The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

5

This will be noted when the problem of such extraction
is not attached to the decimal system, as is the present
usual custom, nor to the sexagesimal system as was the
custom of the Greek astronomers. The fact that tables
of squares and square roots have been discovered on
tablets from the temple library at Nippur1 and from the
library of Sardanapalus IV at Babylon2 has an important
bearing on the methods of solution of the Pell equation.
It is therefore quite possible that solutions of this equation
extend back to the ancient Babylonians four thousand
years ago.
It will not be far out of the way to say that the first
approximations to 42 appeared both in India and in
Greece about four hundred years before Christ. The
younger Pythagoreans (before 410 B.C.) recognized and
proved the incommensurability of the diagonal and side
of a square and set forth certain approximations.3 The
Sulva-sutras in India, which contained approximations
to I-2, are not later than the fourth or fifth century
before Christ.4
The fact that these approximations appeared about
the same time in Greece and India, and that the first
step in these approximations appears so simp]e, indicate
the independence of the Greek and Hindu mathematicians. Furthermore, the processes of root approximaHilprecht, op. cit., p. 13; D. E. Smith, "The mathematical tablets of
Nippur," Bulletin of the American Mathematical Society, vol. XII (2),
p. 394, New York, 1907.
2 M. L. Rodet, "Sur les methodes d' approximation chez les ancients,"
Bulletin de la Societe Mathematique de France, vol. VII, p. 159, Paris, 1879.
3H. Vogt, "Die Entdeckungsgeschichte des Irrationalen nach Plato
und anderen Quellen des 4. Jahrhunderts," Bibliotheca mathematica,
vol. X (3), p. 97, Leipzig, 1909. G. Junge, "Wann haben die Griechen das
Irrationale entdeckt?" Halle, 1907. M. L. Rodet, "Sur une methode
d'approximation des racines carrees, connu dans l'Inde anterieure a la
conquete d'Alexandre," Bulletin de la Societe Mathematique de France,
vol. VII, p. 98, Paris, 1879.
4G. Thibaut, "On the sulvasutras," Journal of the Asiatic Society of
Bengal, vol. XLIV, p. 239, Calcutta, 1875.