# Élie Joseph Cartan

### Quick Info

Dolomieu (near Chambéry), Savoie, Rhône-Alpes, France

Paris, France

**Élie Cartan**worked on continuous groups, Lie algebras, differential equations and geometry. His work achieves a synthesis between these areas. He is one of the most important mathematicians of the first half of the 20C.

### Biography

**Élie Cartan**'s mother was Anne Florentine Cottaz (1841-1927) and his father was Joseph Antoine Cartan (1837-1917) who was a blacksmith. Let us trace these families back one more generation. Anne Cottaz was the daughter of François Cottaz and Françoise Mallen while Joseph Cartan was the son of Benoît Bordel Cartan (who was a miller) and Jeanne Denard. Joseph and Anne Cartan had four children: Jeanne Marie Cartan (1867-1931); Élie Joseph Cartan, the subject of this biography; Léon Cartan (1872-1956), who followed his father and joined the family blacksmith business; and Anna Cartan (1878-1923), who became a teacher of mathematics. Élie lived with his family in a house on Square Champ-de-Mars in Dolomieu. He remembered his childhood spent with the (quoted in [3]):-

... blows of the anvil, which started every morning from dawn. ... his mother, during those rare minutes when she was free from taking care of the children and the house, was working with a spinning wheel.The family were very poor and, as Élie Cartan later said, his parents were (quoted in [3]):-

... unpretentious peasants who during their long lives demonstrated to their children an example of joyful accomplished work and courageous acceptance of burdens.In late 19th century France it was not possible for children from poor families to obtain a university education. It was Élie's exceptional abilities, together with a lot of luck, which made a high quality education possible for him. When he was in primary school he showed his remarkable talents which impressed his teachers M Collomb and M Dupuis. The latter said:-

Élie Cartan was a shy boy, but his eyes shone with an unusual light of great intelligence, and this was combined with an excellent memory.Cartan may never have become a leading mathematician were it not for the young school inspector, later important politician, Antonin Dubost (1844-1921). Dubost was at this time employed as an inspector of primary schools and it was on a visit to the primary school in Dolomieu, in the French Alps, that he discovered the remarkable young Élie. Dubost encouraged Élie to enter the competition for state funds to allow Élie to attend a Lycée. His teacher M Dupuis prepared him to sit the competitive examinations which were held in Grenoble. An excellent performance allowed him to enter the Collège de Vienne which he attended for the five years 1880-1885. Throughout his school career Dubost continued to support the young boy and obtain further financial support for him. After the Collège de Vienne, he then studied at the Lycée in Genoble for the two years 1885-87 before completing his school education by spending one year at the Janson-de-Sailly Lycée in Paris where he specialised in mathematics. The state stipend was extended to allow him to study at the École Normale Supérieure in Paris.

Cartan became a student at the École Normale Supérieure in 1888 where he attended courses by the leading mathematicians of the day including Henri Poincaré, Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard and Édouard Goursat. Cartan graduated in 1891 and then served for a year in the army before continuing his studies for his doctorate at the École Normale Supérieure. While Cartan was in the army, where he reached the rank of sergeant, his friend Arthur Tresse (1868-1958) was studying under Sophus Lie in Leipzig. On his return, Tresse told Cartan about Wilhelm Killing's remarkable work on the structure of finite continuous groups of transformations. Cartan set about completing Killing's classification and he was able to benefit greatly from a six-month visit by Sophus Lie to Paris in 1892. During the two years 1892-94 that Cartan spent working on his doctoral thesis, he was supported by a prestigious bursary from the Peccot Foundation. Cartan's doctoral thesis of 1894 contains a major contribution to Lie algebras where he completed the classification of the semisimple algebras over the complex field which Killing had essentially found. However, although Killing had shown that only certain exceptional simple algebras were possible, he had not proved that in fact these algebras exist. This was shown by Cartan in his thesis when he constructed each of the exceptional simple Lie algebras over the complex field. His first papers, published in 1893, were two notes stating his results on simple Lie groups. Robert Bryant writes in [12] that in the 1893 note:-

