296 g e o f f r e y L l o y d
it is clear that ideas about what counts as a good, or even a proper, method
of doing so diff ered. 6 A t Physics 185a16–17 Aristotle distinguishes between
fallacious quadratures that are the business of the geometer to refute, and
those where that is not the case. In the former category comes a quadrature
‘by way of segments’ which the commentators interpret as lunules and
forthwith associate with the most famous investigator of lunules, whom
I have already mentioned, namely Hippocrates of Chios. Yet even though
there is another text in Aristotle that accuses Hippocrates of some mistake
in quadratures ( On Sophistical Refutations 171b14–16), it may be doubted
whether Hippocrates committed any fallacy in this area. 7 In the detailed
account that Simplicius gives us of his successful quadrature of four specifi c
types of lunules, the reasoning is throughout impeccable. Quite what fallacy
Aristotle detected then remains somewhat of a mystery.
But two other attempts are also referred to by Aristotle and dismissed
either as ‘sophistic’ or as not the job of the geometer to disprove. Bryson
is named at On Sophistical Refutations 171b16–18 as having produced an
argument that falls in the former category: according to the commentators,
it appealed to a principle about what could be counted as equals that was
quite general, and thus far it would fi t Aristotle’s criticism that the reasoning
was not proper to the subject-matter.
Antiphon’s quadrature by contrast is said not to be for the geometer to
refute ( Physics 185a16–17) on the grounds that it breached the geometri-
cal principle of infi nite divisibility. It appears that Antiphon proceeded by
inscribing increasingly many-sided regular polygons in a circle until – so
he claimed – the polygon coincided with the circle (which had then been
squared). Th e particular interest of this procedure lies in its obvious simi-
larity to the so-called but misnamed method of exhaustion introduced by
Eudoxus in the fourth century. Th is too uses inscribed polygons and claims
that the diff erence between the polygon and the circle can be made as small
as one likes. It precisely does not exhaust the circle. If Antiphon did indeed
claim that aft er a fi nite number of steps the polygon coincided with the
circle, then that indeed breached the continuum assumption. But of course
later mathematicians were to claim that the circle could nevertheless be
treated as identical with the infi nitely-sided inscribed rectilinear fi gure.
Other solutions were proposed by other fi gures, by a certain Hippias for
instance and by Dinostratus. Whether the Hippias in question is the famous
sophist of that name has been doubted, precisely on the grounds that the(^6) Mueller 1982 gives a measured account.
(^7) Lloyd 2006a reviews the question.