See also:

• CYCLOID (from Gr. iciidws, circle, and ennos, form)

CYCLOID (from Gr. iciidws, circle, and ennos, See also:

• FORM (Lat. forma)

form)  , in See also:

• GEOMETRY

geometry, the See also:

• CURVE (Lat. curvus, bent)

curve traced out by a point carried on a circle which rolls along a straight See also:

• LINE

line . The name See also:

• CYCLOID (from Gr. iciidws, circle, and ennos, form)

cycloid is now restricted to the curve described when the tracing-point is on the circumference of the circle; if the point is either within or without the circle the curves are generally termed trochoids, but they are also known as the prolate and curtate cycloids respectively . The cycloid is the simplest member of the class of curves known as roulettes . No mention of the cycloid has been found in writings See also:

• PRIOR (from Lat. prior—former, and hence superior, through O. Fr. priour)
• PRIOR, MATTHEW (1664-1721)

prior to the 15th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century . See also:

• FRANCIS
• FRANCIS (FRarrcols) OF SALES, ST (1567-1622)
• FRANCIS (Lat. Franciscus, Ital. Francesco, Span. Francisco, Fr. Francois, Ger. Franz)
• FRANCIS, SIR PHILIP (174o-1818)

Francis Schooten (Commentary on See also:

• DESCARTES, RENE (1596-1650)

Descartes) assigns the invention of the curve to Rene Descartes and the first publication on this subject after Descartes to Marin See also:

• MERSENNE, MARIN

Mersenne . Evangelista 'See also:

• TORRICELLI, EVANGELISTA (1608—1647)

Torricelli, in the first See also:

• REGULAR

regular dissertation on the cycloid (De dimensione cycloidis, an appendix to his De dimensione parabolae, 1644), states that his friend and See also:

• TUTOR (Lat. tutor, guardian, tueri, to watch over, protect)

tutor Galileo discovered the curve about 1599 . See also:

• JOHN
• JOHN (1167–1216)
• JOHN (1290-c. 1320)
• JOHN (1296-1346)
• JOHN (1371–1419)
• JOHN (1468-1532)
• JOHN (1801-1873)
• JOHN (Heb. llni')
• JOHN (ZAPOLYA) (1487-1540)
• JOHN, 4TH MARQUESS OF TWEEDDALE (c. 1695-1762)
• JOHN, DON (1545-1578)
• JOHN, DON (1629–1679)
• JOHN, GOSPEL OF ST
• JOHN, or HAYS (1513–1571)
• JOHN, THE APOSTLE
• JOHN, THE EPISTLES OF

John See also:

• WALLIS, JOHN (1616-1703)

Wallis discussed both the See also:

• HISTORY

history and properties of the curve in a See also:

• TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

tract De cycloide published at See also:

• OXFORD
• OXFORD, EARLS OF
• OXFORD, EDWARD DE VERE, 17TH EARL
• OXFORD, JOHN DE VERE, 13TH EARL OF (1443-1513)
• OXFORD, PROVISIONS OF
• OXFORD, ROBERT DE VERE, 9TH EARL OF (1362-1392)
• OXFORD, ROBERT HARLEY, 1ST

Oxford in 1659 . He there shows that the cycloid was investigated by Carolus Bovillus about 1500, and by See also:

• CARDINAL (Lat. cardinalis)

Cardinal See also:

• CUSANUS, NICOLAUS (NICHOLAS OF CUSA) (1401–1464)

Cusanus (Nicolaus de Cusa) as See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early as 1451 . Honore Fabri (Synopsis geometrica, 1669) treated of the curve and enumerated many theorems concerning it . Many other mathematicians have written on the cycloid—Blaise See also:

• PASCAL, BLAISE (1623-1662)
• PASCAL, JACQUELINE (1625-T661)

Pascal, W . G . See also:

• LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

Leibnitz, the Bernoullis, See also:

• ROGER (d. 1139)
• ROGER (d. 1181)

Roger See also:

• COTES, ROGER (1682-1716)

Cotes and others—and so assiduously was it studied that it was sometimes named the " See also:

• HELEN

Helen of Geometers." The determination of the See also:

• AREA

area was the subject of many investigations and much controversy .

Galileo attempted the evaluation by weighing the curve against the generating circle; this rough method gave only an approximate value, viz., a little less than thrice the generating circle . Torricelli, by employing the " method of indivisibles," deduced that the area was exactly three times that of the generating circle; this result had been previously established in 164o in See also:

• FRANCE
• FRANCE, ANATOLE (1844– )

France by G . P. de See also:

• ROBERVAL, GILLES PERSONNE (or PERSONIER) DE (1602-1675)

Roberval, but his investigation was unknown in See also:

• ITALY
• ITALY (Italia)

Italy . Blaise Pascal determined the area of the See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section made by any line parallel to the See also:

• BASE

base and the volumes and centres of gravity of the solids generated by revolving the curve about its See also:

• AXIS (Lat. for " axle ")

axis and base . Before See also:

• PUBLISHING

publishing his results he proposed these problems for public competition in 1658 under the assumed name of See also:

• AMOS
• AMOS, SHELDON (1835–1886)

Amos Dettonville . John Wallis in See also:

• ENGLAND
• ENGLAND, THE CHURCH OF

England, and A. la Louere in France, accepted the See also:

• CHALLENGE (O. Fr. chalonge, calenge, &c., from Lat. calumnia, originally meaning trickery, from calvi, to deceive, hence a false accusation, a " calumny ")

challenge, but the former could only submit in-correct solutions, while the latter failed completely . Having established his priority, Pascal published his investigations, which occasioned a See also:

• GREAT

great sensation among his contemporaries, and Wallis was enabled to correct his methods . See also:

• SIR

Sir See also:

• CHRISTOPHER, SAINT (Christophorus, Christoferus)

Christopher See also:

• WREN (O. Eng. wrcknna, Mid. Eng. wrenne; Icel. rindill)
• WREN, SIR CHRISTOPHER (1632-1723)

Wren, the famous architect, determined the length of the arc and its centre of gravity, and See also:

• PIERRE

Pierre See also:

• FERMAT, PIERRE DE (1601-1665)

Fermat deduced the See also:

• SURFACE

surface of the spindle generated by its revolution . A famous See also:

• PERIOD
• PERIOD (Gr. irepioSos, a going or way round, circuit, aepi, round, and &Sis, way, road)

period in the history of the cycloid is marked by a See also:

• BITTER, KARL THEODORE FRANCIS (1867– )

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 See also:

• SHORT, FRANCIS JOB (1857– )

short and See also:

• SIMPLE

simple See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution . At the same See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

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 See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation was first given by Wilhelm Gottfried Leibnitz (Ada eruditorum; 1686) in the See also:

• FORM (Lat. forma)

form y=(2x—x2)z+f (2x—x2)idx . Among other early writers on the cycloid were Phillippe de See also:

• LAHIRE, LAURENT DE (1606-1656)

Lahire (164o—1718) and See also:

• FRANCOIS (1558-1614)

Francois See also:

• NICOLE, PIERRE (1625–1695)

Nicole (1683—1758) . The See also:

• MECHANICAL

mechanical properties of the cycloid were investigated by Christiaan See also:

• HUYGENS, CHRISTIAAN (1629-1695)
• HUYGENS, SIR CONSTANTIJN (1596-1687)

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 See also:

• PROPERTY

property he constructed the isochronal pendulum generally known as the cycloidal pendulum . In 1697 John See also:

• BERNOULLI

Bernoulli proposed the famous problem of the See also:

• BRACHISTOCHRONE (from the Gr. /3paxurros, shortest, and xpovos, time)

brachistochrone (see See also:

• MECHANICS

MECHANICS), and it was proved by Leibnitz, See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

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 See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite number of cusps placed along the fixed line and separated by a See also:

• CONSTANT, BENJAMIN (1845-1902)

constant distance equal to the circumference of the See also:

• ROLLING

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 Q line PQ which divides the curve symmetrically is the axis . The e co-ordinates of any point R on the • cycloid are expressible in the form x=a(O+See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

sin 0); y=a (1—See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

cos 0), hP . ~.-- M where the co-See also:

• ORDINATE

ordinate axes are the tangent at the vertex 0 and the the generating circle, and 0 the See also:

• ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position . Eliminating 0 between these two relations the equation is obtained in the form x=(See also:

• LAY

lay—y2)z+a vers-1 y/a . The clumsiness of the relation renders it practically useless, and the two See also:

• SEPARATE

separate relations in terms of a single parameter 6 suffice for the See also:

• DEDUCTION (from Lat. deducere, to take or lead from or out of, derive)

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 R from the vertex the results may be expressed in the forms s=4a sin 20 and s=-(8ay); the See also:

• TOTAL

total length of the curve is 8a . The See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

intrinsic P equation is s=4a sin lb, 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 at any point is readily deduced from the intrinsic equation and has the value p=4 cos 10, and is equal to twice the normal which is 2a cos 20 . The trochoids were studied by Torricelli and F. See also:

• VAN

van Schooten, and more completely by John Wallis, who showed that they possessed properties similar to those of the See also:

• COMMON

common cycloid . The cartesian equation in terms similar to those used above is x=aO+b sin 0; y=a—b cos 0, where a is the See also:

• RADIUS

radius of the generating circle and b the distance of the carried point from the centre of the circle . If s the point is without the circle, i.e. if a< b, then the curve exhibits a See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

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. i, curve c . A The See also:

• COMPANION (through the O. Fr. compaignon or compagnon, from the Late Lat. companio,—cum, with, and panis, bread,—one who shares meals with another; the word has been wrongly derived from the Late Lat. compagnus, one of the same pages or district)

companion to the cycloid is a curve so struction, form and equation to the common cycloid . It is generated as follows: Let See also:

• ABC

ABC be a circle having AB for a See also:

• DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter . Draw any line DE perpendicular to AB and See also:

• MEETING (from " to meet," to come together, assemble, 0. Eng. metals ; cf. Du. moeten, Swed. mota, Goth. gamotjan, &c., derivatives of the Teut. word for a meeting, seen in O. Eng. Wit, moot, an assembly of the people; cf. witanagemot)

meeting the circle in E, and take a point P on DE such that the line DP =arc BE ; then the See also:

• LOCUS (Lat. for " place "; in Gr. rinros)

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=aO, y=a(I—cos 0) and y—a=a sin (x/a—.x); the latter form shows that the locus isthe See also:

• HARMONIC

harmonic curve . For epi- and hypo-cycloids and epi- and hypo-trochoids see EPICYcLolD . REFERRNcEs.—Geometrical constructions See also:

• RELATING

relating to the curves above described are to be found in T . H . Eagles, Constructive Geometry of See also:

• PLANE

Plane Curves . For the mechanical and See also:

• ANALYTICAL

analytical investigation, reference may be made to articles MECHANICS and INFINITESIMAL CALCULUS . A See also:

• HISTORICAL

historical bibliography of these curves is given in Brocard, Notes de bibliographie See also:

• DES

des courtier geometriques (1897) . See also See also:

• MORITZ, KARL PHILIPP (1757-1793)

Moritz Cantor, Geschichle der Mathematik (1894—1901) .