**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* = *a ^{x},*
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 *ay*^{2} - *x*^{2} + *bx*^{2}
= 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 DETERMINANT, EQUATION,
POLYNOMIAL, ROOT, VALUE, VECTOR.** See *Eigenvalue.*

**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, *a ^{x}*,

**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* ; D*xz* = *x*D*z* [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 {E^{n}}
is a sequence of nomempty, closed subsets of E such that E^{n+1} is a
subset of E^{n} for each n, there is at least one element that belongs
to all of the E^{n}'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} + *n*^{2}
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 *a, b* indefinite omnes numeros reales integros inter -

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 *a*^{2} + *b*^{2};
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 *r*_{0} 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 *R*_{1} (the correlation of the first variable with
the other two). Yule (1907) discussed the coefficient of n-fold correlation *R*^{2}_{1(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

(2) If the original function is a vector then *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 ^{2} 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 *F* a scalar function of

*F* is the slope of *F*

*V* *F* is the twirl of the slope which is necessarily
zero

*S* *F* = ^{2}*F* is the convergence of the slope, which is the concentration of *F.*

Also *S*

*V*

Now, the convergence being a scalar if we
operate on it with

The twirl of

Hence, ^{2}

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.*