304 
PLATO 
{() Solution of x 2 + y 2 = z 2 in integers. 
We have already seen (p. 81) that Plato is credited with 
a rule (complementary to the similar rule attributed to Pytha 
goras) for finding a whole series of square numbers the sum 
of which is also a square ; the formula is 
(2 n) 2 + ( / a 2 — l) 2 = (n 2 + l) 2 , 
(77) Incommensurables. 
On the subject of incommensurables or irrationals we have 
first the passage of the Theaetetus record in that Theodorus 
proved the incommensurability of Vs, */5 ... Vl7, after 
which Theaetetus generalized the theory of such ‘roots’. 
This passage has already been fully discussed (pp, 203-9). 
The subject of incommensurables comes up again in the Laws, 
where Plato inveighs against the ignorance prevailing among 
the Greeks of his time of the fact that lengths, breadths and 
depths may be incommensurable as well as commensurable 
with one another, and appears to imply that he himself had 
not learnt the fact till late (dKovaas oyfre 7rore), so that he 
was ashamed for himself as well as for his countrymen in 
general. 1 But the irrationals known to Plato included more 
than mere ‘ surds ’ or the sides of non-squares ; in one place 
he says that, just as an even number may be the sum of 
either two odd or two even numbers, the sum of two irra 
tionals may be either rational or irrational. 2 An obvious 
illustration of the former case is afforded by a rational straight 
line divided ‘ in extreme and mean ratio ’. Euclid (XIII. 6) 
proves that each of the segments is a particular kind of 
irrational straight line called by him in Book X an a2wtome ; 
and to suppose that the irrationality of the two segments was 
already known to Plato is natural enough if we are correct in 
supposing that ‘ the theorems which ’ (in the words of Proclus) 
‘ Plato originated regarding the section ’ 3 were theorems about 
what came to be called the ‘golden section’, namely the 
division of a straight line in extreme and mean ratio as in 
Eucl. II. 11 and VI. 30. The appearance of the latter problem 
in Book II, the content of which is probably all Pythagorean, 
suggests that the incommensurability of the segments with 
1 Laws, 819 n-820 C. 2 Hippias Maior, 303 B, C. 
3 Proclus on Eucl. I, p. 67. 6.