**FACTOR (noun).** Fibonacci (1202) used *factus ex multiplicatione*
(Smith vol. 2, page 105).

*Factor* appears in English in *Algebra* by Kersey: "The Quantities
given to be multiplied one by the other are called Factors."

**FACTOR (verb)** appears in English in 1848 in *Algebra* by
J. Ray: "The principal use of factoring, is to shorten the work, and simplify
the results of algebraic operations." *Factorize* (spelled "factorise")
is found in 1886 in *Algebra* by G. Chrystal (OED2).

**FACTOR GROUP** was used (in German as *factorgruppe*) by Otto
Hölder (1859-1937) in 1889. The term was apparently coined by him.

The term **FACTORIAL** was coined (in French) by Louis François Antoine
Arbogast (1759-1803).

**FEJER KERNEL** appears in 1937 in *Differential and Integral
Calculus,* 2nd. ed. by R. Courant: "The expression *s _{m}*
is called the "Fejér kernel", and is of great importance in the more advanced
study of Fourier series" [James A. Landau].

The term **FERMAT'S LAST THEOREM** appears in the title of Gabriel
Lamé's "Memoire sur le dernier theoreme de Fermat," C. R. Acad. Sci. Paris,
9, 1839, pp. 45-46. Lamé explained the reason for the term:

De tous les theoremes sur les nombres, enonces par Fermat, un seul reste incompletement demontre. [Of all the theorems on numbers stated by Fermat, just one remains incompletely demonstrated (proved).]In his

L'Academie nous a charges, M. Liouville et moi, de lui rendre compte d'un Memoire de M. Lamé sur le dernier theoreme de Fermat. [The Academy has charged us, Mr Liouville and myself, to review memoir of Mr. Lamé on the last theorem of Fermat.]This citation is from C. R. Acad. Sci. Paris, 9, 1839, pp. 359-363.

An early use of the phrase "Last Theorem of Fermat" in English appears in "Application to the Last Theorem of Fermat" (1860), in "Report on the Theory of Numbers", part II, art. 61, addressed by Henry J. S. Smith.

The OED2 has this 1865 citation from *A Dictionary of Science, Literature,
and the Arts,* by William T. Brande and Cox: "Another theorem, distinguished
as Fermat's last Theorem, has obtained great celebrity on account of the
numerous attempts that have been made to demonstrate it."

In May 1816, Carl Friedrich Gauss (1777-1855) wrote a letter to Heinrich Olbers in which he mentioned the theorem. According to an English translation (Singh, p. 105; also an Internet web page), he referred to the theorem as Fermat's Last Theorem. However, in fact Gauss wrote, "Ich gestehe zwar, dass das Fermatsche Theorem als isolierter Satz fuer mich wenig Interesse hat..." (I confess that the Fermat theorem holds little interest for me as an isolated result...)

[Julio González Cabillón and William C. Waterhouse contributed to this entry.]

The term **FIBONACCI SEQUENCE** was coined by Edouard Anatole Lucas
(1842-1891) (*Encyclopaedia Britannica,* article: "Leonardo Pisano").

**FIBONACCI NUMBER** is dated 1890-95 in RHUD2.

**FIELD (neighborhood).** In 1893 in *A treatise on the theory
of functions* J. Harkness and F. Morley used the word *field* in
the sense of an interval or neighborhood:

The function f(x) is said to be continuous at the point c ... if a field (c-h to c+h) can be found such that for all points of this field, |f(x)-f(c)| < epsilon.The term

**FIELD (modern definition).** The term *Zahlkörper* (body of
numbers) is due to Richard Dedekind (1831-1916) (Kline, page 1146). The
term appears in *Stetigkeit und Irrationale Zahlen* (Continuity &
Irrational Numbers). Dedekind did not allow for finite fields; for him,
the smallest field was the field of rational numbers.

Dedekind used Zahlenkörper in Supplement XI of his 4th edition of Dirichlet's
*Vorlesungenueber
Zahlentheorie,* section 160. In a footnote, he explained his choice
of terminology, writing that, in earlier lectures (1857-8) he used the
term 'rationalen Gebietes' and he says that Kronecker (1882) used the term
'Rationalitaetsbereich'.

