Earliest Known Uses of Some of the Words of Mathematics

Earliest Known Uses of Some of the Words of Mathematics

BANACH SPACE. According to an Internet post, Banach wrote (in French) "spaces of type (B)". It has been suggested that he may have thus invited the subsequent term Banach space.

The term Banach space was coined by Maurice Fréchet (1878-1973), according to the University of St. Andrews website.

Banach space is found in National Mathematics Magazine in November 1945 in the title "Integrations of Functions in a Banach Space" by M. S. Macphail.

BANACH-TARSKI is found in Waclaw Sierpinski, "Sur le paradoxe de MM. Banach et Tarski," Fundamenta Mathematicae 33, pp 229-234 (1945) [James A. Landau].

BAR CHART occurs in Nov. 1914 in W. C. Brinton, "Graphic Methods for Presenting Data. IV. Time Charts," Engineering Magazine, 48, 229-241 (David, 1998).

The form of diagram, however, is much older; there is an example from William Playfair's Commercial and Political Atlas of 1786.

BAR GRAPH is dated 1924 in MWCD10.

Bar graph is found in 1925 in Statistics by B. F. Young: "Bar-graphs in the form of progress charts are used to represent a changing condition such as the output of a factory" (OED2).

The term BARYCENTRIC CALCULUS appears in 1827 in the title Der barycentrische calkul by August Ferdinand Möbius (1790-1868).

BASE (of a geometric figure) appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements (OED2).

BASE (in an isosceles triangle) is found in English in 1571 in Digges, Pantom.: "Isoscheles is such a Triangle as hath onely two sides like, the thirde being vnequall, and that is the Base" (OED2).

BASE (in logarithms) appears in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800) by Lacroix: "Et si a désigne la base du système, il en résulte l'équation y = ax, dans laquelle les logarithmes sont les abscisses."

Base is found in the 1828 Webster dictionary, in the definition of radix: "2. In logarithms, the base of any system of logarithms, or that number whose logarithm is unity."

BASE (of a number system). Radix was used in the sense of a base of a number system in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

Base is found in the Century Dictionary (1889-1897): "The base of a system of arithmetical notation is a number the multiples of whose powers are added together to express any number; thus, 10 is the base of the decimal system of arithmetic."

BASE ANGLE is found in 1848 in "On the Formation of the Central Spot of Newton's Rings Beyond the Critical Angle" by Sir George Gabriel Stokes in the Transactions of the Cambridge Philosophical Society [University of Michigan Historic Math Collection].

BASIS (of a vector space). The term basis-system was used by Frobenius and Stickelberger in 1878 in Crelle, according to Moore (1896) [James A. Landau].

BAYES ESTIMATE, BAYES SOLUTION in statistical decision theory. Wald ("Contributions to the Theory of Statistical Estimation and Testing  Hypotheses," Annals of Mathematical Statistics, 10, (1939), 299-326) originally used the term "minimum risk estimate" for what Hodges & Lehmann called a Bayes estimate ("Some Problems in Minimax Point Estimation," Annals of Mathematical Statistics, 21, (1950), 182-197.) Wald had used the term "Bayes solution" (in a more general setting) in his "An Essentially Complete Class of Admissible Decision Functions" Annals of Mathematical Statistics, 18, (1947), 549-555. Hodges & Lehmann (Annals of Mathematical Statistics, 19, 396-407) used the term BAYES RISK for a concept Wald had treated in 1939 without naming it [John Aldrich, based on David (2001)].

BAYES'S RULE is found in 1863 in An outline of the necessary laws of thought: a treatise on pure and applied logic by William Thomson: "The probability that there exist a cause of the reproduction of any event observed several times in succession is expressed by a fraction which has for its denominator the number 2 multiplied by itself as many times as the event has been observed, and for its numerator the same product minus one. This has been called Bayes's rule, and its validity is not so generally admitted as that of the preceding ones" [University of Michigan Historic Math Collection].

BAYES'S THEOREM. Règle de Bayes appears in 1843 in Exposition de la Théorie des Chances et des Probabilités by A. A. Cournot (David, 1998).

Bayes's Theorem appears in English in 1865 in A History of the Mathematical Theory of Probability by Isaac Todhunter (David, 1995).

BAYESIAN is found in 1950 in Contributions to Mathematical Statistics by R. A. Fisher. Fisher, a critic of the Bayesian approach, was distinguishing the probabilities used in the Bayesian argument from those generated by his own fiducial argument. The old names for the Bayesian argument, the "method of inverse probability" or "inverse method," have now disappeared, a change from the time De Morgan (An Essay on Probabilities (1838), p. vii) wrote "This [inverse] method was first used by the Rev. T. Bayes ... [who], though almost forgotten, deserves the most honourable remembrance from all who treat the history of this science" [John Aldrich, using David, 1998].

BELL-SHAPED. Bell-shaped parabola appears in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science. The equation is ay2 - x2 + bx2 = 0.

Bell-shaped parabola appears in an 1860 translation of a Latin work of Isaac Newton, Sir Isaac Newton's Enumeration of lines of the third order, generation of curves by shadows, organic description of curves, and construction of equations by curves [University of Michigan Historic Math Collection].

Bell-shaped curve is found in 1876 in Catalogue of the Special Loan Collection of Scientific Apparatus at the South Kensington Museum by Francis Galton (David, 1998).

J. V. Uspensky, in Introduction to Mathematical Probability (1937), writes that "the probability curve has a bell-shaped form" [James A. Landau].

BELL CURVE is dated ca. 1941 in MWCD10.

BERNOULLI NUMBERS. According to Cajori (vol. 2, page 42), Leonhard Euler introduced the name "Bernoullian numbers."

The term appears in 1769 in the title "De summis serierum numeros Bernoullianos involventium" by Leonhard Euler.

According to the University of St. Andrews website, in its article on Johann Faulhaber, the Bernoulli numbers were "so named by [Abraham] de Moivre" (1667-1754).

BERNOULLI TRIAL is dated 1951 in MWCD10, although James A. Landau has found the phrases "Bernoullian trials" and "Bernoullian series of trials" in 1937 in Introduction to Mathematical Probability by J. V. Upensky.

BESSEL FUNCTION. Franceschetti (p. 56) implies that this term was introduced by Oskar Xavier Schlömilch in 1854.

Bessel'schen Functionen appears in 1868 in the title Studien über die Bessel'schen Functionen by Eugen Lommel.

Philosophical Magazine in 1872 has "The value of Bessel's functions is becoming generally recognized" (OED2).

Bessel function appears in 1894 in Ann. Math. IX. 27 in the heading "Roots of the Second Bessel Function" (OED2).

BETA DISTRIBUTION. Distribuzione [beta] is found in 1911 in C. Gini, "Considerazioni Sulle Probabilità Posteriori e Applicazioni al Rapporto dei Sessi Nelle Nascite Umane," Studi Economico-Giuridici della Università de Cagliari, Anno III, 5-41 (David, 1998).

The term BETTI NUMBER was coined by Henri Poincaré (1854-1912) and named for Enrico Betti (1823-1892), according to a history note by Victor Katz in A First Course in Abstract Algebra by John B. Fraleigh.

BETWEENNESS. The earliest citation in the OED2 for this word is in 1892 Monist II. 243: "In reality there are not two things and, in addition to them a betweenness of the two things."

Betweenness appears in G. B. Halsted, "The betweenness assumptions," Amer. Math. Monthly 9, 98-101.

The OED2 has a 1904 citation which makes reference to "Hilbert's betweenness assumptions."

BEZOUTIANT was "used by Sylvester and later writers" (Cajori 1919, page 249).

BIASED and UNBIASED. Biased errors and unbiased errors (meaning "errors with zero expectation") are found in 1897 in A. L. Bowley, "Relations Between the Accuracy of an Average and That of Its Constituent Parts," Journal of the Royal Statistical Society, 60, 855-866 (David, 1995).

Biased sample is found in 1911 An Introduction to the theory of Statistics by G. U. Yule: "Any sample, taken in the way supposed, is likely to be definitely biassed, in the sense that it will not tend to include, even in the long run, equal proportions of the A’s and [alpha]'s in the original material" (OED2).

Biased sampling is found in F. Yates, "Some examples of biassed sampling," Ann. Eugen. 6 (1935) [James A. Landau].

The term BICURSAL was introduced by Cayley (Kline, page 938).

In 1873 Cayley wrote, "A curve of deficiency 1 may be termed bicursal."

BIJECTION and BIJECTIVE are dated 1966 in MWCD10.

BILLION. See million.

BIMODAL appears in 1903 in S. R. Williams, "Variation in Lithobius Forficatus," American Naturalist, 37, 299-312 (David, 1998).

BINARY ARITHMETIC appears in English in 1796 A Mathematical and Philosophical Dictionary (OED2).

BINOMIAL. According to the OED2, the Latin word binomius was in use in algebra in the 16th century.

Binomial first appears as a noun in English in its modern mathematical sense in 1557 in The Whetstone of Witte by Robert Recorde: "The nombers that be compound with + be called Bimedialles... If their partes be of 2 denominations, then thei named Binomialles properly. Howbeit many vse to call Binomialles all compounde nombers that have +" (OED2).

BINOMIAL COEFFICIENT. According to Kline (page 272), this term was introduced by Michael Stifel (1487-1567) about 1544. However, Julio González Cabillón believes this information is incorrect. He says Stifel could not have used the word coefficient, which is due to Vieta (1540-1603).

Binomial coefficient is found in Rottock, "Ueber Reihen mit Binomialcoefficienten und Potenzen," Pr. d. G. Rendsburg (1868).

Binomial coefficient is found in English in an 1868 paper by Arthur Cayley [University of Michigan Historical Math Collection].

BINOMIAL DISTRIBUTION is found in 1911 in An Introduction to the Theory of Statistics by G. U. Yule: "The binomial distribution,..only becomes approximately normal when n is large, and this limitation must be remembered in applying the table..to cases in which the distribution is strictly binomial" (OED2).

BINOMIAL THEOREM appears in 1742 in Treatise of Fluxions by Colin Maclaurin (Struik, page 339).

In Gilbert and Sullivan's The Pirates of Penzance (1879), the song "I Am The Very Model of a Modern Major-General" includes the lines:

I'm very well acquainted, too, with matters mathematical,
I understand equations, both the simple and quadratical,
About binomial theorem I'm teeming with a lot o' news,
With many cheerful facts about the square of the hypotenuse. [...]
I'm very good at integral and differential calculus;
I know the scientific names of beings animalculous:

BINORMAL. According to Howard Eves in A Survey of Geometry, vol II (1965), "The name binormal was introduced by B. de Saint-Venant in 1845" [James A. Landau].

The term BIOMATHEMATICS was coined by William Moses Feldman (1880-1939), according to Garry J. Tee in "William Moses Feldman: Historian of Rabbinical Mathematics and Astronomy." The term appears in Feldman's textbook Biomathematics published in 1923.

BIOSTATISTICS appears in Webster's New International Dictionary (1909).

BIPARTITE. In 1858, Cayley referred to "bipartite binary quantics."

BIPARTITE CURVE appears in 1879 in George Salmon (1819-1904), Higher Plane Curves (ed. 3): "We shall then call the curve we have been considering a bipartite curve, as consisting of two distinct continuous series of points" (OED2).

BIQUATERNION. Hamilton used the term biquaternion in the sense of a quaternion with complex coefficients.

In the more recent sense, William Kingdon Clifford (1845-1879) coined the term. It appears in 1873 in Proc. London Math. Soc. IV. 386.

BISECT. According to the OED2, bisect is apparently of English formation. The word is dated ca. 1645 in MWCD10.

Bisection appears in 1656 in a translation of Hobbes's Elem. Philos. (1839) 307: "By perpetual bisection of an angle" (OED2).

In 1660, Barrow's translation of Euclid's Elements has "To bisect a right line."

Bisector appears in English in 1864 in The Reader 5 Oct. 483/2: "The internal and external bisectors of the angle" (OED2).

BIT was coined by John W. Tukey (1915-2000).

According to Niels Ole Finnemann in Thought, Sign and Machine, Chapter 6, "After some more informal contacts during the first war years, on the initiative of mathematician Norbert Wiener, a number of scientists gathered in the winter of 1943-44 at a seminar, where Wiener himself tried out his ideas for describing intentional systems as based on feedback mechanisms. On the same occasion J. W. Tukey introduced the term a 'bit' (binary digit) for the smallest informational unit, corresponding to the idea of a quantity of information as a quantity of yes-or-no answers."

Several Internet web pages say Tukey coined the term in 1946. Another web page says, "Tukey records that it evolved over a lunch table as a handier alternative to 'bigit' or 'binit.'"

Bit first appeared in print in July 1948 in "The Mathematical Theory of Communication" by Claude Elwood Shannon (1916-2001) in the Bell Systems Technical Journal. In the article, Shannon credited Tukey with the coinage [West Addison assisted with this entry.]

BIVARIATE is found in 1920 in Biometrika XIII. 37: "Thus in 1885 Galton had completed the theory of bi-variate normal correlation" (OED2).

BOOLEAN is found in 1851 in the Cambridge and Dublin Mathematical Journal vi. 192: "...the Hessian, or as it ought to be termed, the first Boolian Determinant" (OED2).

BOOLEAN ALGEBRA. Boolian algebra appears in the Century Dictionary (1889-1897):

Boolian algebra, a logical algebra, invented by the English mathematician George Boole (1815-64), for the solution of problems in ordinary logic. It has also a connection with the theory of probabilities.

