The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

45

which is the same as that of Alkarkhi, but in the later
editions (1534, 1537, 1542), while there are some examples
involving the formula of Alkarkhi, or rather of Heron of
Alexandria, there are others which show that he possessed
another method which plays an important role in the
theory of numbers, for the following approximations,' all
of which he gives, yield solutions to the Pell equation
x2 - Ay2 = 1:
16 17 659
i128 c 11, 8iS0. 8 129c7 o 17
51' 18' 2820'
25 285 13
1300 3175 7 1 1375 19 135 11
103 109 68186
%175 c 8 156' 756 c 27 0, 611 c 24 8
151 197 1
a231 3 15 761, 800 o 28 169, 4100 c 64.
760 693' 32'
Even if he had done no other work, these approximations
would give the Spanish mathematician a worthy place
in the history of mathematics.
Buteo2 in his approximations to the square root of a
number makes repeated use of methods which could be
represented by the formula3
a2 +b \ a+ 2
For 166 he makes several approximations all of which
1 J. Perrot, "Sur une arithmetique espagnole du seizieme siecle,"
Bullettino di bibliografia e di storia delle scienze matematiche e fisiche,
vol. XV, p. 163, Rome, 1882.
2 "Ioan. Buteonis Logistica, quae et arithmetica vulgo dicitur in libros
quinque digesta," p. 76, Lyons, 1559.
3This method we have seen dates back to Archimedes and Heron
We find it again in both letters of Nicolas Rhabdas. P. Tannery, "Notice
sur les deux lettres arithmetiques de Nicolas Rhabdas (text grec et traduction)," p. 40, 68, Paris, 1886. Later examples of its use are seen in
the works of Luca Paciuolo, Cataldi, Cardan, and Tartaglia.