*Department
of*

Science & Technology Studies

*University College London*
Nicholas Kollerstrom's

Newton's 1702 Lunar Theory

####
A Worked Example

The Cambridge amateur astronomer, Richard Dunthorne, published
in 1739 a book of tables modelled on Newton's 1702 lunar theory, to see
how well it worked. His worked example is here shown. Its seven steps exactly
express Newton's theory. The page below shows everything except how to
find the solar position. This is merely introduced halfway down, being
subtracted from the apogee and node positions to generate their 'annual
arguments,' ie their angles with the Sun.
His worked example was computed for **3.40 pm GMT on
2nd January 1737**. Open the TMM Replica,
and set it for that time and date. It should show an exact agreement with
values given in the historical example, within arcseconds. The Dunthorne
example starts with three 'mean motions' computed, for the Moon, its apogee
and its ascending node (those for the sun have already been found, and
are not shown here). The lunar 'mean motion', ie the longitude of the 'mean
moon' at a given time was given as 2 20 7 8, as signs
(30 degrees), degrees, minutes and seconds. Two 'signs' meant it was in
twenty degrees of Gemini (Aries, Taurus, Gemini), and thus 80 degrees of
celestial longitude..

Its 'Annual Argument' was the Sun-apse angle (strictly,
as can be seen from the example, this was between the first-equated
apse and the equated sun), and its 'mean Anomaly' was the Moon-apse angle,
again using 'equated' values. The seven 'steps of equation' appear clearly
in this worked example. It is evident that the '4th Equation' (ie,
the Equation of Centre) is much the largest, as over five degrees, being
subtracted from the 3rd -equated Moon. After the seventh 'equation' has
been subtracted, then finally the 'reduction' is subtracted to convert
to the ecliptic plane (as if all the foregoing computations had been in
the plane of the lunar orbit), to give the answer 'Moon in ecliptic'.

I used to use this worked example for checking my computer
reconstruction of Newton's theory. It agreed within about ten arcseconds
in all stages. The lunar longitude finally obtained was in error by nearly
seven arcminutes - three times the maximum error claimed by Gregory when
he was publishing the theory in 1702! Dunthorne commented in a 'To the
Reader' that 'the Newtonian numbers are a little deficient...' The
final 'Moon in ecliptic', ie the answer, appears as 14 degrees of Gemini,
ie six degrees have been subtracted from the 'mean' position by the steps
of equation.

Compare this with a modern
program with the true Sun and Moon longitudes for that moment in time.
See if your work agrees with the 7 arcminutes error shown by the theory.

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