The cycloid, the source of so much contention, and on that account compared to the golden apple thrown by Discord among the gods, was again brought into notice by Pascal. This philosopher, not less celebrated for his piety and zeal in defence of the Christian religion, than his mathematical invention, took the cycloid as the subject of his meditation in those sleepless nights which he passed, in consequence of bad health ; and he soon ex tended his discoveries beyond what was then known. He was not of a disposition to boast of his discoveries in geometry ; but some of his pious friends supposed that it would be useful to have it known, that the man who had defended religion and Christianity against infidelity, was perhaps the most profound thinker, and the greatest geometer in Europe. They therefore requested that he would try the skill of the mathematicians his cotempora ries, by proposing publicly his problems on the cycloid. He took their advice; and, under the assumed name of .4. Dettonville, he addressed a circular letter to the geo metricians, in 1658, inviting them to resolve his prob lems, and promising forty pistoles to the author of the first solution, and twenty to that of the next, requiring them also to be sent to Paris by an assigned time. Only two contended for the prize; Lalouerc, a French mathe matician, and our countryman Dr Wallis. As the former had merely made a partial attempt to resolve the prob lems, and had failed, his claims were immediately set aside. Dr Wallis, however, had been more successful, yet he had committed some mistakes, and, on this ac count, the prize was not awarded him. There were others, who, without aspiring to the prize, sent solutions of one or other of the problems to Pascal. Of this num ber were, Slusius; the prelate Ricci ; the celebrated Huygens ; and Sir Christopher Wren, who discovered the rectification of the curve. Pascal published his own solutions in the beginning of the year 1659, in a work enti tled Letters ,from ?. Dettonville to AT. de Carcavi. In the same year, Dr Wallis published a work on the cycloid, and other curves, in which he resolved some of Pascal's problems by his Arithmetic of Infinities ; and, in the year following, Lalouere also published a treatise on the cy cloid; and another work •appeared about the same time from the pen of P. Fabri, the Jesuit.
The cycloid is remarkable, as well on account of its mechanical as its geometrical properties ; and Mr Huy gens discovered some of the most interesting of both kinds. To the latter class belongs the property, which we shall demonstrate in this article, by which he shewed how a pendulum may be made to vibrate in an arc of a cycloid ; and to the former, the very beautiful property, that all vibrations of a pendulum in arcs of a cycloid, are performed in equal times. See Mze HA Sias.
The very curious problem proposed uy John Ber noulli, viz. " to find the path along which a body may roll from one given point to another, in the shortest time possible, the points being supposed neither in the same vertical nor the same horizontal plane," on account of its elegance, engaged the attention of the most cele brated mathematicians in Europe, who found, that the curve required, or the line of swiftest descent, as it is called, must be a cycloid. Sec MECHANICS.
Epicycloids appear to have been first invented by Roe mer, the Danish astronomer, celebrated for having first discovered the progressive motion of light. He turned his attention to the theory of epicyeloids, while at Paris, about the year 1674, not as a speculation purely geome trical, but with a view to improve mechanics, because he found that, by giving to the teeth of wheels the figure of these curves, the action of the moving power on them would be uniform ; and that on this account their friction would be diminished.
There are other branches of mechanics connected with the theory of epicycloids. Huygens found that, supposing the force of gravity to act uniformly in the di rection of parallel lines, a pendulum moving in a cycloid would perform all its vibrations' in equal times, whether it described a greater or a lesser are. But, by extending the hypothesis, and supposing the force of gravity to be directed to the earth's centre, and to be in all places as the distance from the centre, it became a question what curve a pendulum ought in this case to describe, so as to perform unequal vibrations in equal times ? Sir .Isaac Newton shewed that the curve ought to be an epicycloid. See Princitia, lib. i. prop. 51.
The subject of spherical epicycloids was treated by Herman, in the first volume of the Commentaries of the Petersburgh Academy. It appears that a mathemati cian, named Offenburg, had proposed this problem, a to pierce a spherical roof with oval windows, the perime ter of any one of which may be absolutely rectifiable." Herman believed that he could resolve the problem by spherical epicyeloids ; having supposed that, in general, they admitted of an algebraic rectification. He had, how ever, committed a mistake in his reasoning, and thus was led to a wrong conclusion, as was afterwards shewn by John Bernoulli, in a paper on the rectification of these curves, given in the Memoirs of the Academy of Sei Paris, 1732, where he shews that the rectifica tion of the curve proposed by Herman requires the qua drature of the hyperbola.
We shall now give a brief view of the properties of cycloids and epicyeloids.
110. Oev livrnsc, p.