154 A History ofMathematics
dialogue form, and full of artful reasoning and rhetoric. (Misleadingly, the Dover edition of the
second work translates ‘discorsi’ as ‘dialogues’, which gives the two works the same (short) title.
In the text here we refer to it as theDiscourses.) His ‘Euclidean’ bent may seem strange, given
the importance which he is often given as the inventor of ‘instantaneous velocity’, since that idea
at least would seem to need the infinitely small (or some equivalent) to define it—as the ratio
of infinitesimal distance to infinitesimal time, say. However, it relates to his extreme reverence
for Archimedes (again) as a true scientist in opposition to Aristotle. The infinitely small is per-
haps present as a subtext of his discussions of motion in the two texts. However, they tend to be
concealed by a vague description of velocity (which was in some sense Galileo’s favoured term)
as a ‘degree of swiftness’; we know what it is if it is uniform, as he often says, but that is not
really the point. The following exchange shows how, and with how little clarity, the infinite was
introduced:SAGR:^16 A great part of your difficulty consists in accepting this very rapid passage of the moving body through the
infinite gradations of slowness antecedent to the velocity acquired during the given time...
SALV: The moving body does pass through the said gradations, but without pausing in any one of them. So that even
if the passage requires but a single instant of time, still, since a very small time contains infinite instants, we shall not
lack a sufficiency of them to assign to each its own part of the infinite degrees of slowness, though the time be as short
as you please. (Galileo 1967, p. 22)It has been pointed out by those who favour Duhem’s thesis that uniformly accelerated motion
was already introduced in the fourteenth century (at Merton college Oxford), and that Galileo’s key
result (that if acceleration is uniform, the time taken to cover a distance is equal to the time which
would be taken moving constantly at the mean speed) was also known. The crucial contribution
which Galileo made was the observation—which was confirmed by his experiments—thatfree
fallwas uniformly accelerated. This, with the related deductions (e.g. that the path of a projectile
is a parabola) set him quite apart from the fourteenth century discussions of uniformly accelerated
motionin general, however much he may have drawn on them. For the Oxford men, speed was
a ‘quality’, whose intensity could vary, and the difficulties about instants of time which worried
Galileo’s characters seem not to have arisen. The fact that Galileo did return repeatedly to the
infinitely large and small, with varying degrees of sophistication, shows an increasing feeling that
a defence was needed.
Galileo better than most others (probably Kepler in particular) realized the pitfalls of reasoning
with the infinite. In the First Day of theDiscourses, he devotes a long digression to the subject. Can
one divide a continuum into an infinite number of pieces? Why can one use a limiting argument
to show that a circle and a point have the same ‘volume’, although the circle is clearly much
bigger? His classic example is the first instance of what, many years later, would be called a one-one
correspondence between infinite sets. I have given it as Appendix D. The conclusion tends to a
sensible caution: one cannot use the terms ‘equal’, ‘greater’, or ‘less’ for infinite quantities. A great
deal of such caution—which derived partly from Galileo’s respect for the Greek tradition—had to
disappear for further progress to be made.- Both works are presented as ‘dialogues’ between three parties: Simplicio, who is the mouthpiece of Aristotelian views, Salviati,
who represents (roughly) Galileo himself, and an intelligent arbiter Sagredo, who tends to raise objections, while seeing the force of
Galileo’s arguments.