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Dive into the fascinating world of mathematics with our captivating articles, written by passionate experts. Whether you are a curious amateur or a seasoned mathematician, our articles cover a wide range of topics, from fundamental concepts to the latest discoveries, including practical applications and philosophical reflections.
center of gravity
The center of gravity is a point associated with a geometric object or physical body, defined as the point at which the overall effect of a distribution may be regarded as concentrated. In geometry, the center of gravity of a triangle is the point where its three medians intersect. It lies on each median one-third of the way from the opposite side, or equivalently two-thirds of the way from the corresponding vertex. More generally, the center of gravity of a plane figure or three-dimensional solid is the barycenter of its points with uniform weighting, and can be calculated as the average of the coordinates of its constituent points. In physics, the center of gravity of a solid body is the point through which the resultant gravitational force on all parts of the body acts. In a uniform gravitational field, it coincides with the center of mass. The concept of center of gravity is fundamental in statics, solid mechanics, and architecture, where it is used in the study of structural stability.
Center of a conic
The center of a conic is the intersection of its two axes of symmetry. For an ellipse, it is equidistant from the four vertices and is the midpoint of each diameter. For a hyperbola, it is the fixed point about which the curve is symmetric and lies at the intersection of the two principal axes. The circle, a special case of the ellipse whose two axes have the same length, has a unique center equidistant from all points on the curve. A parabola, on the other hand, has no center in the usual sense, since its single axis of symmetry has no perpendicular acting as a second axis; its center is conventionally placed at infinity. In projective geometry, the center of a conic is defined as the pole of the line at infinity with respect to the conic.
Charles Cellérier
Charles Cellérier (1818–1889) was a Swiss mathematician and physicist, born and died in Geneva, where he devoted most of his career to teaching and research in various branches of mathematics. Although little known during his lifetime—because he deliberately kept a low profile—and after his death, his work is of notable historical interest. He is particularly recognised for having constructed, independently of Karl Weierstrass and probably before him, an example of a continuous nowhere differentiable function, a result that remained in his unpublished papers until his death and was published only in 1890 by Charles-Ange Laisant. This discovery illustrates the type of pathological functions that would overturn classical intuitions about analysis in the 19th century. He published a number of articles as well as a course in mechanics.
Maurice Caveing
Maurice Caveing (born 1923) was a French historian of mathematics specializing in ancient mathematics. His historiographical approach seeks to show how, in specific social and cultural contexts, people developed mathematical knowledge that cannot be understood independently of those contexts. This approach, which combines the history of ideas with an analysis of the historical conditions of intellectual production, brings him close to positivism and to a humanism attuned to the human dimension of scientific work. His 1982 thesis, entitled La constitution du type mathématique de l'idéalité dans la pensée grecque, examines the conditions under which Greek thought developed the concept of an ideal mathematical object. His major publications include Essai sur le savoir mathématique dans la Mésopotamie et l'Égypte anciennes (1994) and Le problème des objets dans la pensée mathématique (2004).
Arthur Cayley
Arthur Cayley (1821–1895) was a British mathematician and one of the central figures of 19th-century mathematics. After studying at Cambridge, he practised as a lawyer for fourteen years while publishing approximately 250 mathematical articles. In 1863, he accepted a chair in pure mathematics at Cambridge, less lucrative than his legal career but allowing him to devote most of his time to mathematics. He is regarded as one of the inventors of matrix algebra and the founder of matrix theory as an independent discipline. From 1854 onwards, he laid the foundations of abstract group theory, going beyond the permutation groups then known, and introduced the concept of a vector space. His work in n-dimensional geometry, non-Euclidean geometry and projective geometry opened up perspectives that would permanently shape the development of mathematics and physics. His name is attached to numerous objects and results: the algebra of octonions, or Cayley octaves; the Cayley graph of a group; the Cayley surface (a ruled cubic surface); Cayley's theorem, according to which every group is isomorphic to a subgroup of a symmetric group; and the Cayley–Hamilton theorem, according to which every square matrix satisfies its characteristic polynomial. A member of the Royal Society from 1852, he was one of the founders of the modern British school of pure mathematics.
