Understanding the‘ScientificRevolution’ 151
1.5
1
0.5
0
–0.5^0
–1–1 –0.5 0.5 1–1.50
0Fig. 2Graph of a cubic curve.Europe, even those of Stevin, had been formally set out, either on the Greek model (sequence of
propositions and proofs), or on the model of the abbacus schools, which was also to some extent
that of Diophantus, and of the Chinese (sequence of problems and solutions). It could be said that
the same structure underlies theGéométrie(e.g. the extract I have given asks a question and solves
it); but the whole is absorbed into a smooth narrative which appears to lead on without a break
from one ‘discovery’ to the next, pausing for comments, explanations, or excuses for avoiding them:
But I shall not stop to explain this in more detail, because I should deprive you of the pleasure of mastering it yourself,
as well as the advantage of training your mind by working over it, which is in my opinion the principal benefit to be
derived from this science. (Descartes 1954, p. 10)
The nearest approach to this scheme is Kepler’sAstronomia Nova(see below), which purports to
be an account of his struggles to discover the laws of motion of the planets. The latter, however, is a
story of discovery, whileLa Géométrieis an account of how the reader should proceed. The novelty
lies in the ‘you’ of the sentence quoted above: the reader can be addressed, not in the imperative
(‘Find...’, ‘Draw...’) as a teacher addresses a student, but as an intelligent equal.
The fact that his contemporaries found theGéométriedifficult may help us, all the same, to guard
against an ‘unhistoricist’ approach to his work. We read it from a perspective in which, on the
whole, the translation from curves to equations and back is a familiar one. Descartes was making a
major innovation, and clearly he did not explain it as well, to a seventeenth-century audience, as he
hoped; its absorption took time, although only twenty years later young Isaac Newton was already
(by some reports) finding Descartes more congenial than Euclid.
9. Infinities
Nature is an infinite sphere in which the centre is everywhere, the circumference is nowhere. (Pascal 1966, p. 89
(no. 199))
But let us remember that we are dealing with infinities and indivisibles, both of which transcend our finite
understanding, the former on account of their magnitude, the latter because of their smallness. (Galileo 1954, p. 26)
Around 1600, more or less independently of the work in algebra, we see the first systematic use
of ‘the infinite’ in European mathematics; by mid-century it was becoming frequent, and Pascal, a
mathematical mystic, used it in a number of metaphorical statements (such as his famous ‘wager’),
as well as in an early version of the calculus. The impetus seems mainly to come from physical
applications, and from a recognition that infinities in some sense underlie Archimedes’ work,
although it may be necessary to be more careless than he was in what one allows. And indeed,