Title: The Pell equation, by Edward Everett Whitford.
Author: Whitford, Edward Everett, 1865-
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.