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.The 1890

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 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).

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).

In 1798 *A Course of Mathematics* by Charles Hutton has:
"Add the figures..and find how many nines are contained in their sum.
Reject those nines, and set down the remainder" (OED2).

*Casting out nines* is found in 1811 in *An Elementary
Investigation of the Theory of Numbers* by Peter Barlow [James A. Landau].

**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,"

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(Strictly speaking, the categoricity in itself is not seem in this statement but in itsNand consequently (...) to one another.

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

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

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

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

*Cathetus* occurs in English 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[Contributed by Ken Pledger]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 calledCayley's Theorem,and it might reasonably be regarded as third in order of importance, being preceded only by the theorems of Lagrange and Sylow.

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 June 1871 in "The Vegetable World,
Part II," *The Ladies' repository: a monthly periodical, devoted to
literature, arts, and religion*: "Numerous ovules are inserted in
two series at the central angle of the cells."

*Central angle* is also found the March 1876 issue of *The
Manufacturer and Builder*: "...but this angle is half the central
angle of the octagon...."

**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 the arithmetic sense in
an 1847 *Webster* dictionary.

*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 ruleJames 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

**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 Shapiro:

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.

The term **CHARACTERISTIC EQUATION** (for determinants) was
introduced by Cauchy, *Exercises d'analyse et de physique
mathématique,* 1, 1840, 53 = *Oeuvres,* (2), 11, 76
(Kline, page 801; Eves, page 364).

According to the University of St. Andrews website, Wilhelm Karl Joseph Killing (1847-1923) introduced the term 'characteristic equation' of a matrix.

**CHARACTERISTIC FUNCTION.** 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", not of a random variable, but of a set
*A* with respect to a "superset" *U* is also 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

See also the entry *eigenvector* on this website. [Hans
Fischer, Brian Dawkins, Ken Pledger, Carlos César de
Araújo]

**CHARACTERISTIC ROOT** is found in "One-Parameter Projective
Groups and the Classification of Collineations," Edward B. Van Vleck,
*Transactions of the American Mathematical Society,* Vol. 13,
No. 3. (Jul., 1912).

*Characteristic root* is found in H. F. Blichfeldt,
"Linear groups which contain transformations having only two distinct
characteristic roots,"
*American M. S. Bull.* 23, 207 (1917).

*Characteristic root* is also found in 1933 in *The Theory of
Matrices* by C. C. MacDuffee [Ken Pledger].

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.

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.

**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).

**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 1857 in *Mathematical Dictionary
and Cyclopedia of Mathematical Science.*

**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.* XXXIV: "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*:

... theSome 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 theCochlea,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.

Er hat sie daher dieThe reference for this citation isCochleoïdegenannt, von *cochlea* = Schneckenhaus. [Therefore, it was christened theCochleoid,from *cochlea* = snail's house.]

**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
Woolsey Johnson,
"On the calculation of the Co-factors of alternants of the fourth order,"
*Newcomb Am. J.* (1885).

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

*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 hisArs Combinatoriahe 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 theArs Combinatoria.I have found only two places where Leibniz applies the general term "discerptio" to the special partition into two summands.

*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(These citations were provided by Julio González Cabillón).

f(x+y+ ...) =fx+fy+ ...Les fonctions qui, comme

f,sont telles que la fonction de lasomme(algébrique) d'un nombre quelconque de quantites est égale a la somme des fonctions pareilles de chacune de ces quantités, seront appeléesdistributives.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 pasSin.(

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

fetg,sont telles qu'elles donnent des résultats identiques, quel que soit l'ordre dans lequel on les applique au sujet, seront appeléescommutatives 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 pasSin.

az=aSin.z;Exz=xEz; Dxz=xDz[D = delta]; ...il s'ensuit que le sinus avec le facteur constant, l'état varié ou la difference avec le facteur variable, ... n'appartiennent point a la classe des fonctions commutatives entre elles.

**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 estThis citation was provided by Mark Dunn.compactlorsqu'il ne comprend qu'un nombre fini d'éléments ou lorsque toute infinité de ses éléments donne lieu à au moins un élément limite.

In his 1906 thesis, Fréchet wrote:

A set E is calledAt the end of his life, Fréchet did not remember why he chose the term:compactif, 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.