According to E. V. Hutington in "New Sets of Independent Postulates for the Algebra of Logic with Special Reference to Whitehead and Russell's Principia Mathematica," Trans. Amer. Math. Soc. (1933), the term Boolean algebra was introduced by H. M. Sheffer in the paper "A Set of Five Independent Postulates for Boolean Algebras with Application to Logical Constants", Trans. Amer. Math. Soc., 14 (1913).

In an illuminating passage of "Algebraic Logic", Halmos writes (p. 11):

Terminological purists sometimes object to the Boolean use of the word "algebra". The objection is not really cogent. In the first place, the theory of Boolean algebras has not yet collided, and it is not likely to collide, with the theory of linear algebras. In the second place, a collision would not be catastrophic; a Boolean algebra is, after all, a linear algebra over the field of integers modulo 2. (...) While, to be sure, a shorter and more suggestive term than "Boolean algebra" might be desirable, the nomenclature is so thoroughly established that to change now would do more harm than good.

[Carlos César de Araújo]

BORROW is found in English in 1594 in Blundevil, Exerc.: "Take 6 out of nothing, which will not bee, wherefore you must borrow 60" (OED2).

In October 1947, "Provision for Individual Differences in High School Mathematics Courses" by William Lee in The Mathematics Teacher has: "The Social Mathematics course stresses understanding of arithmetic: 'carrying' in addition, 'regrouping' (not 'borrowing') in subtraction, 'indenting' in multiplication are analyzed and understood rather than remaining mere rote operations to be performed blindly."

BOYER'S LAW appears in H. C. Kennedy, "Boyer's Law: Mathematical formulas and theorems are usually not named after their original discoverers," Amer. Math. Monthly, 79:1 (1972), 66-67.

Boyer's theorem is found in 1968 in History of Mathematics by Barnabas Hughes.

The term BRACHISTOCHRONE was introduced by Johann Bernoulli (1667-1748). Smith (vol. 2, page 326) says the term is "due to the Bernoullis."

The terms BRA VECTOR and KET VECTOR were introduced by Paul Adrien Maurice Dirac (1902-1984). The terms appear in 1947 in Princ. Quantum Mech. by Dirac: "It is desirable to have a special name for describing the vectors which are connected with the states of a system in quantum mechanics, whether they are in a space of a finite or an infinite number of dimensions. We shall call them ket vectors, or simply kets, and denote a general one of them by a special symbol >|. ... We shall call the new vectors bra vectors, or simply bras, and denote a general one of them by the symbol <|, the mirror image of the symbol for a ket vector" (OED2).

BRIGGSIAN LOGARITHM. The phrase Briggs logarithm is found in the 1771 edition of the Encyclopaedia Britannica [James A. Landau].

BROKEN LINE is found in 1852 in Elements of the differential and integral calculus by Charles Davies: "But the arc POM can never be less than the chord PM, nor greater than the broken line PNM which contains it; hence, the limit of the ratio POM/PM = 1; and consequently, the differential of the arc is equal to the differential of the chord."

Broken line is found in 1852 in Legendre, A. M. (Adrien Marie): Elements of geometry and trigonometry, from the works of A. M. Legendre. Revised and adapted to the course of mathematical instruction in the United States, by Charles Davies" "5. A Straight Line is one which lies in the same direction between any two of its points. 6. A Broken Line is one made up of straight lines, not lying in the same direction."

Broken line is found in 1852 in Elements of plane trigonometry, with its application to mensuration of heights and distances, surveying and navigation by William Smyth: "Instead of a broken line, a field is sometimes bounded by a line irregularly curves, as by the margin of a brook, river, or lake. In this case (fig. 60) we run, as before, a chain line as near the boundary as possible, and by means of offsets determine a sufficient number of points in the curve to draw it." [These three citations were found using the University of Michigan Historic Math Collection.]

According to Schwartzman (page 38), the "broken line," meaning a curve composed of connected straight line segments, was adopted "around 1898" by David Hilbert (1862-1943).

BROWNIAN MOTION. In the course of the 20th century the physical phenomenon described by Brown in 1827 was described in mathematical terms and gradually "Brownian motion" came to refer as much to the mathematical formalism as to the phenomenon. Mathematical theories were developed by, inter alia, A. Einstein ("Zur Theorie der Brownschen Bewegung" (1905)). The "Brownian motion process" of J. L. Doob's Stochastic Processes (1954) is a type of stochastic process divested of physical application. Doob states that the process "was first discussed by Bachelier and later, more rigorously by Wiener. It is sometimes called the Wiener process." An earlier term in physics (and mathematics) was "Brownian movement." This slowly gave way to "Brownian motion," although David (2001) reports an early appearance of "Brownian motion" in 1892 in W. Ramsay's Report of a paper read to the Chemical Society, London. Nature, 45, 429/2. (See Wiener process.) [John Aldrich]

BRUN'S CONSTANT was coined by R. P. Brent in "Irregularities in the distribution of primes and twin primes," Math. Comp. 29 (1975), according to Algorithmic Number Theory by Bach and Shallit [Paul Pollack].

The term BYTE was coined in 1956 by Dr. Werner Buchholz of IBM. A question-and-answer session at an ACM conference on the history of programming languages included this exchange:

JOHN GOODENOUGH: You mentioned that the term "byte" is used in JOVIAL. Where did the term come from?
JULES SCHWARTZ (inventor of JOVIAL): As I recall, the AN/FSQ-31, a totally different computer than the 709, was byte oriented. I don't recall for sure, but I'm reasonably certain the description of that computer included the word "byte," and we used it.
FRED BROOKS: May I speak to that? Werner Buchholz coined the word as part of the definition of STRETCH, and the AN/FSQ-31 picked it up from STRETCH, but Werner is very definitely the author of that word.
SCHWARTZ: That's right. Thank you.

CALCULUS. In Latin calculus means "pebble." It is the diminutive of calx, meaning a piece of limestone.

In Latin, persons who did counting were called calculi. Teachers of calculation were known as calculones if slaves, but calculatores or numerarii if of good family (Smith vol. 2, page 166).

The Romans used calculos subducere for "to calculate."

In Late Latin calculare means "to calculate." This word is found in the works of the poet Aurelius Clemens Prudentius, who lived in Spain c. 400 (Smith vol. 2, page 166).

Calculus in English, defined as a system or method of calculating, is dated 1666 in MWCD10.

The earliest citation in the OED2 for calculus in the sense of a method of calculating, is in 1672 in Phil. Trans. VII. 4017: "I cannot yet reduce my Observations to a calculus."

The restricted meaning of calculus, meaning differential and integral calculus, is due to Leibniz.

A use by Leibniz of the term appears in the title of a manuscript Elementa Calculi Novi pro differentiis et summis, tangentibus et quadraturis, maximis et minimis, dimensionibus linearum, superficierum, solidorum, allisque communem calculum transcendentibus [The Elements of a New Calculus for Differences and Sums, Tangents and Quadratures, maxima and minima, the measurement of lines, surfaces and solids, and other things which transcend the usual sort of calculus]. The manuscript is undated, but appears to have been compiled sometime prior to 1680 (Scott, page 157).

Newton did not originally use the term, preferring method of fluxions (Maor, p. 75). He used the term Calculus differentialis in a memorandum written in 1691 which can be found in The Collected Correspondence of Isaac Newton III page 191 [James A. Landau].

Webster's dictionary of 1828 has the following definitions for calculus, suggesting the older meaning of simply "a method of calculating" was already obsolete:

1. Stony; gritty; hard like stone; as a calculous concretion.
2. In mathematics; Differential calculus, is the arithmetic of the infinitely small differences of variable quantities; the method of differencing quantities, or of finding an infinitely small quantity, which, being taken infinite times, shall be equal to a given quantity. This coincides with the doctrine of fluxions.
3. Exponential calculus, is a method of differencing exponential quantities; or of finding and summing up the differentials or moments of exponential quantities; or at least of bringing them to geometrical constructions.
4. Integral calculus, is a method of integrating or summing u moments or differential quantities; the inverse of the differential calculus.
5. Literal calculus, is specious arithmetic or algebra.

The 1890 Funk & Wagnalls Standard Dictionary has: "While calculus is sometimes used in this wide sense, it is commonly used, when without a qualifying word, for the infinitesimal calculus, and includes differential calculus and integral calculus."

The use of calculus without the definite article has become common only in the twentieth century. Some early titles in which "the" appears not to occur are Robinson's Differential and Integral Calculus for High Schools and Colleges (1868), Treatise on Infinitesimal Calculus by Price (1869), Differential Calculus with Numerous Examples by B. Williamson (1872), Calculus of Finite Differences by G. Boole (1872), Integral Calculus by W. E. Byerly (1898), The discovery of Calculus by A. C. Hathaway (1919).

See also differential calculus and integral calculus.

The term CALCULUS OF DERIVATIONS was coined by Arbogast, according to the Mathematical Dictionary and Cyclopedia of Mathematical Science.

CALCULUS OF FINITE DIFFERENCES. An earlier term, method of increments, appears in 1715 in the title Methodus Incrementorum by Brook Taylor.

Method of increments appears in English in 1763 in the title The Method of Increments by W. Emerson.

The phrase finite difference appears in 1807 in the title An Investigation of the General Term of an important Series in the Inverse Method of Finite Differences by J. Brinkley.

Finite difference also appears in Sir John Frederick William Herschel, "On the development of exponential functions, together with several new theorems relating to finite differences," Trans. Phil. Soc., (1814), 440-468; (1816), 25-45.

Calculus of finite differences is found in 1820 in the title A Collection of Examples of the Applications of the Calculus of Finite Differences by Sir John Frederick William Herschel (1792-1871).

The term CALCULUS OF VARIATIONS was introduced by Leonhard Euler in a paper, "Elementa Calculi Variationum," presented to the Berlin Academy in 1756 and published in 1766 (Kline, page 583; DSB; Cajori 1919, page 251). Lagrange used the term method of variations in a letter to Euler in August 1755 (Kline).

Calculus of variations is found in English in 1810 in A Treatise on Isoperimetrical Problems, and the Calculus of Variations by Robert Woodhouse [James A. Landau].

The term CANONICAL FORM is due to Hermite (Smith, 1906).

Canonical form is found in 1851 in the title "Sketch of a Memoir on Elimination, Transformation, and Canonical Forms," by James Joseph Sylvester (1814-1897), Cambridge and Dublin Mathematical Journal 6 (1851).

CARDINAL. Glareanus recognized the metaphor between cardinal numbers and Cardinal, a prince of the church, writing in Latin in 1538.

The earliest citation in the OED2 is by Richard Percival in 1591 in Bibliotheca Hispanica: "The numerals are either Cardinall, that is, principall, vpon which the rest depend, etc."

CARDIOID was first used by Johann Castillon (Giovanni Francesco Melchior Salvemini) (1708-1791) in "De curva cardiode" in the Philosophical Transactions of the Royal Society (1741) [Julio González Cabillón and DSB].

CARMICHAEL NUMBER appears in H. J. A. Duparc, "On Carmichael numbers," Simon Stevin 29, 21-24 (1952).

CARRY (process used in addition). According to Smith (vol. 2, page 93), the "popularity of the word 'carry' in English is largely due to Hodder (3d ed., 1664)."

CARTESIAN COORDINATES. Hamilton used Cartesian method of coordinates in a paper of 1844 [James A. Landau].

Cartesian co-ordinates appears in 1885 in S. Newcomb, Elem. Analytic Geom. in the heading "Cartesian or bilinear co-ordinates" (OED2).

CARTESIAN GEOMETRY was used by Jean Bernoulli "as early as 1692," according to Boyer (page 484).

CARTESIAN PLANE appears in April 1956 in "Graphing in Elementary Algebra" by Max Beberman and Bruce E. Meserve in The Mathematics Teacher: "These axes are usually taken with a common origin, with the first co-ordinate referring to a horizontal axis having its unit point on the right of the origin, and with the second co-ordinate referring to a vertical axis having its unit point above the origin. We call such a co-ordinate plane a Cartesian plane."

CARTESIAN PRODUCT is found in Albert A. Bennett, "Concerning the function concept," The Mathematics Teacher, May 1956: "If A, B are sets, by "A X B" (called the "Cartesian product of A by B") is meant "the set of all ordered pairs (a, b), where a is an element of A, and b of B."

CASTING OUT NINES. Fibonacci called the excess of nines the pensa or portio of the number (Smith vol. 1, page 153).

Pacioli (1494) spoke of it as "corrente mercatoria e presta" (Smith vol. 1, page 153).

"Casting out the nines" is found in the first edition of the Encyclopaedia Britannica (1768-1771) in the article, "Arithmetick."

CATALECTICANT. Catalectic is found in English as early as 1589, describing verse, and meaning "lacking a syllable at the end or ending in an incomplete foot."

The OED2 shows a use of catalectic by James Joseph Sylvester in 1851 in the "The theory of the catalectic forms of functions of the higher degrees of two variables."

Catalecticant was coined by James Joseph Sylvester, who wrote:

Meicatalecticizant would more completely express the meaning of that which, for the sake of brevity, I denominate the catalecticant.

The quotation appears in "On the principles of the calculus of forms," Cambridge and Dublin Mathematical Journal 7 (1852), pp. 52-97, reprinted in Vol. 1 of Sylvester's Collected Papers as Paper 42, pp. 284-327. The quotation appears as a footnote to p. 293.

Bruce Reznick, who provided this quotation, writes, "Sylvester may appear a little pompous to us, but there is a reason for his language: a 'catalectic' verse is one in which the last line is missing a foot. A general homogeneous polynomial p(x,y) of degree 2k can be written as a sum of k+1 linear polynomials raised to the 2k-th power . . . unless its catalecticant vanishes, in which case it needs k linear polynomials, or fewer."