Bonaventura Cavalieri
Bonaventura Francesco Cavalieri (1598–1647) was an Italian mathematician, astronomer and priest, born in Milan. He studied theology at the monastery of San Gerolamo and geometry at the University of Pisa, where he became acquainted with Galileo, whose belief in heliocentrism he shared, then considered heretical by the Church. He maintained an extensive correspondence with him, comprising at least 112 letters. Cavalieri is best known for two fundamental contributions that foreshadow integral calculus. The first is the method of indivisibles, set out in Geometria indivisibilium continuorum nova quadam ratione promota in 1635: this method regards lines as composed of infinitely many points, surfaces as formed of infinitely many lines, and solids as formed of infinitely many surfaces, making it possible to calculate areas and volumes without explicitly resorting to limiting processes. The second is Cavalieri's principle, already stated by Liu Hui in 3rd-century China: two solids have equal volumes if every pair of parallel cross-sections at equal distances from a reference plane has equal areas. Cavalieri was aware of the paradoxes to which his method could lead; Torricelli would propose an improved version. His method was a decisive step towards differential geometry and the integral calculus of Newton and Leibniz. He also contributed to the study of conic sections (1632), plane and spherical trigonometry for astronomical use (1635), and provided rigorous proofs of the theorems attributed to Guldin.
Jean Cavaillès
Jean Cavaillès (1903–1944), French mathematician and philosopher of mathematics. He excelled in his studies in philosophy and placed first in the agrégation examination upon entering the École normale supérieure in 1923, while also obtaining a degree in mathematics. Having passed the agrégation in philosophy in 1927, he taught in secondary schools before preparing a thesis on the philosophical foundations of mathematics, for which he consulted the archives of Paul du Bois-Reymond and studied the correspondence between Richard Dedekind and Georg Cantor. In 1937, he defended two theses simultaneously under Léon Brunschvicg’s supervision: Méthode axiomatique et formalisme and Remarques sur la formation de la théorie abstraite des ensembles. He subsequently taught at the University of Strasbourg. During the Occupation, he joined the Resistance and became one of its most active figures. Arrested, he was executed by firing squad in February 1944. He was posthumously made a Companion of the Liberation, and his remains are interred in the chapel of the Sorbonne. His philosophical work, centred on the relations between formal logic, set theory and transcendental philosophy, had a lasting influence on the epistemology of mathematics in France.
Caustic of a circle
The caustic of a circle is the geometric envelope of light rays from a point source reflected by a circular mirror. The shape of this caustic depends on the position of the light source relative to the circle. When the source lies on the circle itself, the caustic is a cardioid, a curve with one cusp whose shape resembles that of a heart. When the source is placed at infinity—that is, when the light rays arrive parallel to one another—the caustic takes the form of a nephroid, a curve with two cusps belonging to the family of epicycloids. For other positions of the source, at a finite distance but outside the circle, the caustic takes intermediate forms between these two special cases. The caustic of a circle is a classic example illustrating the link between the geometry of plane curves and optical phenomena.
Salomon de Caus
Salomon de Caus (1576–1626), a French engineer and architect, was probably born in Dieppe, in the Pays de Caux. A representative of the Renaissance encyclopaedic ideal, he trained in painting, ancient languages, engineering, architecture and mathematics, with a marked preference for mechanics and machines. Travelling through several European countries, he worked successively for wealthy patrons as an architect, engineer, drawing master and designer of ornamental gardens, where he created hydraulic machines, fountains featuring allegorical statues and various automata. He mastered mechanics, hydraulics, perspective and music, and built organs and sundials. In 1612, he published La Perspective avec la raison des ombres et miroirs. In 1615, his Raisons des forces mouvantes avec diverses machines tant utiles que plaisantes set out a theory concerning the expansion and condensation of water vapour, making him a pioneer in the practical use of steam power. That same year, his Institution harmonique presented an overview of Renaissance musical theories based on harmonic proportions. In 1624, while in the service of Louis XIII, he published the Pratique et la démonstration des horloges solaires, a work devoted to the proportions and construction of sundials. He also devised a system for cleaning streets using water pumps. At the end of his life, he was working on an unfinished translation of Vitruvius. In the 19th century, an apocryphal legend, born of a false letter written by Samuel-Henry Berthoud to illustrate an engraving, portrayed him as a persecuted inventor, confined at Bicêtre for claiming the invention of the steam engine. This mystification, which Berthoud himself found difficult to debunk, took lasting hold in the popular imagination. He nevertheless remains recognised as one of the first to have put steam power to practical use in a hydraulic machine.
caustic
In geometrical optics, a caustic is the envelope of light rays from a light source after reflection or refraction at a surface or curve. When a caustic is produced by reflection, it is called a catacaustic; when it results from refraction, it is called a diacaustic. The source may be at a finite distance, in which case the result is called a point-source caustic, or at an infinite distance, producing a solar caustic. The shape of the caustic depends both on the geometry of the reflecting or refracting surface and on the position of the light source. For a circular surface illuminated by a point source on the circle, the caustic is a cardioid; if the source is at infinity, it is a nephroid. Other shapes, such as logarithmic spirals and catenaries, are also possible depending on the configuration. The concept of a caustic was studied by Ehrenfried Walther von Tschirnhaus in 1681 and subsequently developed by Jacques Bernoulli and Philippe de La Hire. It has applications in instrumental optics, acoustics, and the physics of waves.
