet undiscovered, and especially so in orbit beyond Saturn, it is a little surprising that it never crossed the mind of William Herschel in March 1781 that the ``curious either nebulous star or perhaps a comet'' he had noticed in his telescope might indeed be a major planet. [21] It is often said that this failure of imagination was because of the total novelty of his discovery - that no primary planet had ever been discovered in historic times, which of course is true. But in view of the numerous references we have seen to the ``known'' planets, including that of Bode just cited, it seems more likely that Herschel - an isolated and self-taught amateur - was simply unaware of the professional astronomers' openness to new discoveries among the planets. As early as 4 April the Astronomer Royal, Nevil Maskelyne, wrote to their mutual friend William Watson of Herschel's ``comet or new planet'', and on the 23rd he wrote to Herschel:

How to calculate the orbit, however, was a difficult problem, for the body had been observed for only a very tiny fraction of a complete orbit. If it was a comet, then it would be simplest to assume a parabolic orbit. P.-F.-A. Méchain, a French mathematician who had discovered several comets, being misled by the earliest observations which made it likely that the object was indeed a comet, sent Herschel a letter in which he gave the perihelion distance of the supposed comet as 0.46 AU and perihelion date as 23 May 1781; Anders Johan Lexell, a Finnish-born professor of mathematics at St Petersburg who was visiting England at the time, soon after proposed a perihelion distance of 16 AU with perihelion not to be reached until 1789.

On the other hand, if it was a planet, then it was simplest to assume a circular orbit, and Lexell was one of a number of astronomers who, finding that parabolic orbits were incompatible with the observations, investigated circular orbits. Lexell derived for the radius of the orbit the excellent value of 18.93 AU - that is, with the radius of Saturn's orbit put at 100, a distance that compared well with the prediction of 196 from the Titius-Bode relation. More sophisticated calculations followed, some of them taking into account observations made years earlier when the planet had been mistaken for a star; and soon it was clear that the object was indeed a planet and, moreover, one that fitted well the Titius-Bode relation.

This remarkable confirmation of the relation naturally reinforced Bode's belief, and it likewise persuaded Baron Francis Xaver von Zach, the court astronomer at Gotha. Both men were convinced there was an undiscovered planet between Mars and Jupiter, and in 1787 Zach began to search for it. Not unreasonably, he limited his investigation to the Zodiac, and believing that only a methodical search offered hope of success, he produced for himself a catalogue of zodiacal stars arranged by right ascension; but without success. The autumn of 1799 found him visiting astronomers in Celle, Bremen and Lilienthal, and there the idea of a cooperative attack on the problem emerged:

It was on 21 September the following year that the cooperative attack - probably without precedent in the history of science - became a reality. On that day six astronomers met in Lilienthal: von Zach himself; J.H. Schröter, the chief magistrate of Lilienthal, whose world-famous collection of instruments included a Herschel reflector of 27ft focal length; H.W.M. Olbers, physician from nearby Bremen and longtime collaborator with Schröter; C.L. Harding, who was employed by Schröter and who was himself to discover the third asteroid in 1804; F.A. Freiherr von Ende; and Johann Gildemeister. They decided that even six observers were too few for the task ahead, and nominated instead a group of twenty-four practising astronomers chosen from throughout Europe. Schröter was to be president and Zach secretary. The entire Zodiac was divided up into twenty-four zones each of 15 degrees in longitude, and extending some 7 or 8 degrees north and south of the ecliptic in latitude. The zones were allocated to the members by lot. Each member was to draw up a star chart for his zone, extending to the smallest telescopic stars,

Zach accordingly sent out the invitations to join this society of celestial cops. One of those chosen was, naturally, Giuseppe Piazzi of Palermo, the southernmost of the European observatories. Piazzi had been born in 1746 in Valtellina, in what was then part of Switzerland but is now northern Italy. [24] As a young man, Piazzi joined the Theatine Order, and afterwards taught mathematics in a number of Italian cities. In 1780 he was invited to take the chair of higher mathematics at the Academy of Palermo. Arriving at Palermo, Piazzi, although inexperienced in astronomy, expressed a wish to found an astronomical observatory: Palermo was further south than any existing European observatory and so offered access to regions of the sky inaccessible elsewhere. His royal patron was in favour, and prepared to forego Piazzi's services while he equipped himself and his observatory for the task ahead. Piazzi accordingly set off for England where he might obtain good advice, from such disparate figures as William Herschel and the Astronomer Royal, Nevil Maskelyne, and - equally important - good instruments. Jesse Ramsden was an instrument-maker without peer, but he was notorious for failing to produce on time. Piazzi persuaded him to attempt a 5ft vertical circle of unique design. The circle, which has been described as ``a masterpiece of eighteenth-century technology'', was twice abandoned by Ramsden, and he completed it eventually in August 1789 only because Piazzi himself was present in London - indeed, in Ramsden's workshop, and quite literally breathing down Ramdsen's neck as the work proceeded. The circle gave readings in azimuth by micrometer microscope, and readings in altitude by two diametrically opposed microscopes. The divisions on the circles were illuminated by an inclined silver mirror fixed to each microscope, and the wires in the telescope eyepiece by transmitting light from a small lamp through the hollow tube-axis. [25]

