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.