Augustin-Louis Cauchy
Augustin-Louis Cauchy (1789–1857), a French mathematician, was one of the most important figures in 19th-century mathematics. Raised in a monarchist and religious environment, he was recognised at a very early age by Laplace and Lagrange as an exceptional talent. He entered the École polytechnique at the age of sixteen in 1805, graduated as an engineer in the Corps of Bridges and Roads, and worked in Cherbourg until 1813, when he permanently gave up engineering to devote himself exclusively to mathematics. During the Restoration, he became a professor at the École polytechnique. Refusing to swear allegiance to Louis-Philippe during the Revolution of 1830, he went into exile in Turin, where he was notably involved in educating the young Henri, grandnephew of Charles X. He returned to France in 1838 and regained a position at the École polytechnique. Cauchy's most important contribution to mathematics lies in introducing rigour into analysis: he was the first to formulate precisely the fundamental concepts of limit, continuity, derivative and integral, and to establish rigorous criteria for the convergence of series and sequences. This work, notably collected in his 1821 Cours d'analyse, forms the foundation of modern analysis. Nevertheless, his theory still had gaps, particularly concerning uniform convergence. Furthermore, Cauchy is often criticised for his neglect of the work of Galois and Abel, whose important manuscripts he is said to have misplaced. His work is of considerable scope: he made major contributions to complex analysis, algebra, mechanics and mathematical physics.
catenoid
A catenoid is the surface of revolution obtained by rotating a catenary about its axis. It is the simplest non-planar minimal surface, that is, a surface whose mean curvature is zero at every point. The catenoid has the remarkable property of being the minimum-area surface whose boundary consists of two parallel circles of equal radius, whose centres are aligned along a line perpendicular to the planes of those circles. This property is demonstrated by the soap-film experiment: when two coaxial circular rings are immersed in a soap solution, the film that forms between them takes the shape of a catenoid, provided that the distance between the rings does not exceed approximately 0.66 times their common diameter; beyond this limit, the film breaks. A catenoid can be parametrized using hyperbolic functions and is related to the helicoid by a continuous isometric deformation. It was studied by Euler as early as 1740, in the context of his foundational work on the calculus of variations.
Categorization
Mathematically, categorization is the process of grouping objects, entities, or situations into classes on the basis of shared properties or equivalence relations. It is therefore closely linked to the notion of an equivalence relation, which partitions a set into disjoint classes whose elements share a given property. In cognitive and linguistic contexts, however, categorization extends far beyond the formal framework of mathematics: it is a fundamental mechanism of conceptualization, enabling people to reduce the world's complexity by organizing their knowledge into structured categories. Three types of categorical organization are traditionally recognized. Taxonomic categories, or families, bring together items that resemble one another and share intrinsic properties—animals, plants, and foods—and tend to be relatively stable across individuals. Schematic or thematic categories are based not on similarities among their items but on their co-occurrence in the same scene or everyday event; they may also encompass sequences of actions or events, and their contents vary with individual experience. Finally, perceptual categories group objects according to observable physical properties—size, shape, and color—and are among the first to develop in young children, who naturally use them to organize their environment.
Pietro Cataldi
Pietro Cataldi (1548–1626), Italian mathematician. He taught in turn in Florence, Perugia and Bologna, devoting much of his life to teaching and writing mathematical works, more than thirty of which are attributed to him. His work covered several fields: he studied computing square roots using infinite series and made a significant contribution to the study of continued fractions, of which he was one of the earliest proponents in the European mathematical tradition. He also compiled astronomical tables, notably tables of sunrises in Bologna, reflecting an interest in practical astronomy. In geometry, he devoted himself to the study of geometric transformations and, following the efforts of many mathematicians before him, attempted to prove that Euclid's fifth postulate—the parallel postulate—is a logical consequence of the first four, thus anticipating developments that would lead to non-Euclidean geometries.
category
In mathematics, a category is an abstract structure consisting of objects and morphisms—also called arrows—connecting these objects, such that the composition of morphisms is associative and each object has an identity morphism. Category theory, developed in the 1940s by Samuel Eilenberg and Saunders Mac Lane, provides a unifying framework for expressing and comparing mathematical structures of very different kinds—groups, topological spaces, sets and modules—using a common language. Morphisms between categories that preserve the composition structure are called functors: to each object of a source category, a functor assigns an object of a target category, and to each morphism of the source category, a morphism in the target category, compatibly with composition. Natural transformations, in turn, establish correspondences between functors. Category theory plays a fundamental role in modern mathematics, particularly in homological algebra, algebraic geometry and mathematical logic.
