早期數學字彙的歷史 (C)

Last revision: Aug. 3, 1999

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

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

Instead of "categorical", the term "complete" is sometimes used, chiefly in older texts. The influence, in this case, comes from Hilbert's 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 catenary shown by the OED is by Thomas Jefferson in a letter of Dec. 23, 1788, to Thomas Paine. The two men discussed the design of bridges. Paine had referred to a catenarian arch in his letter to Jefferson of Sept. 15. Jefferson had received a treatise on the equilibrium of arches from Lorenzo Mascheroni.

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 Cayley's Table, and it was used by Cayley in his first article on group theory, Philosophical Magazine, vol. 7 (1854), p. 49. The theorem may be called Cayley's Theorem, and it might reasonably be regarded as third in order of importance, being preceded only by the theorems of Lagrange and Sylow.
[Contributed by Ken Pledger]

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

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) ? f2(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 ? f1(A) ? f 2(A) ? f3(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 rule
James 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 appear to behave in the long run in an unpredictable fashion. [Carlos C?sar de Ara?jo]

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

See also 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:

... the Cochlea, or Spiral about a Cylinder, arising from a Circular motion about an Ax, together with a Rectilinear (in the Surface of the Cylinder) Perpendicular to the Plain of such Circle, (or, if the Cylinder be Scalene at such Angles with the Plain of the Circle, as is the Axis of that Cylinder) both motions being uniform, but not in the same Plain.
Some sources incorrectly attribute the term to Benthan and Falkenburg in 1884. While studying the processes of a mechanism of construction for steam engines, C. Falkenburg, Mechanical Engineer of the Actiengesellschaft Atlas in Amsterdam, rediscovered this curve. On March 25, 1883, he submitted an article titled "Die Cochleo?de", which was published in Archiv der Mathematik und Physik.
Er hat sie daher die Cochleo?de genannt, von *cochlea* = Schneckenhaus. [Therefore, it was christened the Cochleoid, from *cochlea* = snail's house.]
The reference for this citation is Nieuw Archief voor Wiskunde [Amsterdam: Weytingh & Brave], vol. 10, pp. 76-80, 1884. This entry was contributed by Julio Gonz?lez Cabill?n.

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

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

Les fonctions qui, comme f, sont telles que la fonction de la somme (alg?brique) d'un nombre quelconque de quantites est ?gale a la somme des fonctions pareilles de chacune de ces quantit?s, seront appel?es distributives.

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 pas

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

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

4. Soit

fgz = gfz.

Les fonctions qui, comme f et g, sont telles qu'elles donnent des r?sultats identiques, quel que soit l'ordre dans lequel on les applique au sujet, seront appel?es commutatives 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 pas

Sin.az = a Sin.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.

(These citations were provided by Julio Gonz?lez Cabill?n).

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 est compact lorsqu'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.
This citation was provided by Mark Dunn.

In his 1906 thesis, Fr?chet wrote:

A set E is called compact if, when {En} is a sequence of nomempty, closed subsets of E such that En+1 is a subset of En for each n, there is at least one element that belongs to all of the En's.
At the end of his life, Fr?chet did not remember why he chose the term:
... 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 (Origin of the name "mathematical induction", Amer. Math. Monthly, 1918, pp. 197-201) noted an earlier use of the term "vollst?ndige Induktion" in the article "Induction" in Ersch and Gruber’s Encyklop?die (1840), but in an uninteresting and totally different "Aristotelian sense". According to Abraham Fraenkel (1891-1965) (Abstract Set Theory, 1953, p. 253),
[the] term "complete induction" used in most continental languages (...) [stress] the contrast with induction in natural science which is incomplete by its very nature, being based on a finite and even relatively small number of experiments.
This entry was contributed by Carlos C?sar de Ara?jo. See also mathematical induction.

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

...quando campus arithmeticae ad quantitates imaginarias extenditur, ita ut absque restrictione ipsius obiectum constituant numeri formae a + bi, denotantibus i pro more quantitatem imaginariam \/-1, atque a, b indefinite omnes numeros reales integros inter -oo et +oo. Tales numeros vocabimus numeros integros complexos, ita quidem, ut reales complexis non opponantur, sed tamquam species sub his contineri censeatur.
The citation above is from Gauss's paper "Theoria Residuorum Biquadraticorum, Commentatio secunda," Societati Regiae Tradita, Apr. 15, 1831, published for the first time in Commentationes societatis regiae scientiarum Gottingensis recentiones, vol. VII, Gottingae, MDCCCXXXII (1832)]. [Julio Gonz?lez Cabill?n]

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

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

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

CONFIDENCE INTERVAL was coined by Jerzy Neyman (1894-1981) in 1934 in 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.
CONFORMAL MAPPING. The term projectio conformis was introduced by F. T. Schubert in 1789 (DSB, article: "Euler").

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

Shortly afterwards, on September 8, 1679, he included a similar definition in a letter to Hugens (sic) van Zulichem. In his ges. Werke etc. as above, volume 2, p. 22, he illustrates congruence with a pair of triangles, and says that they "peuvent occuper exactement la meme place, et qu'on peut appliquer ou mettre l'un sur l'autre sans rien changer dans ces deux figures que la place." [Ken Pledger and Julio 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.
CONJUGATE. Augustin-Louis Cauchy (1789-1857) used conjugu?es for for a + bi and a - bi in Cours d'Analyse alg?brique (1821) (Smith vol. 2, page 267).

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

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

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

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

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

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

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 continuum hypothesis, and the question as to whether it is true or not is known as the problem of the continuum
(Hausdorff used [alef] to mean the infinity of the continuum.)[James A. Landau]

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 Was sind und Was sollen die Zahlen? (1887, §21) [Carlos C?sar de Ara?jo].

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

C = A X B

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

(This citation contributed by Lee Rudolph.)

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

See also anharmonic ratio and 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 forAn upside-down deltaXF:

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

The vector part [An upside-down deltaXF] I would call the twist of the vector function. Here the word twist has nothing to do with a screw or helix. [T]he word turn ... 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).

This quote was taken from Stein and Barcellos, page 984.

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

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

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

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

A circle has a constant curvature which is inversely proportional to its radius. The largest circle that is tangent to a curve (on its concave side) at a point has the same curvature as the curve at that point. The center of this circle is the "centre of curvature" of the curve at that point.
Curvature appears in English in 1710 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris, in which it is stated that "the Curvatures of different Circles are to one another Reciprocally as their Radii" (OED1).

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

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

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

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

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.

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