five ones, and did not mean once five, or one times five ones, your words would have no meaning. Neither you yourself nor any one else would know what you meant. But when you say once five, and then once ten, you not only express an agreement, you also express a difference between five and ten. Now, the general term for this difference is ἀόριστος δυὰς, and this δυὰς or diversity is said to be ἀόριστος or indefinite, because it varies indefinitely—once 10, once 20, once 30, once 40, once 1000, once 1,000,000—the once term, the μονὰς, never varies, but the other term, the δυὰς, as expressed by 20, 30, 40, 1000, 1,000,000, varies indefinitely, and its variations are inexhaustible; hence it is called ἀόριστος. Perhaps the simplest translation of ἀόριστος would be the indefinite any; ἀόριστος δυὰς any particular number. I conceive that in this way the Pythagorean doctrine, that the μονὰς and the ἀόριστος δυὰς are the elements of number, may be explained. Neither is the number one any exception; it, too, is composed of the μονὰς and the ἀόριστος δυὰς. One, like all other numbers, is different from any other number. In what respect does it differ from all other numbers? It differs from them in being one. In what respect does it agree with them? It agrees with them in being once one, or one times one, or one one. When we say "one," we usually mean "one one;" but we do not always or necessarily mean this, but may just as well mean 100 or 1000. One, viewed strictly, stands for once any number; and therefore, when it stands for the
