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 its centre of gravity, and Pierre Fermat deduced the surface of the spindle generated by its revolution. A famous period in the history of the cycloid is marked by a bitter controversy which sprang up between Descartes and Roberval. The evaluation of the area of the curve had made Roberval famous in France, but Descartes considered that the value of his investigation had been grossly exaggerated; he declared the problem to be of an elementary nature and submitted a short and simple solution. At the same time he challenged Roberval and Fermat to construct the tangent; Roberval failed but Fermat succeeded. This problem was solved independently by Vicenzo Viviani in Italy. The cartesian equation was first given by Wilhelm Gottfried Leibnitz (Acta eruditorum, 1686) in the form y = (2x − x2) + ∫(2x − x2)dx. Among other early writers on the cycloid were Phillippe de Lahire (1640–1718) and François Nicole (1683–1758).

The mechanical properties of the cycloid were investigated by Christiaan Huygens, who proved the curve to be tautochronous. His enquiries into evolutes enabled him to prove that the evolute of a cycloid was an equal cycloid, and by utilizing this property he constructed the isochronal pendulum generally known as the cycloidal pendulum. In 1697 John Bernoulli proposed the famous problem of the brachistochrone (see ), and it was proved by Leibnitz, Newton and several others that the cycloid was the required curve.

The method by which the cycloid is generated shows that it consists of an infinite number of cusps placed along the fixed line and separated by a constant distance equal to the circumference of the rolling circle. The name cycloid is usually restricted to the portion between two consecutive cusps (fig. 1, curve a); the fixed line LM is termed the base, and the line PQ which divides the curve symmetrically is the axis. The co-ordinates of any point R on the cycloid are expressible in the form x = a( + sin ); y = a(1 − cos ), where the co-ordinate axes are the tangent at the vertex O and the axis of the curve, a is the radius of the generating circle, and the angle R′CO, where RR′ is parallel to LM and C is the centre of the circle in its symmetric position. Eliminating between these two relations the equation is obtained in the form x = (2ay − y2) + a vers-¹ y/a. The clumsiness of the relation renders it practically useless, and the two separate relations in terms of a single parameter suffice for the deduction of most of the properties of the curve. The length of any arc may be determined by geometrical considerations or by the methods of the integral calculus. When measured from the vertex the results may be expressed in the forms s = 4a sin and s = &radic;(8ay); the total length of the curve is 8a. The intrinsic equation is s = 4a sin, and the equation to the evolute is s = 4a cos, which proves the evolute to be a similar cycloid placed as in fig. 2, in which the curve QOP is the evolute and QPR the original cycloid. The radius of curvature at any point is readily deduced from the intrinsic equation and has the value = 4 cos, and is equal to twice the normal which is 2a cos.

The trochoids were studied by Torricelli and F. van Schooten, and more completely by John Wallis, who showed that they possessed properties similar to those of the common cycloid. The cartesian equation in terms similar to those used above is x = a + b sin ; y = a − b cos, where a is the radius of the generating circle and b the distance of the carried point from the centre of the circle. If the point is without the circle, i.e. if a &lt; b, then the curve exhibits a succession of nodes or loops (fig. 1, curve b); if within the circle, i.e. if a > b, the curve has the form shown in fig. 1, curve c.

The companion to the cycloid is a curve so named on account of its similarity of construction, form and equation to the common cycloid. It is generated as follows: Let ABC be a circle having AB for a diameter. Draw any line DE perpendicular to AB and meeting the circle in E, and take a point P on DE such that the line DP = arc BE; then the locus of P is the companion to the cycloid. The curve is shown in fig. 3. The cartesian equation, referred to the fixed diameter and the tangent at B as axes may be expressed in the forms x = a, y = a(1 − cos ) and y − a = a sin (x/a − ); the latter form shows that the locus is the harmonic curve.

For epi- and hypo-cycloids and epi- and hypo-trochoids see .

—Geometrical constructions relating to the curves above described are to be found in T. H. Eagles, Constructive Geometry of Plane Curves. For the mechanical and analytical investigation, reference may be made to articles and. A historical bibliography of these curves is given in Brocard, Notes de bibliographie des courbes géométriques (1897). See also Moritz Cantor, Geschichte der Mathematik (1894–1901).

CYCLOMETER (Gr., circle, and  , measure), an instrument used especially by cyclists to determine the distance they have traversed. In a common form a stud attached to one spoke of the wheel engages with a toothed pinion and moves it on one tooth at each revolution. The pinion is connected with a train of clockwork, the gearing of which bears such a ratio to the circumference of the wheel that the distance corresponding to the number of times it has revolved is shown on a dial in miles or other units.

 CYCLONE (Gr. , whirling, from  , a circle), an atmospheric system where the pressure is lowest at the centre. The winds in consequence tend to blow towards the centre, but being diverted according to Ferrel’s law they rotate spirally inwards at the surface of the earth in a direction contrary to the movement of the hands of a watch in the northern hemisphere, and the reverse in the southern hemisphere. The whole system has a motion of translation, being usually carried forward with the great wind-drifts like eddies upon a swift stream. Thus their direction of movement over the British Islands is usually from S.W. to N.E., though they may remain stationary or move in other directions. The strength of the winds depends upon the atmospheric gradients. (See .)

 CYCLOPEAN MASONRY (from the Cyclopes, the supposed builders of the walls of Mycenae), a term in architecture, used, in conjunction with Pelasgic, to define the rude polygonal construction employed by the Greeks and the Etruscans in the walls of their cities. In the earliest examples they consist only of huge masses of rock, of irregular shape, piled one on the other and trusting to their great size and weight for cohesion; sometimes smaller pieces of rock filled up the interstices. The walls and gates of Tiryns and Mycenae were thus constructed. Later, these blocks were rudely shaped to fit one another. It is not always possible to decide the period by the type of construction, as this depended on the material; where stratified rocks could be obtained, horizontal coursing might be adopted; in fact, there are instances in Greece, where a later wall of cyclopean construction has been built over one with horizontal courses.

 CYCLOPES (, the round-eyed, plural of Cyclops), a type of beings variously described in Greek mythology. In Homer they are gigantic cave-dwellers, cannibals having only one eye, living a pastoral life in the far west (Sicily), ignorant of law and order, fearing neither gods nor men. The most prominent among them was Polyphemus. In Hesiod (Theogony, 264) they are the three sons of Uranus and Gaea—Brontes, Steropes and Arges,—storm-gods belonging to the family of the Titans, who furnished Zeus with thunder and lightning out of gratitude for his having released them from Tartarus. They were slain by Apollo for having forged the thunderbolt with which Zeus slew Asclepius. Later legend transferred their abode to Mt Aetna, the Lipari islands or Lemnos, where they assisted Hephaestus at his forge. A third class of Cyclopes are the builders of the so-called “Cyclopean” walls of Mycenae and Tiryns, giants with arms in their belly, who were said to have been brought by Proetus from Lycia to Argos, his original home (Pausanias ii. 16. 5; 25. 8). Like the Curetes and Telchines they are mythical types of prehistoric workmen and architects, and as such the objects of worship.

 CYCLOSTOMATA, or, a group of fishes including the ordinary lampreys and hagfish, and so called from the wide permanently gaping mouth which is without the hinged jaws characteristic of other vertebrates. 