On this page
Before 1705, most people treated each bright comet as a one-off visitor. Edmond Halley argued that one of them kept coming back, and he put a year on its return: 1758. He did not live to see it. This post tells how he reached that prediction, what the calculation involved and what a student can take from the way he worked.
Who was Edmond Halley?
Edmond Halley was born on 8 November 1656 near London and died at Greenwich on 14 January 1742, according to the MacTutor biography at the University of St Andrews. He entered Queen's College, Oxford, in 1673, but left in 1676 to sail to the island of St Helena in the South Atlantic. There, in about eighteen months, he catalogued 341 stars of the southern sky. He was elected a Fellow of the Royal Society in 1678, became Savilian Professor of Geometry at Oxford in 1704, and succeeded John Flamsteed as Astronomer Royal in 1720.
How did Halley help Newton publish the Principia?
MacTutor records that Halley urged Isaac Newton to write the *Principia*, then paid for its publication himself, because the Royal Society could not, and corrected the proofs. That book set out the laws of motion and gravitation that Halley would use on comets.
Halley's own account, in the English translation of his 1705 Synopsis of the Astronomy of Comets, gives Newton the credit for the method. Newton had shown that comets follow the same principles as the planets and had given a geometric way to find a comet's orbit. Halley wrote that, following Newton, he had "attempted to bring the same Method to Arithmetical Calculation". The original was published in Latin in the Philosophical Transactions of the Royal Society in 1705.
How did Halley work out that three comets were one?
He began by collecting data. In his words, "having collected all the Observations of Comets I could, I fram'd this Table, the Result of a prodigious deal of Calculation". The table listed the orbital elements of every comet that had been properly observed: the tilt of its path, the point where it came closest to the Sun and how close it came.
He explained what the table was for. Whenever a new comet appeared, astronomers could compare its elements with the table to see whether it had appeared before, and so find its period and foretell its return (Synopsis, 1705).
Comparing the entries, he found that the comet observed by Peter Apian in 1531, the one described by Johannes Kepler and Longomontanus in 1607 and the one Halley had seen himself in 1682 agreed in all their elements. Only the time between visits differed slightly. You can check the arithmetic yourself: 1607 − 1531 = 76 years, but 1682 − 1607 = 75 years.
It would have been easy to drop the idea there. Halley instead asked whether a known cause could explain a gap of about a year. He pointed out that the motion of Saturn is disturbed by the other planets, especially Jupiter, enough to make Saturn's period uncertain by some days. A comet that travels, as he put it, "almost Four times higher than Saturn" would be disturbed even more. Then came the prediction, in the 1705 English text: "Hence I dare venture to foretell, That it will return again in the Year 1758."
Did the comet come back?
It did. NASA's page on the comet states that "Halley correctly predicted the comet would return in 1758", and MacTutor notes that the calculation was hard because Halley had to allow for the pull of Jupiter. Halley had died in 1742, so he never saw it.
A review in the Philosophical Transactions of the Royal Society A (Hughes, 1987) describes the comet's place in history as "the first periodic comet, the first predicted cometary return", and notes that it was French mathematical astronomers who gave it Halley's name.
NASA gives its period as about 76 years on average and its next return as 2061. On its 1986 visit, an international fleet of spacecraft met it. The European Space Agency's Giotto probe reached it on 14 March 1986 and passed within 596 kilometres of the nucleus.
How big is the comet's orbit?
Kepler's third law gives a quick check. For anything orbiting the Sun, with the period T in years and the semi-major axis a in astronomical units (AU, the Earth–Sun distance), T² = a³. With T = 76: T² = 5776, and the cube root of 5776 is about 17.9. So the comet's average distance from the Sun, half the long axis of its ellipse, is about 18 AU. The far point of an ellipse is always farther from the Sun than the semi-major axis, so the comet's outermost point lies well beyond 18 AU, which fits Halley's remark about Saturn. Our post on Kepler's laws explains where T² = a³ comes from.
What can a student take from Halley's way of working?
Build the table first. Halley's discovery came from setting many cases side by side. An error log of your wrong answers does the same job for exam practice.
Don't discard a near-match too quickly. When one number disagreed, he looked for a cause he already knew about.
Make a claim specific enough to be wrong. "It will return in 1758" could have failed in public.
Help other people's good work reach the world. Without Halley, the *Principia* might have waited much longer.
Orbits and Kepler's laws are easier to understand when you can change an ellipse and watch the period change. The Fermi Simulation Lab has interactive simulations for ideas that are easier to see than to read, which helps before you try orbit questions on paper.
Try the calculation yourself today: find the semi-major axis of a comet with a period of 27 years (the answer is a whole number of AU).
