central conic for the focus, and the first positive pedal of a circle
for any point. The form of the limaçon depends on the ratio of
the two constants; if a be greater
than b, the curve lies entirely outside
the circle; if a equals b, it is known
as a cardioid (q.v.); if a is less than
b, the curve has a node within the
circle; the particular case when
b＝2a is known as the trisectrix
(q.v.). In the figure (1) is a limaçon,
(2) the cardioid, (3) the trisectrix.
Properties of the limaçon may be deduced from its mechanical construction; thus the length of a focal chord is constant and the normals at the extremities of a focal chord intersect on a fixed circle. The area is (b2+a2/2)π, and the length is expressible as an elliptic integral.
LIMASOL, a seaport of Cyprus, on Akrotiri Bay of the south
coast. Pop. (1901) 8298. Excepting a fort attributed to the
close of the 12th century the town is without antiquities of
interest, but in the neighbourhood are the ancient sites of
Amathus and Curium. Limasol has a considerable trade in
wine and carobs. The town was the scene of the marriage of
Richard I., king of England, with Berengaria, in 1191.
LIMB. (1) (In O. Eng. lim, cognate with the O. Nor. and Icel.
limr, Swed. and Dan. lem; probably the word is to be referred
to a root li- seen in an obsolete English word “lith,” a limb, and
in the Ger. Glied), originally any portion or member of the body,
but now restricted in meaning to the external members of the
body of an animal apart from the head and trunk, the legs and
arms, or, in a bird, the wings. It is sometimes used of the lower
limbs only, and is synonymous with “leg.” The word is also
used of the main branches of a tree, of the projecting spurs of a
range of mountains, of the arms of a cross, &c. As a translation
of the Lat. membrum, and with special reference to the church
as the “body of Christ,” “limb” was frequently used by
ecclesiastical writers of the 16th and 17th centuries of a person
as being a component part of the church; cf. such expressions
as “limb of Satan,” “limb of the law,” &c. From the use of
membrum in medieval Latin for an estate dependent on another,
the name “limb” is given to an outlying portion of another,
or to the subordinate members of the Cinque Ports, attached to
one of the principal towns; Pevensey was thus a “limb” of
Hastings. (2) An edge or border, frequently used in scientific
language for the boundary of a surface. It is thus used of the
edge of the disk of the sun or moon, of the expanded part of a
petal or sepal in botany, &c. This word is a shortened form of
“limbo” or “limbus,” Lat. for an edge, for the theological use
of which see Limbus.
LIMBACH, a town in the kingdom of Saxony, in the manufacturing
district of Chemnitz, 6 m. N.W. of that city. Pop. (1905)
13,723. It has a public park and a monument to the
composer Pache. Its industries include the making of worsteds,
cloth, silk and sewing-machines, and dyeing and bleaching.
LIMBER, an homonymous word, having three meanings.
(1) A two-wheeled carriage forming a detachable part of the
equipment of all guns on travelling carriages and having on it a
framework to contain ammunition boxes, and, in most cases,
seats for two or three gunners. The French equivalent is avant-train,
the Ger. Protz (see Artillery and Ordnance). (2) An
adjective meaning pliant or flexible and so used with reference
to a person’s mental or bodily qualities, quick, nimble, adroit.
(3) A nautical term for the holes cut in the flooring in a ship
above the keelson, to allow water to drain to the pumps.
The etymology of these words is obscure. According to the New English Dictionary the origin of (1) is to be found in the Fr. limonière, a derivative of limon, the shaft of a vehicle, a meaning which appears in English from the 15th century but is now obsolete, except apparently among the miners of the north of England. The earlier English forms of the word are lymor or limmer. Skeat suggests that (2) is connected with “limp,” which he refers to a Teutonic base lap-, meaning to hang down. The New English Dictionary points out that while “limp” does not occur till the beginning of the 18th century, “limber” in this sense is found as early as the 16th. In Thomas Cooper’s (1517?–1594) Thesaurus Linguae Romanae et Britannicae (1565), it appears as the English equivalent of the Latin lentus. A possible derivation connects it with “limb.”