... Über die einfachen Transformationgruppen ... he announces, in particular, that he has found examples of Lie groups corresponding to each of the 'exceptional' root systems found by Killing. One of the things that I find remarkable about this work is the way that Cartan found interpretations of the exceptional groups as transformation groups.Cartan published full details of the classification in a third paper which was essentially his doctoral thesis. He obtained his doctorate in 1894 from the Faculty of Science at the Sorbonne. He was then appointed to the University at Montpellier where he lectured from 1894 to 1896. Following this, he was appointed as a lecturer at the University of Lyon, where he taught from 1896 to 1903. In Lyon in 1903 he married Marie-Louise Bianconi (1880-1950), the daughter of Pierre-Louis Bianconi who had been a professor of chemistry but had become an inspector in Lyon. Élie and Marie-Louise Cartan had four children: Henri Paul Cartan; Jean Cartan; Louis Cartan; and Hélène Cartan. The eldest son, Henri Cartan, was to produce brilliant work in mathematics and has a biography in this archive. The two other sons died tragically. Jean, a composer of fine music, died of tuberculosis in 1932 at the age of 25 while their son Louis became a physicist at the University of Poitiers. He was a member of the Resistance fighting in France against the occupying German forces. After his arrest in February 1943 the family received no further news but they feared the worst. Only in May 1945 did they learn that he had been beheaded by the Nazis in December 1943. By the time they received the news of Louis' murder by the Germans, Cartan was 75 years old and it was a devastating blow for him. Their fourth child was a daughter Hélène who became a teacher of mathematics at the Lycée Fénelon.

In 1903 Cartan was appointed as a professor at the University of Nancy but he also taught at the Institute of Electrical Engineering and Applied Mechanics. He remained there until 1909 when he moved to Paris [3]:-

In 1909 Cartan built a house in his home village Dolomieu, where he regularly spent his vacations. In Dolomieu Cartan continued his scientific research but sometimes went to the family smithy and helped his father and brother to blow the blacksmith's bellows.His appointment in 1909 in Paris was as an assistant lecturer at the Sorbonne but three years later he was appointed to the Chair of Differential and Integral Calculus in Paris. From 1915 to 1918, during World War I, he was drafted into the army where he continued to hold his former rank of sergeant. He was able to continue his mathematical career and, at the same time, work in the military hospital attached to the École Normale Supérieure. He was appointed as Professor of Rational Mechanics in 1920, and then Professor of Higher Geometry from 1924 to 1940. He retired in 1940 but did not stop teaching at this point for he went on to teach at the École Normale Supérieure for girls.

Cartan worked on continuous groups, Lie algebras, differential equations and geometry. His work achieved a synthesis between these areas. He added greatly to the theory of continuous groups which had been initiated by Lie. After the work of his thesis on finite continuous Lie groups, he later classified the semisimple Lie algebras over the real field and found all the irreducible linear representations of the simple Lie algebras. He turned to the theory of associative algebras and investigated the structure for these algebras over the real and complex field. Joseph Wedderburn would complete Cartan's work in this area.

He then turned his attention to representations of semisimple Lie groups. His work is a striking synthesis of Lie theory, classical geometry, differential geometry and topology which was to be found in all Cartan's work. He applied Grassmann algebra to the theory of exterior differential forms. He developed this theory between 1894 and 1904 and applied his theory of exterior differential forms to a wide variety of problems in differential geometry, dynamics and relativity. Dieudonné writes in [1]:-

He discussed a large number of examples, treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight and that has baffled two generations of mathematicians.In 1899 Cartan published his first paper on the Pfaff problem

*Sur certaines expressions différentielles et le probleme de Pfaff*Ⓣ. In this paper Cartan gave the first formal definition of a differential form. Victor Katz writes [26]:-

His definition was a "purely symbolic" one; namely, he defined "differential expressions" as homogeneous expressions formed by a finite number of additions and multiplications of the differentials dx, dy, d z , . ., and certain differentiable coefficient functions.Over the following years he wrote several other important papers on this topic including

*Sur l'intégration de certaines systèmes de Pfaff de caractère deux*Ⓣ (1901). In 1936-37 he delivered a series of lectures at the Sorbonne which covered his contributions to the topic. The lectures were published in 1945 in the book

*Les systèmes différentiels extérieurs et leurs applications géométriques*Ⓣ.

