1 The Birth of Science - MSRI

342 11. The Age-Long Recovery
But this left modern science with a serious weak spot. Since its results
originated in the acquisition of external elements, created by a different
civilization and not completely understood, it is not surprising that the
science of Descartes, Kepler and Newton, despite its potential superiority
(due to its applicability to a wider range of phenomena) was poorer than
ancient science in its methodology. In the works of early modern science,
individual pieces of content either recovered from ancient science or derived
from such were plunged in a foreign overarching framework based
on theology and natural philosophy. The crystal sphere of fixed stars —
which, as we recall, was first introduced to explain the rigid nightly motion
of the heavens, then abandoned in the time of Heraclides of Pontus
when the hypothesis of the earth’s rotation was made, and then taken up
again with the end of ancient science — did not disappear with the rise
of heliocentrism: it still surrounded Kepler’s universe. Likewise Newton
tried to frame his “new” science in Aristotelian categories, in particular
preserving a concept of absolute space that is virtually incompatible with
the principle of inertia.
As we remarked on page 324, the eventual evolution of modern science
into true scientific theories was ensured by the fact that its structure was
circumscribed by technical elements that followed closely the surviving
Hellenistic treatises, from which authors continued to draw. Nonetheless,
the level of mathematical rigor remained for a long period far below what page 419
it was in Hellenistic times. Here is how Newton discusses the limit of the
ratio between two infinitesimals (what he calls the “ultimate proportion”
of “evanescent quantities”):
Therefore if hereafter I should happen to consider quantities as made
up of particles . . . I would not be understood to mean indivisibles,
but evanescent divisible quantities[.]
Perhaps it may be objected, that there is no ultimate proportion of
evanescent quantities; because the proportion, before the quantities
have vanished, is not the ultimate, and when they are vanished, is
none. . . . But the answer is easy; . . . by the ultimate ratio of evanescent
quantities is to be understood the ratio of the quantities not before
they vanish, nor afterwards, but with which they vanish. 161
Newton conceives “evanescent” quantities as real objects, which vary
in real time: their ultimate ratio (in our language, the limit of their ratio)
is thus the value the ratio takes at the moment in which the two values
vanish. It is clear that Newton has no awareness of using a mathematical
161 Newton, Philosophiae Naturalis Principia mathematica, Book I, Section I, after scholium to lemma
XI, Motte/Cajori translation.
Revision: 1.11 Date: 2003/01/06 07:48:20