But as I have said, mathematicians have not been content with a computation of this sort. They have calculated the number not to the _eighth_, but to the _six hundred and twentieth_ decimal place. Now, if we remember that each new decimal makes the result ten times more exact, we shall begin to see what a waste of time there has been in this tremendous calculation. We all remember the story of the horse which had twenty-four nails in its shoes, and was valued at the sum obtained by adding together a farthing for the first nail, a halfpenny for the next, a penny for the next, and so on, doubling twenty-four times. The result was counted by thousands of pounds. The old miser who paid at a similar rate for a grave eighteen feet deep (doubling for each foot), killed himself when he heard the total. But now consider the effect of multiplying by ten, six hundred and twenty times. A fraction, with that enormous number for denominator, and unity for numerator, expresses the minuteness of the error which would result if the ‘long value’ of the circumference were made use of. Let an illustration show the force of this:— It has been estimated that light, which could eight times circle the earth in a second, takes 50,000 years in reaching us from the faintest stars seen in Lord Rosse’s giant reflector. Suppose we knew the exact length of the tremendous line which extends from the earth to such a star, and wanted, for some inconceivable purpose, to know the length of the circumference of a circle, of which that line was the radius. The value deduced from the above-mentioned calculation of the relation between the circumference and the diameter would differ from the truth by a length which would be imperceptible under the most powerful microscope ever yet constructed. Nay, the radius we have conceived, enormous as it is, might be increased a million-fold, or a million times a million-fold, with the same result. And the area of the circle formed with this increased radius would be determinable with so much accuracy, that the error, if presented in the form of a minute square, would be utterly imperceptible under a microscope a million times more powerful than the best ever yet constructed by man. Not only has the length of the circumference been calculated once in this unnecessarily exact manner, but a second calculator has gone over the work independently. The two results are of course identical figure for figure. It will be asked then, what _is_ the problem about which so great a work has been made? The problem is, in fact, utterly insignificant; its only interest lies in the fact that it is insoluble—a property which it shares along with many other problems, as the trisection of an angle, the duplication of a cube, and so on.