Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

The Equation of Centre

The astronomers of Restoration England were just beginning to apply Kepler's first two laws in the preparation of planetary and  solar/lunar tables. The problem was that there was no easy way of finding the Equation of Centre. Figure 1: Equation of CentreFigure 1 shows the angle called 'mean anomaly' (M), as between the mean moon (moving with uniform angular velocity around the Earth, at one focus of the ellipse), and the mean apogee. The 'true' Moon was found by applying an 'equation', here represented by the angle theta, so its anomaly value is given by {M - theta}. The Moon is here moving from apogee to perigee, when the true moon lags behind the mean moon. Equation of Centre tables would give values for theta, depending on eccentricity and mean anomaly.

Other methods of finding the Equation of Centre, which did not assume an elliptical shape to the orbit,  were much simpler to solve (so-called equant methods) and were still being used. In the 1670s, Newton was hardly familiar with the idea of using the Kepler second law in an astronomical context (though Bernard Cohen and Curtis Wilson have disagreed on this matter). Tables would give the Equation of Centre for the lunar orbit, as the angle between the mean Moon and its 'equated' ie more correct position on an elliptical orbit, where its magnitude depended upon the eccentricity E and the Moon-apse angle known as the 'anomaly' (A-M, where A and M are the apse line and lunar longitudes).

The first three terms of the modern Equation of Centre are:

(2E-E3/4)sin(A-M) - 5/4E2sin2(A-M) + 13/12E3sin3(A-M) ...

The first two terms (ie, using only E and E2 values) are generally adequate for the computations performed here.

Flamsteed's 'Equation of Centre' tables in his 1681 Doctrine of the Sphere agreed with the values given by this equation, to within six arcseconds, indicating that Kepler's second law was being applied before the Principia was published (Thoren, 1974, Gingerich and Welther, 1974). Earlier, the tables published in 1673 in Horrock's Opera Omnia had only slightly less accurate Equation of Centre tables, with a maximal error (I found) of 13 arcseconds. We are not told who prepared these tables, but it may have been Flamsteed. Finding the the Equation of Centre, Principia Bk.I, Prop.31.This early British use of Kepler's second law has received scant recognition. It is evident that Flamsteed was using a solution of 'Kepler's equation', to give effectively exact Equation of centre values. He may have been the first to do this. The reconstruction of TMM  (TMM Replica) used the modern Equation of Centre, as Newton did not specify any procedure for obtaining it.

This diagram from Newton's Principia (Propn 31, Book 1) show his geometrical approach to finding the equation of centre, where S is the focus of the ellipse and and AB is its apse line.


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