Cartan's papers on differential equations are in many ways his most impressive work. Again his approach was totally innovative and he formulated problems so that they were invariant and did not depend on the particular variables or unknown functions. This enabled Cartan to define what the general solution of an arbitrary differential system really is but he was not only interested in the general solution for he also studied singular solutions. He did this by moving from a given system to a new associated system whose general solution gave the singular solutions to the original system. He failed to show that all singular solutions were given by his technique, however, and this was not achieved until four years after his death.

From 1916 onwards he published mainly on differential geometry. Klein's 'Erlanger Programme' was seen to be inadequate as a general description of geometry by Weyl and Veblen, and Cartan was to play a major role. He examined a space acted on by an arbitrary Lie group of transformations, developing a theory of moving frames which generalises the kinematical theory of Darboux. In fact this work led Cartan to the notion of a fibre bundle although he does not give an explicit definition of the concept in his work.

Cartan further contributed to geometry with his theory of symmetric spaces which have their origins in papers he wrote in 1926. In these he developed ideas first studied by Clifford and Cayley and used topological methods developed by Weyl in 1925. This work was completed by 1932 and so provides [1]:-

... one of the few instances in which the initiator of a mathematical theory was also the one who brought it to completion.Cartan then went on to examine problems on a topic first studied by Poincaré. By this stage his son, Henri Cartan, was making major contributions to mathematics and Élie Cartan was able to build on theorems proved by his son. Henri Cartan said [24]:-

[My father] knew more than I did about Lie groups, and it was necessary to use this knowledge for the determination of all bounded circled domains which admit a transitive group. So we wrote an article on the subject together [Les transformations des domaines cerclés bornés Ⓣ, C. R. Acad. Sci. Paris 192 (1931), 709-712]. But in general my father worked in his corner, and I worked in mine.Cartan discovered the theory of spinors in 1913. These are complex vectors that are used to transform three-dimensional rotations into two-dimensional representations and they later played a fundamental role in quantum mechanics. Cartan published the two volume work

*Leçons sur la théorie des spineurs*Ⓣ in 1938 [37]:-

In the preface to the two volumes ... M Cartan points out that, in their most general mathematical form, spinors were discovered by him in 1913 in his work on linear representations of simple groups, and he emphasises their connection ... with Clifford-Lipschitz hypercomplex numbers. ... M Cartan's book will be indispensable to mathematicians interested in the geometrical and physical aspects of group theory, giving, as it does, a complete and authoritative survey of the algebraic theory of spinors treated from a geometrical point of view.We have given a list, as complete as possible, of all Cartan's French or English books at THIS LINK.

We have given brief extracts from reviews of some of these books at THIS LINK.

As to his teaching abilities, Shiing-Shen Chern and Claude Chevalley write [14]:-

Cartan was an excellent teacher; his lectures were gratifying intellectual experiences, which left the student with a generally mistaken idea that he had grasped all there was on the subject. It is therefore the more surprising that for a long time his ideas did not exert the influence they so richly deserved to have on young mathematicians. This was perhaps partly due to Cartan's extreme modesty. Unlike Poincaré, he did not try to avoid having students work under his direction. However, he had too much of a sense of humor to organize around himself the kind of enthusiastic fanaticism which helps to form a mathematical school.He is certainly one of the most important mathematicians of the first half of the 20th century. Dieudonné writes in [1]:-

