NICOMACHUS 
99 
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interested Nicomachus. If the verbiage is eliminated, thg 
mathematical content can be stated in quite a small com 
pass. Little or nothing in the book is original, and, except 
for certain definitions and refinements of classification, the 
essence of it evidently goes back to the early Pythagoreans. 
Its success is difficult to explain except on the hypothesis that 
it was at first read by philosophers rather than mathemati 
cians (Pappus evidently despised it), and afterwards became 
generally popular at a time when there were no mathemati 
cians left, but only philosophers who incidentally took an 
interest in mathematics. But a success it undoubtedly was; 
this is proved by the number of versions or commentaries 
which appeared in ancient times. Besides the Latin transla 
tion by Apuleius of Madaura (born about a.d. 125), of which 
no trace remains, there was the version of Boetius (born about 
480, died 524 A.D.); and the commentators include lamblichus 
(fourth century), Heronas, 1 Asclepius of Tralles (sixth century), 
Joannes Philoponus, Proclus. 2 The commentary of lamblichus 
has been published, 3 as also that of Philoponus, 4 while that of 
Asclepius is said to be extant in MSS. When (the pseudo-) 
Lucian in his Philopatris (c. 12) makes Critias say to Triephon 
‘ you calculate like Nicomachus ’, we have an indication that 
the book was Avell known, although the remark may be less a 
compliment than a laugh at Pythagorean subtleties. 5 
Book I of the Introductio, after a philosophical prelude 
(cc. 1-6), consists principally of definitions and laws of forma 
tion. Numbers, odd and even, are first dealt with (c. 7); then 
comes the subdivision of even into three kinds (1) evenly-even, 
of the form 2 n , (2) even-odd, of the form 2 (2 n+ 1), and (3) 
odd-even, of the form 2 m+1 (2 n+ 1), the last-named occupying 
a sort of intermediate position in that it partakes of the 
character of both the others. The odd is next divided into 
three kinds : (1) ‘ prime and incomposite ’, (2) £ secondary and 
1 v. Eutoc. in Archim. (ed. Heib. iii, p. 120. 22). 2 v. Suidas. 
3 The latest edition is Pistelli’s (Teubner, 1894). 
* Ed. Hoche, Heft 1, Leipzig, 1864, Heft 2, Berlin, 1867. 
5 Triephon tells Critias to swear by the Trinity (‘One (proceeding) from 
Three and Three from One ’), and Critias replies, ‘You would have me 
learn to calculate, for your oath is mere arithmetic and you calculate 
like Nicomachus of Gerasa. I do not know what you mean by your 
“ One-Three and Three-One ”; I suppose you don’t mean the rerpanrixs 
of Pythagoras or the oybods or the Tpia<ds ? ’ 
H 2