Earliest Known Uses of Some of the Words of Mathematics

DECAGON appears in English in 1571 in A geometrical practise named Pantometria by Thomas Digges (1546?-1595) (although the spelling decagonum is used).

DECILE (in statistics) was introduced by Francis Galton (Hald, p. 604).

Decile appears in 1882 in Francis Galton, Rep. Brit. Assoc. 1881 245: "The Upper Decile is that which is exceeded by one-tenth of an infinitely large group, and which the remaining nine-tenths fall short of. The Lower Decile is the converse of this" (OED2).

DECIMAL is derived from the Latin decimus, meaning "tenth." According to Smith (vol. 2, page 14), in the early printed books numbers which are multiples of 10 were occasionally called decimal numbers, citing Pellos (1492, fol. 4), who speaks of "numbre simple," "nubre desenal," and "nubre plus que desenal" and Ortega (1512, 1515 ed., fols. 4, 5), who has "lo numero simplice," "lo numero decenale," and "lo numero composto."

In 1603 Johann Hartmann Beyer published Logistica Decimalis.

Decimal occurs in English in 1608 in the title Disme: The Art of Tenths, or Decimall Arithmetike. This work is a translation by Robert Norman of La Thiende, by Simon Stevin (1548-1620), which was published in Flemish and in French in 1585.

DECIMAL POINT. In 1617 in his Rabdologia John Napier referred to the period or comma which could be used to separate the whole part from the fractional part, and he used both symbols.

In 1704 the term Separating Point is used in the Lexicon Technicum in the entry "Decimal."

According to Cajori (vol. 1, page 329), "Probably as early as the time of Hutton the expression 'decimal point' had come to be the synonym for 'separatrix' and was used even when the symbol was not a point."

Decimal point appears in the 1771 edition of the Encyclopaedia Britannica in the article "Arithmetick": "The point thus prefixed is called the decimal point" [James A. Landau].

In 1863, The Normal: or, Methods of Teaching the Common Branches, Orthoepy, Orthography, Grammar, Geography, Arithmetic and Elocution by Alfred Holbrook has: "The separatrix is the most important character used in decimals, and no pains should be spared to impress this on the minds of pupils."

DECIMAL SYSTEM is found in English in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

DECISION THEORY (as a term for the approach to statistical inference), inaugurated by Abraham Wald in 1939 ("Contributions to the Theory of Statistical Estimation and Testing  Hypotheses," Annals of Mathematical Statistics, 10, 299-326) and developed in his book Statistical Decision Functions (1950), came into use around 1950. The phrase appears in Lehmann’s "Some Principles of the Theory of Testing Hypotheses," Annals of Mathematical Statistics, 21, (1950), 1-26 [John Aldrich].

The term DEFINITE INTEGRAL is defined by Sylvestre-François Lacroix (1765-1843) in Traité du calcul différentiel et integral (Cajori 1919, page 272).

Augustin Louis Cauchy (1789-1857) used the term. It is found in his Oeuvres (2), IV, 125: "This limit is called a definite integral."

Definite integral is found in English in 1831 in Elements of the Integral Calculus (1839) by J. R. Young:

Thus, if we know that for x = a the value of the integral Fx + C ought to be A, then we have Fa + C = A, therefore the value of the constant is in that case C = A - Fa, so that the definite integral, as it is then called, is Fx + A - Fa.

DEGREE (angle measure) is found in English in about 1386 in Chaucer's Canterbury Tales: "The yonge sonne That in the Ram is foure degrees vp ronne" (OED2). He again used the word in about 1391 in A Treatise on the Astrolabe: "9. Next this folewith the cercle of the daies, that ben figured in manere of degres, that contenen in nombre 365, dividid also with longe strikes fro 5 to 5, and the nombre in augrym writen under that cercle."

DEGREE (of a polynomial). See order.