Cartan's recognition as a first rate mathematician came to him only in his old age; before 1930 Poincaré and Weyl were probably the only prominent mathematicians who correctly assessed his uncommon powers and depth. This was due partly to his extreme modesty and partly to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since then his influence has been steadily increasing, and with the exception of Poincaré and Hilbert, probably no one else has done so much to give the mathematics of our day its present shape and viewpoints.J H C Whitehead writes [48]:-

Élie Cartan is one of the great architects of contemporary mathematics.The authors of [6] write:-

Cartan was one of the leading mathematicians of his generation, particularly influential for his work on geometry and the theory of Lie Algebras. In the bleak years after World War I he was one of the most prominent mathematicians in France. He eventually became a notable influence on the Bourbaki group, of which his son Henri, another distinguished mathematician, was one of the seven founder members.William Hodge considers Cartan as [23]:-

... a great mathematical genius taking in the scene in a broad survey, and picking out the essentials, so that with a master-stroke he goes straight to the heart of a problem. His knowledge of innumerable special cases, and his mastery of intricate argument, enabled him to advance his subject by giant strides, and make a lasting mark on the vast range of mathematical endeavour. By his death, the world has indeed lost one of the great architects of modern mathematicsRobert Hermann writes [22]:-

Cartan is certainly one of the greatest and most original minds of mathematics, whose work on Lie groups, differential geometry, and the geometric theory of differential equations is at the foundation of much of what we do today. In my view, his place in mathematics is similar to that of the great turn-of-the-century masters in other areas of intellectual life. Just as Freud was influenced by the mechanistic world view of 19th century science, but used this background to create something new and revolutionary which has profoundly influenced 20th century thought, so Cartan built, on a foundation of the mathematics which was fashionable in the 1890's in Paris, Berlin and Göttingen, a mathematical edifice whose implications we are still investigating. His work was highly intuitive and geometric, but was also based on a formidable combination of original methods of calculation and analysis, ranging in mathematical expertise from algebra to topology.For his outstanding contributions Cartan received many honours, but as Dieudonné explained in the above quote, these did not come until late in career. He received honorary degrees from the University of Liege in 1934, and from Harvard University in 1936. In 1947 he was awarded three honorary degrees from the Free University of Berlin, the University of Bucharest and the Catholic University of Louvain. In the following year he was awarded an honorary doctorate by the University of Pisa. He was elected to the Polish Academy of Sciences in 1921, the Norwegian Academy of Science and Letters in 1926, the Accademia dei Lincei in 1927 and elected a Fellow of the Royal Society of London on 1 May 1947. Elected to the French Academy of Sciences on 9 March 1931 he was vice-president of the Academy in 1945 and President in 1946. He became an honorary member of the London Mathematical Society in 1939. A crater on the moon is named for him.

A celebration was held on 18 May 1939 in the Sorbonne to celebrate Cartan's 70th birthday. Many tributes were made by friends and colleagues who described his contributions to a wide range of different areas of mathematics. In 1969, to celebrate the 100th anniversary of Cartan's birth, a conference was held in Bucharest. The proceedings was published [5] and our list of references contains several papers delivered at that conference, namely [17], [18], [19], [30], [31], [33], [45], and [46]. The conference 'The Mathematical Heritage of Élie Cartan' was held in Lyon, France from 25 June to 29 June 1984 to celebrate the 115th anniversary of Cartan's birth.

