The Pell equation, by Edward Everett Whitford.

6

THE PELL EQUATION

tion were developed differently by the Hindus and the
Greeks, and this adds probability to the idea of independent discovery.' Indeed, we fail to find any definite
connection between the Greek and the Hindu algebra.
Each starts with a simple approximation to the square
root of a number and seeks from this a closer approximation. We shall see what different results are obtained.
The Hindus were skilful calculators but mediocre theorists, while the genius of the Greeks tended more towards
geometry.
One of the monuments of the old Hindu mathematicians is that curious work, the Sulva-sutras, or "Precepts
of line." These books contain rules to be observed by
the Brahmins in the construction of their altars, and give
close approximations for a/2. Baudhayana, the author
of the oldest of these works, uses first the approximation
17/12.2 Both Baudhayana3 and Apastamba4, another
of the early writers, give this rule in relation to the approximation for the diagonal of a square. "Increase the
measure by its third part and this third by its own fourth
less the thirty-fourth part of that fourth. The sutras
are characterized by an enigmatical shortness, but the
rule evidently means that
1 1 1
a2 3 +343.4 4.34'
This result, 577/408, furnishes a solution of the Pell
equation
X2 - 2y2 = 1.
1 P. Tannery, "Sur la mesure du cercle d' Archimede," Memoires de la
Societe des Sciences de Bordeaux, vol. IV (2), p. 313, Paris, 1882. S.
Gunther, "Die quadratischen Irrationalitaten der Alten und deren Entwickelungsmethoden," Zeitschrift fur Mathematik und Physik, Abhandlungen zur Geschichte der Mathematik, vol. XXVII, p. 43, Leipzig, 1882.
2 M. Simon, "Geschichte der Mathematik in Altertum," p. 157, Berlin,
1909.
3 G. Thibaut, loc. cit.
4 A. Biirk, "Das Apastamba-Sulba Sfitra," Zeitschrift der Deutschen
Morgenlindischen Gesellschaft, vol. LV, p. 543, Leipzig, 1901.