=1852.= If it were required to determine inclined planes of varying inclinations of such lengths that a free rolling body would descend on them in equal times, any one who understands the mechanical laws involved would admit that this would necessitate sundry preparations. But in the circle the proper arrangement takes place of its own accord for an infinite variety of positions yet with the greatest accuracy in each individual case. For all chords which meet the vertical diameter whether at its highest or lowest point, and whatever their inclinations, have this in common: that the free descent along them takes place in equal times. I remember, one bright pupil, who, after I had stated and demonstrated this theorem to him, and he had caught the full import of it, was moved as by a miracle. And, indeed, there is just cause for astonishment and wonder when one beholds such a strange union of manifold things in accordance with such fruitful rules in so plain and simple an object as the circle. Moreover, there is no miracle in nature, which because of its pervading beauty or order, gives greater cause for astonishment, unless it be, for the reason that its causes are not so clearly comprehended, marvel being a daughter of ignorance.--KANT. _Der einzig mögliche Beweisgrund zu einer Demonstration des Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 137._ =1853.= These examples [taken from the geometry of the circle] indicate what a countless number of other such harmonic relations obtain in the properties of space, many of which are manifested in the relations of the various classes of curves in higher geometry, all of which, besides exercising the understanding through intellectual insight, affect the emotion in a similar or even greater degree than the occasional beauties of nature.--KANT. _Der einzig mögliche Beweisgrund zu einer Demonstration des Daseins Gottes; Werke (Hartenstein), Bd. 2, p. 138._ =1854.= 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 honor 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.--DODGSON, C. L.