Halley thus became, at a very early age, a full member of Britain’s scientific elite. For over sixty years he played a highly active role in the Royal Society as a Fellow, as Clerk for a few years, and as editor of the Philosophical Transactions from 1685 until 1693. Halley contributed no fewer than eighty-four papers to the journal, largely devoted to astronomy (star charts, lunar theory, comets, eclipses, transits of Venus, the dimensions of the solar system, novae and nebulae), although questions of geophysics (terrestrial magnetism, trade winds, monsoons, tidal theory) are also prominent. Other papers touch on areas of pure mathematics, optics, ballistics, hydrostatics, pneumatics and social statistics – Halley was a pioneer in the use of mortality tables for calculating annuities.

The great practical problem facing the astronomers of Halley’s age was that of determining longitude at sea. The Admiralty had offered a prize for the first workable solution, and Halley spent much of his life pursuing three different approaches to the problem. A sufficiently accurate lunar theory would, in principle, provide a solution, the observed motions of the Moon against the background of the stars serving as a sort of great clock. But lunar theory was not yet up to the job. The periodic motions of the satellites of Jupiter provided another possible method, but observing those bodies required a long telescope which could not be kept steady on a tossing ship.

A quite different method would be to make use of the variation in the magnetic declination (i.e. the angle between magnetic and geographical north) at different points on the Earth’s surface. From 1698 to 1700 Halley served as a naval captain plotting degrees of magnetic declination throughout much of the Atlantic Ocean, and plotting isogonics (later renamed ‘Halleyan lines’) connecting points of equal declination. The intersection of one of these lines with a line of latitude would, it was hoped, enable sailors to fix their position. None of Halley’s three methods provided, within his lifetime, an adequate solution to the problem; in his later years, he would see the early models of Harrison’s famous marine chronometer succeed where astronomical and geophysical methods had failed.

Early in 1684 Halley found himself speculating about the mechanics of the solar system. In company with Robert Hooke and Christopher Wren, he had hit on the idea that the attractive force that keeps the planets in their orbits varies as the inverse square of their distance from the Sun. None of the three men, however, could solve the mathematical problem of deriving Kepler’s laws of planetary motion from the proposed inverse square law. Halley decided to call on Isaac Newton (already well known to members of the Royal Society as a mathematician of formidable powers) and put the problem to him.

The momentous visit took place in August 1684. Newton told Halley that he had already solved the problem years ago, and promised to send him the proof. On receiving the desired proof, Halley, fully aware of its significance, returned to Cambridge for a second meeting with Newton. It was during the course of this second visit that Halley realized the extent of Newton’s achievement, and persuaded him to publish it in book form. The idea of the Principia was born.

Halley not only had to persuade Newton to publish the Principia; he also found himself having to see it through the printers at his own expense. (The Royal Society was going through one of its periodic financial crises.) Worse still, a priority dispute sprang up, with the irascible Hooke claiming credit for the inverse square law, and Newton threatening to suppress the whole of Book 3 of his masterpiece, the great System of the World. Halley, fortunately, was on good terms with both these difficult men, and was able, by a combination of diplomacy and hard labour, to see Newton’s ‘divine treatise’ through the press in July 1687. Rarely in the history of science can one man’s masterpiece have owed such a debt to the labours of another.

As a Preface to the first edition of the Principia, Halley composed an ‘Ode to Newton’, in which he refers to the order of the solar system as ‘eternal’ – this may have been a mere lapse on Halley’s part, or a little poetic licence, but it was enough to excite the wrath of the clergy. He was accused of the heresy of ‘eternalism’, i.e. of denying both the creation of our physical universe ex nihilo and its eventual destruction in the universal conflagration prophesied in Scripture. There were also reports, spread by enemies such as Flamsteed and William Whiston, of Halley as a ‘sceptic and banterer of religion’.

In 1691, when he applied for the Savilian Professorship of Astronomy at Oxford, he was examined by Edward Stillingfleet and Bentley, on the suspicion of holding materialist views. On this occasion, he did not get the job. When the second edition of the Principia was being edited by Bentley in 1713, the ‘eternalist’ sections of Halley’s ode were quietly deleted. The charge of irreligion seems to have stuck: it is widely believed that the ‘infidel mathematician’ to whom Berkeley addressed his Analyst (1734) was Halley.

Was Halley really an ‘eternalist’? According to Kubrin (1971), he was speculating, at around this period, on the possible role of comets in the economy of the solar system. An idea suggested in an unpublished paper of 1694 is that our Earth is in decline, and will continue to decay until a catastrophic collision with a comet restores it to its pristine vigour. This would entail a sort of cyclical cosmogony, with an endless succession of ‘Earths’, each no doubt furnished with its proper inhabitants. Small wonder that the paper was not published until 1724, when Halley felt his position to be much more secure.

