, is a curve generated by the revolution of a point of the periphery of a circle, which rolls along or upon the circumference of another circle, either on the convex or concave side of it.

When a circle rolls along a straight line, a point in its circumference describes the curve called a cycloid. | But if, instead of the right line, the circle roll along the circumference of another circle, either equal to the former or not, then the curve described by any point in its circumference is what is called the Epicycloid.

If the generating circle roll along the convexity of the circumference, the curve is called an Upper, or Exterior Epicycloid; but if along the concavity, it is called a Lower, or Interior Epicycloid. Also the circle that revolves is called the Generant; and the arc of the other circle along which it revolves, is called the Base of the Epicycloid. Thus, ABC or BLV is the Generant; DPVE the Exterior Epicycloid, its axis BV; DPUE the Interior Epicycloid; and DBE their common Base.

For the Length of the Curve.

The length of any part of the curve of an Epicycloid, which any given point in the revolving circle has described, from the position where it touched the circle upon which it revolved, is to double the versed side of half the arc which all the time of revolving touched the quiescent circle, as the sum of the diameters of the circles, is to the semidiameter of the quiescent circle in the Exterior Cycloid; or as the difference of the diameters is to that semidiameter, for the Interior one.

For the Area of the Epicycloid.

Dr. Halley has given a general proposition for the measuring of all cycloids and Epicycloids: thus, the area of a cycloid, or Epicycloid, either primary, or contracted, or prolate, is to the area of the generating circle; and also the areas of the parts generated in those curves, to the areas of analogous segments of the circle; as the sum of double the velocity of the centre and the velocity of the circular motion, is to this velocity of the circular motion. See the Demonstr. in the Philos. Trans. number 218.

Spherical Epicycloids are formed by a point of the revolving circle, when its plane makes a constant angle with the plane of the circle on which it revolves. Messrs. Bernoulli, Maupertuis, Nicole, and Clairaut, have demonstrated several properties of these Epicycloids, in Hist. Acad. Sci. for 1732.

Parabolic, Elliptic, &c. Epicycloids.

If a parabola roll upon another equal to it; its focus will describe a right line perpendicular to the axis of the quiescent parabola: also the vertex of the rolling parabola will describe the cissoid of Diocles; and any other point of it will describe some one of Newton's defective hyperbolas, having a double point in the like point of the quiescent parabola.

In like manner, if an ellipse revolve upon another ellipse, equal and similar to it, its focus will describe a circle, whose centre is in the other focus, and consequently the radius is equal to the axis of the ellipsis; and any other point in the plane of the ellipse will describe a line of the 4th order.

The same may be said also of an hyperbola, revolving upon another, equal and similar to it; for one of the foci will describe a circle, having its centre in the other focus, and the radius will be the principal axis of the hyperbola; and any other point of the hyperbola will describe a line of the 4th order.

Concerning these lines, see Newton's Principia, lib. 1; also De la Hire's Memoires de Mathematique &c, where he shews the nature of this line, and its use in Mechanics; see also Maclaurin's Geometria Organica.

EP<*>HANY, a christian festival, otherwise called the Manifestation of Christ to the Gentiles, observed on the 6th of January, in honour of the appearance of our Saviour to the three magi or wise men, who came to adore him and bring him presents.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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