THE TRACT ON INDIVISIBLE LINES
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became a definite doctrine. ' There is plenty of evidence for
this 1 ; Proclus, for instance, tells us of ‘ a discourse or argu
ment by Xenocrates introducing indivisible lines ’. 2 The tract
On indivisible lines was no doubt intended as a counterblast
to Xenocrates. It can hardly have been written by Aristotle
himself ; it contains, for instance, some expressions without
parallel in Aristotle. But it is certainly the work of some
one belonging to the school ; and we can imagine that, having
on some occasion to mention ‘ indivisible lines Aristotle may
well have set to some pupil, as an exercise, the task of refuting
Xenocrates. According to Simplicius and Philoponus, the
tract was attributed by some to Theophrastus 3 ; and this
seems the most likely supposition, especially as Diogenes
Laertius mentions, in a list of works by Theophrastus, £ On
indivisible lines, one Book’. The text is in many places
corrupt, so that it is often difficult or impossible to restore the
argument. In reading the book we feel that the writer is
for the most part chopping logic rather than contributing
seriously to the philosophy of mathematics. The interest
of the work to the historian of mathematics is of the slightest.
It does indeed cite the equivalent of certain definitions and
propositions in Euclid, especially Book X (on irrationals), and
in particular it mentions the irrationals called ‘ binomial ’ or
‘ apotome ’, though, as far as irrationals are concerned, the
writer may have drawn on Theaetetus rather than Euclid.
The mathematical phraseology is in many places similar to
that of Euclid, but the writer shows a tendency to hark back
to older and less fixed terminology such as is usual in
Aristotle. The tract begins with a section stating the argu
ments for indivisible lines, which we may take to represent
Xenocrates’s own arguments. The next section purports to
refute these arguments one by one, after which other con
siderations are urged against indivisible lines. It is sought to
show that the hypothesis of indivisible lines is not reconcilable
with the principles assumed, or the conclusions proved, in
mathematics ; next, it is argued that, if a line is made up
of indivisible lines (whether an odd or even number of such
lines), or if the indivisible line has any point in it, or points
1 Cf. Zeller, ii. I 4 , p. 1017. 2 Proclus on Eucl. I, p. 279. 5.
3 See Zeller, ii. 2 s , p. 90, note.