Department of
Science & Technology Studies
University College London

Nicholas Kollerstrom's
Newton's 1702 Lunar Theory  

Quest for a Lunar Theory

The main contenders for finding longitude, at the dawn of the eighteenth century, were the Moon, the moons of Jupiter, or a chronometer. There were dire problems in detecting the second of these on board a tossing ship, though they had the advantage of exact adherence to Kepler's laws of planetary motion in their elliptical motions. Tables of their appearances from behind the orb of Jupiter were published. Marine chronometers came into use from the mid-eighteenth-century onwards, but remained prohibitively expensive for most vessels.

variant of Helvetius's Lunar diagramThe Principia had made the lunar sidereal period of 27.321 days the crux of its argument about gravity theory. Could that orbit serve as a universal clock, by comparison with which the longitude could be found?  The problem was that the inequalities in its orbit seemed to defy analysis. As Halley wrote in his ode prefacing the Principia:
 

At last we learn wherefore the silver moon
Once seemed to travel with unequal steps
As if she scorned to suit her pace to numbers
Till now made clear to no astronomer.

All of the proposed methods for finding longitude involved comparing local time with a version of Universal time: each hour difference between these two indicated fifteen degrees of longitude. The Earth revolves on its axis twenty-seven times faster than the Moon takes to orbit. This meant that there was a twenty-seven fold error-multiplication factor in the lunar method, so that a one arcminute error in lunar position gave a 27 arcminute error in terrestrial longitude. So, to find terrestrial longitude within one degree, lunar longitude had to be found within two arcminutes, ignoring other sources of error (from, eg, atmospheric refraction and parallax).

The 1714 Act of Longitude offered cash prizes for any method for finding longitude at sea to within one degree or less. England was the only nation that ever paid out such a cash prize, awarding part of it posthumously to Tobias Mayer, or rather to his family, and the rest to John Harrison, the watchmaker.

Local apparent time could be found on a ship, from times of sunrise and sunset, and a clock could easily keep that time during a day. Local mean time was then found by applying an 'Equation of time' - 'the Aequation of the Naturall Days' as Flamsteed called it, which had a maximum value of seventeen minutes. No such reliable Equation of time existed until 1673, when Flamsteed published a table of it in a postscript to Horrock's Opera, then in 1707 Whiston published an improved version from Flamsteed.
 

The quadrant was invented in 1731, using a pair of mirrors to gauge the Sun's height above the horizon. From around 1730 it began to look as if one of the methods was going to work. To quote from Dava Sobell,  'In longitude determination, a realm of endeavour where nothing had worked for centuries, suddenly two rival approaches of apparently equal merit ran neck and neck'. Perfection of the two methods blazed parallel trails of development down the decades from the 1730s to the 1760s. John Harrison, the watchmaker, paid a visit to Edmund Halley in 1730 for advice on the sea-going chronometer he planned to construct.

The British Nautical Almanac was published from 1767 onwards, giving lunar longitude positions at three-hourly intervals. Merchant vessels came to adopt the Greenwich meridian for their reference, as the positions given in the Almanac were for Greenwich time. Ephemerides of other nations reproduced these positions, as being the best available (Sadler, 1976). Positional data of the first page of the Nautical almanac had a mean error of 16"±17" I found, which is in accord with what was then believed about these tables, ie that they gave longitude within about 1° (Howse, Nov.1993 p.4).   

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