Eugène Catalan
Eugène Charles Catalan (1814–1894) was a Belgian mathematician, born in Bruges and died in Liège. From 1833, he studied at the École polytechnique in Paris, alongside Liouville among others. He subsequently held a post in Châlons-sur-Marne before returning to the École polytechnique as professor of descriptive geometry. His academic career in France was hindered by his left-wing political activism and his participation in the Revolution of 1848, after which he settled in Belgium and obtained a chair at the University of Liège. In 1874, he founded the Nouvelle correspondance mathématique with Joseph Neuberg, a journal that played an important role in disseminating mathematics within the French-speaking community. His name is associated with two major contributions: a now-famous conjecture according to which the only consecutive integers that are both proper powers are 8 and 9—a conjecture proved in 2002 by Preda Mihailescu—and a family of semiregular polyhedra, the Catalan polyhedra, which are the duals of the Archimedean solids.
Catala i Poch Maria Assumpcio
Maria Assumpció Català i Poch (1925–2009) was a Spanish mathematician, astronomer and astrophysicist from Catalonia. She studied at the University of Barcelona and obtained her degree in mathematics in 1953, driven by a long-standing passion for astronomy. She directed her research towards sunspots, dynamical stellar systems and the Oort cloud. In 1970, she earned a doctorate in mathematics devoted to the dynamics of stellar systems with cylindrical symmetry, becoming the first woman to obtain this degree in Barcelona. Throughout her career, she conducted systematic observation programmes and specialised in orbit determination and the study of ellipses. She taught at the University of Barcelona, the Universitat Politècnica de Catalunya and the Institut Henri Poincaré in Paris. She represented Spain in the International Astronomical Union.
Guido Castelnuovo
Guido Castelnuovo (1865–1952) was an Italian mathematician and one of the leading representatives of the Italian school of algebraic geometry. Born in Venice to a father who was a novelist and activist for Italian unification, he studied in Padua under Giuseppe Veronese, who instilled in him a taste for geometry. At this time, he began a fruitful collaboration with Corrado Segre. In 1891, he was appointed to the chair of analytic and projective geometry at the University of Rome, where he worked alongside Luigi Cremona and with Federigo Enriques. The bulk of his scientific contributions concerned algebraic geometry: the study of algebraic curves, linear systems of plane curves, and birational invariants. With Enriques, he developed, from 1892 onwards, a thorough theory of algebraic surfaces, which remains one of the major achievements of Italian geometry. He was also interested in algebraic functions, Abelian integrals, statistics, and probability. Having founded the School of Statistics at the University of Rome in 1927, he exerted considerable influence on the mathematicians of the following generation. Dismissed from the university in 1935 under the antisemitic laws of the Fascist regime, he continued to teach Jewish students clandestinely during the Second World War. After the Liberation, he became involved in reorganising Italian scientific institutions through the National Research Council and was elected senator in 1949. His publications include Geometria analitica e proiettiva (1903), Calcolo della probabilità e applicazioni (1919), and Le origini del calcolo infinitesimale nell'era moderna.
Ernst Cassirer
Ernst Cassirer (1874–1945) was a German philosopher born in Breslau who died in Princeton. Trained in the neo-Kantian tradition, he is chiefly known as one of the foremost commentators on and successors to Kant's thought, whose epistemological foundations he explored with particular rigour. He taught in Germany until 1933, when he left the country to flee Nazism, and went on to teach at Oxford, Gothenburg and Yale. His philosophy is organised around the notion of symbolic form, which enables him to analyse the various expressions of human culture—language, myth, art and science—as modes of symbolically shaping the world. Within this framework, mathematics occupies a central place, especially because of its constitutive role in the natural sciences. Studying Felix Klein's Erlangen programme, Cassirer reconsidered Kant's a priori categories by reformulating them not as absolute and fixed forms, but as universal structures specific to each field of knowledge, particularly mathematics. This reassessment enabled him to relate Kantian transcendental philosophy to contemporary developments in mathematics and physics.
Paul de Casteljau
Paul de Faget de Casteljau, born in 1930, was a French mathematician and physicist. He is known for introducing in 1959 the notion of polar forms and a recursive algorithm for evaluating and representing polynomial curves expressed in the Bernstein basis, an algorithm that now bears his name. This algorithm, discovered while he was working for the automobile manufacturer Citroën on computer-aided design, was kept secret for nearly sixteen years before being made public in 1975. It is one of the foundations of computational geometry and is closely linked to Bézier curves, which are a direct application of it. De Casteljau also contributed to unifying the theory of Bézier curves and that of splines under the broader theory of blossoms, thereby providing a coherent algebraic approach to geometric modelling. His final research concerned quaternions and metric geometry.