1 The Birth of Science - MSRI

11.4 A Late Disciple of Archimedes 307
At the end of the third day of the Dialogues — the day that contains
Galileo’s greatest contribution to the science of dynamics — there is a passage
of great significance, a profession of what the author, now old, thinks
was his greatest scientific achievement. Galileo congratulates himself by
putting in Sagredo’s mouth the following words:
I do think we can grant our Academician [Galileo] that it is no idle
boast when he says, at the beginning of this treatise of his, that he
has founded a new science dealing with a very old subject. And seeing
with what ease and clarity he deduces from one very simple
principle the demonstrations of so many propositions, I marvel at
how such a matter could have been left untouched by Archimedes,
Apollonius, Euclid and so many other famous mathematicians and
philosophers; the more so because plenty of thick books have been
written about motion. 65
Thus Galileo’s highly ambitious scientific aim was the recovery, after
so many centuries of oblivion, of the the Hellenistic scientific method,
consisting in the creation of hypothetico-deductive systems where natural
phenomena can fit. With his well-honed critical mind, Galileo unequivocally
names his own models — the great scientists of the golden period —
without letting his obvious admiration blur into reverence for an undifferentiated
“Antiquity”. Indeed, he does not hesitate to take issue with both
Aristotle and Ptolemy.
Galileo did succeed in breathing new life into two legacies from his distant
masters, the experimental method and the deductive method. Yet he
still lacked a command of the more refined Hellenistic mathematical tools.
While he could use Euclidean proof techniques and geometric algebra,
he never did grasp the so-called “method of exhaustion” and the theory page 379
of proportions (and indeed nobody would for another two centuries and
more).
The crux of Euclid’s definition of proportions is that it is equivalent
to a construction of the notion of the ratio between magnitudes; 66 thus
it is altogether foreign to a Platonist understanding of mathematics and
definitions. If ratios between magnitudes are conceived of as preexisting,
equality between them cannot but seem a self-evident notion, and Euclid
is guilty of introducing an abstruse and superfluous complication — one
which Galileo felt able to dispense with easily:
65
Galileo Galilei, Discorsi e dimostrazioni. . . , end of third day = [Galileo: Opere], vol. VIII, pp. 266–
267.
66
See page 40, where the definition is given, and page 157.
Revision: 1.11 Date: 2003/01/06 07:48:20