In Latin, persons who counted 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 above sense,
is *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. Newton did not originally use the
term, preferring *method of fluxions* (Maor, p. 75).

See also *differential calculus* and *integral calculus.*

**CALCULUS OF FINITE DIFFERENCES** appears in 1860 in the title
*Calculus
of Finite Differences* by George Boole [James A. Landau].

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).

**CANONICAL FORM** is found in 1851 in the title *On a remarkable
discovery in the Theory of Canonical forms of Hyperdeterminants* by
James Joseph Sylvester (1814-1897) (OED2).

**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 Gonz?lez Cabill?n 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 COORDINATE** is dated ca. 1888 in MWCD10. However, Hamilton
used *Cartesian method of coordinates* in a paper of 1844 [James A.
Landau].

**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 *Vollst?ndigkeitsaxiom*
("completeness axiom") in his *Grundlagen der Geometrie* (1899) and
*?ber 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 C?sar de Ara?jo.]

**CATENARY.** According to 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.

The OED shows a use of *catenarian arch* 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.The first use of

**CATHETUS.** Nicolas Chuquet (d. around 1500), writing in French,
used the word *cath?te* (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) and 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 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 Fr?chet (1878-1973)
(Katz). The term is dated ca. 1949 in MWCD10.

**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.

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.

The term **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]. According to an
Internet web page, Polya coined the term in the 1920s.

**CENTRAL TENDENCY** is dated ca. 1928 in MWCD10.

**CENTROID** appears in the *Century Dictionary* (1889-1897).
The word is dated 1875-1880 in RHUD2.

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 *c?vienne* in 1888 by Professor
A. Poulain (Facult? catholique d'Angers, France). The word honors the Italian
mathematician Giovanni Ceva (1647?-1734) [Julio Gonz?lez Cabill?n].

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* (R)
*S*. According to him (§37), a "system" (his name for "set")
*K*
? *S* is a *chain* (under f)
when f (*K* ) ?
*K*. (Incidentally, from such a "chain" one really gets a descending
*chain* -in one of the more modern uses of this word -, namely, ...?
f ^{3}(*K*) ?
f^{2}(*K*) ?
f ^{1}(*K*) ?
*K*.) Soon after (§44), he fixes *A* ?
*S* and defines the "chain of the system *A*" (under f
) as the intersection of all chains (under f
) *K* ?*S* such that *A* ?*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* ? f^{1}(*A*)
? f ^{2}(*A*)
? f^{3}(*A*)
? ..., 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 L?wenheim 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 C?sar de
Ara?jo.]

**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).

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

Peter Flor has found *Kettenregel* in *H?here 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."

*Chain rule* occurs in English in 1847 and 1909 *Webster*
dictionaries in its arithmetic sense. The latter dictionary says the rule
is also called *Rees's rule,* "for K. F. de Rees, its inventor."

*Chain rule* occurs in English in the Second English Edition of
R. Courant *Differential and Integral Calculus* translated by E. J.
McShane (n.p.: Interscience Publishers, a division of John Wiley &
Sons, 1937), chapter III section 4.1 and appendix to Chapter III section
3.3.

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 two citations, 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** 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 Gonz?lez
Cabill?n].

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 H?non family (from periodic to strange attractor) is regarded as chaos - unpredictability - and the very existence of the H?non 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

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..."

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

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
Poincar? (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", 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 Vall? Poussin (1866-1962) in *Int?grales 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 (specially 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 C?sar de Ara?jo]

**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).

The term 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.

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* [James A. Landau].

**CHORD** is first used in English in 1551 by Robert Recorde in
*The
Pathwaie to Knowledge* (OED2).

**CIRCLE GRAPH** is dated 1928 in MWCD10.

**CIRCLE OF CONVERGENCE** appears in Edward Burr Van Vleck, "On An
Extension of the 1894 Memoir of Stieltjes," *Transactions of the American
Mathematical Society* 4 (Jul., 1903).

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

**CIRCULAR FUNCTION** appears in the 1872 second edition of *A
Treatise on the Calculus of Finite Differences* by George Boole [James
A. Landau].

