EPICYCLI D, in geometry, a curve generated by the revolution of a point in the circumfe•,nce of a circle, while it is moved round the circumference of another quiescent circle in the same plane, so that in each circle, the distance between the point of contact at the commencement of the motion, and the point of contact at any instant while in motion, is equal one to the other. Hence if the circum ferences are equal, all parts of the circumference of the moving circle will have been in contact with all the parts of the circumference of the quiescent circle.

It' the generating circle proceed along the convexity of the periphery, it is called an upper, or exterior epicycloid : if along the concavity, a lower, or interior cpicycloid.

The part of the quiescent circle which the generating circle moves along, is called the base.

The length of any part of the enrve. which any given point in the revolving circle has described from the time of its first in contact with the quiescent circle, is, to double the versed sine of half the arc, as the sum of the diameters of the circles, to the semi-diameter of the quiescent circle ; m the circumference of the moving circle be carried along the convex side of the quiescent circle ; but it' upon the concave side, as the ditkrenee of the diameters to the said semi-diameter.

Dr. Halley gives us a general proposition for the measuring of all cydoids and epicydoids : thus, the area of a cycloid or epicycloid, either primary, contracted, or prolate, is to the area of the generating circle, and also to the areas of the parts generated in these curves, to the areas of the analogous segments of the circle, as the sum of double the velocity of the centre, and the velocity of the circular motion, to the velocity of the circular motion. The demonstration hereof', sec Phil. Trans. No. :21S.

The areas of epicycloid s may be determined by the tbllow ing proportion: As the radius of the circle of the base three times that of the radius, together with twice that of the generating circle, so is the circular segment to the epicycloidal sector, or the whole generating circle to the whole area of the epicvdoid.

As to the tangents, it is known from the time of Descartes, that a line drawn from any point to that of the base, which touches the circle, whilst this point is described, is perpen dicular to the curve, and consequently to the tangent.

Maupertius, discussing this subject, conceived a polygon to revolve upon another, the sides of which are respectively equal ; one of the angles described a curve, the periphery of which is formed of arcs of circles, and the area is composed of circular sectors, and right-lined triangles. Ile determined the proportion of the area, and of the periphery of this figure to those of the generating polygon. lie also supposed those polygons to become circles, the figure described to become an epicycloid, and the above-mentioned proportion, modified agreeably to this supposition, gave him the area and periphery of the epieycloid. Jfem. de l'Acad. 17:27.

It does not appear that any writer published an account of epicyeloids, before the celebrated Sir Isaac Newton, who, in the first book of his Principia, proposed a general, and a very simple method of rectifying these curves. After him, Bernouilli, during his residence at Paris, showed how, by means of the integral and differential calculus, to determine their area and rectification. The invention of epicycloids is, however, ascribed to Reaumur, the celebrated Danish philosopher, during his residence at Paris, about the year 16'74. These curves appeared to him to be such as best suited the teeth of wheels, constructed so as to diminish their mutual friction, and to render the action of the power more uniform ; hence he was led to consider them, and to this purpose they have been applied. llowever. M. de la Hire makes no mention of ReauMn•, and seems to claim the merit of this geometrical and mechanical invention. But M. Leibnitz, who resided at Paris in 1674, and the two following years, says, that the invention of epieyeloids, and their application to mechanics, was the work of this Danish mathematician, and that he was esteemed the author of it.