Once the Ramsden circle was installed in Palermo, Piazzi found himself in a privileged situation. He had an instrument of unique quality, a good climate, and the southernmost latitude of any European observatory. He very rightly set to work to exploit these advantages in the compilation of a star catalogue better than any that had gone before. A feature of his painstaking work was the repeated measurement of stellar positions on different nights, so that the final coordinates were accurate to a few seconds of arc. The first of Piazzi's two great catalogues, with the coordinates of some 6,748 stars, was to appear in 1803. [26] The accuracy of his work gave astronomers once more the confidence to tackle the question of stellar parallax, which had been largely in abeyance since it became clear around 1730 that parallax could not be much more than a second of arc.

The beginning of 1801 found Piazzi patiently at work on the star catalogue. As he wrote a few months later,

Piazzi had in fact measured the position of the object on a total of 24 nights between 1 January and 11 February, though some positions were marked as `doubtful' or even `very uncertain'. On 24 January, Piazzi had announced his discovery in letters to fellow astronomers, among them his fellow-countryman, Barnaba Oriani of Milan. In it, Piazzi confided to him that

To the others, he claimed nothing more than the discovery of a comet, though making it clear that the `comet' had no nebulosity or tail.

When after 11 February he could no longer see the object, Piazzi set to work to investigate its orbit, though such mathematical investigations were not his strength. He began with the assumption that it was indeed a comet, and fitted a parabola to three of the observations to see if the orbit would account for the others. It did not. A second attempt with a different group of three observations likewise failed:

But how to recover the now-lost planet at some future date? In Piazzi's opinion, the best hope lay in identifying some past occasion when the planet had been observed in the belief that it was a star. He thinks it may well have been the object observed by Bode in 1772, and that it had probably been listed at some time or other by la Caille or by Tobias Mayer:

We can well believe Piazzi when he says that not only Oriani, but more especially Bode of the celestial cops, ``were instantly of the opinion that it was a new planet; and settled nearly the same elements of its orbit, as I have done''. One can imagine the German's delight that the hoped-for planet had been found, even if the discovery owed nothing to the celestial cops themselves.

But now Piazzi himself was beginning to have doubts. He had estimated the size of the object from the fact that it was almost, but not quite, covered by one of the wires of his telescope, and his conclusion was that it was larger than the Earth. However, it would seem that in the hazy nights that followed, the true (and much smaller) size of the object became more evident to the Palermo astronomer, who began to think that the object was diminishing in size and therefore moving rapidly away, so that it must be a comet after all:

Eventually, in April, when illness had prevented him from making further progress in the investigation of the object's orbit, Piazzi sent his complete observations to Oriani, Bode, and Lalande in Paris, together with his suspicions that it might be a comet after all. And with that, Piazzi's own role in the story comes to an end, save for his naming the body - should it ever be recovered - Ceres Ferdinandea, Ceres for the patron goddess of Sicily, and Ferdinandea for Piazzi's royal patron.

It is one of the problems of writing history, that no story ever has a tidy ending. One would wish to go on to discuss the mathematical analysis of Piazzi's observations by Gauss that enabled Zach to recover it at the end of the year; the discovery of Pallas by Olbers in March 1802; the announcement by William Herschel [32] in May that these bodies were tiny compared to the planets - he estimated Ceres had a diameter of only 162 English miles (though this is perhaps a quarter of what we consider the true value), and proposed that these bodies should be termed asteroids rather than planets, much to Piazzi's annoyance; the discovery of Juno by Harding in 1804, and of Vesta by Olbers in 1807; and indeed the role of Bode's Law (or better, as we have seen, the Titius-Bode Law) in the discovery of Neptune in 1846. But in this celebration of the bicentenary of Palermo Observatory, we are perhaps justified in ending this story with the most famous discovery ever made here - but a discovery made possible by the fine instrument Piazzi had managed to acquire, and by Piazzi's dedication in using it towards the compilation of his two great star catalogues - catalogues that raised European standards of precision astronomy in the opening years of the new century.