CONCHOID (Gr. κόγχη, shell, and εἶδος, form), a plane curve invented by the Greek mathematician Nicomedes, who devised a mechanical construction for it and applied it to the problem of the duplication of the cube, the construction of two mean proportionals between two given quantities, and possibly to the trisection of an angle as in the 8th lemma of Archimedes. Proclus grants Nicomedes the credit of this last application, but it is disputed by Pappus, who claims that his own discovery was 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 θ ± 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), Count Della Penna, Marshal d’Ancre, 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, due 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.