The Pell equation, by Edward Everett Whitford.

4

THE PELL EQUATION

found in the dimensions of ancient structures, such as the
Egyptian pyramids, the Parthenon and other temples on
the Acropolis at Athens, and altars and temples in many
places. For example, the principal room, called the
King's chamber,1 in the pyramid of Cheops, has for the
ratio of its height to its breadth about 1.117 or nearly
1 /5 showing that the architect must have known the
approximate value of this surd; and it is suggestive, to
say the least, when we note that this is one half the ratio
x: y of some of the solutions of
x2 - 5y2 = 1.
The ratio 17: 12 is found many times on the Acropolis,
and
x= 17, y = 12
satisfies
x2 - 2y2 = 1.
These solutions are interwoven with the number theory
of Pythagoras and his followers, and the ancient tradition
that Pythagoras had obtained his knowledge of numbers
by a sojourn upon the Euphrates is strengthened by recent
discoveries.2 These solutions are found in connection with
the mystic Platonic number,3 and a profound consideration
of such numbers no doubt gave impulse to the researches of
Theon of Smyrna in connection with his side- and diagonalnumbers.
There is a very intimate mathematical relation between
the Pell equation and the extraction of square roots.
1H. A. Naber, "Das Theorem des Pythagoras," p. 48, Haarlem, 1908.
2 H. V. Hilprecht, "Mathematical, meteorological and chronological
tablets from the temple library at Nippur," vol. XX, part 1 of series A,
cuneiform texts, University of Pennsylvania, Phila., 1906. M. Cantor,
"Uber die alteste indische Mathematik," Archiv der Mathematik und
Physik, vol. VIII (3), Heft 1, p. 63, Leipzig, 1904. Cantor sees no ground
for supposing Pythagoras a pupil of India but rather holds Egypt to be
the source of his mathematical knowledge. He refers to L. von Schroeder,
"Pythagoras und die Inder," 1884.
3 S. Giinther, "Die platonische Zahl," p. 5, Dresden, 1882.