Department
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:
-
slightly altered the mean motions adding 10" to the mean
Moon, eg, and1'40" to the mean apse.
-
altered the sequence, so that TMM's 6th equation became its
4th;
-
omitted the small seventh equation. The 'Variation' inequality,
which Newton put as the fifth step, thereby became his last equation, the
sixth.
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:
'...the Faults of the Computus
formed therefrom rarely exceed a quarter Part of what is found in the best
Lunar Tables before that time extant…By this it was evident that Sir Isaac
had spared no Part of that Sagacity and Industry peculiar to himself, in
settling the Epoches, and other Elements of the Lunar Astronomy, the Result
many times, for whole Months together, rarely differing two Minutes of
Motion from the Observations themselves..' (Halley. 1731/2. Phil. Trans.
1731/2, Vol.37, p.191)
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
