1 The Birth of Science - MSRI

11.9 Ancient Science and Modern Science 341
the relative standing that algebra and geometry had held since Hellenistic
times. The bridge between geometric and algebraic problems was the
assignment of coordinates to points. This, too, was not a radically new
idea: Apollonius had already used what came to be called Cartesian coordinates.
The novelty was that the systematic use of the algebraic form
reduced the drawing, now a mere “sketch” of the curve under study, to a
subsidiary role, and primacy went to the curve’s equation, from which the
desired results could be obtained through numerical computation. This
allowed the study of a much broader mathematical phenomenology.
Shall we then conclude that the superiority of modern mathematics is
based on a new idea, logarithms? Certainly not. That writing numbers as
powers of the same base allows the reduction of time-consuming operations
to easier operations on exponents is lucidly explained (en passant,
as it were) in Archimedes’ Arenarius. 160a Nor was the practice of compiling
numerical tables new, since Hellenistic astronomers made use of trigonometric
tables. But in the seventeenth century we start seeing the compilation
of numerical tables to a hitherto unmatched degree of precision
and extension. Only the new, detailed tables of logarithms could make
the ancient geometric calculation methods obsolete; but their preparation
requires tremendous labor, hardly to be undertaken unless the expected
use of the product exceeds a certain threshold. Moreover if it were not for page 418
printing it would not be possible to keep the tables reliable, and printing in
turn only makes sense where the reading public is sufficiently wide. Thus
the essential novelty is not to be found in new ideas, but in the achievement
of a critical mass of interested individuals.
I believe that this discussion of methods of calculation exemplifies a
much more general pattern: the factors that made modern science take
off do not rest on radically new ideas, but rather on there being again, in
early modern Europe, an opportunity for remnants of ancient culture to
interact and develop, with the advantage of extension to a much wider
social base. When there started to be mutual interaction between scientific
and industrial development, the existence of wide markets became even
more important.
160a Archimedes, Arenarius, 147, 27 – 148, 26 (ed. Mugler, vol. II). To get an efficient numerical
method from this it is of course necessary to take a geometric progression not of natural numbers
like the one considered in the Arenarius, but of noninteger magnitudes whose ratio is close
to unity (thus Napier’s table involved a geometric progression of ratio 0.9999999, while a table
of decimal logs to, say, three decimal places involves a progression whose ratio is the thousandth
root of 10). It is to be supposed that this step was within the reach of Hellenistic mathematicians,
given the parallel development of the theory of proportions for integers and for magnitudes in the
Elements and elsewhere. If it was not taken the reason is presumably to be sought in a relative lack
of interest in numerical methods.
Revision: 1.11 Date: 2003/01/06 07:48:20