The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

15

mean, as its name implies, is derived from a problem of
music. Tannery1 says of the numbers of Theon, "I
conceive this series as obtained upon the numbers themselves, without any immediate generalization, without any
foresight, and without perhaps any demonstration of this
characteristic property, but I believe there is a simpler
means of explaining their invention, and this means is
intimately connected with the consideration of the harmony of Pythagoras which gives the degree of approximation p/q = 3/2, 2q/p = 4/3."
As the solutions of
X2 - 2y2 = = 1
were evidently made in order to obtain approximations
to the ~/2, so when we find Archimedes" giving without
explanation the fractions 265/153 and 1351/780 as approximations to a/3, the most natural hypothesis is that he
obtained them from similar equations with 3 substituted
for 2. Archimedes was the first Greek mathematician
who was not content to speculate over irrationals but
handled them in computation. He expressed the sides
of regular inscribed and circumscribed polygons in terms
of the radius of the circle. In particular for a regular
hexagon he gave the proportion
r: 3
*2
and then set
1351 265
780 153'
There has been much speculation as to how these approximations were obtained.3 Let us look at one of the solu1 P. Tannery, "Du r6le de la musique grecque dans le developpement
de la mathematique pure," Bibliotheca mathematica, vol. III (3), pt. 161,
Leipzig, 1902.
2 T. L. Heath, "Works of Archimedes," p. LXXX, Cambridge, 1897.
3 P. Tannery, "Sur la mesure du cercle d'Archimede," Memoires de la
Societe des sciences de Bordeaux, vol. IV (2), p. 303, Paris, 1882. H. G.
Zeuthen, "Nogle hypotheser on Arkhimedes kvadratsrodsheregning,"
Tidsskrift for Mathematik, vol. III (4), p. 145, Copenhagen, 1879.