TheCalculus 177
are complicated. As has been pointed out for example, by Hall, the work would have been doubly
unfamiliar to its readers if it had used both a new physical and a new mathematical language:To the few skilled mathematicians in Europe around 1685 who were capable of understanding Newton’s mathem-
atical arguments at all, however expressed, the form he actually adopted inPrincipiawas far more convenient and
familiar than either the method of fluxions—known to no one but Newton—or the Leibnizian differential calculus...
(Hall 1980, p. 30)And there was already enough that they could and did disbelieve in the physics—notably the idea
of a force of gravitation acting at a distance through a vacuum, which seemed pure mystification
in contrast to Descartes’s ideas. ThePrincipiais intended to follow the model of Archimedes’
physical texts—theStaticsandOn Floating Bodies, although it is far longer; with a set of first
principles and rigorous deductions from them. Furthermore, Newton, who had as we have seen
been an enthusiastic ‘modernist’ and follower of Descartes in the 1660s, had changed his position
radically, for reasons which were mostly, it seems, concerned with his interests outside physics and
mathematics. He now (whatever his private practice) took every occasion to attack the modern
algebraic school of geometry to which he had once belonged, and scribbled ‘Not Geometry’ in the
margin of his copy of Descartes. During the 1670s, he had immersed himself in his studies on
alchemy and on the meaning and chronology of the Old Testament, on which his views were very
unorthodox. Rather like Stevin 100 years earlier, he had come to believe in a golden age of ‘first
knowledge’, which the Greeks had corrupted; that, for example, the rotation of the earth round
the sun was known to the Egyptians and to Pythagoras. He planned additions to thePrincipia
which would have explained this ancient learning (he had worked hard on reconstructing lost
Greek texts); had they been published, the work would have been seen as definitely eccentric, and
would probably have attracted much less admiration. For example, he drafted an explanation of
the ‘occult’ force of gravitation which would have convinced none of the sceptics:Thus far I have explained the properties of gravity. But by no means do I consider its cause. However I will say in
what sense the Ancients theorized about it. Thales held that all bodies were animate, inferring this from magnetic and
electrical attractions...He taught that everything was full of Gods, and by Gods he meant animate bodies.^8We need to consider one example to clarify how thePrincipiamight have appeared—and indeed
how it appears to us, when we can read it. I have reproduced as Appendix C the crucial deduction
of Kepler’s area law, which is often cited as an example. The statement is Newton’s version of the
area law; as he had found, the law (equal areas are swept out in equal times) follows simply from
the supposition that the body—always supposed to be a ‘particle’, concentrated at a point—moves
under a force which is directed to the immovable centre S from which the areas are calculated.
The proof is in two parts. The first part is good Greek geometry, if physically untrue. We think
of the body as being moved in a succession of jerks (‘by a great impulse’) at equal time intervals,
rather than moving smoothly, the impulses being directed to the centre S. We find (as in Fig. 5)
that it moves in a polygon, and that the successive triangles have equal areas. We now suppose the
number of triangles increased‘in infinitum’, which is no longer Greek geometry at all, but the now
familiar argument that an infinite-sided polygon is a curve. Newton asserts that the force now acts
continually, and areas remain proportional to times.
- Gregory MS fo. 13, cited Iliffe (1995, p. 172). Leibniz’s ideas on the nature of atoms or ‘monads’ and the souls which animated
them were equally strange; but they were not linked to an ideology which tried to validate ideas by their antiquity.