Many have supposed that there exists some exact relation between the circumference and the diameter of the circle, and that the problem to be solved is the determination of this relation. Suppose, for example, that the approximate relation discovered by Archimedes (who found, that if a circle’s diameter is represented by _seven_, the circumference may be almost exactly represented by _twenty-two_) were strictly correct, and that Archimedes had proved it to be so; then, according to this view, he would have solved the great problem; and it is to determine a relation of some such sort that many persons have set themselves. Now, undoubtedly, if any relation of this sort could be established, the problem would be solved; but as a matter of fact no such relation exists, and the solution of the problem does not require that there should be any relation of the sort. For example, we do not look on the determination of the diagonal of a square (whose side is known) as an insoluble, or as otherwise than a very simple problem. Yet in this case no exact relation exists. We cannot possibly express both the side and the diagonal of a square in whole numbers, no matter what unit of measurement we adopt: or, to put the matter in another way, we cannot possibly divide both the side and the diagonal into equal parts (which shall be the same along each), no matter how small we take the parts. If we divide the side into 1,000 parts, there will be 1,414 such parts, _and a piece over_ in the diagonal; if we divide the side into 10,000 parts, there will be 14,142, and still a little piece over, in the diagonal; and so on for ever. Similarly, the mere fact that no exact relation exists between the diameter and the circumference of a circle is no bar whatever to the solution of the great problem. Before leaving this part of the subject, however, I may mention a relation which is very easily remembered, and is very nearly exact—much more so, at any rate, than that of Archimedes. Write down the numbers 113,355, that is, the first three odd numbers each repeated twice over. Then separate the six numbers into two sets of three, thus,—113) 355, and proceed with the division thus indicated. The result, 3·1415929‍‍..., expresses the circumference of a circle whose diameter is 1, correctly to the sixth decimal place, the true relation being 3·14159265.