CYCLOID. 705 CYCLOPEAN ARCHITECTURE, around the outside of the fixed circumference is called an epicycloid. On the other hand, that produced by rolling the generating circle ini the inside of the fixed circle is called a hypocyeloid. These curves belong to a general class *illcd 'roulettes.' The construction for the cycloid was known to Bouvelles (1503), but its name is due to Galilei (q.v.), who in a letter to Torri- eclli (103!)) recommends it for bridge arches. The term trochoid is due to Roberval (q.v.), and the term roulette (1C.')9) to Pascal (q.v.). Eoberval also effected (1U34) 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 revolution about its axis. Descartes constructed its tangents, and Pascal (1058) determined the length of its arc, 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 cii'cle. If Ai ( Fig. 1 ) be taken as the origin of coordinates, and a be the radius of the generating circle, © the equation of the cycloid is a? = a vers— i — ■^/'2ay^l/■. It is simpler, however, to use the expressions for x and y separately ; viz. a; = a ( e — sin d) , y = a (1 — cos 6). The equations of the hypocyeloid are x = (a — b) cos9 -f- Jcos - — - — -0, 2/=: (a — 6) sin 6 — 6sm— r — <?, where a and 6 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 liypocy- cloid is x'--y'j=a', 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 ])article from any point on its inverted arc 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. an object acted upon by the force of gravity, and setting out from an}' 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 brxjchistocJirone (Gr. Ppuxia- Tof, brachistos, shortest, and XP'^'"'C> chronos, time. The problem of finding the brachisto- ehrone was proposed by Jean (Johann) Ber- noulli in 109C, 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, Aperc-ii hisiorique sur Vorigine et le d^veloppenicnt dcs mcihodea en g^otnvtrie (Paris, 1875) ; de Groningue, Histoire de la cy- clo'ide (Hambiirg. 1701) : Tannery, "La cycloi'de dans I'antiquitr." in Bulletin des sciences mathe- matiques (Paris, 1883). CY'CLOID FISHES. One of the four orders of lishes pruposc<l by Ag.issi/i, ba.sed on the fliar- acter of the scah^s. Cycloid scales have the ])0S- terior or free margin smooth and not s])iMous. Cycloid-like ctenoid (<l.v.) scales are not covered with enamel, and belong to many of the present as well as many fossil fishes. The chub and its allies are examples. CYCLOM'ETER (Gk. /cikAoc, Icyklos, circle -- /liriioit. iititnm, measure). An instrument for recording the revolutions of any rjtating object, as of a carriage-wheel, bicyele-wheid, or certain parts of machinery. A similar instru- ment for measuring distances traversed is an odometer. This instrument, invented by Hud- son, is e.xtensively u.sed by surveyor-- in collect- ing data for maps. It is commonly attached to the wheel of a wagon, or to a light vehicle drawn by hand. The term cyclometry is often applied to the method of loeasuriiig circumfer- ences or areas of circles (q.v. ), but more generally it refers to the theory of circular functions. See P'ONCTIOXS. CY'CLONE. See Storm. CY'CLOP.a:'DIA. See Enctclop.edia. CY'CLOPE'AN ARCHITECTURE, or Ma- sonry. The name frequently used for an ancient w^all of large, irregular stones, rudely hewn or quite unwrought. The term originated in Greece, where structures of this kind were fabled to have been the work of the Cj'clopes. The ancients also attributed them to the Pelasgians (q.v.), whence such walls are sometimes called Pelasgian. The walls of Tiryns (q.v.), near Nauplia, arc an ex- ample of the ruder stylo of Cyclopean masonry. They are of irregular unshajjen stones, from to 10 feet long, from 3 to 4 feet wide, and from 2 to 3 feet deep; the interstices are filled up by small stones, and clay mortar was emplojed to bind them, though it has now been washed away to a great extent. The walls of ]M3'eena^ arc in part of the saiiie rude construction as at Tiryns, but near the Lion Gate they are faced with huge reetangular blocks, fitted in rudely horizontal courses, and the same style of masonry (but more carefully executed) is e?nployed in the great beehive tombs. A portion of the wall of -Myeente — probably of later date — is built of polygonal stones, carefully fitted so as to leave no interstices. Walls of the same general char- acter are found in Asia Minor and Italy, where they surround many of the old Etruscan towns, though here the walls are more commonly in the ruilo ashlar masonry found at Mycenip. While in Greece these walls belong for the most part to the Jlycen.Tan period, and are ]irobably to be attributed to the Aeh;ran domination, it is not likely that this style of fortification was peculiar to any one race, as similar masonry has been found in China, and also in Peru, and on a smallei scale in the British Isles. Polygonal masonry, composed of carefully hewn and fitted blocks, is common in Greek works of later times, and the early walls of Troy, as well as the tombs already mentioned, show that the Myccna'an civi- lization was capable of building walls of hewn and fitted stones, as good as. or better than, those erected in the classical times. The Cyclopean architecture is discussed in histories of archi- tecture (q.v.), or in works dealing with the Mycenaean Age (see Arch.TvOLOGY) , or Etruria (q.v.). For a descrijition of the remains