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

**CIRCUMCIRCLE** appears in 1885 in *Treatise on Analytic Geometry*
by John Casey (OED2).

**CIRCUMFERENCE.** *Circumferentia* is a Latin translation of
the earlier Greek term *periphereia,* which was used by Euclid. An
earlier use of *periphereia* is by Heraclitus: "The beginning and
end join on the circumference of the circle (kuklou periphereias)" (D.
V. 12 B 103) (Michael Fried).

**CISSOID** appears in Proclus (in *Euclid,* p.111, 152, 177...)
where Proclus refers it to Geminus (c. 130 BC - c. 70 BC), whose original
work we do not have. 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, it became associated with a curve described
by Diocles in his work, *On Burning Mirrors* (there is, by the way,
a translation from the Arabic translation by Toomer).

*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*)," but Michael Fried, who contributed this entry, says
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).

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 (containing all its limit points)** is found in 1902
*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).

**CLOSED (elements produced by an operation are in the set)** is
defined in *Webster's New International Dictionary* (1909).

**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 isCochleo?degenannt, 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:
*Encyclop?die des sciences
math?matiques,* Tome I, Vol. 2, 1907, p. 2. According to Smith (vol.
2, page 393), Vieta coined the term.

The expression **COEFFICIENT OF CORRELATION** apparently was originated
by Edgeworth in 1892, according to "Notes on the History of Correlation,"
by Pearson and Kendall. The term appears in an 1892 paper "Correlated Averages"
by Edgeworth (Stigler, page 319) [James A. Landau].

In 1888, Francis Galton (1822-1911) used the term *index of co-relation*
in *Proc. R. Soc.* (OED2).

The expression *correlation coefficient* appears in a paper published
in 1895 [James A. Landau].

The term **COEFFICIENT OF VARIATION** is due to Karl Pearson (Cajori
1919, page 382). He introduced the term in "Regression, Heredity, and Panmixia"
(1896), according to the DSB.

**COFACTOR** is found in "On the Invariants of Quadratic Differential
Forms, II," Charles Nelson Haskins, *Transactions of the American Mathematical
Society,* Vol. 5, No. 2. (Apr., 1904).

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

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

The term is found 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.

The term **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").

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 Gonz?lez Cabill?n].

**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** is found (with the abbreviation "Com. log.")
in *A new manual of logarithms to seven places of decimals* edited
by Carl Christian Bruhns (1830-1881) and published in German, English,
and Italian in 1870.

The term also appears in 1881 in *Elements of Algebra* by G. A.
Wentworth, which also uses the terms *decimal logarithm* and
*logarithm
in the common system* [James A. Landau].

**COMMUTATIVE** and **DISTRIBUTIVE** were used (in French) by
Fran?ois 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 Gonz?lez Cabill?n).

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

f,sont telles que la fonction de lasomme(alg?brique) d'un nombre quelconque de quantites est ?gale a la somme des fonctions pareilles de chacune de ces quantit?s, seront appel?esdistributives.Ainsi, parce que

a(x+y+ ...) =ax+ay+ ...;E(x+y+ ...) =Ex+Ey+ ...; ...le facteur 'a', l'?tat vari?

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 r?sultats identiques, quel que soit l'ordre dans lequel on les applique au sujet, seront appel?escommutatives entre elles.Ainsi, parce que qu'on a

abz=baz;aEz=Eaz; ...les facteurs constans 'a', 'b', le facteur constant 'a' et l'?tat vari?

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'?tat vari? ou la difference avec le facteur variable, ... n'appartiennent point a la classe des fonctions commutatives entre elles.

**COMPACT** was introduced by Maurice Ren? Fr?chet (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'?l?ments ou lorsque toute infinit? de ses ?l?ments donne lieu ? au moins un ?l?ment limite.

