Halley as Savilian Professor of Geometry at Oxford, c. 1690Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

Halley's Version

Halley used the version of TMM given in the second edition of the Principia (1713) with three adjustments. He used it on an almost daily basis for eighteen years after 1720, a complete Saros cycle. The error-values he found on each computation were finally published in 1749, posthumously. His method: There was a further adjustment which Halley recommended to Newton in the 1690s, namely that the epicycle used in the Horroxian theory (defining the varying eccentricity and second apse equation) should itself expand and contract on a yearly basis. Newton incorporated this in his 1713 version of the theory. However, my reconstruction of Halley's two worked examples, shows that he did not in fact bother to use this! It would not have improved matters.

In all, I found that these several adjustments made Halley's version of TMM less accurate than, say, LeMonnier's tables, which perfectly embodied the 1702 instructions.

Halley's crisis

Halley's mature and final opinion on the accuracy of TMM was given in 1731, when he was Britain's foremost astronomer and both Newton (d. 1727) and Flamsteed (d. 1719) were mere memories for him. Then, after consulting both his own lunar tables as the Astronomer Royal and those of his predecessor, his view of TMM three decades after its composition was that: Halley developed a 'saros' method of allowing for the errors in the theory: every 18 years and 11 days, the same errors would recur, and so could be allowed for. This was a valid method, as the same (maximal) errors do turn up a Saros Cycle later or earlier. Regretably, Halley never published his Saros Cycle method while he was alive and it was little appreciated. (See, Halley and the Saros).

This was the reason why he spent his entire tenure as Astronomer Royal observing lunar transits and computing therefrom the error-values generated by the Newtonian theory (TMM), or rather his version of it. The graph shows a month of these eror- values, after Halley had been plodding along doing this for ten years. From the year 1732, this was around the time he gave the progress report to the Royal Society, above quoted. It can be seen that a sizeable baseline-drift has occurred, of about two arcminutes, in mean motion error; the error-values fluctuate to something resembling the lunar month; and they are generally within several arcminutes of that dispaced-zero position. For comparison, two further continuous lines have been added: the errors that would have been generated by using TMM over this period (red line), and the larger errors that would be generated by using Halley's tinkered-with version thereof (blue line).

A second graph shows just the same things plotted for one year later. It must have been a shock, as Halley found himself recording up to eight arcminutes of error, from Newton's lunar theory, repeating at 30-day intervals. About 2.5 arcminutes of this was due to cumulative drift in the mean-motion values, i.e., a systematic error. Still, the theory was coming up with errors far larger than he had indicated two years earlier to be possible. Alas we lack any comment from Halley on this matter. One can see how he was regularly taking the meridian-transit observations on the downward slope, as the theory-error was increasing, then left off each month after it had peaked - to get some sleep!

Halley was the first astronomer to be overtaken by the pace of progress: by the time his Tables were published (finally in 1749), they were obsolete and the era of the Newtonian Lunar Theory had ended. The new theories of Mayer and Euler arriving from the Continent, truly based on Newton's theory of gravitation, superceded it.

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