Meicatalecticizant probably did not appear anywhere in print again until Reznick used it in his monograph "Sums of even powers of real linear forms," which appeared as a Memoir of the American Mathematical Society, No. 463 in 1992.

In a letter to Thomas Archer Hirst dated Dec. 19, 1862, Sylvester wrote, "On further relfexion I retract my opinion expressed yesterday evening and reocmmend the continuance [illegible] of the word 'Catalecticant.' This sort of invariant is so important and stands in such close relation to the Canonizant that we cannot afford to let it go unnamed and as this name has been used by Cayley as well as myself it may as well remain. ... I took the Idea of the name from the Iambicus Trimeter Catalecticus."

CATASTROPHE THEORY is found in Thomas F. Banchoff, "Polyhedral catastrophe theory. I: Maps of the line to the line," Dynamical Syst., Proc. Sympos. Univ. Bahia, Salvador 1971, 7-21 (1973).

CATEGORICAL (AXIOM SYSTEM). This term was suggested by John Dewey (1859-1952) to Oswald Veblen (1880-1960) and introduced by the latter in his A system of axioms for geometry, Trans. Amer. Math. Soc. 5 (1904), 343-384, p. 346. Since then, the term as well as the notion itself has been attributed to Veblen. Nonetheless, the first proof of categoricity is due to Dedekind: in his Was sind und Was sollen die Zahlen? (1887) it was in fact proved that the now universally called "Peano axioms" are categorical - any two models (or "realizations") of them are isomorphic. In Dedekind's words:

132. Theorem. All simply infinite systems are similar to the number-series N and consequently (...) to one another.

(Strictly speaking, the categoricity in itself is not seem in this statement but in its proof.)

Instead of "categorical", the term "complete" is sometimes used, chiefly in older texts. The influence, in this case, comes from Hilbert's Vollständigkeitsaxiom ("completeness axiom") in his Über den Zahlbegriff (1900). Other names that were proposed for this concept are "monomorphic" (for categorical and consistent in Carnap's Introduction to symbolic logic, 1954) and "univalent" (Bourbaki), but these did not attain popularity. (It goes without saying that there is no connection with "Baire category", "category theory" etc.) The concept was somewhat shaken when Thoralf Skolem discovered (1922) that first-order set theory is not categorical. Facts like this have caused some confusion among mathematicians. Thus in his The Loss of Certainty (1980, p. 271) Morris Kline wrote:

Older texts did "prove" that the basic systems were categorical; (...) But the "proofs" were loose (...) No set of axioms is categorical, despite "proofs" by Hilbert and others.

This remark was corrected by C. Smorynski in an acrimonious review:

The fact is, there are two distinct notions of axiomatics and, with respect to one, the older texts did prove categoricity and not merely "prove".

[This entry was contributed by Carlos César de Araújo.]

CATENARY. According to E. H. Lockwood (1961) and the University of St. Andrews website, this term was first used (in Latin as catenaria) by Christiaan Huygens (1629-1695) in a letter to Leibniz in 1690.

According to Schwartzman (page 41) and Smith (vol. 2, page 327), the term was coined by Leibniz.

Maor (p. 142) shows a drawing by Leibniz dated 1690 which Leibniz labeled "G. G. L. de Linea Catenaria."

Huygens wrote "Solutio problematis de linea catenaria" in the Acta Eruditorum in 1691.

In 1727-41, Ephraim Chambers' Cyclopedia or Universal Dictionary of Arts and Sciences uses the Latin form catenaria in the article on the tractrix (OED2).

The OED shows a use of catenarian curve in English in 1751.

The 1771 edition of the Encyclopaedia Britannica uses the Latin form catenaria:

CATENARIA, in the higher geometry, the name of a curve line formed by a rope hanging freely from two points of suspension, whether the points be horizontal or not. See FLUXIONS.

In a letter to Thomas Jefferson dated Sept. 15, 1788, Thomas Paine, discussing the design of a bridge, used the term catenarian arch:

Whether I shall set off a catenarian Arch or an Arch of a Circle I have not yet determined, but I mean to set off both and take my choice. There is one objection against a Catenarian Arch, which is, that the Iron tubes being all cast in one form will not exactly fit every part of it. An Arch of a Circle may be sett off to any extent by calculating the Ordinates, at equal distances on the diameter. In this case, the Radius will always be the Hypothenuse, the portion of the diameter be the Base, and the Ordinate the perpendicular or the Ordinate may be found by Trigonometry in which the Base, the Hypothenuse and right angle will be always given.

In a reply to Paine dated Dec. 23, 1788, Thomas Jefferson used the word catenary:

You hesitate between the catenary, and portion of a circle. I have lately received from Italy a treatise on the equilibrium of arches by the Abbé Mascheroni. It appears to be a very scientifical work. I have not yet had time to engage in it, but I find that the conclusions of his demonstrations are that 'every part of the Catenary is in perfect equilibrium.'

The earliest citation for catenary in the OED2 is from the above letter.

CATHETUS. Nicolas Chuquet (d. around 1500), writing in French, used the word cathète (DSB).

Cathetus occurs in English in 1571 in A Geometricall Practise named Pantometria by Thomas Digges (1546?-1595) (although it is spelled Kathetus).

Cathetus is found in English in the Appendix to the 1618 edition of Edward Wright's translation of Napier's Descriptio. The writer of the Appendix is anonymous, but may have been Oughtred.

CAUCHY-SCHWARTZ INEQUALITY. Caucy-Schwarz inequality, Schwarz's inequality, and Schwarz's inequality for integrals appear in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant [James A. Landau].

CAUCHY CONVERGENCE TEST. Cauchy's integral test is found in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "Cauchy's integral test for the convergence of simple series can be extended to double series."

Cauchy's convergence test and Cauchy test appear in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant. Courant writes that the test is also called the general principle of convergence [James A. Landau].

The term CAUCHY SEQUENCE was defined by Maurice Fréchet (1878-1973) (Katz). The term is dated ca. 1949 in MWCD10.

CAUCHY'S THEOREM appears in 1868 in Genocchi, "Intorno ad un teorema di Cauchy," Brioschi Ann.

The term also appears in the title "Sur un théorème de Cauchy présenté par M. Hermite" (1868).

Cauchy's theorem appears in the third edition of An Elementary Treatise on the Theory of Equations (1875) by Isaac Todhunter.

CAYLEY'S SEXTIC was named by R. C. Archibald, "who attempted to classify curves in a paper published in Strasbourg in 1900," according to the St. Andrews University website.

CAYLEY'S THEOREM is found in J. W. L. Glaisher, "Note on Cayley's theorem," Messenger of Mathematics (1878).

Cayley's theorem, referring to a theorem given by Cayley in 1843, appears in 1897 in Abel's Theorem and the Allied Theory Including the Theory of the Theta Functions by H. F. Baker (1897).

The term Cayley's theorem (every group is isomorphic to some permutation group) was apparently introduced in 1916 by G. A. Miller. He wrote Part I of the book Theory and Applications of Finite Groups by Miller, Blichfeldt and Dickson. He liked the idea of listing the most important theorems, with names, so when this theorem had no name he introduced one. His footnote on p. 64 says:

This theorem is fundamental, as it reduces the study of abstract groups uniquely to that of regular substitution groups. The rectangular array by means of which it was proved is often called Cayley's Table, and it was used by Cayley in his first article on group theory, Philosophical Magazine, vol. 7 (1854), p. 49. The theorem may be called Cayley's Theorem, and it might reasonably be regarded as third in order of importance, being preceded only by the theorems of Lagrange and Sylow.

[Contributed by Ken Pledger]

The terms CEILING FUNCTION and FLOOR FUNCTION were coined by Kenneth E. Iverson, according to Integer Functions by Graham, Knuth, and Patashnik.

CENTRAL ANGLE is found in 1851 in The field practice of laying out circular curves for rail-roads by John Cresson Trautwine: "The deflexion angle of any curve is equal to the angle t c u, or t c s, ^c., at the centre of the circle, subtended by one of the equal chords t u, or t s. This angle at the centre, so subtended, is called the central angle. The tangential angle being always half the deflexion angle, is, of course, always half the central angle" [University of Michigan Digital Library].

CENTRAL LIMIT THEOREM. In 1919 R. von Mises called the limit theorems Fundamentalsätze der Wahrscheinlichkeitsrechnung in a paper of the same name in Math Z. 4, 1-97.

Central limit theorem appears in the title "Ueber den zentralen Grenzwertsatz der Wahrscheinlichkeitsrechnung," Math. Z., 15 (1920) by George Polya (1887-1985) [James A. Landau]. Polya apparently coined the term in this paper.

Central limit theorem appears in English in 1937 in Random Variables and Probability Distributions by H. Cramér (David, 1995).

CENTRAL TENDENCY is dated ca. 1928 in MWCD10.

Central tendency is found in 1929 in Kelley & Shen in C. Murchison, Found. Exper. Psychol. 838: "Some investigators have often preferred the median to the mean as a measure of central tendency" (OED2).

CENTROID is found in 1882 in Minchin, Unipl. Kinemat.: "To find..the position of the Centroid ('centre of gravity') of any plane area" (OED2).

The term CEPSTRUM was introduced by Bogert, Healey, and Tukey in a 1963 paper, "The Quefrency Analysis of Time Series for Echoes: Cepstrum, Pseudoautocovariance, Cross-Cepstrum, and Saphe Cracking." The word was created by interchanging the letters in the word "spectrum."

CEVIAN was proposed in French as cévienne in 1888 by Professor A. Poulain (Faculté catholique d'Angers, France). The word honors the Italian mathematician Giovanni Ceva (1647?-1734) [Julio González Cabillón].

An early use of the word in English is by Nathan Altshiller Court in the title "On the Cevians of a Triangle" in Mathematics Magazine 18 (1943) 3-6.

CHAIN. In his ahead-of-time Was sind und Was sollen die Zahlen? (1887), Richard Dedekind introduced the term chain (kette) with two related senses. Improving on his notation and style somewhat, let us take a function f : S ® S. According to him (§37), a "system" (his name for "set") K Ì S is a chain (under f) when f (K ) Ì K.  (Incidentally, from such a "chain" one really gets a descending chain -in one of the more modern uses of this word -, namely, ...Ì f 3(K) Ì f 2(K) Ì f 1(K) Ì K.) Soon after (§44), he fixes A Ì S and defines the "chain of the system A" (under f ) as the intersection of all chains (under f ) K Ì S such that A Ì K. This formulation sounds familiar today, but in Dedekind's time it was a breakthrough! Now, it is easy to see (and he did it in §131) that the "chain of A" (under f ) is simply the union of iterated images A È f 1(A) È f 2(A) È f 3(A) È ..., a result which would yield a simpler definition. But what are the numbers 1, 2, 3, ...? This was precisely the question he intended to answer once and for all through his concept of chain! Gottlob Frege (in his Begriffsschrift, 1879) had similar ideas but his notation was strange and his terminology repulsively philosophic.

Dedekind's "theory of chains" would come to be quoted or used in many places: in proofs of the "Cantor-Bernstein" theorem (Dedekind-Peano-Zermelo-Whittaker), in Keyser's "axiom of infinity" (Bull. A. M. S., 1903, p. 424-433), in Zermelo's second proof of the well-ordering theorem (through his "q -chains", 1908) and in Skolem's first proof of Löwenheim theorem (1920) - to name only a few. All that said, it is simply wrong to say that "Dedekind's approach was so complicated that it was not accorded much attention." (Kline, Mathematical Thought from Ancient to Modern Times, p. 988.) Quite the contrary: the term "chain" in that sense did not survive, but the concept paved the way for the more general notion of closure (hull, span) of a set under an entire structure. [This article contributed by Carlos César de Araújo.]

CHAIN RULE. This term originally referred to a rule for calculating an equivalence in different units of measure when an intermediate unit of measure was involved.

In early Dutch books, it is called the chain rule, Den Kettingh-Regel and Den Ketting Reegel (Smith vol. 2, page 573).

Other names in various Dutch and Dutch-French books of the 17th and 18th century are Regula conjuncta, Regel conjoinct, Te Zamengevoegden Regel, Regel van Vergelykinge, and De Gemenghde Regel (Smith vol. 2, page 573).

Chain rule is found in English in 1842 in the title The Chain Rule; a Commercial Arithmetic by Charles L. Schönberg. [This title is listed in the 1850 Catalogue of the Mercantile library in New York, which was viewed at the University of Michigan Digital Library].

Kettenregel is found in 1877 in W. Simerka, "Die Kettenregel bei Congruenzen," Casopis.

In German, R. Just in Kaufmännisches Rechnen, I (1901) has "Gleichsam wie die Glieder einer 'Kette'" (Smith vol. 2, page 573).

In Differential and Integral Calculus (1902) by Virgil Snyder and J. I. Hutchinson, the calculus rule is shown but is not named.

In 1909, a Webster dictionary says the rule (in arithmetic) is also called Rees's rule, "for K. F. de Rees, its inventor."

In 1912 in Advanced Calculus by Edwin Bidwell Wilson, the calculus rule is referred to as "the rule for differentiating a function of a function."

