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 original. The conchoid has been employed by later mathematicians, notably Sir Isaac Newton, in the construction of various cubic curves.

The conchoid is generated as follows:—Let O be a fixed point and BC a fixed straight line; draw any line through O intersecting BC in P and take on the line PO two points X, X′, such that PX = PX′ = a constant quantity. Then the locus of X and X′ is the conchoid. The conchoid is also the

locus of any point on a rod which is constrained to move so that it always passes through a fixed point, while a fixed point on the rod travels along a straight line. To obtain the equation to the curve, draw AO perpendicular to BC, and let AO = a; let the constant quantity PX = PX′ = b. Then taking O as pole and a line through O parallel to BC as the initial line, the polar equation is r = a cosec ± b, the upper sign referring to the branch more distant from O. The cartesian equation with A as origin and BC as axis of x is x2y2 = (a + y)2 (b2 − y2). Both branches belong to the same curve and are included in this equation. Three forms of the curve have to be distinguished according to the ratio of a to b. If a be less than b, there will be a node at O and a loop below the initial point (curve 1 in the figure); if a equals b there will be a cusp at O (curve 2); if a be greater than b the curve will not pass through O, but from the cartesian equation it is obvious that O is a conjugate point (curve 3). The curve is symmetrical about the axis of y and has the axis of x for its asymptote.

 CONCIERGE (a French word of unknown origin; the Latinized form was concergius or concergerius), originally the guardian of a house or castle, in the middle ages a court official who was the custodian of a royal palace. In Paris, when the Palais de la Cité ceased about 1360 to be a royal residence and became the seat of the courts of justice, the Conciergerie was turned into a prison. In modern usage a “concierge” is a hall-porter or janitor.  CONCINI, CONCINO (d. 1617),, Italian adventurer, minister of King Louis XIII. of France, was a native of Florence. He came to France in the train of Marie de’ Medici, and married the queen’s lady-in-waiting, Leonora Dori, known as Galigai. The credit which his wife enjoyed with the queen, his wit, cleverness and boldness made his fortune. In 1610 he had purchased the marquisate of Ancre and the position of first gentleman-in-waiting. Then he obtained successively the governments of Amiens and of Normandy, and in 1614 the bâton of marshal. From then first minister of the realm, he abandoned the policy of Henry IV., compromised his wise legislation, allowed the treasury to be pillaged, and drew upon himself the hatred of all classes. The nobles were bitterly hostile to him, particularly Condé, with whom he negotiated the treaty of Loudun in 1616, and whom he had arrested in September 1616. This was done on the advice of Richelieu, whose introduction into politics was favoured by Concini. But Louis XIII., incited by his favourite Charles d’Albert, duc de Luynes, was tired of Concini’s tutelage. The baron de Vitry received in the king’s name the order to imprison him. Apprehended on the bridge of the Louvre, Concini was killed by the guards on the 24th of April 1617. Leonora was accused of sorcery and sent to the stake in the same year.

In 1767 appeared at Brescia a De Concini vita, by D. Sandellius. On the rôle of Concini see the Histoire de France, published under the direction of E. Lavisse, vol. vi. (1905), by Mariéjol.

 CONCLAVE (Lat. conclave, from cum, together, and clavis, a key), strictly a room, or set of rooms, locked with a key; in this sense the word is now obsolete in English, though the New English Dictionary gives an example of its use so late as 1753. Its present loose application to any private or close assembly, especially ecclesiastical, is derived from its technical application to the assembly of cardinals met for the election of the pope, with which this article is concerned.

Conclave is the name applied to that system of strict seclusion to which the electors of the pope have been and are submitted, formerly as a matter of necessity, and subsequently as the result of a legislative enactment; hence the word has come to be used of the electoral assembly of the cardinals. This system goes back only as far as the 12th century.

Election of the Popes in Antiquity.—The very earliest episcopal nominations, at Rome as elsewhere, seem without doubt to have been made by the direct choice of the founders of the apostolic Christian communities. But this exceptional method was replaced at an early date by that of election. At Rome the method of election was the same as in other towns: the Roman clergy and people and the neighbouring bishops each took part in it in their several capacities. The people would signify their approbation or disapprobation of the candidates more or less tumultuously, while the clergy were, strictly speaking, the electoral body, met to elect for themselves a new head, and the bishops acted as presidents of the assembly and judges of the election. The choice had to meet with general consent; but we can well imagine that in an assembly of such size, in which the candidates were acclaimed rather than elected by counting votes, the various functions were not very distinct, and that persons of importance, whether clerical or lay, were bound to influence the elections, and sometimes decisively. Moreover, this form of election lent itself to cabals; and these frequently gave rise to quarrels, sometimes involving bloodshed and schisms, i.e. the election of antipopes, as they were later called. Such was the case at the elections of Cornelius (251), Damasus (366), Boniface (418), Symmachus (498), Boniface II. (530) and others. The remedy for this abuse was found in having recourse, more or less freely, to the support of the civil power. The emperor Honorius upheld Boniface against his competitor Eulalius, at the same time laying down that cases of contested election should henceforth be decided by a fresh election; but this would have been a dangerous method and was consequently never applied. Theodoric upheld Symmachus against Laurentius because he had been elected first and by a greater majority. The accepted fact soon became law, and John II. recognized (532) the right of the Ostrogothic court of Ravenna to ratify the pontifical elections. Justinian succeeded to this right together with the kingdom which he had destroyed; he demanded, together with the payment of a tribute of 3000 golden solidi, that the candidate elected should not receive the episcopal consecration till he had obtained the confirmation of the emperor. Hence arose long vacancies of the See, indiscreet interference in the elections by the imperial officials, and sometimes cases of simony and venality. This bondage became lighter in the 7th century, owing rather to the weakening of the imperial power than to any resistance on the part of the popes.

9th to 12th Centuries.—From the emperors of the East the power naturally passed to those of the West, and it was exercised after 824 by the descendants of Charlemagne, who claimed that the election should not proceed until the arrival of their envoys. But this did not last long; at the end of the 9th century, Rome, torn by factions, witnessed the scandal of the posthumous condemnation of Formosus. This deplorable state of affairs lasted almost without interruption till the middle of the 11th century. When the emperors were at Rome, they presided over the elections; when they were away, the rival factions of the barons, the Crescentii and the Alberici especially, struggled for the spiritual power as they did for the temporal. During this period were seen cases of popes imposed by a faction rather than elected, and then, at the mercy of sedition, deposed, poisoned and thrown into prison, sometimes to be restored by force of arms.

The influence of the Ottos (962–1002) was a lesser evil; that of the emperor Otto III. was even beneficial, in that it led to the election of Gerbert (Silvester II., in 999). But this was only a temporary check in the process of decadence, and in 1146 Clement II., the successor of the worthless Benedict IX., admitted that henceforth not only the consecration but even the election of the Roman pontiffs could only take place in presence of the