EPICYCLOI D, in Geometry, is the curve generated by any point in the plane of a moveable circle, which rolls either on the inside or the outside of the circum ference of a fixed circle. If the circles be both in the same plane, the curve generated will be the plane epi cycloid.
If again the moveable and fixed circles be in different planes, and the former be the base of a right cone, that rolls on the surface of another right cone, the base of which is the latter, so that the vertices of the cones are at the same point ; then, in this case, the curve generated by any point on the. plane of the moveable circle is call ed a spherical epicycloid; because the generating point being always at the same distance from the common ver tex of the cones, the curve described by it will be on the surface of a sphere.
If a circle roll along a straight line, any point in the plane of the circle will generate a curve, which is called a cycloid. The three classes of curves evidently belong to the same family ; for if we the cones by which the spherical epicycloid is generated to change to cy linders, by their axes becoming infinitely long ; then, the spherical will change to a plane epicycloid : and if again we suppose one of the cylinders to change to a plane, by the radius of its base becoming infinite, the epicycloid will be changed to a cycloid.
Galileo appears to have been the first that noticed particularly the cycloid, about the year 1599. From the elegance of its form, he was led to regard it as a proper figure for the arches of a bridge : he endeavoured to find its area, but did not succeed, on account of the imper fect state of mathematical analysis at that time. It was he that gave the curve the name by which it is now commonly known.
Mersennus, a learned Frenchman, also turned his at tention to the cycloid, ahout the year 1615 ; but neither was he able to sqtrAre it. Happening, however, in the year 1628 to become aquainted with Roberval, he pro posed the problem of squaring the curve to him ; but he also was unable to accomplish it. However, this cir
cumstance led Roberval to study the works of the Greek mathematicians, particularly the writings of Archime des, and about six years afterwards, when Mersennus again proposed the problem to him, he effected its so lution. His success was communicated by Mcrsennus to Descartes, as a thing of importance, but he did not think there was .much merit in the discovery. He was not made acquainted with the demonstration ; but in his answer to Mersennus, he sent a sketch of one. After wards, when Roberval, mortified by the opinion of Des cartes, retaliated by saying, that he had discovered his demonstration by his knowing beforehand what ought to be the result, the latter investigated the method of draw ing tangents, o the curve, and sent his solution to iNier sennus, challenging Roberval, and also Fermat, with whom he then had a dispute, to resolve the same prob lem. Roberval made various vain attempts, and sent five or six different solutions ; and it is even supposed that in the end he availed himself of the true solution of Fermat, which had come into the hands of his friend Mersennus, as Descartes called on him, but in vain, for a demonstration.
Galileo having been informed by Mersennus about the year 1639, that the question of the area of the cycloid was then greatly agitated among the French mathemati cians, but, as it appears, without having been made ac quainted with what had been found, he requested Caval lerius to attempt its solution. Cavallerius tried it, but did not succeed ; however, after the death of Galileo, which happened in 1642, his disciples Torricelli and Viviani, were more successful ; the former found the area, and the latter the method of drawing tangents to the curve. The claim of Torricelli to the honour of his discovery was contested by Roberval ; but the charge of plagiarism, which he brought against the Italian mathe matician, has not been believed by his countryman \Ion tucla, who has discussed the controversy in his History of 'Mathematics, second edition.