Julio González Cabillón believes that Eliakim Hastings Moore (1862-1932)
was the first person to use the English word *field* in its modern
sense and the first to allow for a finite field. He coined the expressions
"field of order *s*" and "Galois-field of order *s* = *q ^{n}*."
These expressions appeared in print in December 1893 in the

Perhaps because of the older mathematical meaning of the English word3. Galois-field of order s = q^{n}Suppose that we have a system of symbols or

marks,µ_{1}, µ_{2}... µ_{s}, in numberss,and suppose that thesesmarks may be combined by the four fundamental operations of algebra ... and that when the marks are so combined the results of these operations are in every case uniquely determined and belong to the system of marks. Such a system ofsmarks we call afield of order s.The most familiar instance of such a field, of order

s=q=aprime, is the system ofqincongruous classes (moduloq) of rational integral numbersa.[...]It should be remarked further that every field of order

sis in fact abstractly considered a Galois-field of orders=q^{n}.

At any event, a decade later Edward V. Huntington wrote:

Closely connected with the theory of groups is the theory of fields, suggested by GALOIS, and due, in concrete form, to DEDEKIND in 1871. The wordThe following footnote makes it clear that termfieldis the English equivalent for DEDEKIND's termKörper;.KRONECKER's termRationalitätsbereich,which is often used as a synonym, had originally a somewhat different meaning. The earliest expositions of the theory from the general or abstract point of view were given independently by WEBER and by Moore, in 1893, WEBER's definition of an abstract field being substantially as follows: [...]The earliest sets of

independentpostulates for abstract fields were given in 1903 by Professor Dickson and myself; all these sets were the natural extensions of the sets of independent postulates that had already been given for groups.

The most familiar and important example of an infinite field is furnished by the rational numbers, under the operations of ordinary addition and multiplication. In fact, a field may be briefly described as a system in which the rational operations of algebra may all be performed (excluding division by zero). A field may be finite, provided the number of elements (called the order of the field) is a prime or a power of a prime.The quote above is from a paper by Huntington presented to the AMS on December 30, 1904, and received for publication on February 9, 1905.

[Information for this article was contributed by Julio González Cabillón, Heinz Lueneburg, William Tait, and Sam Kutler].

**FINITE** appears in English in 1570 in Sir Henry Billingsley's
translation of Euclid's *Elements* (OED2).

**FINITE CHARACTER.** This set-theoretic term - usually applied to
"properties" ("collections") of sets - was introduced by John Tukey (1915-
) in *Convergence and uniformity in topology,* Annals of Math. Studies,
No 2, Princeton University Press, 1940, p. 7. *Tukey's lemma* (a useful
equivalent of the axiom of choice) states that every non-empty collection
of finite character has a maximal set with respect to inclusion. This result
is also known as the "Teichmüller-Tukey lemma" because Oswald Teichmüller
(1913-1943) had arrived at it independently in *Braucht der Algebriker
das Answahlaxiom?,* Deutsche Mathematik, vol. 4 (1939), pp. 567-577
[Carlos César de Araújo].

**FIRST DERIVATIVE, SECOND DERIVATIVE, etc.** Christian Kramp (1760-1826)
used the terms *premiére dérivée* and *seconde dérivée* (first
derivative and second derivative) (Cajori vol. 2, page 67).

However, the DSB implies Joseph Louis Lagrange (1736-1813) introduced
these terms in his *Théorie des fonctions.*

**FIXED-POINT (arithmetic)** was used in 1955 by R. K. Richards in
*Arithmetic
Operations in Digital Computers* [James A. Landau].

**FLOATING-POINT** is dated 1948 in MWCD10.

The terms **FLUXION** and **FLUENT** were introduced by Isaac
Newton (1642-1727). Newton used *fluent* in 1665 to represent any
relationship between variables (Kline, page 340). The word *fluxion*
appears once, perhaps by an oversight, in the *Principia* (Burton,
page 377).

The term **FOCUS** (of an ellipse) was introduced by Johannes Kepler
(1571-1630).

**FOLIUM OF DESCARTES.** Roberval, through an error, was led to believe
the curve had the form of a jasmine flower, and he gave it the name *fleur
de jasmin,* which was afterwards changed (Smith vol. 2, page 328).

*Folium of Descartes* was used in 1848 in *Differential Calculus*
(1852) by B. Price (OED2).

The curve is also known as the *noeud de ruban.*

**FORMULA** appears in the phrase "an algebraic formula" in 1796
in *Elements of Mineralogy* by Richard Kirwan (OED2).

**FOURIER'S THEOREM** is found in English 1834 in *Rep. Brit. Assoc.*
(OED2).

**FOURIER TRANSFORM** is found in English in 1923 in the *Proceedings
of the Cambridge Philosophical Society* (OED2).

**FOURTH DIMENSION.** Nicole Oresme (c. 1323-1382) wrote, "I say
that it is not necessary to give a fourth dimension" in *Quaestiones
super geometriam Euclides* [James A. Landau].

