1 The Birth of Science - MSRI

316 11. The Age-Long Recovery
teach us that God is also the author of all our actions, in so far as
they exist and in so far as they have some goodness, but that it is the
various dispositions of our wills that can render them evil. 96
Descartes, like Kepler, must have read the De facie and studied carefully
the passage we quoted on page 245. But where Plutarch talks about bodies
“turned aside by something else”, Descartes finds it natural to link the
something else, the cause of deviations from straightness, with the powers
of evil, the cause of misdeeds. His method strikes today’s reader as much
more “scientific” when, writing about geometry, his source is not Plutarch,
but the tradition of Euclid and Pappus. 97
Historians of mathematics are quick to point out how much Cartesian
mathematics differs from its Hellenistic counterpart. C. Boyer and others
place the novelty of Descartes’s method in that he abandoned the homogeneity
principle, 98 which prevented “Greek mathematics” from dealing
with expressions such as x 2 + x or x 2 + x 3 :
In one essential respect, he [Descartes] broke from Greek tradition,
for instead of considering x 2 and x 3 , for example, as an area and
a volume, he interpreted them also as lines. This permitted him to
abandon the principle of homogeneity, at least explicitly, and yet retain
geometrical meaning. 99
But abandoning the homogeneity principle was not an unprecedented
step. Neugebauer writes:
[T]hat Heron adds areas and line segments can no longer be viewed
as a novel sign of the rapid degeneration of the so-called Greek spirit,
but simply reflects the algebraic or arithmetic tradition of Mesopotamia.
100
In truth Heron’s method differs somewhat from Descartes; Heron, to add
x 2 and x, represents the summands as a square and a segment, using these
as graphical tokens for algebraic quantities, the fundamental entities in the
Mesopotamian tradition. By contrast, Descartes, to whom the fundamental
entities remain geometric, can only add after making both summands
into segments. It seems to me that our current procedure is in substance
closer to Heron’s than to Descartes’. In any case it is clear that, clichés page 390
aside, judgements about what mathematical procedures are “modern” are
both subjective and liable to fluctuate.
96 Le monde de M. Descartes, ou Le traité de la lumière, chapter 8, at 80% (Gaukroger translation).
97 The main source of Descartes’ Geometry is Pappus’ Collectio.
98 See note 47 on page 39.
99 [Boyer], p. 371 (1st ed.), pp. 337–338 (2nd ed.).
100 [Neugebauer: ESA], p. 146.
Revision: 1.11 Date: 2003/01/06 07:48:20