on the tire of a wheel running on a straight, level road, will describe in the air a series of peculiar arches, called the cycloid. The law of its formation is simple; the law of its curvature is also simple. The path in which the spot moves curves exactly in proportion to its nearness to the lowest point of the wheel. By the simplicity of its law, it ought, according to the canon, to be a beautiful curve. Now, although artists have not shown any admiration for the cycloid, as they have for the ellipse, yet the mathematicians have gazed upon it with great eagerness, and found it rich in intellectual treasures. Chasles, in his History, says that the cycloid interweaves itself with all the great discoveries of the seventeenth century.
A curve which fulfils more perfectly the demands of our dictum is that of an elastic thread, to which we have already alluded. If the two ends of a straight steel hair be brought towards each other by simple pressure, the intervening spring may be put into a series of