THE TRACT ON INDIVISIBLE LINES 
347 
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.