that we find more and more baffling the further we go. The “perfect numbers” form a series that may be as full of interest, for all that I know, as the primes. The properties of the “Arithmetical Triangle” are linked in the most unexpected fashion with the laws of our statistical science, and with the nature of certain orderly combinations of vast importance in other branches of mathematical inquiry. Countless other combinations of numbers form topics, not only of numerous well-known plays and puzzles, but of scientific investigations whose character is actually adventurous, — so arduous is their course, and so full of unexpected bearings upon other branches of knowledge has been their outcome. Nobody amongst us can pretend to fathom the value for concrete science, and for life, that has yet to be derived from advances in the Theory of Numbers.
These, then, are mere hints of the inexhaustible properties of the number-series. I speak still as layman; but I am convinced that these significant properties are quite as inexhaustible as the number-series itself. Now, the value of such properties you can never tell until you see what they are. Their meaning in the life of reason can only be estimated when they are present. Hence, you can never wisely decide not to know them until you have first known them. But they are not to be known merely as the endless repetitions of the same over and over. Hence it is wholly vain to say, “Numbers come from counting, and counting is vain repetition of the same over and over.” Whoever views the numbers merely thus, knows not whereof he speaks. It is not “counting, with nothing to count”; it is finding what Order means, that is the task of a true Theory of Numbers.
As a fact, then, the number-series in its wholeness seems to be a realm not only of inexhaustible truth, but of a truth that possesses an everywhere relatively individual type. And its validity has relations that we, at present, but imperfectly know, and a rational value that appears to be fundamental in every orderly inquiry.
We can, then, neither assert that to all the varieties which
