irrationals a deep mystery, a symbol of the unspeakable? We are told that the one who first divulged the theory of irrationals, which the Pythagoreans kept secret, perished in consequence in a shipwreck. Its discovery is ascribed to Pythagoras, but we must remember that all important Pythagorean discoveries were, according to Pythagorean custom, referred back to him. The first incommensurable ratio known seems to have been that of the side of a square to its diagonal, as . Theodorus of Cyrene added to this the fact that the sides of squares represented in length by , , etc., up to , and Theætetus, that the sides of any square, represented by a surd, are incommensurable with the linear unit. Euclid (about 300 B.C.), in his Elements, X. 9, generalised still further: Two magnitudes whose squares are (or are not) to one another as a square number to a square number are commensurable (or incommensurable), and conversely. In the tenth book, he treats of incommensurable quantities at length. He investigates every possible variety of lines which can be represented by , a and b representing two commensurable lines, and obtains 25 species. Every individual of every species is incommensurable with all the individuals of every other species. "This book," says De Morgan, "has a completeness which none of the others (not even the fifth) can boast of; and we could almost suspect that Euclid, having arranged his materials in his own mind, and having completely elaborated the tenth book, wrote the preceding books after it, and did not live to revise them thoroughly."[9] The theory of incommensurables remained where Euclid left it, till the fifteenth century.
Euclid devotes the seventh, eighth, and ninth books of his Elements to arithmetic. Exactly how much contained in these books is Euclid's own invention, and how much is borrowed from his predecessors, we have no means of knowing.
