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 quote$ 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 ^th of his life; his beard grew after ^th 
• more; he married after -i^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, 
|x + Y2 x + \x + 5 + %x + 4 = x, 
which gives «=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.