Peter Flor has found Kettenregel in Höhere Mathematik (1921) by Hermann Rothe, where it is used in the calculus sense slightly differing from the present use, viz. only for composites of three or more functions. Flor writes, "Here the word 'chain' ('Kette', in German) is suggestive. I tried, rather perfunctorily, to pursue the term further back in time, without success. It seems that around 1910, most authors of textbooks as yet saw no problem in computing dz/dx = (dz/dy)*(dy/dx). On the other hand, when I was a student in Vienna and Hamburg (1953 and later), the word Kettenregel was a well-established part of elementary mathematical terminology, in German, for the rule on differentiating a composite of two functions. I guess that its use must have become general around 1930."

In 1922 in Introduction to the Calculus by William F. Osgood, the rule in calculus is not named.

Chain rule occurs in English in the calculus sense in 1937 in the Second English Edition of R. Courant, Differential and Integral Calculus, translated by E. J. McShane. Presumably the term appears in the German original, as well as in the 1st English edition of 1934.

Kettenregel appears in Differential und Integralrechnung by v. Mangoldt and Knopp in 1938 but is used only for composites of three or more functions.

Also in 1938, another classic appeared, the textbook of analysis by Haupt and Aumann, in which Kettenregel is used for the rule for the derivative of any composite function, exactly as we do now [Peter Flor].

Charles Hyman, ed., German-English Mathematical Dictionary, New York: Interlanguage Dictionaries Publishing Corp, 1960, has on page 59 the entry

kettenregel (f), kettensatz (m) [= English] chain rule

James A. Landau, who provided the last citation, suggests that "chain rule" is a German term which was at some point translated into English, possibly by Courant and McShane.

Chain rule appears with a different meaning in N. Chater and W. H. Chater, "A chain rule for use with determinants and permutations," Math. Gaz. 31, 279-287 (1947).

CHAOS appears in 1938 in Norbert Wiener, "The homogeneous chaos," Am. J. Math. 60, 897-936.

Chaos was coined as a mathematical term by James A. Yorke and Tien Yien Li in their classic paper "Period Three Implies Chaos" [American Mathematical Monthly, vol. 82, no. 10, pp. 985-992, 1975], in which they describe the behavior of some particular flows as chaotic [Julio González Cabillón].

It should be stressed that some mathematicians do not feel comfortable with the term "chaos". As an example we quote Paul Halmos in his Has Progress in Mathematics Slowed Down? (Am. Math. Monthly, 1990, p. 563):

Why the word "chaos" is used? The reason seems to be (...) a subjective (not really a mathematical) reaction to an unexpected appearance of discontinuity. A possible source of confusion is that the startling discontinuity can occur at two different parts of the theory. Frequently a dynamical system depends on some parameters (...), and, of course, (...) on the initial point. The startling change of the Hénon family (from periodic to strange attractor) is regarded as chaos - unpredictability - and the very existence of the Hénon strange attractor, not obviously visible in the definition of the dynamical system, is regarded as chaos - unpredictability. I would like to register a protest vote against the attitude that the terminology implies. The results of nontrivial mathematics are often startling, and when infinity is involved they are even more likely to be so. It's not easy to tell by looking at a transformation what its infinite iterates will do - but just because different inputs sometimes produce discontinuously outputs doesn't justify describing them as chaotic.

Probably having in mind such reservations, many prefer to use the term "deterministic chaos". That is to say, one is dealing with deterministic systems (such as a non-linear differential equation) which appear to behave in the long run in an unpredictable fashion. [Carlos César de Araújo]

CHARACTER (group character) appears in title of the paper "Uber die Gruppencharactere" by Ferdinand Georg Frobenius (1849-1917), which was presented to the Berlin Academy on July 16, 1896.

According to Shapir

However, in Dedekind's edition of Dirichlet's Vorlesungen ueber Zahlentheorie in 1894, Dedekind included a footnote in which he singled out the notion of "character," defined it explicitly, and denoted it by chi(n). [6.55]. However, he did not give the function a name. Weber's Lehrbuch der Algebra, II, 1899, defined the function chi(A) as a "Gruppencharakter," and developed some of its elementary properties. . . E. Landau's use of the symbol chi(n) in his texts, together with the terms "charakter and Hauptcharakter" most probably led to the subsequent widespread acceptance of the notation and terminology. Landau credited G. Torelli, 1901, with playing a major role in applying the theory of functions to the study of prime numbers [6.56]. Landau's treatment of characters [6.5.7] suggests that it was Torelli's use of notation that led to Landau's. This is further supported by a 1918 paper of Landau [6.58], where chi(n) is introduced in connection with a discussion of Torelli's results.

[Paul Pollack]

The term CHARACTERISTIC (as used in logarithms) was introduced by Henry Briggs (1561-1631), who used the term in 1624 in Arithmetica logarithmica (Cajori 1919, page 152; Boyer, page 345).

According to Smith (vol. 2, page 514), the term characteristic "was suggested by Briggs (1624) and is used in the 1628 edition of Vlacq." In a footnote, he provides the citation from Vlacq: "...prima nota versus sinistram, quam Characteristicam appellare poterimus..."

Scott (page 136) provides the following citation from Vlacq's Tabulae Sinuum, Tangentium et Secantium: "Here you will note that the first figure of the logarithm, which is called the characteristic is always less by unity than the nuber of figures in the number whose logarithm is taken" (p. xvii).

Scott (page 137) also provides this citation from Adriani Vlacq, Tabulae Sinuum, Tangentium et Secantium, et Logarithmorum. Sinuum, Tangentium et Numerorum ab Unitate ad 100000: "Si datur numerus 3.567894 = 3 567894/1000000 vel 35 67894/100000 vel 356 7894/10000 Logarithmi eorum iidem sunt, qui numeri integri 3567894, escepta tantum Characteristica aut prima figura, et modus eos inveniendi prorsus est idem." [Scott shows the decimal points as raised dots.]

The term index was another early term for the characteristic of a logarithm.


CHARACTERISTIC FUNCTION (1) of a random variable. The first person to apply characteristic functions was Laplace in 1810. Cauchy was probably the first to apply a name to the functions, using the term fonction auxiliaire. In 1919 V. Mises used the term komplexe Adjunkte.

The term characteristic function was first used by Jules Henri Poincaré (1854-1912) in Calcul des Probabilites in 1912. He wrote "fonction caracteristique." Poincare's usage corresponds with what is today called the moment generating function. This information is taken from H. A. David, "First (?) Occurrence of Common Terms in Mathematical Statistics," The American Statistician, May 1995, vol 49, no 2 121-133.

In 1922 P. Levy used the term characteristic function in the title Sur la determination des lois de probabilite par leurs fonctions characteristiques.

Characteristic function appears in English in 1934 in S. Kullback, "An Application of Characteristic Functions to the Distribution Problem of Statistics," Annals of Mathematical Statistics, 5, 263-307 (David, 1995).

CHARACTERISTIC FUNCTION (2) of a set A with respect to a "superset" U is widely used to designate the function from U to {0, 1} that is 1 on A and 0 on its complement. The name explains the common choice of the Greek letter [chi] (chi, which represents kh or ch) for this function. With this meaning, the term seems to have been introduced for the first time by C. de la Vallé Poussin (1866-1962) in Intégrales de Lebesgue, Fonctions d'ensemble, Classes de Baire (Paris, 1916), p. 7. This information is supported by references in Hausdorff's Set Theory (2d ed., Chelsea, 1962, pp. 22, 341, 342), where this function is denoted "simply by [A], omitting the argument x and thus emphasizing only its dependence on A."

Probably to avoid confusion with the other meaning (especially in probability theory, where both notions are useful), some prefer to use the term "indicator function". Besides, it is interesting no note that many logicians turn the usual order of things upside-down: for them, "characteristic function" of a set A (of natural numbers, 0 included) refers to the characteristic function of the complement! In his Foundations of mathematics (1968), W. S. Hatcher explains (p. 215):

In analysis, the characteristic function is usually 1 on the set and 0 off the set, but we generally reverse the procedure in number theory [[more precisely, in recursion theory]]. The reason stems from the minimalization rule and the fact that, when we treat characteristic functions in this way, a given problem often reduces to finding the zeros of some function. In analysis, we want the characteristic functions to be 1 on the set so that the measure of a set will be the integral of its characteristic function.

What is worse, the "characteristic function" of A in this sense is also called the "representing function" by many other logicians. The first logician to use this term seems to be Gödel in his Princeton lectures of 1934 (On undecidable propositions of formal mathematical systems, notes by S. C. Kleene and Barkeley Rosser). Having defined his (primitive) "recursive functions", he goes on to say that an n-place relation (essentially, a set of n-tuples of natural numbers) is "recursive" if its corresponding "representing function" is "recursive".

See also indicator function. [Hans Fischer, Brian Dawkins, Ken Pledger, Carlos César de Araújo]

The term CHARACTERISTIC TRIANGLE was used by Leibniz and apparently coined by him, as triangulum characteristicum.

The term CHINESE REMAINDER THEOREM is found in 1929 in Introduction to the theory of numbers by Leonard Eugene Dickson [James A. Landau].

CHI SQUARE. Karl Pearson introduced the chi-squared test and the name for it in an article in 1900 in The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. Pearson had been in the habit of writing the exponent in the multivariate normal density as -1/2 chi-squared [James A. Landau, John Aldrich].

CHORD is found in English in 1551 in The Pathwaie to Knowledge by Robert Recorde:

Defin., If the line goe crosse the circle, and passe beside the centre, then is it called a corde, or a stryngline.

CHURCH'S THESIS. Martin Davis believes the term thesis first occurs in this connection in 1943 in Stephen Cole Kleene, "Recursive Predicates and Quantifiers," Transactions of the American Mathematical Society 53: "... led Church to state the following thesis ... Thesis I. Every effectively calculable function ... is general recursive."

Wilfried Sieg believes the first use of Church's thesis occurs in 1952 in Introduction to Metamathematics by Stephen Cole Kleene (1909-1994).

CIRCLE. According to Todhunter's translation of Euclid, Book 1 Def. 15 says "a circle is a plane figure bounded by one line, which is called the circumference ..." However Proposition 1 assumes circles consist of their circumferences: "From the point C, at which the circles cut one another, draw the straight lines ..." Heath's translation has the same problems: Def 15 "A circle is a plane figure contained by one line such that...", Prop 1 "... and from the point C, in which the circles cut one another, to the points A, B let the straight lines..." [John Harper].

A Mathematical and Philosophical Dictionary (1796) has, "The circumference or periphery itself is called the circle, though improperly, as that name denotes the space contained within the circumference."

Modern geometry texts define a circle as the set of points in a plane equidistant from a given point; the term disk is used for the circle and its interior.

CIRCLE GRAPH is dated 1928 in MWCD10.

CIRCLE OF CONVERGENCE appears in the Century Dictionary (1889-1897).

Circle of convergence appears in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley in the heading "The circle of convergence."

Circle of convergence also appears in 1898 in Introduction to the theory of analytic functions by Harkness and Morley: "Hence there is a frontier value R such that when |x| > R there is divergence. That is, with the circle (R) the series is absolutely convergent and without the circle it is divergent. The circle (R) is called the circle of convergence.

The term CIRCULAR COORDINATES was used by Cayley. Later writers used the term "minimal coordinates" (DSB).

CIRCULAR FUNCTION. Lacroix used fonctions circulaires in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).

Circular function appears in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "Thus, ax, a log x, sin x, &c., are transcendental functions: the first is an exponential function, the second a logarithmic function, and the third a circular function" [James A. Landau]

CIRCUMCENTER appears in the Century Dictionary (1889-1897).

CIRCUMCIRCLE was used in 1883 by W. H. H. Hudson in Nature XXVIII. 7: "I beg leave to suggest the following names: circumcircle, incircle, excircle, and midcircle" (OED2).

CIRCUMFERENCE. Periphereia was used by Heraclitus: "The beginning and end join on the circumference of the circle (kuklou periphereias)" (D. V. 12 B 103) (Michael Fried).

Periphereia was also used by Euclid.

Circumferentia is a Latin translation of the earlier Greek term periphereia.

Circumference is found in modern translations of the Bible, in 2 Chronicles 4:2, Jeremiah 52:21, and Ezekiel 48:35. However, the word does not appear in the King James version.

CIRCUMSCRIBE is found in English in 1570 in Billingsley's translation of Euclid: "How a triangle ... may be circumscribed about a circle" (OED2).

CIRCUMSCRIBED is found in its modern sense in 1571 in Digges Pantom.: "Circumscribed and inscribed bodies" (OED2).

CIRCUMSCRIBED POLYGON is found in 1850 in Lectures on the philosophy of arithmetic and the adaptation of that science to the business purposes of life: with numerous problems, curious and useful, solved by various modes; with explanations designed to make the study and application of arithmetic pleasant and profitable to such as have not the aid of a teacher; as well as to exercise advanced classes in schools by Uriah Parke: "The operator can know when he is correct to any given number of places by the length of the inscribed and circumscribed polygon coinciding to such extent; for as the circumference of the circle is between them it must be thus far the same. In VAN CEULEN's proportion given above, the first number gives the length of the inscribed, and the second the circumscribed polygon; the circle being between them" [University of Michigan Digital Library].

CISSOID. This term is mentioned by Geminus (c. 130 BC - c. 70 BC), according to Proclus, although the original work of Geminus does not survive.

Cissoid appears in Proclus (in Euclid, p.111, 152, 177...). It is not completely clear what curve Proclus was calling the cissoid (see W. Knorr, The Ancient Tradition in Geometric Problems, New York: Dover Publications, Inc., pp.246ff for a detailed discussion).

In the 17th century, cissoid became associated with a curve described by Diocles in his work, On Burning Mirrors.

