The Pell equation, by Edward Everett Whitford.

2

THE PELL EQUATION

the contributions of Pell and Brouncker.1 Wallis2 gives
Pell credit for certain researches in indeterminate analysis,
but where
Ay2 + 1 = x2
is discussed only Brouncker's methods are set forth. The
assertion of Hankel,3 for example, that Pell treated of the
equation
Ay2 + 1 = x2
rests upon a misunderstanding. Hankel said, "Pell has
done it no other service than to set it forth again in a
much read work," i.e., in the notes to the English translation which Brancker,4 in 1668, published of the "Teutschen Algebra" of Rahn.5 Konen4 says that in the only
copy which he could obtain of this work there is nothing
relative to this equation, and he thinks that it is very
probable that Pell never considered it. Enestr6m6 holds
the same opinion, basing it upon the examination of three
copies of Rahn's algebra. Nevertheless it seems not improbable that Pell solved the equation, for we find it discussed in Rahn's algebra7 under the form
x = 12yy - ZZ.
This shows that Pell had some acquaintance with the Pell
equation, and that Euler was not so far out of the way
when he attributed to Pell work upon it.
1 G. Wertheim, "Uber den Ursprung des Ausdruckes 'Pellsche Gleichung,"' Bibliotheca mathematica, vol. II (3), p. 360, Leipzig, 1901.
2 J. Wallis, "Opera mathematica," vol. II, chap. 98, p. 418, Oxford, 1693.
3 H. Hankel, "Zur Geschichte der Mathematik in Alterthum und
Mittelalter," p. 203, Leipzig, 1874.
4H. Konen, "Geschichte der Gleichung t2 - Du2 = 1," Leipzig, 1901.
Konen confuses Thomas Brancker and William Brouncker.
5G. Wertheim, "Die Algebra des Johann Heinrich Rahn (1659),"
Bibliotheca mathematica, vol. III (3), p. 125, Leipzig, 1902.
6 G. Enestr6m, "Uber den Ursprung der Bennenung Pell'sche Gleichung," Bibliotheca mathematica, vol. III (3), p. 204, Leipzig, 1902.
7 J. H. Rahn, "An introduction to algebra, translated out of the High
Dutch into English by Thomas Brancker, M.A. Much altered and augmented by D. P.," p. 143, London, 1668.