The Seventeenth Century. There was one thing that
the mathematics of the Renaissance era did not
achieve. It did not continue creatively the mathematics
of Euclid, Archimedes, and Apollonius, even if it did
begin to translate and study their works. The exploita-
tion in depth of Greek mathematics was achieved only
in the seventeenth century, in the era of the Scientific
Revolution. It is this development that created the
infinitesimal calculus, and thus molded the world image
which we have inherited today. For instance, Johannes
Kepler anticipated the infinitesimal calculus in two
ways; by his approach to Archimedean calculations of
volumes, and also by his manner of reasoning which
led him to conclude that planetary orbits are not circles
but ellipses; and without his intimate knowledge of
Apollonius this conclusion would have hardly come
about. Next, René Descartes' avowed aim in his La
Géométrie was to solve or re-solve geometrical prob-
lems of the great Greek commentator Pappus (third
century A.D.) who was groping for topics and attitudes
in geometry that were beyond his pale. Now, Des-
cartes' technical equipment was the operational appa-
ratus of Viète, and it was this fusion of Viète with
Pappus that created our coordinate and algebraic ge-
ometry. Next, Isaac Barrow, the teacher of Isaac
Newton, perceived the importance of Euclid's 5th book
(theory of proportion in lieu of our theory of positive
real numbers) for the eventual unfolding of analysis.
And, finally, Newton himself, by an extraordinary
tour-de-force, pressed his forward-directed
Principia
(
Principles of Natural Philosophy) into the obstructive
mold of backward-directed but powerful Archimedism.
That this impressment was even harder on Newton's
readers than on Newton is attested to by a memorable
statement (in the nineteenth century) of William
Whewell:
The ponderous instrument of synthesis [meaning Archimed-
ism], so effective in his hands, has never since been grasped
by one who could use it for such purposes; and we gaze
at it with admiring curiosity, as on some gigantic implement
of war, which stands idle among the memorials of ancient
days, and makes us wonder what manner of man he was
who could wield as a weapon what we can hardly lift as
a burden
(Whewell, I, 408).
To sum it up, it was not the Renaissance proper of
the fifteenth and sixteenth centuries that saw the
“rebirth” of antiquity as much as it was the succeeding
Scientific Revolution of the seventeenth century. What
the Renaissance did was to contribute a prerequisite
algebraic symbolism, and this was something that, from
our retrospect, antiquity might have additionally pro-
duced out of itself but did not.