**FRACTAL** was coined by Benoit Mandelbrot (1924- ) in 1975 in Les
Objets Fractals according to the OED2, which shows a use of the word (as
an adjective in "the idea of the fractal dimension") in the Nov. 1975 *Scientific
American.* The OED2 also shows a use of the word by Mandelbrot in *Fractals*
in 1977:

Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that..classical geometry..is hardly of any help in describing their form... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals - or fractal sets.Johnson (page 155) says the term was coined by Mandelbrot in an article "Intermittent Turbulence and Fractal Dimension" published in 1976.

In *The Fractal Geometry of Nature* Mandelbrot wrote:

I coinedAccording to John Conway, Mandelbrot originally definedfractalfrom the Latin adjectivefractus.The corresponding Latin verbfrangeremeans "to break:" to create irregular fragments. It is therefore sensible -- and how appropriate for our needs! -- that, in addition to "fragmented" (as infraction or refraction), fractusshould also mean "irregular," both meanings being preserved infragment.The proper pronunciation is frac'tal, the stress being placed as

infraction.

The word **FRACTION** is from the Latin *frangere* (to break).
Some writers called fractions "broken numbers."

In the 12th century Adelard of Bath used *minuciae* in his *Regulae
abaci.* However in the translation of al-Khowarizmi attributed to Adelard,
*fractiones*
is used (Smith vol. 2, page 218).

Johannes Hispalensis in his *Liber Algorismi de practica arismetrice*
used *fractiones* (Smith vol. 2, page 218).

Fibonacci (1202) generally used *fractio.*

In English, the word was used by Geoffrey Chaucer (1342-1400) (and spelled
"fraccions") about 1391 in *A treatise on the Astrolabe* (OED2).

Baker (1568) spoke of "fractions or broken numbers."

The term **FREQUENTIST** (one who believes that the probability of
an event should be defined as the limit of its relative frequency in a
large number of trials) was used by M. G. Kendall in 1949 in Biometrika
XXXVI. 104: "It might be thought that the differences between the frequentists
and the non-frequentists (if I may call them such) are largely due to the
differences of the domains which they purport to cover" (OED2).

**FRUSTUM** first appears in English in 1658 in *The Garden of
Cyrus* by Sir Thomas Browne (OED2). [This word is commonly misspelled
as *frustrum.*]

The term **FUCHSIAN FUNCTION** was coined by Henri Poincaré (1854-1912)
(Smith vol. I and *Encyclopaedia Britannica,* article: "Poincaré").
He used *Fuchsian* and *Kleinean functions* for automorphic functions
of one complex variable, which he discovered (DSB).

The word **FUNCTION** first appears in a Latin manuscript "Methodus
tangentium inversa, seu de fuctionibus" written by Gottfried Wilhelm Leibniz
(1646-1716) in 1673. Leibniz used the word in the non-analytical sense,
as a magnitude which performs a special duty. He considered a function
in terms of "mathematical job"--the "employee" being just a curve. He apparently
conceived of a line doing "something" in a given *figura* ["aliis
linearum in figura data functiones facientium generibus assumtis"]. From
the beginning of his manuscript, however, Leibniz demonstrated that he
already possessed the idea of function, a term he denominates *relatio.*

A paper "De linea ex lineis numero infinitis ordinatim..." in the *Acta
Eruditorum* of April 1692, pp. 169-170, signed "O. V. E." but probably
written by Leibniz, uses *functiones* in a sense to denote the various
'offices' which a straight line may fulfil in relation to a curve, viz.
its tangent, normal, etc.

In the *Acta Eruditorum* of July 1694, "Nova Calculi differentialis..."
(page 316), Leibniz used the word *function* almost in its technical
sense, defining *function* as "a part of a straight line which is
cut off by straight lines drawn solely by means of a fixed point, and of
a point in the curve which is given together with its degree of curvature."
The examples given were the ordinate, abscissa, tangent, normal, etc. [Cf.
page 150 of Leibniz' "Mathematische Schriften," vol. III, edited by C.
I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63.]

