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80 come to him. And the law of circularity, thus formulated, was henceforth master within him, and governed his appreciation of things. He did not test his Ideal circle by comparing it with the sun or with an apple; on the contrary, he tested the circularity of a fruit by comparing it with his abstract or Ideal circle. His circle then was an algebraization of the round outline of the sun or moon, or of a fruit.

In the same way we discover a law of number, first, by thinking of some particular numbers; but as soon as we know the Law, we can state it Algebraically, i.e., in a manner which conveys no information as to what were the particular numbers of which we happened to be thinking when we discovered it. The particular-numbers suggested the law to our consciousness; they do not prove it to our reason. When once it has been suggested, it carries its own evidence, independently of particular numbers. And as soon as we have formulated a Law thus algebraically, it is henceforth master within us. Particular statements about number are referred to it; and our opinion as to the truth of those statements is controlled by it. For instance, the law that one number multiplied by a second always comes to the same result as is obtained by multiplying the second by the first was of course suggested to the consciousness first by the observation of some particular pairs of numbers; but it is not proved by reference to any special numbers; it is general and algebraic. And no student thoroughly understands it as a law of number until he understands it in its algebraic statement: ab=ba. As soon as he understands the algebraic statement, it becomes master of his thought.