See also:

• CATENARY (from Lat. eaten¢, a chain)

CATENARY (from See also:

• LAT

Lat. eaten¢, a See also:

• CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain)  , in See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics, the See also:

• CURVE (Lat. curvus, bent)

curve assumed by a See also:

• UNIFORM

uniform See also:

• CHAIN (through the O. Fr. citable, chcene, &c., from Lat. catena)

chain or See also:

• STRING

string See also:

• HANGING

hanging freely between two supports . It was investigated by Galileo, who erroneously determined it to be a See also:

• PARABOLA

parabola; Jungius detected Galileo's See also:

• ERROR (Lat. error, from errare, to wander, to err)

error, but the true See also:

• FORM (Lat. forma)

form was not discovered until 1691, when See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James See also:

• BERNOULLI

Bernoulli published it as a problem in the Acta Erudilorum . Bernoulli also considered the cases when (1) the chain was of variable See also:

• DENSITY (Lat. densus, thick)

density, (2) extensible, (3) acted upon at each point by a force directed to a fixed centre . These curves attracted much See also:

• ATTENTION (from Lat. ad-tendo, await, expect; the condition of being " stretched " or " tense ")

attention and were discussed by 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 Bernoulli, See also:

• LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

Leibnitz, See also:

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

Huygens, See also:

• DAVID
• DAVID (a Hebrew name meaning probably beloved 1)
• DAVID, FELICIEN (1810—1876)
• DAVID, GERARD [GHEERAERT DAVIT], (?=1523)
• DAVID, PIERRE JEAN (1789–1856)
• DAVID, ST (Dewi, Sant)

David See also:

• GREGORY
• GREGORY (Gregorius)
• GREGORY (Grigorii) GRIGORIEVICH ORLOV, COUNT (1734-1783)
• GREGORY, EDWARD JOHN (1850-19o9)
• GREGORY, OLINTHUS GILBERT (1774—1841)
• GREGORY, ST (c. 213-C. 270)
• GREGORY, ST, OF NAZIANZUS (329–389)
• GREGORY, ST, OF NYSSA (c.331—c. 396)
• GREGORY, ST, OF TOURS (538-594)

Gregory and others . Essais de critique generale (2nd ed.), La Logique, i. pp . 184, 190, 207-225 . Discussions, p . 577 . ' See also:

• LOGIC (Xoy1K7, sc. rixvrt, the art of reasoning)

Logic, i . 83 ; cf . See also:

• BAIN, ALEXANDER (1818-1903)
• BAIN, ANDREW GEDDES (1797-1864)

Bain, Ded . See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

Log., App .

C . The See also:

• MECHANICAL

mechanical properties of the curves are treated in the See also:

• ARTICLE (from Lat. articulus, a joint)

article See also:

• MECHANICS

MEcHANIcs,where various forms are illustrated . The See also:

• SIMPLE

simple See also:

• CATENARY (from Lat. eaten¢, a chain)

catenary is shown in the figure . The cartesian See also:

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

equation referred to the See also:

• AXIS (Lat. for " axle ")

axis and directrix is y=c cosh (x/c) or y= 2c(ez/°+e-,/,) ; other forms are s=c sinh (x/c) and y2=c2+s2, s being the arc measured from the vertex; the See also:

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

intrinsic equation is s=c tan tt . The See also:

• RADIUS

radius of curvature and normal are each equal to c See also:

• SECT

sect ' . The See also:

• SURFACE

surface formed by revolving the catenary about its directrix is named the alysseide . It is a minimal surface, i.e. the catenary solves the problem : to find a curve joining two given points, which when revolved about a See also:

• LINE

line co-planar with the points traces a surface of minimum See also:

• AREA

area (see See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

VARIATIONS, CALCULUS OF) . x The involute of the catenary is called the tractory, tractrix or antifriction curve; it has a See also:

• CUSP (Lat. cuspis, a spear, point)

cusp at the vertex of the catenary, and is asymptotic to the directrix . The cartesian equation is x=d (c'—y') +;clog[[c- d (c2_y2)}/(c+jl (c2+y2)}), and the curve has the geometrical See also:

• PROPERTY

property that the length of its tangent is See also:

• CONSTANT, BENJAMIN (1845-1902)

constant . It is named the tractory, since a See also:

• WEIGHT

weight placed on the ground and See also:

• DRAWN

drawn along by means of a flexible string by a See also:

• PERSON, OFFENCES AGAINST THE

person travelling in a straight line, the weight not being in this line, describes the curve in question . It is named the antifriction curve, since a See also:

• PIVOT (Fr. pivot; probably connected with Ital. pivolo, peg, pin, diminutive of piva, pipa, pipe)

pivot and step having the form of the surface generated by revolving the curve about its See also:

• VERTICAL (from Lat. vertex, highest point)

vertical axis See also:

• WEAR

wear away equally (see MECHANICS: Applied) .