UraniaDepartment of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Flamsteed's 1681 Lunar Theory 

Flamsteed's Lunar Theory, 1681

To download the
"Flamsteed's Theory of the Moon's Motion"
package click here

Though the Reverend John Flamsteed was employed to find longitude, as Britain's first Astronomer Royal, to the end that the King's ships would be safe at sea, he soon realised the inherent hopelessness of his position. In 1683 he published an article advocating use of the moons of Jupiter as the more promising method, as the Moon's motion was so erratic and unpredictable. He found that the Jupiter moons exactly followed Kepler's second law and so were quite reliable as timekeepers (Phil. Trans., 13, pp.404-8). But this posed the practical problem that, clearly, no one expected to track these on board a tossing ship.The Observatory at Greenwich

As a North-countryman, Flamsteed had come down to London with details of the work of Horrocks and his co-workers Gascoigne, Townsley, Crabtree - initiators of the British tradition in astronomy. The first British Keplarians and the first to use new technology such as eyepiece micrometers, they combined theoretical and practical insights. Above all, they used a new lunar theory devised by the 22-year old Jeremiah Horrocks.

Horrock's theory used the time between solar conjunctions with the mean apse (six and a half months) to drive an epicycle, and that epicycle defined a varying eccentricity and an oscillation in the apse line, the two being out of phase with each other. This comprised the second of his three 'equations', the first being an annual equation and the third the Variation - followed by what was called the Reduction, which was an adjustment to convert from the lunar orbit onto the plane of the ecliptic.

This Horroxian model was first published in 1673 (by John Wallis, as Horrox's Opera Posthuma), thirty-eight years after Horrock's death. Flamsteed improved the Horroxian model in three ways:

Earlier published 'equation of time' tables--e.g., by Streete--to convert apparent to mean time, had five arc-minute errors, while those of Flamsteed's tables did not exceed ten arc-seconds (I and Bernard Yallop of the Nautical Almanac Office, RGO, ascertained this in Kollerstrom and Yallop 1995). Flamsteed published his "Doctrine of the Sphere" (De Sphaera) in 1681, containing his version of the Lunar theory--with the worked example here.

Flamsteed, with assistant, observe the 1715 solar eclipseIn 1694 Newton and Gregory visited Flamsteed at Greenwich. The latter may have been rather frosty, as Gregory had recently published Newton's lunar 'theory', based entirely on the many observations which Flamsteed had supplied, without making acknowledgement of this -- despite two written promises by Newton that he would do so. Yet, Flamsteed showed them a table of error-values, perhaps the first ever in this context, comparing observed lunar longitudes with theoretically computed longitudes based on his Horroxian model, the values going up to around 8 arcminutes. This became the stimulus for Newton's great endeavour to develop a lunar theory. The longitude positions derived from observation were accurate to around half an arcminute (Kollerstrom and Yallop, 1995).

Close-up of Flamsteed, with assistant: notice clock at 9:00pm and scroll with 15 May 1715In general, errors from Flamsteed's version of the Horroxian theory could easily go up to twelve arcminutes, as Newton pointed out to him. For about six months through the winter of 1694/5 there is a marvellous amity and mutual respect in the correspondence between the two of them, as data was sent from Greenwich to Cambridge. Eventually it broke off, as Flamsteed's migraine attacks caused delays in reply and the mathemaician grew impatient. An opportunity was discerned of shifting 'the blame' for not developing the hoped-for lunar theory onto the astronomer, on the grounds that he had not sent some data, where it has remained ever since.

A Clandestine Affair

Flamsteed did clearly compose a 'Newtonian' lunar theory in the first decade of the eighteenth-century, as his January 1703 letter to Sharp indicated, but despite his hopes this was not included in his posthumous Historia Coelestis. A version of it was published some decades later by Pierre Lemonnier in Paris. LeMonnier announced, somewhat as a sales blurb to his Institutions Astronomiques of 1745, that he had new, accurate and original tables from Flamsteed. In the maner of one announcing a scoop who is not at liberty to disclose his source, he explained that the Flamsteed tables contained all he had needed except the procedure for finding latitude.

It seems reasonable to conjecture, as did Curtis Wilson in General History of Astronomy, that Halley took these Flamsteed tables on becoming Astronomer Royal and later gave them to Lemonnier. LeMonnier did develop and improve the Flamsteed tables somewhat and was far from being merely a plagiarist, as may be seen by comparing his published tables with the thirty-seven quarto pages of manuscript of the tables in Flamsteed's hand now at the University Library, Cambridge.

Biographers of Halley usually claim that his lunar tables were prepared by 1619, whereas existing correspondence indicates that they were rather composed in the early 1620s -- i.e., in the immediate aftermath of Flamsteed's death in 1619. Curiously, the only surviving copy of Halley's tables lies in the Logan Library in Philadelphia: James Logan purchased a copy of the Halley tables, as apparently the only copy sold, in 1723 - and complained about the price. (He paid a guinea for them!) But we digress.

The contents of this page remain the copyrighted, intellectual property of Nicholas Kollerstrom.  Details. rev: May 1998