308 
PLATO 
i- _ 
point of view lies in the terms ‘ rational ’ and ‘ irrational 
diameter of 5 A fair approximation to /2 was obtained 
by selecting a square number such that, if 2 be multiplied by 
it, the product is nearly a square ; 25 is such a square number, 
since 25 times 2, or 50, only differs by 1 from 7 2 ; conse 
quently is an approximation to V2, It may have been 
arrived at in the tentative way here indicated ; we cannot 
doubt that it was current in Plato’s time ; nay, we know that 
the general solution of the equations 
x 2 —2 y 2 = +1 
by means of successive ‘ side- ’ and ‘ diameter- ’ numbers was 
Pythagorean, and Plato was therefore, here as in so many 
other places, ‘ Pythagorizing 
The diameter is again mentioned in the Politicus, where 
Plato speaks of £ the diameter which is in square (&Wpei) 
two feet’, meaning the diagonal of the square with side 
1 foot, and again of the diameter of the square on this 
diameter, i. e. the diagonal of a square 2 square feet in area, 
in other words, the side of a square 4 square feet in area, 
or a straight line 2 feet in length. 1 
Enough has been said to show that Plato was abreast of 
the mathematics of his day, and we can understand the 
remark of Proclus on the influence which he exerted upon 
students and workers in that field ; 
‘ he caused mathematics in general and geometry in particular 
to make a very great advance by reason of his enthusiasm 
for them, which of course is obvious from the way in which 
he filled his books with mathematical illustrations and every 
where tries to kindle admiration for these subjects in those 
who make a pursuit of philosophy.’ 2 
Mathematical ‘ arts 
Besides the purely theoretical subjects, Plato recognizes the 
practical or applied mathematical ‘ arts ’ ; along with arith 
metic, he mentions the art of measurement (for purposes of 
trade or craftsmanship) and that of weighing 3 ; in the former 
connexion he speaks of the instruments of the craftsman, 
the circle-drawer (roproy), the compasses (¿¿a/Spr^y), the rule 
1 Politicus, 266 b. 2 Proclus on Eucl. I, p. 66. 8-14. 
3 Philebus, 55 e-56 e.