Mathematics Dictionary (1949) by James says "the cissoid was first studied by Diocles about 200 B. C., who gave it the name 'Cissoid' (meaning ivy)"; however, according to Michael Fried, Diocles himself does not call his curve a cissoid.

The term CLASS (of a curve) is due to Joseph-Diez Gergonne (1771-1859). He used "curve of class m" for the polar reciprocal of a curve of order m in Annales 18 (1827-30) (Smith vol. I and DSB).

CLASS FIELD. The modern concept of a class field is due to Teji Takagi (1875-1960).

Leopold Kronecker (1823-1891) used the terminology "species associated with a field k."

Class field was introduced by Heinrich Weber (1842-1913) in Elliptische Funktionen und algebraische Zahlen in 1891. He originally only used the term for the Kronecker class field, but in 1896 enlarged the concept of a class field to fields K associated with a congruence class group in k, but only in the second edition of his Lehrbuch der Algebra was the term class field used to designate a general class field. (Günther Frei in "Heinrich Weber and the Emergence of Class Field Theory")

The terms CLASSICAL GROUP and CLASSICAL INVARIANT THEORY were coined by Hermann Weyl (1885-1955) and appear in The classical groups, their invariants and representations (1939).

CLASSICAL PROBABILITY. This term for probability as defined by Laplace and earlier writers came into use in the 1930s when alternative definitions were widely canvassed. J. V. Uspensky (Introduction to Mathematical Probability, 1937, p. 8) gave the "classical definition," which he favored, and criticized the "new definitions" (von Mises) and "the attempt to build up the theory of probability as an axiomatic science" (Kolmogorov) [John Aldrich].

CLASSICAL statistical inference. The polar pair "classical" and "Bayesian" have figured in discussions of the foundations of statistical inference since the 1960s. The body of work to which "classical" was attached went back only to the 1920s and -30s but, as Schlaifer wrote in 1959 (Probability and Statistics for Business Decisions, p. 607), "it is expounded in virtually every course on statistics [in the United States] and is adhered to by the great majority of practicing statisticians." Schlaifer and a few others were sponsoring a rejuvenated Bayesian alternative. The "classical" tag may have derived some authority from Neyman's "Outline of a Theory of Statistical Estimation based on the Classical Theory of Probability" (Philosophical Transactions of the Royal Society, 236, (1937), 333-380), one of the classics of classical statistics. The non-classical possibility Neyman had in mind and rejected was the Bayesian theory of Jeffreys. Confusingly Neyman's "classical theory of probability" has more to do with Kolmogorov and von Mises than with Laplace [John Aldrich].

CLELIA was coined by Guido Grandi (1671-1742). He named the curve after Countess Clelia Borromeo (DSB).

CLOSED (elements produced by an operation are in the set). Closed cycle appears in Eliakim Hastings Moore, "A Definition of Abstract Groups," Transactions of the American Mathematical Society, Vol. 3, No. 4. (Oct., 1902): "For in any finite set of elements with multiplication-table satisfying (1, 2) there exists a closed cycle of (one or more) elements, each of which is the square of the preceding element in the cycle...."

The phrase "closed under multiplication" appears in Saul Epsteen, J. H. Maclagan-Wedderburn, "On the Structure of Hypercomplex Number Systems," Transactions of the American Mathematical Society, Vol. 6, No. 2. (Apr., 1905).

CLOSED CURVE. In 1551 in Pathway to Knowledge Robert Recorde wrote, "Defin., Lynes make diuerse figures also, though properly thei maie not be called figures, as I said before (vnles the lines do close)" (OED2).

Closed curve is found in 1855 in An elementary treatise on mechanics, embracing the theory of statics and dynamics, and its application to solids and fluids by Augustus W. Smith: "Since the above principle is true, whatever be the number of sides of the polygon, it is true when the number becomes indefinitely great, or when the base becomes a continued closed curve, as a circle, an ellipse, &c.; or, the center of gravity of a cone, right or oblique, and on any base, is one fourth the distance from the center of gravity of the base to the vertex" [University of Michigan Digital Library].

CLOSED SET. Georg Cantor (1845-1918) in "De la puissance des ensembles parfaits de points," Acta Mathematica IV, March 4, 1884, introduced (in French) the concept and the term "ensemble fermé [Udai Venedem].

Closed is found in English in 1902 in Proc. Lond. Math. Soc. XXXI "Every example of such a set [of points] is theoretically obtainable in this way. For..it cannot be closed, as it would then be perfect and nowhere dense" (OED2).

CLUSTER ANALYSIS is found in 1939 in Cluster Analysis by R. C. Tryon [James A. Landau].

COCHLEOID (or COCHLIOID). In 1685 John Wallis referred to this curve as the cochlea:

... the Cochlea, or Spiral about a Cylinder, arising from a Circular motion about an Ax, together with a Rectilinear (in the Surface of the Cylinder) Perpendicular to the Plain of such Circle, (or, if the Cylinder be Scalene at such Angles with the Plain of the Circle, as is the Axis of that Cylinder) both motions being uniform, but not in the same Plain.

Some sources incorrectly attribute the term to Benthan and Falkenburg in 1884. While studying the processes of a mechanism of construction for steam engines, C. Falkenburg, Mechanical Engineer of the Actiengesellschaft Atlas in Amsterdam, rediscovered this curve. On March 25, 1883, he submitted an article titled "Die Cochleoïde", which was published in Archiv der Mathematik und Physik.

Er hat sie daher die Cochleoïde genannt, von *cochlea* = Schneckenhaus. [Therefore, it was christened the Cochleoid, from *cochlea* = snail's house.]

The reference for this citation is Nieuw Archief voor Wiskunde [Amsterdam: Weytingh & Brave], vol. 10, pp. 76-80, 1884. This entry was contributed by Julio González Cabillón.

COEFFICIENT. Cajori (1919, page 139) writes, "Vieta used the term 'coefficient' but it was little used before the close of the seventeenth century." Cajori provides a footnote reference: Encyclopédie des sciences mathématiques, Tome I, Vol. 2, 1907, p. 2. According to Smith (vol. 2, page 393), Vieta coined the term.

The term COEFFICIENT OF VARIATION appears in 1896 in Karl Pearson, "Regression, Heredity, and Panmixia," Philosophical Transactions of the Royal Society of London, Ser. A. 187, 253-318 (David, 1995). The term is due to Pearson (Cajori 1919, page 382). According to the DSB, he introduced the term in this paper.

CO-FACTOR is found in 1849 in Trigonometry and Double Algebra by Augustus De Morgan: "When an expression consists of terms, let them be called co-terms; when of factors, co-factors [University of Michigan Historic Math Collection].

The word COMBINANT was coined by James Joseph Sylvester (DSB).

The word appears in a paper by Sylvester in 1853 in Camb. & Dublin Math. Jrnl. VIII. 257: "What I term a combinant" (OED2).

COMBINATION was used in its present sense by both Pascal and Wallis, according to Smith (vol. 2, page 528).

In a letter to Fermat dated July 29, 1654, Pascal wrote a sentence which is translated from French:

If from any number of letters, as 8 for example, A, B, C, D, E, F, G, H, you take all the possible combinations of 4 letters and then all possible combinations of 5 letters, and then of 6, and then of 7, of 8, etc., and thus you would take all possible combinations, I say that if you add together half the combinations of 4 with each of the higher combinations, the sum will be the number equal to the number of the quaternary progression beginning with 2 which is half of the entire number.

This translation was taken from A Source Book in Mathematics by David Eugene Smith.

Combinations is found in English in 1673 in the title Treatise of Algebra...of the Cono-Cuneus, Angular Sections, Angles of Contact, Combinations, Alternations, etc. by John Wallis (OED2).

Leibniz used complexiones for the general term, reserving combinationes for groups of three.

Eberhard Knobloch writes in "The Mathematical Studies of G. W. Leibniz on Combinatorics," Historia Mathematica 1 (1974):

Leibniz's terminology for partitions, just as for symmetric functions, is not consistent. In his Ars Combinatoria he speaks of "discerptiones, Zerfällungen" as mentioned above, and defines them as special cases of "complexiones" (combinations). The Latin term "discerptio" he uses most, and it appears in numerous manuscripts up to his death. When he wants to refer to specific partitions into 1, 2, 3, 4 ... summands, he writes "uniscerptiones, biscerptiones, triscerptiones, quadriscerptiones..." and sometimes also "1scerptiones, 2scerptiones..." evidently following his former usage for combinations of certain sizes in the Ars Combinatoria. I have found only two places where Leibniz applies the general term "discerptio" to the special partition into two summands.

COMBINATORICS. Combinatorial was first used in the modern mathematical sense by Gottfried Wilhelm Leibniz (1646-1716) in his Dissertatio de Arte Combinatoria (Dissertation Concerning the Combinational Arts) (Encyclopaedia Britannica, article: "Combinatorics and Combinatorial Geometry").

Combinatorial analysis is found in English in 1818 in the title Essays on the Combinatorial Analysis by P. Nicholson (OED2).

An early use of the term combinatorics is by F. W. Levi in an essay entitled "On a method of finite combinatorics which applies to the theory of infinite groups," published in the Bulletin of the Calcutta Mathematical Society, vol. 32, pp. 65-68, 1940 [Julio González Cabillón].

COMMENSURABLE is found in English in 1557 in The Whetstone of Witte by Robert Recorde (OED2).

COMMON DIFFERENCE and COMMON RATIO are found in the 1771 edition of the Encyclopaedia Britannica in the article "Algebra" [James A. Landau].

COMMON FRACTION. Thomas Digges (1572) spoke of "the vulgare or common Fractions" (Smith vol. 2, page 219).

COMMON LOGARITHM appears in 1798 in Hutton, Course Math.: "When the radix r is = 10, then the index n becomes the common or Briggs's log. of the number N" (OED2).

Common system of logarithms appears in the 1828 Webster dictionary, in the definition of radix: "Thus in Briggs', or the common system of logarithms, the radix is 10; in Napier's, it is 2.7182818284."

Common logarithm appears in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson: "Thus, in the common logarithms, in which 10, the radix of the decimal system of notation, is the base, we have...."

Common logarithm appears in a footnote in Elementary Illustrations of the Differential and Integral Calculus (1899) by Augustus de Morgan. This book is largely a reprint of a series of articles which appeared in the Library of Useful Knowledge in 1832, and thus the term may appear there.

COMMUTATIVE and DISTRIBUTIVE were used (in French) by François Joseph Servois (1768-1847) in a memoir published in Annales de Gergonne (volume V, no. IV, October 1, 1814). He introduced the terms as follows (pp. 98-99):

3. Soit

f(x + y + ...) = fx + fy + ...

Les fonctions qui, comme f, sont telles que la fonction de la somme (algébrique) d'un nombre quelconque de quantites est égale a la somme des fonctions pareilles de chacune de ces quantités, seront appelées distributives.

Ainsi, parce que

a(x + y + ...) = ax + ay + ...; E(x + y + ...) = Ex + Ey + ...; ...

le facteur 'a', l'état varié E, ... sont des fonctions distributives; mais, comme on n'a pas

Sin.(x + y + ...) = Sin.x + Sin.y + ...; L(x + y + ...) = Lx + Ly + ...;

...les sinus, les logarithmes naturels, ... ne sont point des fonctions distributives.

4. Soit

fgz = gfz.

Les fonctions qui, comme f et g, sont telles qu'elles donnent des résultats identiques, quel que soit l'ordre dans lequel on les applique au sujet, seront appelées commutatives entre elles.

Ainsi, parce que qu'on a

abz = baz ; aEz = Eaz ; ...

les facteurs constans 'a', 'b', le facteur constant 'a' et l'état varié E, sont des fonctions commutatives entre elles; mais comme, 'a' etant toujours constant et 'x' variable, on n'a pas

Sin.az = a Sin.z ; Exz = xEz ; Dxz = xDz [D = delta]; ...

il s'ensuit que le sinus avec le facteur constant, l'état varié ou la difference avec le facteur variable, ... n'appartiennent point a la classe des fonctions commutatives entre elles.

(These citations were provided by Julio González Cabillón).

COMPACT was introduced by Maurice René Fréchet (1878-1973) in 1906, in Rendiconti del Circolo Matematico di Palermo vol. 22 p. 6. He wrote:

Nous dirons qu'un ensemble est compact lorsqu'il ne comprend qu'un nombre fini d'éléments ou lorsque toute infinité de ses éléments donne lieu à au moins un élément limite.

This citation was provided by Mark Dunn.

In his 1906 thesis, Fréchet wrote:

A set E is called compact if, when {En} is a sequence of nomempty, closed subsets of E such that En+1 is a subset of En for each n, there is at least one element that belongs to all of the En's.

At the end of his life, Fréchet did not remember why he chose the term:

... jai voulu sans doute éviter qu'on puisse appeler compact un noyau solide dense qui n'est agrémenté que d'un fil allant jusqu'à l'infini. C'est une supposition car j'ai complétement oubliè les raisons de mon choix!" [Doubtless I wanted to avoid a solid dense core with a single thread going off to infinity being called compact. This is a hypothesis because I have completely forgotten the reasons for my choice!] (Pier, p. 440)

Some mathematicians did not like the term "compact." Schönflies suggested that what Fréchet called compact be called something like "lückenlos" (without gaps) or "abschliessbar" (closable) (Taylor, p. 266).

Fréchet's "compact" is the modern "relatively sequentially compact," and his "extremal" is today's "sequentially compact" (Kline, page 1078).

Compact is found in Paul Alexandroff and Paul Urysohn, "Mémoire sur les espaces topologiques compacts," Koninklijke Nederlandse Akademie van Vetenschappen te Amsterdam, Proceedings of the section of mathematical sciences) 14 (1929).

