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