### References (show)

- J Dieudonne, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - Biography in
*Encyclopaedia Britannica*. http://www.britannica.com/biography/Elie-Joseph-Cartan - M A Akivis and B Rosenfeld,
*Élie Cartan (1869-1951)*(Amer. Math. Soc., Providence R.I., 1993). - R Debever (ed.),
*Élie Cartan-Albert Einstein : letters on absolute parallelism, 1929-1932*(Princeton, 1979). *Élie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975).- T Gowers, J Barrow-Green and I Leader,
*The Princeton Companion to Mathematics*(Princeton University Press, Princeton, 2008). - T Hawkins,
*Emergence of the theory of Lie groups*(Springer-verlag, New York, 2000). *Notice sur les travaux scientifiques de M Êlie Cartan*(Gauthier-Villars, Paris, 1931).*Selecta; Jubilé scientifique de M Êlie Cartan*(Gauthier-Villars, Paris, 1939).- M Audin, Cartan, Lebesgue, de Rham et l'analysis situs dans les années 1920. Scènes de la vie parisienne,
*Gaz. Math. No.***134**(2012), 49-75. - M Biezunski, Inside the coconut: the Einstein-Cartan discussion on distant parallelism, in
*Einstein and the history of general relativity, North Andover, MA, 1986*(Birkhäuser Boston, Boston, MA, 1989), 315-324. - R L Bryant,
*Élie Cartan and geometric duality*(Institut d'Élie Cartan, 19 June 1998). - É Cartan, Notice sur les travaux scientifiques,
*Selecta*(1939), 219-272. - S-S Chern and C Chevalley, Élie Cartan and his mathematical work,
*Bull. Amer. Math. Soc.***58**(1952), 217-250. - R Chorlay, Passer au global: le cas d'Élie Cartan, 1922-1930,
*Rev. Histoire Math.***15**(2) (2009), 231-316. - R Debever, Publication de la correspondance Cartan-Einstein (French),
*Acad. Roy. Belg. Bull. Cl. Sci.*(5)**64**(3) (1978), 61-63. - G de Rham, L'oeuvre d'Élie Cartan et la topologie, in
*Élie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), 11-20. - J Dieudonné, Les travaux de Élie Cartan sur les groupes et algèbres de Lie, in É
*lie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), 29-31. - Élie Cartan et l'Académie Roumanie, in É
*lie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), 75-116. - A Finzi, Obituary: Élie Cartan (Hebrew),
*Riveon Lematematika***8**(1954), 76-80. - T Hawkins, Frobenius, Cartan, and the problem of Pfaff,
*Arch. Hist. Exact Sci.***59**(4) (2005), 381-436. - R Hermann, Review: The Theory of Spinors, by Élie Cartan,
*Amer. Math. Monthly***90**(10), 719-720. - W V D Hodge, Obituary: Élie Cartan,
*J. London Math. Soc.***28**(1953), 115-119. - A Jackson, Interview with Henri Cartan [b. 1904],
*Notices Amer. Math. Soc.***46**(7) (1999), 782-788. http://www.ams.org/notices/199907/fea-cartan.pdf - M Javillier, Notice nécrologique sur Élie Cartan (1869-1951),
*C. R. Acad. Sci. Paris***232**(1951), 1735-1791. - V J Katz, Differential forms - Cartan to de Rham,
*Arch. Hist. Exact Sci.***33**(4) (1985), 321-336. - V J Katz, Change of variables in multiple integrals: Euler to Cartan,
*Math. Mag.***55**(1) (1982), 3-11. - K Kenmotsu, É Cartan in the Bonnet problem (Japanese), Geometry of submanifolds (Japanese) Kyoto, 2001,
*Surikaisekikenkyusho Kokyuroku No.***1206**(2001), 45-54. - M S Knebelman, Review: Leçons sur la Géométrie des Espaces de Riemann (1928), by Elie Cartan,
*Amer. Math. Monthly***36**(10) (1929), 528-530. - J-L Koszul, L'oeuvre d'Élie Cartan en géométrie différentielle, in É
*lie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), 39-45. - M Kuranishi, Differential systems, equivalence problem, and infinite Lie groups, in
