ARITHMETIC IN THEON OF SMYRNA 113 
lamblichus states the same facts in a slightly different form. 1 
The truth of these statements can be seen in the following 
way. 2 Since any number m must have one of the following 
forms 
6 Jc, 6/c+l, 6/v + 2, 6/v + 3, 
any square m 2 must have one or other of the forms 
36/c 2 , 12k +1, 36/c 2 + 24/v + 4, 36& 2 +36& +9. 
For squares of the first type ~ and — are both integral, 
for those of the second type 
are both integral, 
for those of the third type 
m 2 — 1 , 
—-— and 
— are both integral, 
and for those of the fourth type 
m 2 
~3 
and 
m 2 — 1 
are both 
integral; which agrees with Theon’s statement. Again, if 
the four forms of squares be divided by 3 or 4, the remainder 
is always either 0 or 1; so that, as Theon says, no square can 
be of the form 3n + 2, 491 + 2, or 4n+3. We can hardly 
doubt that these discoveries were also Pythagorean. 
Iamblichus, born at Chalcis in Coele-Syria, was a pupil of 
Anatolius and Porphyry, and belongs to the first half of the 
fourth century A.D, He wrote nine Books on the Pythagorean 
Sect, the titles of which were as follows: I. On the Life of 
Pythagoras; II. Exhortation to philosophy (TIpoTpenTiKbs 
hu (pL\o(ro(f)Lav); III. On mathematical science in general; 
IY. On Nicomachus’s Introductio Arithmetica; V. On arith 
metical science in physics; YI. On arithmetical science in 
ethics; VII. On arithmetical science in theology; VIII. On 
the Pythagorean geometry; IX. On the Pythagorean music. 
The first four of these books survive and are accessible in 
modern editions; the other five are lost, though extracts 
from VII. are doubtless contained in the Theologumena 
arithmetices. Book IV. on Nicomachus’s Introductio is that 
which concerns us here ; and the few things requiring notice 
are the following. The first is the view of a square number 
1 Iambi, in Nicom., p. 90. 6-11. 
2 Cf. Loria, Le scienze esatte nell' antica Greeia, p. 834.