98 
PYTHAGOREAN ARITHMETIC 
Introductio arithmetica we- find the form of exposition 
entirely changed. Numbers are represented in Euclid by 
straight lines with letters attached, a system which has the 
advantage that, as in algebraical notation, we can work with 
numbers in general without the necessity of giving them 
specific values ; in Nicomachus numbers are no longer de 
noted by straight lines, so that, when different undetermined 
numbers have to be distinguished, this has to be done by 
circumlocution, which makes the propositions cumbrous and 
hard to follow, and it is necessary, after each proposition 
has been stated, to illustrate it by examples in concrete 
numbers. Further, there are no longer any proofs in the 
proper sense of the word ; when a general proposition has been 
enunciated, Nicomachus regards it as sufficient to show that 
it is true in particular instances ; sometimes we are left to 
infer the general proposition by induction from particular 
cases which are alone given. Occasionally the author makes 
a quite absurd remark through failure to distinguish between 
the general and the particular case, as when, after he has 
defined the mean which is ‘ subcontrary to the harmonic ’ as 
being determined by the relation ^—- =-, where a>h>c, 
0 c ct 
and has given 6, 5, 3 as an illustration, he goes on to observe 
that it is a property peculiar to this mean that the product of 
the greatest and middle terms is double of the product of the 
middle and least, 1 simply because this happens to be true in 
the particular case ! Probably Nicomachus, who was not 
really a mathematician, intended his Introduction to be, not 
a scientific treatise, but a popular treatment of the subject 
calculated to awaken in the beginner an interest in the theory 
of numbers by making him acquainted with the most note 
worthy results obtained up to date ; for proofs of most of his 
propositions he could refer to Euclid and doubtless to other 
treatises now lost. The style of the book confirms this hypo 
thesis ; it is rhetorical and highly coloured ; the properties of 
numbers are made to appear marvellous and even miraculous 5 
the most obvious relations between them are stated in turgid 
language very tiresome to read. It was the mystic rather 
than the mathematical side of the theory of numbers that 
1 Nicom. ii. 28. 8.