The Pell equation, by Edward Everett Whitford.

10

THE PELL EQUATION

taken as a line 5 units in length and upon it a square was
constructed. This square contained, accordingly, 25
square units. The diagonal of this square was /2 *25
linear units. This diagonal was called by the Pythagoreans the appr)To &dlt'CeTpos of the number 5. In conjunction with this j/50 - 1 was called the pT'} 8td/LierTpo9
which equaled 7 linear units, i.e., the rational number
whose square differs from the square of the irrational
diagonal by an integral minimum. And 7, 5, is a solution
of
x2- 2y = - 1.
That the pr1T7 BL&tdCe/poq had a general significance to
the Pythagoreans has become evident since the publication of Kroll'sl text of Proclus's commentary on Plato.
It is based on an application of Euclid II, 10,2 which was
common property before Plato and was known to the
Pythagoreans as early as the middle of the fifth century
before Christ. If C (Fig. 1) is the mid-point of the base
A C D B
FIG. 1.
of an isosceles right triangle AEB, and D any point on
AB, then
AD2 + DB2 = 2AC2 + 2CD.
For if Z is the point where the perpendicular at D meets
1Proclus, "In Platonis rempublicam commentarii," editor Kroll, vol.
II, c. 27, Leipzig, 1901.
2 F. Hultsch, loc. cit. H. G. Zeuthen, "Geschichte der Mathematik
im Altertum und Mittelalter," p. 59, Copenhagen, 1896. Also the French
edition, H. G. Zeuthen, "Histoire des mathematiques dans l'antiquite et le
moyen age," J. Mascart, ed., p. 47, Paris, 1902. M. Cantor, op. cit.,
vol. I, p. 249, 2d ed.