DEGREES OF FREEDOM. (See also chi-squared, F-distribution and Student's t-distribution.) Fisher introduced degrees of freedom in connection with Pearson's test in the 1922 paper "On the Interpretation of from "Contingency Tables, and the Calculation of P," J. Royal Statist. Soc., 85, pp. 87-94. He applied the number of degrees of freedom to distributions related to chi-squared--Student's distribution and his own z distribution in his 1924 paper, "On a Distribution Yielding the Error Functions of Several well Known Statistics," Proceedings of the International Congress of Mathematics, Toronto, 2, 805-813 [John Aldrich].

DEL (as a name for the symbol) is found in 1901 in Vector Analysis, A text-book for the use of students of mathematics and physics founded upon the lectures of J. Willard Gibbs by Edwin Bidwell Wilson (1879-1964):

There seems, however, to be no universally recognized name for it, although owing to the frequent occurrence of the symbol some name is a practical necessity. It has been found by experience that the monosyllable del is so short and easy to pronounce that even in complicated formulae in which (the symbol) occurs a number of times no inconvenience to the speaker or hearer arises from the repetition (OED2).

According to Stein and Barcellos (page 836), this is the first appearance in print of the word del.

The term DELTAHEDRON was coined by H. Martyn Cundy. The word may occur in Mathematical Models (1961), by him and A. P. Rollett.

DEMOIVRE'S THEOREM. In his Synopsis Palmariorum Matheseos (1707), W. Jones refers to a "Theorem" of "that Ingenious Mathematician, Mr. De Moivre."

Theorema Moivræanum appears in a review of Jones's book in Acta Eruditorum (1707).

However, the above use of the term refers to a different theorem from the one now associated with this term, and according to Smith in A Source Book in Mathematics, "The terms De Moivre's Formula, De Moivre's Theorem, applied to the formula we are considering, do not seem to have come into general use till the early part of the nineteenth century." The remaining citations pertain to what is now known as Demoivre's formula.

Demoivre's formula appears in A. L. Crelle, Lebrbuch der Elemente der Geometrie und der ebenen und spbärischen Trigonometrie, Berlin, vol. I, 1826, according to Tropfke.

Theorem of De Moivre appears in 1840 in Mathematical Dissertations, for the use of Students in the Modern Analysis, with Improvements in the Practice of Sturm's Theorem, in the Theory of Curvature, and in the Summation of Infinite Series (1841) by J. R. Young [James A. Landau].

Demoivre's formula appears in English in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson.

Demoivre's theorem appears in 1859(?) in Plane Trigonometry by the Right Rev. J.W. Colenso [University of Michigan Historic Math Collection].

Demoivre's theorem appears in English in the third edition of An Elementary Treatise on the Theory of Equations (1875) by Isaac Todhunter.

DENSE is found in English in 1902 in Proc. Lond. Math. Soc. XXXI "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."

DEPENDENT VARIABLE. Subordinate variable appears in English in the 1816 translation of Differential and Integral Calculus by Lacroix: "Treating the subordinate variables as implicit functions of the indepdndent [sic] ones" (OED2).

Dependent variable appears in in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "On account of this dependence of the value of the function upon that of the variable the former, that is y, is called the dependent variable, and the latter, x, the independent variable" [James A. Landau].

DERIVATIVE and DIFFERENTIAL COEFFICIENT. The term derivative was not used by Isaac Newton, who instead used the term fluxion.

Julio González Cabillón believes that derivative was first used in the calculus sense (in Latin, derivata) by Gottfried Wilhelm Leibniz (1646-1716) around 1677:

Aequationem differentialem voco talem qua valor ipsius dx exprimitur, quaeque ex alia derivata est, qua valor ipsius x exprimebatur [cf. page 156 of Leibniz' "Mathematische Schriften," vol. I, edited by C. I. Gerhardt, Verlag von A. Asher & Comp., Berlin, 1849].

John Conway points out that, in the above, it could be argued that Leibniz is merely using the word "derived" in its ordinary sense.

