The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

9

use of the formula just mentioned,1 thus:
3 26
"3 C 15'
and
- 26 - 1 1351
43 15 +2.26 15 780'
both 26, 15, and 1351, 780, being solutions of the Pell
equation
2 - 3y2 = 1.
It is worth noting that the methods2 used are identical with those of finding integral solutions of the Pell
equation. The simplicity of these calculations which
are capable of being continued indefinitely, naturally
provoked efforts to calculate other square roots. These
efforts, by simple methods and by experiments natural
for the calculators to make without any further arithmetic
theory led up to the rules of Brahmagupta and Bhaskara.
With the same starting point, WA oA p/q, the Greeks
were led by somewhat different rules to closer approximations of IA, but still by methods in which the approximations satisfy the equations
x2 - Ay2 = 1.
Plato, who derived his number theory from the Pythagoreans, was acquainted with the approximation 7/5 for
the ratio of the diagonal and side of a square,3 so that
this value dates back at least to the middle of the fifth
century before Christ. The Greeks considered every
rational integer4 first of all as a straight line, so that 5 was
1 P. Tannery, "Sur la mesure... " loc. cit.
2 Plato, Republic (VII, 546 be.). See M. Cantor, "Vorlesungen fiber
der Geschichte der Mathematik," Vol. I, 2d ed., p. 210, Leipzig, 1894.
3 G. Friedlein, "Procli Diadochi in primum Euclidis Elementorum
librum commentarii," p. 427, Leipzig, 1873.
4 F. Hultsch, "Die Pythagoreischen Reihen der Seiten und Diagonalen
von Quadraten und ihre Umbildung zu einer Doppelreihe ganzer Zahlen,"
Bibliotheca mathematica, vol. I (3), p. 8, Leipzig, 1900.