Page:Scientific Monthly, volume 14.djvu/302



IRCLES to square and cubes to double would give a man excessive trouble." Thus sings old Matthew Prior, indicating that "many knotty points there are, which all discuss but few can clear." Indeed, hundreds of would-be pathfinders of the intellect, from the time of Anaxagoras down to ours, have gone into ecstasy in the belief that they had solved the impossible problem of the squaring of the circle, only perhaps later to be cast into the depths of disappointment upon learning of their failure. Many have died under the delusion that they had accomplished the impossible.

The problem of the quadrature of the circle presents a strange phenomenon in the history of thought. A geometric construction is to be effected on very definite assumptions and restrictions—the use of a pair of compasses and an ungraduated ruler.

One element of strangeness lies in the fact that the problem has no bearing whatever on practical life. The mathematician and engineer ean compute the area of a circle to any desired degree of approximation; if they wish, they can carry approximations to hundreds of decimal places, although five or six places suffice in practice.

Another feature constitutes a source of pride to present-day mathematicians. Unlike some philosophical questions which are as far from solution now as they were aeons ago, the circle-problem, after thousands of years of intellectual effort, has been finally and definitely conquered; in 1882 it was proved by conclusive demonstration accepted by all trained mathematicians, that, under the restrictions imposed, the circle can not be squared.

One item of interest, in connection with the quadrature of the circle, is not generally known. The problem suggested an illustrated title-page which is perhaps the most unique that has ever adorned a mathematical book. In 1647 the Flemish Jesuit mathematician. Gregory St. Vincent, published a wonderful geometry, containing genuine pearls of new geometric truth, but unfortunately also a false diamond, the quadrature of the circle. We reproduce the title-page, which presents to the eye the nature of