Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

From Horrocks to Halley:
Recomputing their Work

This page contains a how-to-use-it set of instruction for moving from their apparent time to our GMT (ephemeris time), with a program for the modern equations kindly provided by Jean Meeus specially for the "Newton and the Moon" Web site.

This hands-on section shows how to reconstruct past solar and lunar positions, as degrees of celestial longitude, measured from the 'vernal point' as zero. This can involve three different types of time transform, moving between:

* LAT, local apparent time;
* LMT, local mean time;
* GMT or Greenwich mean time ('Universal time'); and
* DT or TDT (Dynamical time), formerly called Ephemeris time.

The good news is that most of the worked examples we deal with in the early eighteenth-century are given in mean time, often GMT, and the delta-t adjustment to convert to ephemeris time is negligibly small for any times after, say, the mid-seventeenth century.

The last of these is the one which flows evenly (NB, do not attempt to understand this) while GMT or 'universal time' differs in past ages owing to the gradual deceleration of the Earth's rotation, due to tidal friction. The modern 'Dynamical time', formerly Ephemeris Time, is what one feeds into the astronomical computer programs. The first three kinds of time are solar-defined, by the solar days and solar years.

There is a difference, which we may call  delta-t, that bridges the last two of these times. The definitions were set up to make this of zero or only a few seconds magnitude after 1700, which means that one can forget it for Newton's lunar theory, but not for those of Flamsteed or Horrocks. It starts increasing sharply pre-1650: about one minute in the 1640s, two minutes around Galileo's time in 1600, then over two hours for Ptolemy's era. (The RGO's Explanatory Supplement to the Astronomical Almanac gives this function as far back as 1620.) The conversion to GMT is:

Ephemeris Time (or, TDT) = GMT +  delta-t

To convert from the local 'apparent time' (sundial time, or 'God's time') -- LAT to LMT -- one requires the Equation of Time, what Flamsteed called the 'Aequation of the Naturall Days'). This function has a maximum value of seventeen minutes and becomes zero four times a year. A program of this is given here (kindly supplied by Bernard Yallop, of the RGO's Nautical Almanac Office: see Hughes, Yallop & Hohenkirk, 1989), that reliably goes back into past centuries.

A lunar and solar-longitude program is also available. One sets it to Julian ('Old Style') or Gregorian ('New Style') time, bearing in mind that Julian time was used by most of Europe in the seventeenth century, though not adopted by Britain until 1752. There is a difference of ten days between these in the seventeenth century and 11 days in the eighteenth. (It may help to remember this by recalling the fact that when the conversion was made in September of 1752 there were riots in the streets of London with the slogan 'give us back our eleven days'.) The equation here is,

New Style date = Old Style date + 11 days

So, to reconstruct ancient solar & lunar longitude values for a given time:

  1. convert the local apparent time (LAT) given to local mean time (LMT), by adding the Equation of Time.
  2. convert LMT into GMT according to its latitude (four minutes of time per degree of longitude).
  3. convert GMT into Ephemeris time, using the delta-t function. Use this time value for the modern longitude program.
The 1989 paper by Yallop et. al. gave a reliable Equation of Time, and the recent papers by Lesley Morrison et. al. establish the delta-t value. My thesis reconstructed solar-lunar mean motions, though this WebSite is not concerned with such, and Jean Meeus advised me that one should use the formulae given in his 1991 opus -- as it included the latest estimates of Earth-rotation rates and not use his earlier (1985) values as were based on an earlier theory. Thus, it is only recently that the reliable reconstruction of the past has become feasible.

Such reconstruction of the historical past was pioneered by Owen Gingerich and Barbara Welther, and their graphs show error-patterns of the Nautical Almanac over 1779 - 1787 (1983, p.xi, reproduced in Wilson, GHA, 1989 p.187-9). But, these show errors of between one and two minutes of arc for lunar longitude, as would have been of little use for finding the longitude, whereas I found that the Nautical Almanac of 1765 had an error of less than one-third this amount. The low accuracy of this reconstruction, by Gingerich and Welther, could just have been due to lack of a decent Equation of Time.
Below are three additional worked examples for the lunar theories:

  1. one from Horrock's 1673 Opera Omnia
  2. one from Lemonnier's 1746 version of TMM
  3. one from Halley's version of TMM
These supplement the worked examples by Flamsteed and Dunthorne.


Horrock's 1673 Opera Omnia

This is in the Old Style calendar. The date is 13 February, 1672 at 11 hours and 35 minutes, London time (nb. this means 23 hours as the time from noon).

This calculation begins with mean motion (Mot. med.), subtracts an annual equation (Argumentum annuum), then the equation of centre (use Flamsteed program to locate this), then the variation (Var. add), and then the reduction (Red. sub.). The answer is given as Lun. in E. (i.e., the Moon's longitude in the ecliptic).

For more on Horrocks, click here.

from Horrock's Opera Omnia published by John Wallace in 1673


LeMonnier's 1746 version of Newton's TMM

Two time adjustments are required here. Nine miuntes from GMT correction for Paris time. Also, this is in New Style. The date is 4 August 1739, at 5 hours (i.e. 5 pm) and 55 minutes.

The seven steps of Newton's lunar theory are clearly discernable.

For more on LeMonnier, click here.

Pierre Lemmonier 1746, his method (supposedly) based on Flamsteed's work

Halley's version of TMM

The date is 5 December 1725 at 9 pm GMT. Note the seventh equation has been omitted, and Newton's sixth equation has become the fourth, i.e. it is added on before the equation of centre.

Halley slightly adjusted Newton's mean motions. He also slightly simplifed the second equation.

Also note this is close to a lunar eclipse, as the Sun and Moon are in opposition.

For more on Halley, click here.

Edmond Halley

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