Some writers attribute the word derivative to Joseph Louis Lagrange (1736-1813), who used derivée de la fonction and fonction derivée de la fonction as early as 1772 in "Sur une nouvelle espece de calcul relatif a la différentiation et a l'integration des quantités variables," Nouveaux Memoires de l'Academie royale des Sciences etBelles-Lettres de Berlin. Lagrange states, for instance (first pages):

...on designe de même par u'' une fonction derivée de u' de la même maniére que u' l'est de u, et par u''' une fonction derivée de même de u'' et ainsi, ...

... les fonctions u, u', u'', u''', u^IV, ... derivent l'une de l'autre par une même loi de sorte qu'on pourra les trouver aisement par une meme operation répetée. [the functions u, u', u'', u''', u^IV, ... are derived one another from the same law, such that ...]

The term differential coefficient was first used by Sylvestre-François Lacroix (1765-1843) in Traité du calcul différentiel et integral (Cajori 1919, page 272).

An 1816 English translation of Lacroix has: "The limit of the ratio of the increments, or the differential coefficient, will be obtained" (OED2).

In 1891 Differential and Integral Calculus by George A. Osborne prefers the term differential coefficient but has: "The differential coefficient is sometimes called the derivative."

DESCARTES' RULE OF SIGNS. In A Compendious Tract on the Theory of Solutions of Cubic and Biquadratic Equations, and of Equations of the Higher Orders (1833) by Rev. B. Bridge, the rule is called Des Cartes' rule.

In the second edition of Theory and Solution of Algebraical Equations (1843), J. R. Young refers to "the rule of signs."

Descartes' rule of signs is found in English in 1855 in An elementary treatise on mechanics, embracing the theory of statics and dynamics, and its application to solids and fluids. Prepared for the undergraduate course in the Wesleyan university by Augustus W. Smith: "Now, since the degree of the equation is even and the absolute term is negative, there are at least two possible roots, one positive and the other negative. The other two roots may be real or imaginary. If real, Descartes' rule of signs indicates that three will be positive and one negative" [University of Michigan Digital Library].

The term DESCRIPTIVE GEOMETRY occurs in the title Géométrie Descriptive (1795) by Gaspard Monge (1746-1818).

The term DESMIC was coined by Cyparissos Stephanos (1857-1918) in "Sur les systemes desmiques de trois tetraedres," published in Darboux's Bulletin ser 2, vol 3 (1879), pp 424-456 [Julio González Cabillón, Michael Lambrou].

DETERMINANT (discriminant of a quantic) was introduced in 1801 by Carl Friedrich Gauss in his Disquisitiones arithmeticae:

Numerum bb - ac, a cuius indole proprietates formae (a, b, c) imprimis pendere in sequentibus docebimus, determinantem huius formae uocabimus.

(Cajori vol. 2, page 88; Smith vol. 2, page 476).

Laplace had used the term resultant in this sense (Smith, 1906).

DETERMINANT (modern sense). Augustin-Louis Cauchy (1789-1857) was apparently the first to use determinant in its modern sense (Schwartzman, page 70). He employed the word in "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs egales et des signes contraires par suite des transpositions operees entre les variables qu'elles renferment", addressed on November 30, 1812, and first published in Journal de l'Ecole Poytechnique, XVIIe Cahier, Tome X, Paris, 1815:

M. Gauss s'en est servi avec avantage dans ses Recherches analytiques pour decouvrir les proprietes generales des formes du second degre, c'est a dire des polynomes du second degre a deux ou plusieurs variables, et il a designe ces memes fonctions sous le nom de determinants. Je conserverai cette denomination qui fournit un moyen facile d'enoncer les resultats; j'observerai seulement qu'on donne aussi quelquefois aux fonctions dont il s'agit le nom de resultantes a deux ou a plusieurs lettres. Ainsi le deux expressions suivantes, determinant et resultante, devront etre regardees comme synonymes.