In *Mathematische Schriften* is a letter of Sept. 1694 from Johann
Bernoulli to Leibniz saying "quantitatem quomodocunque formatam ex indeterminatis
et constantibus," but without explicit reference to the Latin term *functio.*

On July 5, 1698, Johann Bernoulli, in another letter to Leibniz, for
the first time deliberately assigned a specialized use of the term *function*
in the analytical sense, writing "earum [applicatarum] quaecunque functiones
per alias applicatas *PZ* expressae." (Cajori 1919, page 211) [Cf.
page 507 of Leibniz' "Mathematische Schriften," vol. III, edited by C.
I. Gerhardt, Berlin-Halle (Asher-Schmidt), 1849-63. Also see pages 506-510
and 525-526] At the end of that month, Leibniz replied (p. 526), showing
his approval.

(Information for this entry was provided by Julio González Cabillón and the OED2.)

The phrase **FUNCTION OF x** was introduced by Leibniz (Kline,
page 340).

The term **FUNCTIONAL CALCULUS** was introduced in French by Jacques-Salomon
Hadamard (1865-1963) in the preface of his "Leçons sur le calcul des variations"
[Lessons on the Calculus of Variations], Paris: Librairie Scientifique
A. Hermann et Fils, 1910, p. vii:

Le Calcul des variations n'est autre chose qu'un premier chapitre de la doctrine qu'on nomme aujourd'hui le Calcul Fonctionnel ... [The variational calculus is nothing but a first chapter of the doctrine which one calls today "Functional Calculus"...]For functional, Vito Volterra (1860-1940) used the term "functions of other functions," according to Kramer (p. 550). He used "line function," according to the DSB.

This entry was contributed by Julio González Cabillón.

The term **FUNCTIONAL ANALYSIS** was introduced by Paul P. Lévy (1886-1971)
(Kline, p. 1077; Kramer, p. 550).

**FUNCTOR** was coined by the German philosopher Rudolf Carnap (1891-1970),
who used the word in *Logische Syntax der Sprache,* published in 1934.
For Carnap, a functor was not a kind of mapping, but a function sign -
a syntactic entity. In *Introduction to Symbolic Logic and its Applications*
(Dover, 1958) he defined a n-place functor as "any sign whose full expressions
(involving n arguments) are not sentences". This definition implies that
a functor must stand before its argument-expressions, so that the sign
+, for example, is not a functor when used as an infix sign ^ although
it has the same logical character as a functor. According to him, the function
designated by a functor is the "intension" of that functor, while its "extension"
is the "value-distribution" of the function.

As used in category theory, *functor* was introduced by Samuel
Eilenberg and Saunders Mac Lane, borrowing the term from Carnap's *Logische
Syntax der Sprache.* In his *Categories for the Working Mathematician*
(1972) Mac Lane says (pp. 29-30):

Categories, functors, and natural transformations were discovered by Eilenberg-Mac Lane (...) Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: "Category" from Aristotle and Kant, "Functor" from Carnap (...).In

Perhaps the custom they [S. Eilenberg and S. Mac Lane] had adopted of systematically using notations such as (...) for the various groups they defined in their 1942 paper, suggested to them that they were defining each time a kind of "function" which assigned a commutative group to an arbitrary commutative group (or to a pair of such groups) according to a fixed rule. Perhaps to avoid speaking of the "paradoxical" "set of all commutative groups", they coined the word "functor" for this kind of correspondence; (...)[Carlos César de Araújo]

**FUNDAMENTAL EQUATION.** This term was used by Leopold Alexander
Pars (1896-1985) for the theorem of Lagrange.

The term **FUNDAMENTAL FUNCTIONS** (meaning eigenfunctions) is due
to Poincaré, according to the University of St. Andrews website.

The term **FUNDAMENTAL GROUP** is found in "The Invariant Theory
of the Inversion Group: Geometry Upon a Quadric Surface," Edward Kasner,
*Transactions
of the American Mathematical Society,* Vol. 1, No. 4. (Oct., 1900).

**FUNDAMENTAL SYSTEM.** According to the DSB, Immanuel Lazarus Fuchs
(1833-1902) "introduced the term 'fundamental system' to describe *n*
linearly independent solutions of the linear differential equation *L*(*u*)
= 0."

The term **FUNDAMENTAL THEOREM OF ALGEBRA** "appears to have been
introduced by Gauss" (Smith, 1929, and Burton, page 512).

The term **FUZZY LOGIC** was coined in 1965 by Iranian computer scientist
Lofti Zadeh of the University of California at Berkeley, although the core
concepts go back to the work of Polish mathematician Jan Lukasiewicz in
the 1920s (Hutchinson Encyclopedia).