Science & Technology Studies

Nicholas Kollerstrom's

Newton's 1702 Lunar Theory

The
*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:*

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).

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the copyrighted, intellectual property of Nicholas Kollerstrom. Details.
*rev: May 1998*