A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

HERONIAN INDETERMINATE EQUATIONS 447
and b. The method employed is to take the sum of the area
and the perimeter S + 2 s, separated into its two obvious
factors s(r+2), to
put s(r+2) = A (the given number), and
then to separate A into suitable factors to which s and r + 2
may be equated. They must obviously be such that sr, the
area, is divisible by 6. To take the first example where
A = 280 :' the possible factors are 2 x 140, 4 x 70, 5 x 56, 7 x 40,
8 x 35, 10 x 28, 14 x 20. The suitable factors in this case are
r+2 = 8, s = 35, because r is then equal to 6, and rs is
a multiple of 6.
The author then says that
a= \ [6 + 35- \/ {(6 + 35) 2 -8. 6. 35}] = £(41-1) = 20,
6 = £(41 + 1)=21,
c = 35-6 = 29.
The triangle is« therefore (20, 21, 29) in this case. The
triangles found in the other three cases, by the same method,
are (9, 40, 41), (8, 15, 17) and (9, 12, 15).
Unfortunately there is no guide to the date of the problems
just given. The probability is that the original formulation
of the most important of the problems belongs to the period
between Euclid and Diophantus. This supposition best agrees
with the fact that the problems include nothing taken from
the great collection in the Arithmetica. On the other hand,
it is strange that none of the seven problems above mentioned
is found in Diophantus. The five relating to rational rightangled
triangles might well have been included by him ;
thus
he finds rational right-angled triangles such that the area plus
or minus one of the perpendiculars is a given number, but not
the rational triangle which has a given area ; and he finds
rational right-angled triangles such that the area plus or minus
the sum of two sides is
a given number, but not the rational
triangle such that the sum of the area and the three sides is
a given number. The omitted problems might, it is true, have
come in the lost Books ; but, on the other hand, Book VI would
have been the appropriate place for them.
The crowning example of a difficult indeterminate problem
propounded before Diophantus's time is the Cattle-Problem
attributed to Archimedes, described above (pp. 97-8).
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.