A History of Mathematics From Mesopotamia to Modernity

Understanding the‘ScientificRevolution’ 153

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Fig. 3Kepler’s diagram from ‘Astronomia Nova’.

This is a relatively early stage in the research; Kepler is still following a ‘wrong’ theory, and
finds a result which fails to agree with his data; and even before this he has to adjust his method
because his ideas on summing triangles do not work out. However, he has made a very bold
statement about Archimedes: that his proofs by contradiction are a way of concealing infinite
methods. This is not what the ancient Greek texts say (as far as we know), and it did not conform
to the orthodox view of them in Kepler’s time. (TheMethodin which Archimedes did use a sort of
infinite process was unknown at the time (see Chapter 3).) This ‘misreading’ of Archimedes was
useful to Kepler at the stage he had reached; and the idea that a circle could be thought of as a
polygon with an infinite number of sides had already been used by the mystic Nicholas of Cusa,
whose mathematical/theological thinking certainly influenced him. It recurs in his work on the
measurement of wine-barrels, which I reproduce in Appendix C—there the recourse to the infinite
is justified as simply being quicker.

10. Galileo

SAGR. But I, Simplicio, who have made the test can assure you that a cannon ball weighing one or two hundred
pounds, or even more, will not reach the ground by as much as a span ahead of a musket ball weighing only half a
pound, provided both are dropped from a height of 200 cubits. (Galileo 1954, p. 62)

And so, finally, we return to Galileo as innovator or revolutionary; he who (as in the passage
above) overthrew the authority of Aristotle by appeal to experiment. As we have already suggested,
Galileo, however committed he was to mathematics as the language of the universe, was math-
ematically on the conservative side. Despite learning from the artisans of the Venetian shipyards
(like Tartaglia) and writing his major works in the vernacular (like both Tartaglia and Descartes),
he continued to grind out propositions in the Euclidean style whose proof was by appeal to clas-
sical geometry. His two major works are valued as brilliant examples of Italian literary style; the
Dialogue on the Two Major World-Systemsand theDiscourses on Two New Sciencesare both cast in