A history of Greek mathematics - Wilbourhall.org

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448 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
Numerical solution of quadratic equations.
The geometrical algebra of the Greeks has been in evidence
all through our history from the -Pythagoreans downwards,
and no more need be said of it here except that its arithmetical
application was no new thing in Diophantus. It is probable,
for example, that the solution of the quadratic equation,
discovered first by geometry, was applied for the purpose of
finding numerical values for the unknown as early as Euclid,
if not earlier still. In Heron the numerical solution of
equations is
well established, so that Diophantus was not the
first to treat equations algebraically. What he did was to
take a step forward towards an algebraic notation.
The date of Diophantus can now be fixed with fair certainty.
He was later than Hypsicles, from whom he quotes a definition
of a polygonal number, and earlier than Theon of Alexandria,
who has a quotation from Diophantus's definitions. The
possible limits of date are therefore, say, 150 B.C. to A.D. 350.
But the letter of Psellus already mentioned says that Anatolius
(Bishop of Laodicea about a.d. 280) dedicated to Diophantus
a concise treatise on the Egyptian method of reckoning
hence Diophantus must have been a contemporary, so that he
probably flourished A.D. 250 or not much later.
An epigram in the Anthology gives some personal particulars
his boyhood lasted Jth of his life ; his beard grew after x^th
more ; he married after -|th more, and his son was born 5 years
later ; the son lived to half his father's age, and the father
died 4 years after his son. Thus, if x was his age when
he died,
which gives x = 84.
Works of Diophantus.
The works on which the fame of Diophantus rests are :
(1) the Arithmetica (originally in thirteen Books),
(2) a tract On Polygonal Numbers.