Department
of
Science & Technology Studies
University College London
Nicholas Kollerstrom's
Newton's 1702 Lunar Theory
Why Seven Steps?
'...and the Moon's Place
will be equated a seventh time, and this is her Place in her proper Orbit’.
Newton
(1702), "Theory of the Moon's Motion"
The number seven meant a lot to Newton. He sensed its mythic
resonance, for example, is his seven colours of the rainbow  where he
added an extra colour, indigo, that noone could see in order to make up
a 'complete spectrum' of seven (Cohen, 1980. The Newtonian Revolution,
p. 204). Likewise his Optics (1704) found seven steps of colouration
in the 'Newton's rings', and was itself composed in seven sections (both
these observations were made by D.Castillejo (1986. The Expanding Force
in Newton's Cosmos, p. 97). His more esoteric writings concerning the
decoding of the apocalypse also show much sevenfold symbolism.
The sevenfold structure became a distinctive hallmark
of the various 'Newtonian' ephemerides that used TMM. To quote Craig Waff,
'...nearly all new lunar
tables constructed during the first half of the eighteenth century utilised
in some fashion his [Newton's] tabular theory.' (quoted in Cohen, 1975,
p.79.)
The theory had a symmetry about the Equation
of Centre (the ellipse function) as the central, 4th equation, larger
than any other step.
It would appear that Newton had found just about all of
the lunar equations available to be found, at the level of accuracy available
to him (see, Comparison with Modern Equations). For comparison, Mayer's
lunar theory of 1754 was said to have 13 equationsteps.
Expressing the Newtonian sequence
of lunar equations in their simplest possible form, without any amplitudemodulation
or steps of equation, and with amplitudes expressed in arcminutes:

1st: Annual Equation
12sin (HS)

2nd: Eqn
4sin2(AS)

3rd: Eqn
1sin2(NS)

4th: Eqn Centre

5th: Variation
35sin2(MS)

6th: Eqn
2sin (SM+HA)

7th: Eqn
2sin (SM)
Thus, the second equation goes through two cycles per Sunapse
cycle, as the Variation has two cycles per lunar month. I found that the
four new equations all worked, in that TMM worked better with each of them
than without, as was not implied by earlier commentators (Baily, 1835,
Whiteside,1976, Wilson 1989).When all this arrived in the Principia,
of 1713 (Scholium to Propn. 39, Book III), it was formulated in seven paragraphs.
The contents of this page remain
the copyrighted, intellectual property of Nicholas Kollerstrom. Details.
rev: May 1998