LIMBORCH, PHILIPP VAN (1633–1712), Dutch Remonstrant
theologian, was born on the 19th of June 1633, at Amsterdam,
where his father was a lawyer. He received his education at
Utrecht, at Leiden, in his native city, and finally at Utrecht
University, which he entered in 1652. In 1657 he became a
Remonstrant pastor at Gouda, and in 1667 he was transferred to
Amsterdam, where, in the following year, the office of professor
of theology in the Remonstrant seminary was added to his
pastoral charge. He was a friend of John Locke. He died at
Amsterdam on the 30th of April 1712.
His most important work, Institutiones theologiae christianae, ad praxin pietatis et promotionem pacis christianae unice directae (Amsterdam, 1686, 5th ed., 1735), is a full and clear exposition of the system of Simon Episcopius and Stephan Curcellaeus. The fourth edition (1715) included a posthumous “Relatio historica de origine et progressu controversiarum in foederato Belgio de praedestinatione.” Limborch also wrote De veritate religionis Christianae amica collatio cum erudito Judaeo (Gouda, 1687); Historia Inquisitionis (1692), in four books prefixed to the “Liber Sententiarum Inquisitionis Tolosanae” (1307–1323); and Commentarius in Acta Apostolorum et in Epistolas ad Romanos et ad Hebraeos (Rotterdam, 1711). His editorial labours included the publication of various works of his predecessors, and of Epistolae ecclesiasticae praestantium ac eruditorum virorum (Amsterdam, 1684), chiefly by Jakobus Arminius, Joannes Uytenbogardus, Konrad Vorstius (1569–1622), Gerhard Vossius (1577–1649), Hugo Grotius, Simon Episcopius (his grand-uncle) and Gaspar Barlaeus; they are of great value for the history of Arminianism. An English translation of the Theologia was published in 1702 by William Jones (A Complete System or Body of Divinity, both Speculative and Practical, founded on Scripture and Reason, London, 1702); and a translation of the Historia Inquisitionis, by Samuel Chandler, with “a large introduction concerning the rise and progress of persecution and the real and pretended causes of it” prefixed, appeared in 1731. See Herzog-Hauck, Realencyklopädie.
LIMBURG, one of the many small feudal states into which the
duchy of Lower Lorraine was split up in the second half of the
11th century. The first count, Walram of Arlon, married Judith
the daughter of Frederick of Luxemburg, duke of Lower Lorraine
(d. 1065), who bestowed upon him a portion of his possessions
lying upon both sides of the river Meuse. It received its name
from the strong castle built by Count Walram on the river Vesdre,
where the town of Limburg now stands. Henry, Walram’s son
(d. 1119), was turbulent and ambitious. On the death of Godfrey
of Bouillon (1089) he forced the emperor Henry IV. to recognize
him as duke of Lower Lorraine. He was afterwards deposed
and imprisoned by Count Godfrey of Louvain on whom the ducal
title had been bestowed by the emperor Henry V. (1106). For
three generations the possession of the ducal title was disputed
between the rival houses of Limburg and Louvain. At length
a reconciliation took place (1155); the name of duke of Lower
Lorraine henceforth disappears, the rulers of the territory on the
Meuse become dukes of Limburg, those of the larger territory to
the west dukes of Brabant. With the death of Duke Walram IV.
(1280) the succession passed to his daughter, Irmingardis,
who was married to Reinald I., count of Guelders. Irmingardis
died without issue (1282), and her cousin, Count Adolph of Berg,
laid claim to the duchy. His rights were disputed by Reinald,
who was in possession and was recognized by the emperor. Too
weak to assert his claim by force of arms Adolph sold his rights
(1283) to John, duke of Brabant (q.v.). This led to a long and
desolating war for five years, at the end of which (1288), finding
the power of Brabant superior to his own Reinald in his turn sold
his rights to count Henry III. of Luxemburg. Henry and Reinald,
supported by the archbishop of Cologne and other allies, now
raised a great army. The rival forces met at Woeringen (5th of
June 1288) and John, duke of Brabant (q.v.) gained a complete victory.
It proved decisive, the duchies of Limburg and Brabant passing
under the rule of a common sovereign. The duchy comprised
during this period the bailiwicks of Hervé, Montzen, Baelen,
Sprimont and Wallhorn, and the counties of Rolduc, Daelhem
and Falkenberg, to which was added in 1530 the town of