Obviously disappointed by his failure in 1691, Halley set out to establish his orthodoxy in a number of papers refuting eternalism. If the planets are moving through a very subtle aetherial medium, he suggests, the drag of this medium will eventually cause the solar system to collapse. Halley claimed to have observational evidence of the existence of such a gradual retardation of the planets’ motions. Evidence of the finite age of our Earth can also be found in the oceans. If the rivers bring so much salt per annum to the sea, one can in principle project the process back to estimate the age of the Earth. Eternalism will thus be refuted – but so too will be a literal interpretation of scriptural chronology. The Earth will turn out to be very much older than the Bible tells us.

Halley’s protestations of orthodoxy, aided by the efforts of powerful friends at court, enabled him to succeed in 1703 where he had failed in 1691, when he became Savilian Professor of Geometry at Oxford. As Savilian Professor one of his great achievements was a scholarly Latin edition of Apollonius’s treatise on Conic Sections, reconstructed from fragments of the original Greek and of an Arabic translation. For Book 8 there was no manuscript text, and Halley had to reinvent its probable contents. The Conics was not merely of historical interest to Halley: it is a work of crucial importance to any mathematician wanting to master Newton’s Principia.

The scientific achievement for which Halley is best known today is, of course, his work on comets. In the first edition of the Principia, Newton had applied his principles to the motions of comets, but had supposed that they follow parabolic paths about the focal point of the Sun. (A parabola and an ellipse are almost indistinguishable over the small parts of its orbit for which a comet can be observed.) It was Halley who delved through the historical records of the appearances of comets in an attempt to substantiate his thesis that they move in elliptical orbits of high eccentricity, and can thus be expected to return with a regular periodicity. And it was Halley’s prediction, in his Synopsis of the Astronomy of Comets (1705) – that a given comet, with a period of about seventy-six years, could be expected to return in 1758 – that provided the Newtonian theory with one of its most spectacular triumphs. Hitherto objects of mystery and superstitious dread, comets had been shown to be subject to the same universal laws as the rest of the solar system. One could hardly ask for a clearer illustration of the predictive power of Newtonian mechanics.

One unresolved question of Newtonian cosmology was that of the finitude or infinitude of the universe. Newton had of course discussed the issue in his famous correspondence with Bentley; Halley would no doubt have been familiar with Newton’s views on the subject. In his own paper, published in the Philosophical Transactions for 1720, Halley argues for an infinite universe with an even distribution of stars. If the universe had a central point, he argues, all its matter would collapse towards that point; to maintain stability, there must therefore be an equilibrium of forces, which in turn requires an infinity of stars. Halley’s theory raises both gravitational and optical problems which he could not adequately resolve, but he deserves credit none the less as one of the pioneers of modern physical cosmology.

Like Boyle and Newton, Halley was an atomist in his theory of matter. In October 1689 he presented to the Royal Society a paper on the size of atoms, calculated on the basis of some observations on the thinness of gold leaf. If one cannot see through gold leaf, he reasoned, it must be at least one atom thick (probably more). But the film was, he estimated, only one 134,560th of an inch thick. This entails that the atoms ‘are necessarily less than 1/2433000000 part of the cube of the hundredth part of an inch, and probably many times lesser, if the united surface of the gold without pores or interstices be considered’ (quoted from Ronan, p. 98). This may appear to be a sort of quantitative measure of the size of atoms; in reality, however, it only sets an upper bound – the conclusion is only that atoms are unimaginably minute.

Halley’s relations with John Flamsteed, the first Astronomer-Royal, had initially been friendly, but soon deteriorated into open hostility. Flamsteed was compiling a great map of the stars, the Historia coelestis, but his progress was very slow, and many influential people blamed him for his tardiness and reluctance to release his results. Halley was a leading member of the committee appointed by Prince George of Denmark to take charge of Flamsteed’s tables and see them through the press. This ‘pirated’ edition of the Historia coelestis appeared, against Flamsteed’s will, in 1712; the ‘official’ version only saw the light of day much later, after Flamsteed’s death.

In 1720, Halley succeeded his arch-rival, becoming the second Astronomer-Royal. He seems to have had more success than his predecessor in getting funds for instruments, while a visit from Queen Caroline in 1729 led to a pay-rise. For much of his last twenty years Halley laboured, at the Observatory at Greenwich, to perfect lunar astronomy, but he was never to win the sought-after prize for the solution of the problem of longitude – applying Newtonian principles to this notorious three-body problem continued to perplex mathematicians of the calibre of Euler, Clairaut and d’Alembert.