CYCLOID (Gk. slisXoct3775, kyklocidns, circle like, from KbriXos, loklos, circle + c‘oos, cidos, form). A plane curve, the locus of a point on the circumference of a circle which rolls along a straight line. If, in Fig. 1, circle 0 rolls on the line the point P traces the arc of the cycloid If the generating point is taken at Q, within the circle, the resulting curve is called a prolate cycloid. If the generat ing point is taken at II. in the plane of the circle, the resulting curve is a curtate cycloid or trochoid. If the generating circle rolls on a fixed circle, instead of a straight line, curves like those in Fig. 2 are produced (see CARDIOID) The curve formed by rolling the generating circle around the outside of the fixed circumference is called an epicycloid. On the other band, that produced by rolling the generating circle on the inside of the fixed circle is called. a hypoeyVoid. These curves belong to a general class called `roulettes.' The construction for the cycloid was known to Bouvelles (1503), but its name is incI to Galilei (q.v.). who in a letter to Tiirri eelli (1639) recommends it fur bridge arches. The term trochoid is due to Itoberval (q.v.), and the term roulette (1659) to Pascal (q.v.). Roberval also effected (1634) the quadrature of the cycloid, showing that it equals three times the area of the generating circle, and he deter mined the volume obtained by about its axis. Descartes constructed its tangents, and Pascal (1658) determined the length of its are, and the centre of gravity of its surface and of the corresponding solid of revolution. The length of one branch of the cycloid is four times the diameter of the generating circle, and its area is three times that of the same circle. if (Fig. 1) be taken as the origin of coordinates, and a be the radius of the generating circle, the equation of the cycloid is n= a —V it is simpler, however, to use the expressions for x and p separately; viz. x = a (0—sin 0), y = a ( 1—eos 0). The equations of
the hypoeyeloid are x = (a—b) cos° (a—b) bcos 0, a—b) y = (a— b) siu b '0, where a and b are the radii of the fixed and rolling circles. If the radius of the fixed circle is four times that of the rolling circle, the equation of the hypocy cloid is -l-p5=e, a being the radius of the fixed circle, as in Fig. 2. Because of the elegance of its properties and because of its numerous applica tions in mechanics, the cycloid is the most im portant of the transcendental curves. One of its most interesting properties is that the time of descent from rest of a particle from any point on its inverted are to the lowest point is the same; that is, the cycloid is an isochronous curve. Thus, on an ideally hard and smooth sur face whose sections are cycloids, the particle, hav ing reached the lowest point, will, through the momentum received in its fall, ascend the oppo site branch to a height equal to that through which it fell, losing velocity at the same rate as it acquired it. The cycloid is also the curve of quickest descent: i. e. au object acted upon by the force of gravity, and setting out from any point of the cycloid, will reach any other point of this curve in shorter time than by follow ing any other path. The cycloid is therefore referred to as the brachistoeh•onc (Gr. roc, brachistos, shortest, and xporoc, chronos, time. The problem of finding the brachisto chrone was proposed by Jean (Johann) Ber noulli in 1696, and formed the first important step in the calculus of variations. It was solved by Bernoulli himself, by Leibnitz, Newton, L'HOpital, and Jacques (Jakob) Bernoulli. For the interesting history of the cycloid, consult any of the best histories of mathematics, and also: Chasles, Apercit lasloriyue sir at le (11veloppement des methodes en y(lowe'trie (Paris, 1875) de Groningue. Histoire dc la cy cloide (Hamburg, 1701) ; "La cycloble dans Pantiquitis." in Bulletin des sciences inathe matiques (Paris, 1883).