COMPLEMENT. "Complement of a parallelogram" appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

COMPLEMENTARY FUNCTION is found in 1841 in D. F. Gregory, Examples of Processes of Differential and Integral Calculus: "As operating factors of the form (d/dx)2 + n2 very frequently occur in differential equations, it is convenient to keep in mind that the complementary function due to it is of the form C cos nx + C' sin nx (OED2).

COMPLETE INDUCTION (vollständige Induktion) was the term employed by Dedekind in his Was sind und Was sollen die Zahlen? (1887) for what is nowadays called "mathematical induction", and whose "scientific basis" ("wissenschaftliche grundlage") he claimed to have established with his "Theorem of complete induction" (§59). Dedekind also used occasionally the phrase "inference from n to n + 1", but nowhere in his booklet did he try to justify the adjective "complete".

In Concerning the axiom of infinity and mathematical induction (Bull. Amer. Math. Soc. 1903, pp. 424-434) C. J. Keyser referred to "complete induction" as

a form of procedure unknown to the Aristotelian system, for this latter allows apodictic certainty in case of deduction only, while it is just characteristic of complete induction that it yields such certainty by the reverse process, a movement from the particular to the general, from the finite to the infinite.

Florian Cajori ("Origin of the name "mathematical induction," Amer. Math. Monthly, 1918, pp. 197-201) noted an earlier use of the term "vollständige Induktion" in the article "Induction" in Ersch and Gruber’s Encyklopädie (1840), but in an uninteresting and totally different "Aristotelian sense". According to Abraham Fraenkel (1891-1965) (Abstract Set Theory, 1953, p. 253),

[the] term "complete induction" used in most continental languages (...) [stress] the contrast with induction in natural science which is incomplete by its very nature, being based on a finite and even relatively small number of experiments.

This entry was contributed by Carlos César de Araújo. See also mathematical induction.

COMPLETE SOLUTION. The term complete solution or complete integral is due to Lagrange (Kline, page 532).

The term COMPLETENESS was used by Dedekind in 1872, both to describe the closure of a number field under arithmetical operations and as a synonym for "continuity" (Burn 1992).

COMPLETING THE SQUARE is found in 1806 in Hutton, Course Math.: "The general method of solving quadratic equations, is by what is called completing the square" (OED2).

COMPLEX FRACTION is found in English in 1827 in Hutton, Course Math.: "A Complex Fraction, is one that has a fraction or a mixed number for its numerator, or its denominator, or both" (OED2).

COMPLEX NUMBER. Most of the 17th and 18th century writers spoke of a + bi as an imaginary quantity. Carl Friedrich Gauss (1777-1855) saw the desirability of having different names for ai and a + bi, so he gave to the latter the Latin expression numeros integros complexos. Gauss wrote:

...quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i pro more quantitatem imaginariam \/-1, atque a, b indefinite omnes numeros reales integros inter - oo et + oo. Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeatur.

The citation above is from Gauss's paper "Theoria Residuorum Biquadraticorum, Commentatio secunda," Societati Regiae Tradita, Apr. 15, 1831, published for the first time in Commentationes societatis regiae scientiarum Gottingensis recentiones, vol. VII, Gottingae, MDCCCXXXII (1832)]. [Julio González Cabillón]

The term complex number was used in English in 1856 by William Rowan Hamilton. The OED2 provides this citation: Notebook in Halberstam & Ingram Math. Papers Sir W. R. Hamilton (1967) III. 657: "a + ib is said to be a complex number, when a and b are integers, and i = [sqrt] -1; its norm is a2 + b2; and therefore the norm of a product is equal to the product of the norms of its factors."

COMPOSITE NUMBER (early meaning). According to Smith (vol. 2, page 14), "The term 'composite,' originally referring to a number like 17, 56, or 237, ceased to be recognized by arithmeticians in this sense because Euclid had used it to mean a nonprime number. This double meaning of the word led to the use of such terms as 'mixed' and 'compound' to signify numbers like 16 and 345." Smith differentiates between "composites" and "articles," which are multiples of 10.

COMPOSITE NUMBER (nonprime number). The OED2 shows numerus compositus Isidore III. v. 7.

Napier used the term numeri compositi.

Composite number appears in English in a dictionary of 1730-6 (OED2).

CONCAVE appears in English in 1571 in A Geometricall Practise named Pantometria by Thomas Digges (1546?-1595) (OED2).

CONCAVE POLYGON. Fibonacci referred to such a polygon as a figura barbata in Practica geomitrae.

Re-entering polygon is found in 1851 in Problems in illustration of the principles of plane coordinate geometry by William Walton [University of Michigan Historic Math Collection]. Another term is re-entrant polygon.

Concave polygon appears in 1899 in New Plane Geometry by Wooster Woodruff Beman and David Eugene Smith: "The word polygon is understood, in elementary geometry, to refer to a convex or concave polygon unless the contrary is stated" [University of Michigan Historic Math Collection].

CONCHOID (also known as CONCHLOID). Nicomedes (fl. ca. 250 BC) called various curves the first, second, third, and fourth conchoids (DSB). Pappus says that the conchoids were explored by Nicomedes in his work On Conchoid Lines [Michael Fried].

CONDITIONALLY CONVERGENT SERIES. Semi-convergent series appears in 1872 in J. W. L. Glaisher, "On semi-convergent series," Quart. J.

Conditionally convergent series and semi-convergent series appear in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "A series which converges, but does not converge absolutely, is called semi-convergent. ... A convergent series which is subject to the commutative law is said to be unconditionally convergent; otherwise it is said to be conditionally convergent. ... Semi-convergence implies conditional convergence."

CONDITIONAL PROBABILITY is found in J. V. Uspensky, Introduction to Mathematical Probability, New York: McGraw-Hill, 1937, page 31:

Let A and B be two events whose probabilities are (A) and (B). It is understood that the probability (A) is determined without any regard to B when nothing is known about the occurrence or nonoccurrence of B. When it is known that B occurred, A may have a different probability, which we shall denote by the symbol (A, B) and call 'conditional probability of A, given that B has actually happened.'

[James A. Landau]

CONE is defined in Euclid's Elements, XI, def.18, and it appears in a mathematical context in the presocratic atomist Democritus of Abdera, who wrote:

If a cut were made through a cone parallel to its base, how should we conceive of the two opposing surfaces which the cut has produced -- as equal or as unequal? If they are unequal, that would imply that a cone is composed of many breaks and protrusions like steps. On the other hand if they are equal, that would imply that two adjacent intersection planes are equal, which would mean that the cone, being made up of equal rather than unequal circles, must have the same appearance as a cylinder; which is utterly absurd (D. V. 55 B 155, translation by Philip Wheelwright in The Presocratics, Indianapolis: The Bobbs-Merrill Company, Inc., 1960, p.183).

(This entry was contributed by Michael Fried.)

CONFIDENCE INTERVAL was coined by Jerzy Neyman (1894-1981) in 1934 in "On the Two Different Aspects of the Representative Method," Journal of the Royal Statistical Society, 97, 558-625:

The form of this solution consists in determining certain intervals, which I propose to call the confidence intervals..., in which we may assume are contained the values of the estimated characters of the population, the probability of an error is a statement of this sort being equal to or less than 1 - (epsilon), where (epsilon) is any number 0 < (epsilon) < 1, chosen in advance.

CONFORMAL MAPPING. The term projectio conformis was introduced by F. T. Schubert in 1789 (DSB, article: "Euler").

Gauss used the term conforme Abbildung.

Cayley used the term orthomorphosis.

In 1956, Albert A. Bennett wrote: "Thus a function of one argument, or a mapping, is simply a one-valued, two-term relation. The term 'mapping' thus includes 'functional,' 'projectivity,' and so forth. Although the phrase 'conformal mapping' is old, the general use here mentioned is very recent and may be due to van der Waerden, 1937." (This quotation was taken from "Concerning the function concept," The Mathematics Teacher, May 1956.)

CONGRUENT (geometric figures). Congruere (Latin, "to coincide") was used by geometers of the sixteenth century in their editions of Euclid in quoting Common Notion 4: "Things which coincide with one another are equal to one another." ["Ea ... aequalia sunt, quae sibi mutuo congruunt."]

For instance, in 1539, Christoph Clavius (1537?-1612) writes:

...Hinc enim fit, ut aequalitas angulorum ejusdem generis requirat eandem inclinationem linearum, ita ut lineae unius conveniant omnino lineis alterius, si unus alteri superponatur. Ea enim aequalia sunt, quae sibi mutuo congruunt.

[Cf. page 363 of Clavius's "Euclidis", vol. I, Romae: Apvd Barthdomaevm Grassium, 1589]

As a more technical term for a relation between figures, congruent seems to have originated with Gottfried Wilhelm Leibniz (1646-1716), writing in Latin and French. His manuscript "Characteristica Geometrica" of August 10, 1679, is in his Gesammelte Werke, dritte Folge: mathematische Schriften, Band 5. On p. 150 he says that if a figure can be applied exactly to another without distortion, they are said to be congruent:

Quodsi duo non quidem coincidant, id est non quidem simul eundem locum occupent, possint tamen sibi applicari, et sine ulla in ipsis per se spectatis mutatione facta alterum in alterius locum substitui queat, tunc duo illa dicentur esse congrua, ut A.B et C.D in fig.39 ...

His Figure 39 shows two radii of a circle, with the center labelled both A and C. Later (p.154) he points out that "congruent" is the same as "similar and equal." He used "congruent" in the modern (Hilbert) sense, applied to line segments and various other things as well as triangles.

Shortly afterwards, on September 8, 1679, he included a similar definition in a letter to Hugens (sic) van Zulichem. In his ges. Werke etc. as above, volume 2, p. 22, he illustrates congruence with a pair of triangles, and says that they "peuvent occuper exactement la meme place, et qu'on peut appliquer ou mettre l'un sur l'autre sans rien changer dans ces deux figures que la place." [Ken Pledger and Julio González Cabillón]

In English, writers commonly refer to geometric figures as equal as recently as the nineteenth century. In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has:

Two triangles are equal, when an angle and the two sides which contain it, in the one, are respectively equal to an angle and the two sides which contain it, in the other.

CONGRUENT (in modular arithmetic) was defined by Carl Friedrich Gauss (1777-1855) in 1801 in Disquisitiones arithmeticae: "Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur."

CONIC SECTION is found in the title De sectionibus conicis by Claude Mydorge (1585-1647).

The term is also found in the title Essay on Conic Sections by Blaise Pascal (1623-1662) published in February 1640.

CONJECTURE. Isaac Newton used the term conjecture in 1672 in Phil. Trans. VII. 5084, although whether or not the term was used in a mathematical context is not clear: "I shall refer him to my former Letter, by which that conjecture will appear to be ungrounded."

Jacob Steiner (1796-1863) referred to a result of Poncelet as a conjecture. Poncelet showed in 1822 that in the presence of a given circle with given center, all the Euclidean constructions can be carried out with ruler alone (DSB, article: "Mascheroni").

In Récréations Mathématiques, tome II, Note II, Sur les nombres de Fermat et de Mersenne (1883), É. Lucas referred to "la conjecture de Fermat."

In his article "Conjecture" (Synthese 111, pp. 197-210, 1997), Barry Mazur writes (bottom of page 207):

Since I am not a historian of Mathematics I dare not make any serious pronouncements about the historical use of the term, but I have not come across any appearance of the word Conjecture or its equivalent in other languages with the above meaning [i.e., an opinion or supposition based on evidence which is admittedly insufficient] in mathematical literature except in the twentieth century. The earliest use of the noun conjecture in mathematical writing that I have encountered is in Hilbert's 1900 address, where it is used exactly once, in reference to Kronecker's Jugendtraum.

CONJUGATE. Augustin-Louis Cauchy (1789-1857) used conjuguées for for a + bi and a - bi in Cours d'Analyse algébrique (1821) (Smith vol. 2, page 267).

CONJUGATE ANGLE appears in the Century Dictionary (1889-1897).

The term also appears in Plane and Solid Geometry (1913) by Wentworth and Smith, and it may occur in the earlier 1888 edition, which has not been consulted.

CONSERVATIVE EXTENSION. Martin Davis believes the term was first used by Paul C. Rosenbloom. It appears in The Elements of Mathematical Logic, 1st ed., New York: Dover Publications, 1950.

CONSISTENCY. The term consistency applied to estimation was introduced by R. A. Fisher in "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922). Fisher wrote: "A statistic satisfies the criterion of consistency, if, when it is calculated from the whole population, it is equal to the required population."

In the modern literature this notion is usually called Fisher-consistency (a name suggested by Rao) to distinguish it from the more standard notion linked to the limiting behavior of a sequence of estimators. The latter is hinted at in Fisher's writings but was perhaps first set out rigorously by Hotelling in the "The Consistency and Ultimate Distribution of Optimum Statistics," Transactions of the American Mathematical Society (1930). [This entry was contributed by John Aldrich, based on David (1995).]

CONSTANT was introduced by Gottfried Wilhelm Leibniz (1646-1716) (Kline, page 340).

CONSTANT OF INTEGRATION. In 1807 Hutton Course Math. has: "To Correct the Fluent of any Given Fluxion .. The finding of the constant quantity c, to be added or subtracted with the fluent as found by the foregoing rules, is called correcting the fluent.

In 1831 Elements of the Integral Calculus (1839) by J. R. Young refers to "the arbitrary constant C."