In his 1906 thesis, Fr?chet wrote:

A set E is calledAt the end of his life, Fr?chet 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 ?viter qu'on puisse appeler compact un noyau solide dense qui n'est agr?ment? que d'un fil allant jusqu'? l'infini. C'est une supposition car j'ai compl?tement oubli? 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." Sch?nflies suggested that what Fr?chet called compact be called something like "l?ckenlos" (without gaps) or "abschliessbar" (closable) (Taylor, p. 266).

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

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

**COMPLETE INDUCTION** (vollst?ndige 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 (

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

**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.

**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** and **CONVEX** appear in English in 1571 in *A Geometricall
Practise named Pantometria* by Thomas Digges (1546?-1595) (OED2).

Another term for a concave polygon is a *re-entrant* polygon. Fibonacci
referred to such a polygon as a *figura barbata* in
*Practica geomitrae.*

**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** is found in Florian Cajori, "Divergent
and Conditionally Convergent Series Whose Product is Absolutely Convergent,"
*Transactions of the American Mathematical Society* 2 (Jan., 1901).

The term *semiconvergent,* which may be older, is found in a 1909
*Webster* dictionary.

**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 *Journal of the Royal Statistical Society.*

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.

The word **CONGRUENT** (in Latin for "coincide") was already in use
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 Gonz?lez Cabill?n]

**CONGRUENT** (in modular arithmetic) was defined by Carl Friedrich
Gauss (1777-1855) in 1801 (E. T. Bell, *The Development of Mathematics*).

**CONJECTURE.** 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 *R?cr?ations Math?matiques,* tome II, Note II, Sur les nombres
de Fermat et de Mersenne (1883), ?. 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.

**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.

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

**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].

**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 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.)[James A. Landau]continuum hypothesis,and the question as to whether it is true or not is known as theproblem of the continuum

*Continuum hypothesis* appears in the title "The consistency of
the axiom of choice and of the generalized continuum-hypothesis" by Kurt
G?del, *Proc. Nat. Acad. Sci.,* 24, 556-557 (1938).

**CONTRAPOSITIVE** was used in 1870 by William Stanley Jevons in
*Elementary
Lessons in Logic* (1880).

*Contrapose* and *contraposite* are older words.

**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.

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).

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. 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.

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

Ball (page 243) and Smith (vol. 2, page 622) say the term
*cosecant*
seems to have been first used by Georg Joachim von Lauchen Rheticus (1514-1574)
in his *Opus Palatinum de triangulis,* published posthumously in 1596.

Other 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."

The term **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 1931 in A. L. Bailey, *Journal Amer.
Statist. Assn.* XXXVI 424, in the heading "The analysis of covariance"
(OED2).

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).

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

**CRITICAL POINT** is dated ca. 1889 in MWCD10. The earliest meaning
of this term was "any singular point of a function in the plane of the
complex variable, esp. a branch point, at which two function values become
equal." This definition appears in
*Webster's New International Dictionary,*
1909, and a similar one appears in the 1934 edition of this dictionary.

*Critical point* occurs in "On the Theory of Improper Definite
Integrals," Eliakim Hastings Moore, *Transactions of the American Mathematical
Society,* Vol. 2, No. 4. (Oct., 1901).

The term is defined in the sense involving the derivative of a function
in the *Mathematics Dictionary* of James and James in the 1940s.

**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 *Doppelverh?ltniss.*

**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,** as a term in English, is dated ca. 1696 in MWCD10.

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

**CURL.** In a letter to Peter Guthrie Tait written on Nov. 7, 1870,
James Clerk Maxwell offered some names for**XF**:

Here are some rough-hewn names. Will you like a good Divinity shape their ends properly so as to make them stick?....This quote was taken from Stein and Barcellos, page 984.The vector part [

XF] I would call the twist of the vector function. Here the word twist has nothing to do with a screw or helix. [T]he wordturn... 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).

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.

**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).

*Cyclic group* is found in *Finite Groups* (1908) by Harold
Hilton [Avinoam Mann].

**CYCLIC QUADRILATERAL** was used in 1862 by George Salmon (1819-1904)
in *A treatise on analytic geometry of three dimensions,* published
in 1874 (OED1).

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.*