(Smith vol. 2, page 477; Julio González Cabillón.)

According to Katz, Arthur Cayley (1821-1895) introduced the word determinant as a replacement for several older terms.

DIAGONAL. Julio González Cabillón says, "Heron of Alexandria is probably the first geometer to define the term diagonal (as the straight line drawn from angle to angle)."

DIALYTIC was used by James Joseph Sylvester (1814-1897) in 1853 in Phil. Trans. CXLIII. i. 544:

Dialytic. If there be a system of functions containing in each term different combinations of the powers of the variables in number equal to the number of the functions, a resultant may be formed from these functions, by, as it were, dissolving the relations which connect together the different combinations of the powers of the variables, and treating them as simple independent quantities linearly involved in the functions. The resultant so formed is called the Dialytic Resultant of the functions supposed; and any method by which the elimination between two or more equations can be made to depend on the formation of such a resultant is called a dialytic method of elimination.

DIAMETER. According to Smith (vol. 2, page 278), "Euclid used the word 'diameter' in relation to the line bisecting a circle and also to mean the diagonal of a square, the latter term being also found in the works of Heron."

DIFFERENTIAL (noun) appears in the title of a manuscript of Sept. 10, 1690, by Leibniz, "Methodus pro differentialibus, ponendo z = dy:dx et quaerendo dz ["A method for differentials, positing z = dy/dx and seeking dz"]. He may have used the term earlier, since he used the terms "differential equation" and "differential calculus" earlier (see below).

Differential appears in English as a noun in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris (OED2).

DIFFERENTIAL CALCULUS. The term calculus differentialis was introduced by Leibniz in 1684 in Acta Eruditorum 3. Before introducing this term, he used the expression methodus tangentium directa (Struik, page 271).

Leibniz wrote [source uncertain]: "Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method."

DIFFERENTIAL EQUATION. Gottfried Wilhelm Leibniz (1646-1716) introduced the term in 1676, according to Franceschetti (p. 401).

Leibniz used the Latin aequationes differentiales in Acta Eruditorum, October 1684. See the entry "algorithm" for the context.

The term DIFFERENTIAL GEOMETRY was first used by Luigi Bianchi (1856-1928) in 1894 (Kline, page 554).

DIFFERENTIATE appears in English in 1816 in LaCroix's Differential and Integral Calculus (OED2).

DIGIT. According to Smith (vol. 2, page 12), the late Roman writers seem to have divided the numbers below 100 into digiti (fingers), articuli (joints), and compositi (composites of fingers and joints).

In English, Robert Recorde in the 1558 edition of the Ground of Artes wrote, "A diget is any numbre vnder 10."

The term DIGITADDITION was coined by D. R. Kaprekar, according to an Internet web page.

DIGRAPH was used in 1955 by F. Harary in Transactions of the American Mathematical Society. The term directed graph also occurs there (OED2).

DIHEDRAL appears in 1799 in George Smith, The laboratory: or school of arts: "Terminating in dihedral pyramids" (OED2).

DIHEDRAL ANGLE appears in 1826 in Henry, Elem. Chem.: "Variations of temperature produce a ... difference in ... a crystal of carbonate of lime.... As the temperature increases, the obtuse dihedral angles diminish ...so that its form approaches that of a cube" (OED2).

DIOPHANTINE ANALYSIS appears in 1811 in the title An Elementary Investigation of the Theory of Numbers, with its application to the indeterminate and diophantine analysis by Peter Barlow (OED2).

DIOPHANTINE EQUATION appears in English in 1893 in Eliakim Hastings Moore (1862-1932), "A Doubly-Infinite System of Simple Groups," Bulletin of the New York Mathematical Society, vol. III, pp. 73-78, October 13, 1893 [Julio González Cabillón].