Constant of integration is found in 1846 in "On the Rotation of a Solid Body Round a Fixed Point" by Arthur Cayley in the Cambridge and Dublin Mathematical Journal [University of Michigan Historical Math Collection].

In 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson, it is called the "constant quantity annexed."

CONTINGENCY TABLE was introduced by Karl Pearson in "On the Theory of Contingency and its Relation to Association and Normal Correlation," which appeared in Drapers' Company Research Memoirs (1904) Biometric Series I:

This result enables us to start from the mathematical theory of independent probability as developed in the elementary text books, and build up from it a generalised theory of association, or, as I term it, contingency. We reach the notion of a pure contingency table, in which the order of the sub-groups is of no importance whatever.

This citation was provided by James A. Landau.

The CONTINUED FRACTION was introduced by John Wallis (1616-1703) (DSB, article: "Cataldi").

Wallis used continue fracta in 1655 in Arithmetica Infinitorum Prop. CXCI.

The phrase "Esto igitur fractio eiusmode continue fracta quaelibet sic deignata..." is found in volume I of Opera Mathematica, a collection of Wallis' mathematical and scientific works published in 1693-1699.

The phrase "fractio, quae denominatorem habeat continue fractum" is found in Opera, I, 469 (Smith vol. 2, page 420).

In 1685 Wallis referred to Brouncker's continued fraction as "a fraction still fracted continually" in A Treatise of Algebra [Philip G. Drazin, David Fowler, James A. Landau, Siegmund Probst].

Continued fraction is found in English in 1811 An Elementary Investigation in the Theory of Numbers by Peter Barlow [James A. Landau].

CONTINUOUS. Euler defined a continuous curve in the second volume of his Introductio in analysin infinitorum (Katz, page 580).

CONTINUOUS CURVE is found in English in 1852 in Elements of the differential and integral calculus by Charles Davies [University of Michigan Digital Library].

CONTINUOUS FUNCTION is found in English in 1871 in A manual of spherical and practical astronomy, embracing the general problems of spherical astronomy, the special applications to nautical astronomy, and the theory and use of fixed and portable astronomical instruments, with an appendix on the method of least squares by William Chauvenet: "our geometrical representation should strictly consist of a number of isolated points; but, as these points will be more and more nearly represented by a continuous curve as we increase the accuracy of the observations, and thus diminish the intervals between the successive ordinates, we may, without hesitation, adopt such a continuous curve as expressing the law of error. We shall therefore regard [Greek letter capital delta] as a continuous variable, and [Greek letters small phi followed, no space, by capital delta] as a continuous function of it" [University of Michigan Digital Library].

CONTINUUM. According to the DSB, the term continuum appeared as early as the writings of the Scholastics, but the first satisfactory definition of the term was given by Cantor.

CONTINUUM HYPOTHESIS. In his 1900 Paris lecture, Hilbert titled his first problem "Cantor's Problem of the Power of the Continuum."

In his 1901 doctoral dissertation writen under Hilbert, Felix Bernstein used the term "Cantor's Continuum Problem" ("das Cantorsche Continuumproblem"). This is the first time the term "Cantor's Continuum Problem" appeared in print, according to Cantor's Continuum Problem by Gregory H. Moore, which also states that "presumbly Bernstein obtained the name 'Continuum Problem' by abbreviating Hilbert's title."

Continuum problem also appears in Felix Bernstein, "Zum Kontinuumproblem," Mathematische Annalen 60 (1905); Julius König, "Zum Kontinuum-Problem," Verhandlungen des dritten internat. Math.- Kongress (1905); and Julius König, "Zum Kontinuumproblem," Mathematische Annalen 60 (1905).

In his essay "On the Infinite" (1925) Hilbert referred to the question of whether continuum hypothesis is true as the "famous problem of the continuum"; the word "hypothesis" is not used. Two years later, in "The foundations of mathematics" (1927) he referred to "the proof or refutation of Cantor's continuum hypothesis."

Carlos César de Araújo believes that the use of "hypothesis" here became more popular and well-established only after the 1934 monograph of Sierpinski, "Hypothése du continu."

In the 1962 Chelsea translation of the 1937 3rd German edition of Hausdorff's Mengenlehre pp 45f is the following:

A conjecture that was made at the beginning of Cantor's investigations, and that remains unproved to this day, is that [alef] is the cardinal number next larger than [alef-null]; this conjecture is known as the continuum hypothesis, and the question as to whether it is true or not is known as the problem of the continuum

(Hausdorff used [alef] to mean the infinity of the continuum.)

Continuum hypothesis appears in Waclaw Sierpinski, "Sur deux propositions, dont l'ensemble équivaut à l'hypothése du conntinu," Fundamenta Mathematicae 29, pp 31-33 (1937).

Continuum hypothesis appears in the title "The consistency of the axiom of choice and of the generalized continuum-hypothesis" by Kurt Gödel, Proc. Nat. Acad. Sci., 24, 556-557 (1938) [James A. Landau, Carlos César de Araújo].

CONTRAPOSITIVE. Boethius wrote: " Est enim per contrapositionem conversio, ut si dicas omnis homo animal est, omne non animal non homo est."

Contraposition is found in English in 1551 in T. Wilson, Logike: "A conuersion by contraposition is when the former part of the sentence is turned into the last rehearsed part, and the last rehearsed part turned into the former part of the sentence, both the propositions being uniuersall, and affirmatiue, sauing that in the second proposition there be certaine negatiues enterlaced" (OED2).

Contrapose and contraposite other older terms in English.

De Morgan used the adverb contrapositively in 1858 in Trans. Camb. Philos. Soc. (OED2).

Contrapositive appears as an adjective in the preface to The Elements of Plane Geometry (1868) by R. P. Wright. The preface was written by T. A. Hirst (1830-1892): "The two theorems are, in fact, contrapositive forms, one of the other; the truth of each is implied, when that of the other is asserted, and to demonstrate both geometrically is more than superfluous; it is a mistake, since the true relation between the two is thereby masked."

Contrapositive was used as a noun in 1870 by William Stanley Jevons in Elementary Lessons in Logic (1880): "Convert and show that the result is the contrapositive of the original" (OED2).

CONVERGENCE (of a vector field) was coined by James Clerk Maxwell (Katz, page 752; Kline, page 785). It is the negative of the divergence, q.v.

See curl.

The terms CONVERGENT and DIVERGENT were used by James Gregory in 1667 in his Vera circuli et hyperbolae quadratura (Cajori 1919, page 228). Gregory wrote series convergens.

However, according to Smith (vol. 2, page 507), the term convergent series is due to Gregory (1668) and the term divergent series is due to Nicholas I Bernoulli (1713). In a footnote, he cites F. Cajori, Bulletin of the Amer. Math. Soc. XXIX, 55.

CONVERSE is first found in English in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

CONVEX (curved outward) appears in English in 1571 in A Geometricall Practise named Pantometria by Thomas Digges (1546?-1595) (OED2).

CONVEX POLYGON. In 1828 in Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) is found the following:

We thought it better to restrict our reasoning to those lines which we have named convex, and which are such that a straight line cannot cut them in more than two points.

Convex polygon is found in English in 1852 in an adaptation by Charles Davies of Elements of geometry and trigonometry by Adrien Marie Legendre: "When this proposition is applied to polygons which have re-entrant angles, each re-entrant angle must be regarded as greater than two right angles. But to avoid all ambiguity, we shall henceforth limit our reasoning to polygons with salient angles, which are named convex polygons. Every convex polygon is such, that a straight line, drawn at pleasure, cannot meet the sides of the polygon in more than two points" [University of Michigan Digital Library].

In 1857 Mathematical Dictionary and Cyclopedia of Mathematical Sciences has convex polygon and the synonymous term salient polygon.

See also salient angle.

CONVOLUTION. In his 1933 book The Fourier Integral and Certain of its Applications Norbert Wiener uses convolutions (both the integral and sum types) but calls them by the German name Faltung, stating (p. 45) that there is no good English term [Yaakov Stein].

In 1934, Amer. Jrnl. Math. LVI. 662 has "Bernoulli convolutions" (OED2).

In 1935, Trans. Amer. Math. Soc. XXXVIII. 48 has "Distribution functions and their convolutions ('Faltungen')" (OED2).

The word COORDINATE was introduced by Gottfried Wilhelm Leibniz (1646-1716). He also used the term axes of co-ordinates. According to Cajori (1919, pages 175 and 211), he used the terms in 1692; according to Ball, he used the terms in a paper of 1694.

Leibniz used the term in "De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata, easque omnes tangente, ac de novo in ea re Analysis infinitorum usu," in Acta Eruditorum, vol. 11 (1692), pp. 168-171. On p. 170: "Verum tam ordinata quam abscissa, quas per x & y designari mos est (quas & coordinatas appellare soleo, cum una sit ordinata ad unum, altera ad alterum latus anguli, a duabus condirectricibus comprehensi) est gemina seu differentiabilis." The article is also printed in Leibniz, Mathematische Schriften (ed. Gerhardt), vol. 5, pp. 266-269 [Siegmund Probst].

Descartes did not use the term coordinate (Burton, page 350).

The term COORDINATE GEOMETRY is dated 1815-25 in RHUD2. An early use of the term is by Matthew O'Brien (1814-1855) in A treatise on plane co-ordinate geometry; or, The application of the method of co-ordinates to the solution of problems in plane geometry, Part 1, Cambridge: Deighton, 1844.

COPLANAR appears in Sir William Rowan Hamilton, Lectures on Quaternions (London: Whittaker & Co., 1853): "In that particular case, there was ready a known signification [36] for the product line, considered as the fourth proportional to the unit-line (assumed here on the last-mentioned axis), and to the two coplanar factor-lines" [James A. Landau].

COROLLARY. From the Latin corolla, a small garland. In an essay entitled "The Essence of Mathematics" (see James R. Newman’s anthology The world of mathematics), Charles Saunders Peirce (1839-1914) wrote:

(...) while all the "philosophers" follow Aristotle in holding no demonstration to be thoroughly satisfactory except what they call a "direct demonstration", or a "demonstration why" (...) the mathematicians, on the contrary, entertain a contempt for that style of reasoning, and glory in what the philosophers stigmatize as "mere indirect demonstrations", or "demonstrations that". Those propositions which can be deduced from others by reasoning of the kind that the philosophers extol are set down by mathematicians as "corollaries". That is to say, they are like those geometrical truths which Euclid did not deem worthy of particular mention, and which his editors inserted with a garland, or corolla, against each in the margin, implying perhaps that it was to them that such honor as might attach to these insignificants remarks was due. (...) we may say that corollarial, or "philosophical" reasoning is reasoning with words; while theorematic, or mathematical reasoning proper, is reasoning with specially constructed schemata.

[Carlos César de Araújo]

CORRELATION, CORRELATION COEFFICIENT and COEFFICIENT OF CORRELATION. Francis Galton introduced the measurement of correlation (Hald, p. 604). The index of co-relation appears in 1888 in his "Co-Relations and Their Measurement," Proc. R. Soc., 45, 135-145: "The statures of kinsmen are co-related variables; thus, the stature of the father is correlated to that of the adult son,..and so on; but the index of co-relation ... is different in the different cases" (OED2). "Co-relation" soon gave way to "correlation" as in W. F. R. Weldon's "The Variations Occurring in Certain Decapod Crustacea-I. Crangon vulgaris," Proc. R. Soc., 47. (1889 - 1890), pp. 445-453.

The term coefficient of correlation was apparently originated by Edgeworth in 1892, according to Karl Pearson's "Notes on the History of Correlation," (reprinted in Pearson & Kendall (1970). It appears in 1892 in F. Y. Edgeworth, "Correlated Averages," Philosophical Magazine, 5th Series, 34, 190-204.

Correlation coefficient appears in a paper published in 1895 [James A. Landau].

The OED2 shows a use of coefficient of correlation in 1896 by Pearson in Proc. R. Soc. LIX. 302: "Let r0 be the coefficient of correlation between parent and offspring." David (1995) gives the 1896 paper by Karl Pearson, "Regression, Heredity, and Panmixia," Phil. Trans. R. Soc., Ser. A. 187, 253-318. This paper introduced the product moment formula for estimating correlations--Galton and Edgeworth had used different methods.

Partial correlation. G. U. Yule introduced "net coefficients" for "coefficients of correlation between any two of the variables while eliminating the effects of variations in the third" in "On the Correlation of Total Pauperism with Proportion of Out-Relief" (in Notes and Memoranda) Economic Journal, Vol. 6, (1896), pp. 613-623. Pearson argued that partial and total are more appropriate than net and gross in Karl Pearson & Alice Lee "On the Distribution of Frequency (Variation and Correlation) of the Barometric Height at Divers Stations," Phil. Trans. R. Soc., Ser. A, 190 (1897), pp. 423-469. Yule went fully partial with his 1907 paper "On the Theory of Correlation for any Number of Variables, Treated by a New System of Notation," Proc. R. Soc. Series A, 79, pp. 182-193.

Multiple correlation. At first multiple correlation referred only to the general approach, e.g. by Yule in Economic Journal (1896). The coefficient arrives later. "On the Theory of Correlation" (J. Royal Statist. Soc., 1897, p. 833) refers to a coefficient of double correlation R1 (the correlation of the first variable with the other two). Yule (1907) discussed the coefficient of n-fold correlation R21(23...n). Pearson used the phrases "coefficient of multiple correlation" in his 1914 "On Certain Errors with Regard to Multiple Correlation Occasionally Made by Those Who Have not Adequately Studied this Subject," Biometrika, 10, pp. 181-187, and "multiple correlation coefficient" in his 1915 paper "On the Partial Correlation Ratio," Proc. R. Soc. Series A, 91, pp. 492-498.

