It may well be doubted whether, in all the range of Science, there is any field so fascinating to the explorer—so rich in hidden treasures—so fruitful in delightful surprises—as that of Pure Mathematics. The charm lies chiefly, I think, in the absolute certainty of its results: for that is what, beyond almost all mental treasures, the human intellect craves for. Let us only be sure of something! More light, more light! Ἐν δὲ φάει, καὶ ὀλέσσον. 'And, if our fate be death, give light and let us die!' This is the cry that, through all the ages, is going up from perplexed Humanity, and Science has little else to offer, that will really meet the demands of its votaries, than the conclusions of Pure Mathematics. Most other Sciences are in a state of constant flux—the precious truths of one generation being smiled at as paradoxes by the second generation, and contemptuously swept away as childish nonsense by the third. If you would see a specimen of the rapidity of this process of decomposition, take Biology for a sample: quote, to any distinguished Biologist you happen to meet, some book published thirty years ago, and observe his pitying smile! [Pg xvi] But neither thirty years, nor thirty centuries, affect the clearness, or the charm, of Geometrical truths. Such a theorem as 'the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the sides' is as dazzlingly beautiful now as it was in the day when Pythagoras first discovered it, and celebrated its advent, it is said, by sacrificing a hecatomb of oxen—a method of doing honour to Science that has always seemed to me slightly exaggerated and uncalled-for. One can imagine oneself, even in these degenerate days, marking the epoch of some brilliant scientific discovery by inviting a convivial friend or two, to join one in a beefsteak and a bottle of wine. But a hecatomb of oxen! It would produce a quite inconvenient supply of beef. Now this field of Mathematical research, with all its wealth of hidden treasure, is all too apt to yield nothing to our research: for it is haunted by certain ignes fatui—delusive phantoms, that float before us, and seem so fair, and are all but in our grasp, so nearly that it never seems to need more than one step further, and the prize shall be ours! Alas for him who has been turned aside from real research by one of these spectres—who has found a music in its mocking laughter—and who wastes his life and energy in the desperate chase! One of these—the chief one, I think—is the old old problem of 'Squaring the Circle,' which has certainly wasted many a human life. Whether it has actually driven any one mad, I know not—most of its victims were, I fancy, partly crazed before they entered on the quest—but it clearly has the power of demolishing such slender reasoning powers as they may ever have chanced to possess. [Pg xvii] With two of these misguided visionaries I have myself corresponded.