*Department
of*

Science & Technology Studies

*University College London*
Nicholas Kollerstrom's

Newton's 1702 Lunar Theory

####
Gravity Theory: The Tenuous Link

Newton's original manuscript for TMM is dated February 1700,
which is the month Newton was confirmed as Master of the Mint in London.
It was published in David Gregory's (1702) *Astronomia Physicae*.
In the 18th century, it was reprinted four times in Latin and 13 times
in English (Cohen, 1975, p.8). Enigmatically, the
manuscript contains no allusion to a theory of gravity. In 1975, Bernard
Cohen posed the challenge:
'It would be most useful
to have a careful analysis of Newton's attempts to produce a satisfactory
lunar theory (in the 1690's), and the stages whereby he either partially
or totally abandoned the program of deriving such a theory by mathematical
methods applied to gravitational celestial mechanics' (p.80).

Cohen offered no comment upon either the accuracy of the
theory - whether it was an improvement upon those available - or, to what
extent if any it was based upon a theory of gravitation. Craig Waff commented
in his review of Cohen's book upon the historical irony, that the brief
1702 essay "Theory of the Moon's Motion" was 'probably the most obscure
of Newton's publications', and yet it 'appeared in print during the early
eighteenth century more times than anything else which left the hand of
Newton.' Waff made a claim which I have confirmed:
'Newton's "rules" have been
wholly or partially used by nearly a dozen astronomers or other interested
individuals in order to construct lunar tables.'

Of TMM's composition, Bernard Cohen wrote:
'In book three of the *Principia*
Newton developed at some length the concept of the three-body problem in
the special case of sun, earth, and moon. He then allowed a revised statement
of his solution of the problem of lunar motion to be published in 1702,
as a part of David Gregory's textbook on astronomy.' (1980, p.276)

I rather doubt whether resolution of a three-body problem
appeared in that 1702 thesis. It was only years after abandoning such an
endeavour that Newton produced instead a recipe for finding lunar celestial
longitude. His calculations on lunar theory (at Cambridge University Library)
are in Latin for the *Principia *calculations, while those for TMM
tended to be in English - his reasoning on the more practical side of the
problem took place in English.
I agreed with Dr Waff, that these notes showed:

'a total absence of any
evidence of theoretical deduction of these new equations...there are no
references to or uses of the theory of gravitation found amongst them (1977,
p.70)

The *Principia *of 1687 did well account for the lunar
inequality known as the Variation, by modelling
it on a yearly-revolving elliptical orbit with Earth at its centre. The
French astronomer D'Alembert remarked that this derivation of the Variation
from gravity theory was done 'avec beaucoup de clart' et precision' - whereas,
in contrast, he doubted whether Newton's derivation of the 'annual equation'
was sound:
'... il en est quelques-unes
que M.Newton dit avoir calcul'es par la Th'orie de le gravitation, mais
sans nous apprendre le chemin qu'il a pris pour y parvenir Telles sont
celles de 11°49' qui d'pend de l'equation du centre du soleil... (1754,
I, pp. xiii,xiv) ['There are several things Monsieur Newton claims to have
calculated by the theory of gravity, but without showing us the way he
took to come by them: for example the 11°49' for the solar equation
of centre'.]

Bernard Cohen commented on these three-body problems: 'He
[Newton] was really successful only in accounting for the variation and
the nodal motion.' (1980, p.276)
A problem with these three-body problems is that scholars
expounding them never mention the two or thee-hundred percent error in
lunar mass that was embedded in these weighty computations (See Lunar
Mass Error page). For a computation in celestial mechanics, this should
be at least be mentioned. In 1713 the bulk of TMM was incorporated into
a scholium of the *Principia*. It was still a recipe for finding longitude,
but it was also averred that each of its steps had been derived from gravity
theory: a dual-track policy, so to speak.

In the Third Edition of 1726, the former goal was abandoned,
by dropping the crucial paragraph stating that lunar longitude could thereby
be found, so that the text became solely a tract about gravity theory.
As the only version of the *Principia* translated into English, it
gave no clue that one of its scholia contained the remains of a recipe
for finding longitude. (See Return of the Epicycles,
and The Forgetting of TMM)

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