*Élie Cartan, 1869-1951. Hommage de l'Académie de la République Socialiste de Roumanie à l'occasion du centenaire de sa naissance. Comprenant les communications faites aux séances du 4e Congrès du Groupement des Mathématiciens d'Expression Latine, tenu à Bucarest en 1969*(Editura Academiei Republicii Socialiste Romania, Bucharest, 1975), 33-37. - P Libermann, Elie Cartan (1869-1951) (French), Travaux mathématiques,
*Centre Universitaire de Luxembourg, Fas.***8**(1996), 115-158. - A Lichnerowicz, Élie Cartan, in
- R Miyaoka, The past and the present of isoparametric hypersurfaces-Élie Cartan and the 21st century (Japanese), Geometry of submanifolds (Japanese) Kyoto, 2001,
*Surikaisekikenkyusho Kokyuroku No.***1206**(2001), 32-44. - R Miyaoka, Achievements of Élie Cartan-commentary of S S Chern and C Chevalley, (Japanese), Geometry of submanifolds (Japanese) Kyoto, 2001,
*Surikaisekikenkyusho Kokyuroku No.***1206**(2001), 1-31. - P Nabonnand, La notion d'holonomie chez Élie Cartan,
*Rev. Hist. Sci.***62**(1) (2009), 221-245. - H S Ruse, Review: Leçons sur la théorie des spineurs, by Élie Cartan,
*The Mathematical Gazette***23**(255) (1939), 320-323. - N Saltykow, La vie et l'oeuvre de Élie Cartan (Serbo-Croat),
*Bull. Soc. Math. Phys. Serbie***4**(3-4) (1952), 59-64. - E Scholz, H Weyl's and E Cartan's proposals for infinitesimal geometry in the early 1920s,
*Eur. Math. Soc. Newsl. No.***84**(2012), 22-30. - F Simonart, De Gauss à Cartan,
*Acad. Roy. Belgique. Bull. Cl. Sci.*(5)**36**(1950), 1010-1025. - J M Thomas, Review: Les systèmes différentiels extérieurs et leurs applications géométriques, by Élie Cartan,
*Bull. Amer. Math. Soc.***53**(3) (1945), 261-266. - J A Todd, Review: La méthode du repère mobile, la théorie des groupes continus et les espaces généralisés, by Élie Cartan,
*The Mathematical Gazette***19**(233) (1935), 154. - A Trautman, Comments on the paper by Élie Cartan: 'On a generalization of the notion of Riemann curvature and spaces with torsion', in
*Cosmology and gravitation, Bologna, 1979*(Plenum, New York-London, 1980), 493-496. - J L Vanderslice, Review: Leçons sur la Théorie des Espaces à Connexion Projective, by Élie Cartan
**,***Bull. Amer. Math. Soc.***44**(1) (1938), 11-13. - G Vrănceanu, Introduction, in
- G Vrănceanu, L'influence de l'oeuvre d'Élie Cartan sur les mathématiques roumaines, in
- H Weyl, Review: La Théorie des Groupes Finis et Continus et la Géométrie Différentielle traitées par la Méthode du Repère Mobile, by Élie Cartan,
*Bull. Amer. Math. Soc.***44**(9) (1938), 598-601. - J H C Whitehead, Obituary: Elie Joseph Cartan. 1869-1951,
*Obituary Notices of Fellows of the Royal Society***8**(21) (1952), 71-95. - J H C Whitehead, Review: La Topologie des Groupes de Lie, by Elie Cartan,
*The Mathematical Gazette***23**(255) (1939), 318.

### Additional Resources (show)

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### Honours (show)

### Cross-references (show)

- History Topics:
- History Topics: A history of Quantum Mechanics
- History Topics: African women with a doctorate in mathematics 1
- Societies: French Mathematical Society
- Societies: Zurich Scientific Research Society
- Other: 1900 ICM - Paris
- Other: 1924 ICM - Toronto
- Other: 1928 ICM - Bologna
- Other: 1932 ICM - Zurich
- Other: 1936 ICM - Oslo
- Other: 1954 ICM - Amsterdam
- Other: 1986 ICM - Berkeley
- Other: Earliest Known Uses of Some of the Words of Mathematics (D)
- Other: Earliest Known Uses of Some of the Words of Mathematics (F)

Written by J J O'Connor and E F Robertson

Last Update November 2014

Last Update November 2014