[This entry was largely contributed by John Aldrich.]

The term CORRELOGRAM was introduced by H. Wold in 1938 (A Study in the Analysis of Stationary Time Series). There is a plot of empirical serial correlations, i.e. an empirical correlogram, in Yule's "Why Do We Sometimes Get Nonsense Correlations between Time-series ..." Journal of the Royal Statistical Society, 89, (1926), 1-69 (David 2001).

COSECANT. The Latin cosecans appears in Opus Palatinum de triangulis ("The Palatine Work on Triangles"), which was written by Georg Joachim von Lauchen Rheticus (1514-1574). This treatise was published after his death by his pupil Valentin Otto in 1596. According to Ball (page 243) and Smith (vol. 2, page 622), the term seems to have been first used by Rheticus.

The cosecant was called the secans secunda by Magini (1592) and Cavalieri (1643) (Smith vol. 2, page 622).

Some sources say the word cosecant was introduced by Edmund Gunter (1581-1626). This seems to be incorrect, as his use would likely have occurred after that of Rheticus.

COSET was used in 1910 by G. A. Miller in Quarterly Journal of Mathematics.

COSINE. Plato of Tivoli (c. 1120) used chorda residui for cosine.

Regiomontanus (c. 1463) used sinus rectus complementi.

Pitiscus wrote sinus complementi.

Rhaeticus (1551) used basis.

In 1558 Francisco Maurolyco used sinus rectus secundus for the cosine.

Vieta (1579) used sinus residuae.

Magini (1609) used sinus secundus (Smith vol. 2, page 619).

Cosine was coined in Latin by Edmund Gunter (1581-1626) in 1620 in Canon triangulorum, sive, Tabulae sinuum et tangentium artificialium ad radium 100000.0000. & ad scrupula prima quadrantis, Londini: Excudebat G. Iones, 1620. According to Smith (vol. 2, page 619), "Edmund Gunter (1620) suggested co.sinus, a term soon modified by John Newton (1658) into cosinus, a word which was thereafter received with general favor."

COTANGENT. Bradwardine used the term umbra recta.

Magini (1609) used tangens secunda.

Cotangent was coined in Latin by Edmund Gunter (1581-1626) in 1620 in Canon Triangulorum, or Table of Artificial Sines and Tangents. Gunter wrote cotangens.

The term COUNTABLE was introduced by Georg Cantor (1845-1918) (Kline, page 995). According to the University of St. Andrews website, he introduced the word in a paper of 1883.

COUNTING NUMBER is dated ca. 1965 in MWCD10.

COVARIANCE is found in 1930 in The Genetical Theory of Natural Selection by R. A. Fisher (David, 1998).

Earlier uses of the term covariance are found in mathematics, in a non-statistical sense.

COVARIANT was used in 1853 by James Joseph Sylvester (1814-1897) in Phil. Trans.: "Covariant, a function which stands in the same relation to the primitive function from which it is derived as any of its linear transforms do to a similarly derived transform of its primitive" (OED2).

According to Karen Hunger Parshall in James Joseph Sylvester: Life and Work in Letters, Sylvester coined this term.

Cayley at first used the term hyperdeterminant in this sense.

The term COVARIANT DIFFERENTIATION was introduced by Ricci and Levi-Civita (Kline, page 1127).

COVERING (Belegung, from the verb Belegen = cover) was used by Georg Cantor in his last works (1895-97) on set theory, as shown in the following passage from Philip Jourdain's translation (Contributions to the founding of the theory of transfinite numbers, Dover, 1915, p. 94):

By a "covering of the aggregate N with elements of the aggregate M," or, more simply, by a "covering of N with M," we understand a law by which with every element n of N a definite element of M is bound up, where one and the same element of M can come repeatedly into application. The element of M bound up with n is (...) called a "covering function of n". The corresponding covering of N will be called f (N).

Curiously, at the end of his Introduction Jourdain says that

The introduction of the concept of "covering" is the most striking advance in the principles of the theory of transfinite numbers from 1885 to 1895, (...)

Nevertheless, as everybody nowadays can see, a "covering of N with M" in Cantor's terminology is just a function f : N -> M; and his "covering of N" is nothing more than the direct image of N under f - a concept which was introduced for the first time (at least, in a mathematically recognizable form) in Dedekind's Was sind und Was sollen die Zahlen? (1887, §21) [Carlos César de Araújo].

CRAMER-RAO INEQUALITY in the theory of statistical estimation. The inequality was obtained independently by at least three authors in the 1940s. The name "Cramér-Rao inequality" appears in Neyman & Scott (Econometrica, 1948) and recognises the English language publications of Cramér (1946 Mathematical Methods of Statistics) and Rao (1945 Bull. Calcutta Math. Soc. 37, 81-91). L. J. Savage (Foundations of Statistics 1954) drew attention to the work of Fréchet (1943) and Darmois (1945) and "tentatively proposed" the impersonal information inequality. However the name "Cramér-Rao inequality" has remained popular, though the "Fréchet-Darmois-Cramer-Rao inequality" figures in some French literature. [This entry contributed by John Aldrich, with some information from David (1995).]

CRITERION OF INTEGRABILITY is found in 1816 in Edin. Rev. XXVII: "The theorem, which is called the Criterion of Integrability" (OED2).

The term CRITERION OF SUFFICIENCY was used by Sir Ronald Aylmer Fisher in his paper "On the Mathematical Foundations of Theoretical Statistics," in Philosophical Transactions of the Royal Society, April 19, 1922: "The complete criterion suggested by our work on the mean square error (7) is: -- That the statistic chosen should summarise the whole of the relevant information supplied by the sample. This may be called the Criterion of Sufficiency" [James A. Landau].

CRITICAL POINT is found in 1871 in A General Geometry and Calculus by Edward Olney [University of Michigan Historic Math Collection].

CRITICAL REGION is dated 1951 in MWCD10.

CROSS PRODUCT is found on p. 61 of Vector Analysis, founded upon the lectures of J. Willard Gibbs, second edition, by Edwin Bidwell Wilson (1879-1964), published by Charles Scribner's Sons in 1909:

The skew product is denoted by a cross as the direct product was by a dot. It is written

C = A X B

and read A cross b. For this reason it is often called the cross product.

(This citation contributed by Lee Rudolph.)

CROSS-RATIO. According to Taylor (p. 257), cross-ratio first appeared in Elements of Dynamic, Part 1, Kinematic (1878), p. 42, by William Kingdon Clifford (1845-1879). Clifford wrote "The ratio ab.cd : ac.bd is called a cross-ratio of the four points abcd ..."

See also anharmonic ratio and Doppelverhältniss.

CUBE. The word "cube" was used by Euclid. Heron used "hexahedron" for this purpose and used "cube" for any right parallelepiped (Smith vol. 2, page 292).

CUBE ROOT is found in English in 1679 in Moxon, Math. Dict. "Cube Root, the Root or Side of the third Power: So if 27 be the Cube, 3 is the Side or Root" (OED2).

The word CUBOCTAHEDRON was coined by Kepler, according to John Conway.

CUMULANT was used by James Joseph Sylvester in Phil. Trans. (1853) 1. 543: "The denominator of the simple algebraical fraction which expresses the value of an improper continued fraction" (OED2).

In statistics, cumulant is found in 1931 in R. A. Fisher and J. Wishart, "The Derivation of the Pattern Formulae of Two-Way Partitions from Those of Simpler Patterns," Proceedings of the London Mathematical Society, Ser. 2, 33, 195-208 (David, 1995).

Cumulant is a contraction of cumulative moment function, which Fisher used when he first discussed these quantities in his "Moments and Product Moments of Sampling Distributions," Proceedings of the London Mathematical Society, Series 2, 30, 199-238 (1929). The cumulative moment function of a particular order is a function of moments of the same and lower orders which motivates the name. The term "cumulant" was suggested by Hotelling (see J. Amer. Stat Assoc., 28, 1933, 374 and David 2001). Hald (pp. 344-9) describes how several earlier authors had used these quantities, most notably T. N. Thiele [John Aldrich].

CURL. James Clerk Maxwell wrote the following letter to Peter Guthrie Tait on Nov. 7, 1870:

Dear Tait,

What do you call this? Atled?

I want to get a name or names for the result of it on scalar or vector functions of the vector of a point.

Here are some rough-hewn names. Will you like a good Divinity shape their ends properly so as to make them stick?

(1) The result of An
upside-down deltaapplied to a scalar function might be called the slope of the function. Lamé would call it the differential parameter, but the thing itself is a vector, now slope is a vector word, wheras parameter has, to say the least, a scalar sound.

(2) If the original function is a vector then An upside-down deltaapplied to it may give two parts. The scalar part I would call the Convergence of the vector function, and the vector part I would call the Twist of the vector function. Here the word twist has nothing to do with a screw or helix. If the word turn or version would do they would be better than twist, for twist suggests a screw. Twirl is free from the screw notion and is sufficiently racy. Perhaps it is too dynamical for pure mathematicians, so for Cayley's sake I might say Curl (after the fashion of Scroll). Hence the effect of An upside-down deltaon a scalar function is to give the slope of that scalar, and its effect on a vector function is to give the convergence and the twirl of that function. The result of An

upside-down delta2 applied to any function may be called the concentration of that function because it indicates the mode in which the value of the function at a point exceeds (in the Hamiltonian sense) the average value of the function in a little spherical surface drawn round it.

Now if lower case sigmabe a vector function of rhoand F a scalar function of rho,
An upside-down
deltaF is the slope of F
An upside-down
delta. An upside-down
deltaF is the twirl of the slope which is necessarily zero
An upside-down
delta. An upside-down
deltaF = An
upside-down delta2F is the convergence of the slope, which is the concentration of F.
Also S
An upside-down
delta is the convergence of
An upside-down
delta is the twirl of .

Now, the convergence being a scalar if we operate on it with An upside-down delta , we find that it has a slope but no twirl.

The twirl of is a vector function which has no convergence but only a twirl.

Hence, An upside-down
delta 2 , the concentration of , is the slope of the convergence of together with the twirl of the twirl of , the sum of two vectors.

What I want to ascertain from you if there are any better names for these things, or if these names are inconsistent with anything in Quaternions, for I am unlearned in quaternion idioms and may make solecisms. I want phrases of this kind to make statements in electromagentism and I do not wish to expose either myself to the contempt of the initiated, or Quaternions to the scorn of the profane.

In 1873 by Maxwell wrote in A Treatise on Electricity and Magnetism "I propose (with great diffidence) to call the vector part...the curl."

CURRIED FUNCTION. According to an Internet web page, the term was proposed by Gottlob Frege (1848-1925) and first appears in "Uber die Bausteine der mathematischen Logik", M. Schoenfinkel, Mathematische Annalen. Vol 92 (1924). The term was named for the logician Haskell Curry.

CURVATURE. Nicole Oresme assumed the existence of a measure of twist called curvitas. Oresme wrote that the curvature of a circle is "uniformus" and that the curvature of a circle is proportional to the multiplicative inverse of its radius.

A translation of Isaac Newton in Problem 5 of his Methods of series and fluxions is:

A circle has a constant curvature which is inversely proportional to its radius. The largest circle that is tangent to a curve (on its concave side) at a point has the same curvature as the curve at that point. The center of this circle is the "centre of curvature" of the curve at that point.

Curvature appears in English in 1710 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris, in which it is stated that "the Curvatures of different Circles are to one another Reciprocally as their Radii" (OED1).

CURVE FITTING appears in a 1905 paper by Karl Pearson. A footnote therein references a paper "Systematic Fittings of Curves" in Biometrika which may also contain the phrase [James A. Landau].

CURVE OF PURSUIT. The name ligne de poursuite "seems due to Pierre Bouguer (1732), although the curve had been noticed by Leonardo da Vinci" (Smith vol. 2, page 327).

CYCLE (in a modern sense) was coined by Edmond Nicolas Laguerre (1834-1886).

CYCLIC GROUP. The term cyclical group was used by Cayley in "On the substitution groups for two, three, four, five, six, seven, and eight letters," Quart. Math. J. 25 (1891).

The term also appears in 1898 in Introduction to the theory of analytic functions by J. Harkness and F. Morley: "Such a group is called a cyclic group and S is called the generating substitution of the group."

CYCLIC PERMUTATION. Permutation circulaire is found in Cauchy's 1815 memoir "Sur le nombre des valeurs q'une fonction peut acquérir lorsqu'on permute de toutes les manières possibles les quantités qu'elle renferme" (Journal de l'Ecole Polytechnique, Cahier XVII = Cauchy's Oeuvres, Second series, Vol. 13, pp. 64--96.) This usage was found by Roger Cooke, who believes this is the first use of the term.

CYCLIC QUADRILATERAL. Inscriptible polygon is found in about 1696 in Scarburgh, Euclid (1705): "Polygons do arise, that are mutually with a Circle, or with one another Inscriptible and Circumscriptible" (OED2).

Inscribable is found in the 1846 Worcester dictionary.

Inscriptible quadrilateral is found in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

Cyclic quadrilateral is found in 1888 in Casey, Plane Trigonometry (OED2).

The CYCLOID was named by Galileo Galilei (1564-1642) (Encyclopaedia Britannica, article: "Geometry"). According to the website at the University of St. Andrews, he named it in 1599.

CYCLOTOMY and CYCLOTOMIC were used by James Joseph Sylvester in 1879 in the American Journal of Mathematics.

CYLINDER was used by Apollonius (262-190 BC) in Conic Sections.