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

EPIGRAMS IN THE GREEK ANTHOLOGY 443
+{ 1(71 1)} 2 +1)}2j
1
Biblioiheca mathematlca, viii s , 1907-8, pp. 118-34. See now Geom.
2A. 1-13 in Heron, vol. iv (ed. Heiberg), pp. 414-26.
444 ALGEBRA: DIOPHANTUS OF ALEXANDRIA
fcions of this type with the same number of unknown quantities
where n is any odd number, and Plato with another formula
which was given by Thymaridas, an early Pythagorean, and
of the same sort, namely (2n) 2 + (n — 2 l) = 2 (n 2 +\) 2 . Euclid
was called the e7rdu0rj/xa, flower ' ' or bloom ' ' of Thymaridas
(Lemma following X. 28) finds the following more general
(see vol. i, pp. 94-6). (3) Six more are problems of the usual
formula
type about the filling and emptying of vessels by pipes ; e.g.
(xiv. 130) one pipe fills the vessel in one day, a second in two
m 2 u 2 p 2 q
= 2 { \ (mnp 2 + mnq 2 2
—
) } { \ (mnp 2 — mnq 2 )
2
.
and a third in three ; how long will all three running together
The Pythagoreans too, as we have seen (vol. i, pp. 91-3),
take to fill it? Another about brickmakers (xiv. 136) is of
solved another indeterminate problem, discovering, by means
the same sort.
of the series of side- ' ' and diameter-numbers ' ', any number
Indeterminate equations of the first degree.
of successive integral solutions of the equations
2x 2 2
— y
= + 1.
The Anthology contains (4) two indeterminate equations of
Diophantus does not particularly mention this equation,
the first degree which can be solved in positive integers in an
but from the Lemma to VI. 15 it is clear that he knew how
infinite number of ways (xiv. 48, 144) ; the first is a distribution
of apples, 3x in number, into parts satisfying the equation
seeing that 2 x 2 —
to find any number of solutions when one is known. Thus,
2 1 = y is satisfied by x = 1 , y == 1 , he would
x — Sy = y t
where y is not less than 2 ; the second leads to
put
three equations connecting four unknown quantities
2 ( 1 + x) 2 — 1 = a. square
x + y = x x + yu
= (px-1) 2 , say;
x=
whence x = (4 +
2yv
2p)/(p 2 -2).
x 1
= Take the value 3y,
p — 2, and we have x = 4, and x+ I = 5 ;
in this case 2 . 5 — 2 1 = 49 = 7 2 . Putting x + 5 in place of x,
the general solution of which is x = 4&, y — k, x x
= 3 k,
we can find a still higher value, and so on.
= 2 k. These very equations, which, however, are
y1
made
determinate by assuming that x + y = x 1 + y 1
— 100, are solved
Indeterminate equations in the Heronian collections.
in Dioph. I. 12.
Enough has been said to show that Diophantus was not
Some further Greek examples of indeterminate analysis are
the inventor of Algebra. Nor was he the first to solve indeterminate
now available. They come from the Constantinople manuscript
problems of the second degree.
(probably of the twelfth century) from which Schone edited
the Metrica of Heron ; they have been published and translated
Indeterminate equations of second degree
by Heiberg, with comments by Zeuthen. 1
Two of the problems
before
(thirteen in number) had been published in a less complete
Diophantus.
form in Hultsch's Heron (Geeponicus, 78, 79)
; the others
Take first the problem (Dioph. II. 8) of dividing a square
are new.
number into two squares, or of finding a right-angled triangle
I. The first problem is to find two rectangles such that the
with sides in rational numbers. We have already seen that
perimeter of the second is three times that of the first, and
Pythagoras is credited with the discovery of a general formula
for finding such triangles, namely,
^ 2_ ={ 1^2
the area of the first is three times that of the second. The