The Pell equation, by Edward Everett Whitford.

14

THE PELL EQUATION

Theon adds by way of conclusion, and this is also
pointed out by Proclus, that the sum of the squares of
all the fprval 8dfaeTpot will be twice as large as the sum
of the squares of all the sides. If this sum is limited to
a finite number of terms of the double series, then the
number of terms must be even.
Hultschl takes the view that Proclus did his work
independently of Theon, but that both spring from a
common origin, for he says, "If Proclus, as many signs
indicate, has drawn from the tua67 ldT7rwv e(wDpi'a of Geminos,
so Theon has done so also, not directly but through a
second or third hand."
Guinther2 gives a different explanation whereby Theon
might have arrived at his numbers. From
y + P (x + y), y - q (x - y),
q p
it follows, if
p2 + 2pq - q2 = z2,
4q2 + z = 2q 2q 42 z q2 - z 4q 12q2 z
y=, x=,
z z
and when z = 1, any number q is found for which 2q2 + z
is a perfect square, this solution gives an infinite number
of integral values of y and x; and the solution of the
general equation
(a2 b2)y2 - = X2
is accomplished.
Besides the geometric motive which in the last half of
the fourth century before Christ, led to the discovery of
approximations for J2, and for such numbers as those of
Theon at about the same time, the problems of music
also led to new researches to determine as exactly as
possible J12. The Pythagorean concept of the harmonic
Hultsch, loc. cit.
2S. Gunther, "Uber einen Specialfall der Pell'schen Gleichung,"
Blatter fur das bairische Gymnasial- und Realschulwesen, vol. XVIII, p.
19, Munich, 1882.