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GREECE 35
divorced from geometry, his treatment is purely analytical.
He is the first to say that " a number to be subtracted multiplied
by a number to. be subtracted gives a number to be
added." This is applied to differences, like (2 a; — 3) (2 a; — 3),
the product of which he finds without resorting to geometry.
Identities like (a -\-by = a^-\-2 ab -\- b^, which are elevated
by Euclid to the exalted rank of geometric theorems, with
Diophantus are the simplest consequences of algebraic laws of
operation. Diophantus represents the unknown quantity x by
s', the square of the unknown a? by 8", a^ by k", x* by 88".
His sign for subtraction is j?i, his symbol for equality, i. Addition
is indicated by juxtaposition. Sometimes he ignores these
symbols and describes operations in words, when the symbols
would have answered better. In a polynomial all the positive
terms are written before any of the negative ones. Thus,
a? — 5aP-{-8x — 1 would be in his notation,^ K'a.^°''rj//iS'e/ji.'a..
Here the numerical coefficient follows the x.
To be emphasized is the fact that in Diophantus the fundamental
algebraic conception of negative numbers is wanting.
In 2 as —10 he avoids as absurd all cases where 2 a; < 10.
Take Probl. 16 Bk. I. in his Arithmetica: " To find three numbers
such that the sums of each pair are given numbers." If
a, b, c are the given numbers, then one of the required numbers
is ^(a 4- 6 -f c) — c. If c> ^(a -\-b + c), then this result is unintelligible
to Diophantus. Hence he imposes upon the problem
the limitation, " But half the sum of the three given numbers
must be greater than any one singly.'' Diophantus does not
give solutions general in form. In the present instance the
special values 20, 30, 40 are assumed as the given numbers.
In problems leading to simultaneous equations Diophantus
adroitly uses only one symbol for the unknown quantities.
1 Heath, op. cit., p. 72.