... 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,"

[the] term "complete induction" used in most continental languages (...) [stress] the contrast with induction inThis entry was contributed by Carlos César de Araújo. See alsonatural sciencewhich is incomplete by its very nature, being based on a finite and even relatively small number of experiments.

**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 quantitatesThe citation above is from Gauss's paper "Theoria Residuorum Biquadraticorum, Commentatio secunda," Societati Regiae Tradita, Apr. 15, 1831, published for the first time inimaginariasextenditur, ita ut absque restrictione ipsius obiectum constituant numeri formaea+bi,denotantibusipro more quantitatem imaginariam , atquea, bindefinite omnes numeros reales integros inter - et +. Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeatur.

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. and the use of the term in English
in a dictionary of 1730-6.

**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 1857 in *Mathematical
Dictionary and Cyclopedia of Mathematical Science.* Another
term is *re-entrant polygon.*

*Concave polygon* appears in 1900 in *The Teaching of
Elementary Mathematics* by David Eugene Smith.

**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[James A. Landau]AandBbe two events whose probabilities are (A) and (B). It is understood that the probability (A) is determined without any regard toBwhen nothing is known about the occurrence or nonoccurrence ofB.When itisknown thatBoccurred,Amay have a different probability, which we shall denote by the symbol (A, B) and call 'conditional probability ofA,given thatBhas actually happened.'

**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(This entry was contributed by Michael Fried.)The Presocratics,Indianapolis: The Bobbs-Merrill Company, Inc., 1960, p.183).

**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.

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 esseHis 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.congrua,ut A.B et C.D in fig.39 ...

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.

**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 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."

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

In German, *Integrationconstante* appears in 1859 in "Ueber die
Anzahl der Primzahlen unter einer gegebenen Grösse" by Bernhard
Riemann. The word may occur much earlier.

*Constant of integration* occurs in 1891 in *Differential and
Integral Calculus* by George A. Obsorne: "In general [a
formula], where *c* denotes an arbitrary constant called the
*constant of integration.*"

**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,This citation was provided by James A. Landau.contingency.We reach the notion of a pure contingency table, in which the order of the sub-groups is of no importance whatever.

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).

**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(Hausdorff used [alef] to mean the infinity of the continuum.)continuum hypothesis,and the question as to whether it is true or not is known as theproblem 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 namedconvex,and which are such that a straight line cannot cut them in more than two points.

The same dictionary also has 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.]

**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

**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 dated ca. 1889 in MWCD10.

*Critical point* appears in 1893 in *A Treatise on the Theory
of Functions* by James Harkness and Frank Morley:

Those points of thez-plane at which two or more branches are equal, or at which one or more branches are infinite, are namedcritical points.

In 1912 *Advanced Calculus* by Edwin Bidwell Wilson has: "Points
where *f*'(*z*) = 0 are called *critical points* of
the function."

**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(This citation contributed by Lee Rudolph.)C = A X B

and read A

crossb. For this reason it is often called thecrossproduct.

**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 Mahtmeatical 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. 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,In 1873 by Maxwell wrote inWhat 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 applied to a scalar function might be called the slope of the function. Lamé would call it the differential parameter, but the thing itself is a vector, now slope is a vector word, wheras parameter has, to say the least, a scalar sound.

(2) If the original function is a vector then applied to it may give two parts. The scalar part I would call the Convergence of the vector function, and the vector part I would call the Twist of the vector function. Here the word twist has nothing to do with a screw or helix. If the word

turnorversionwould 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 on a scalar function is to give the slope of that scalar, and its effect on a vector function is to give the convergence and the twirl of that function. The result of^{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 be a vector function of and

Fa scalar function of ,

Fis the slope ofF

V.Fis the twirl of the slope which is necessarily zero

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

AlsoSis the convergence of

Vis the twirl of .Now, the convergence being a scalar if we operate on it with , we find that it has a slope but no twirl.

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

Hence,

^{2}, the concentration of , is the slope of the convergence of together with the twirl of the twirl of , the sum of two vectors.What I want to ascertain from you if there are any better names for these things, or if these names are inconsistent with anything in Quaternions, for I am unlearned in quaternion idioms and may make solecisms. I want phrases of this kind to make statements in electromagentism and I do not wish to expose either myself to the contempt of the initiated, or Quaternions to the scorn of the profane.

**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.

**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 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.*