早期數學字彙的歷史 (F)

Last revision: July 31, 1999


F DISTRIBUTION is found in Leo A. Aroian, "A study of R. A. Fisher's z distribution and the related F distribution," Ann. Math. Statist. 12, 429-448 (1941). The term is named for Fisher.

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 H闤der (1859-1937) in 1889. The term was apparently coined by him.

The term FACTORIAL was coined (in French) by Louis Fran蔞is Antoine Arbogast (1759-1803).

FEJER KERNEL appears in 1937 in Differential and Integral Calculus, 2nd. ed. by R. Courant: "The expression sm is called the "Fej廨 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 Lam's "Memoire sur le dernier theoreme de Fermat," C. R. Acad. Sci. Paris, 9, 1839, pp. 45-46. Lam 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 Rapport sur un memoire de M. Lam, Cauchy remarks:
L'Academie nous a charges, M. Liouville et moi, de lui rendre compte d'un Memoire de M. Lam sur le dernier theoreme de Fermat. [The Academy has charged us, Mr Liouville and myself, to review memoir of Mr. Lam 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 Gonz嫮ez Cabill鏮 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 is not defined therein, suggesting the authors believed it was a common usage.

FIELD (modern definition). The term Zahlk顤per (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 Zahlenk顤per 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 Gonz嫮ez Cabill鏮 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 = qn." These expressions appeared in print in December 1893 in the Bulletin of the New York Mathematical Society III. 75. The paper was presented to the Congress of Mathematics at Chicago on Aug. 25, 1893:

3. Galois-field of order s = qn

Suppose that we have a system of symbols or marks, 1, 2 ... s, in numbers s, and suppose that these s marks 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 of s marks we call a field of order s.

The most familiar instance of such a field, of order s = q = a prime, is the system of q incongruous classes (modulo q) of rational integral numbers a. [...]

It should be remarked further that every field of order s is in fact abstractly considered a Galois-field of order s = qn.

Perhaps because of the older mathematical meaning of the English word field, Moore seems to have been very careful in systematically referring to a field of order s and not the shorter term field.

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 word field is the English equivalent for DEDEKIND's term K顤per;. KRONECKER's term Rationalit酹sbereich, 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 independent postulates 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 following footnote makes it clear that term field already had the same mathematical meaning at the turn of the century as it does now:
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 Gonz嫮ez Cabill鏮, 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 "Teichmller-Tukey lemma" because Oswald Teichmller (1913-1943) had arrived at it independently in Braucht der Algebriker das Answahlaxiom?, Deutsche Mathematik, vol. 4 (1939), pp. 567-577 [Carlos C廥ar de Arajo].

FIRST DERIVATIVE, SECOND DERIVATIVE, etc. Christian Kramp (1760-1826) used the terms premi廨e d廨iv嶪 and seconde d廨iv嶪 (first derivative and second derivative) (Cajori vol. 2, page 67).

However, the DSB implies Joseph Louis Lagrange (1736-1813) introduced these terms in his Th廩rie 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 coined fractal from the Latin adjective fractus. The corresponding Latin verb frangere means "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), fractus should also mean "irregular," both meanings being preserved in fragment.

The proper pronunciation is frac'tal, the stress being placed as infraction.

According to John Conway, Mandelbrot originally defined fractal to mean "having a possibly fractional dimension." Now it is used most often to describe the self-similarity property that many fractal sets have.

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 Poincar (1854-1912) (Smith vol. I and Encyclopaedia Britannica, article: "Poincar"). 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 Gonz嫮ez Cabill鏮 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 "Le蔞ns 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 Gonz嫮ez Cabill鏮.

The term FUNCTIONAL ANALYSIS was introduced by Paul P. L憝y (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 A History of Algebraic and Differential Topology 1900-1960 Jean Dieudonn wrote (p. 96):
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 C廥ar de Arajo]

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 Poincar, 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).


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