Henry B. Fine writes in The Number System of Algebra (1902):

The designation "Diophantine equations," commonly applied to indeterminate equations of the first degree when investigated for integral solutions, is a striking misnomer. Diophantus nowhere considers such equations, and, on the other hand, allows fractional solutions of indeterminate equations of the second degree.

DIOPHANTINE PROBLEM. The phrase "Diophantus Problemes" appears in 1670 [James A. Landau].

The OED2 has the citation in 1700: Gregory, Collect. (Oxf. Hist. Soc.) I. 321: "The resolution of the indetermined arithmetical or Diophantine problems."

DIRECT VARIATION. Directly is found in 1743 in W. Emerson, Doctrine Fluxions: "The Times of describing any Spaces uniformly are as the Spaces directly, and the Velocities reciprocally" (OED2).

Directly proportional is found in 1796 in A Mathematical and Philosophical Dictionary: "Quantities are said to be directly proportional, when the proportion is according to the order of the terms" (OED2).

Direct variation is found in 1856 in Ray's higher arithmetic. The principles of arithmetic, analyzed and practically applied by Joseph Ray (1807-1855):

Variation is a general method of expressing proportion often used, and is either direct or inverse. Direct variation exists between two quantities when they increase togeether, or decrease together. Thus the distance a ship goes at a uniform rate, varies directly as the time it sails; which means that the ratio of any two distances is equal to the ratio of the corresponding times taken in the same order. Inverse variation exists between two quantities when one increases as the other decreases. Thus, the time in which a piece of work will be done, varies inversely as the number of men employed; which means that the ratio of any two times is equal to the ratio of the numbers of men employed for these times, taken in reverse order.

This citation was taken from the University of Michigan Digital Library [James A. Landau].

The term DIRECTION COEFFICIENT (for cos x + i sin x) is due to Hankel (1867) (Smith, 1906).

Argand used the term direction factor (Smith, 1906).

Cauchy used the term reduced form (l'expression réduite) (Smith, 1906).

DIRECTIONAL DERIVATIVE is found in 1912 in Advanced Calculus by Edwin Bidwell Wilson: "The derivative (13) is called the directional derivative of f in the direction of the line."

DIRECTRIX. According to the DSB, Jan de Witt (1625-1672) "is credited with introducing the term 'directrix' for the parabola, but it is clear from his derivation that he does not use the term for the fixed line of our focus-directrix definition."

DIRICHLET'S PROBLEM is found in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "The basis for Riemann's work is a famous proposition known among continental mathematicians as Dirichlet's Principle, or Problem."

DISCRETE appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "Two contrary kynds of quantity; quantity discrete or number, and quantity continual or magnitude" (OED2).

DISCRETE MATHEMATICS occurs in 1971 in the title of the journal Discrete Mathematics.

DISCRIMINANT was introduced by James Joseph Sylvester (1814-1897) in 1852 in the Cambridge and Dublin Mathematical Journal, vol. I, 52. He used the word "for determinant, which is still found occasionally," according to the OED2, which attributes this information to H. T. Gerrans.

In 1876 George Salmon used discriminant in its modern sense in Mod. Higher Algebra (ed. 3): "The discriminant is equal to the product of the squares of all the differences of the differences of any two roots of the equation" (OED2).

DISCRIMINANT ANALYSIS is found in Palmer O. Johnson, "The quantification of qualitative data in discriminant analysis," J. Am. Stat. Assoc. 45, 65-76 (1950).

See also W. G. Cochran and C. I. Bliss, "Discriminant functions with covariance," Ann. Math. Statist. 19 (1948) [James A. Landau].

DISJOINT is found in 1914 in Projective Geometry by George Ballard Mathews: "Taking any two elements of different names (plane, point; plane, line; point, line), we may distinguish them as being disjoint or conjoint [University of Michigan Historic Math Collection].

Disjoint, referring to sets, is found in the phrase "two disjoint closed sets" in 1937 in Transactions of the American Mathematical Society (OED2).

DISJUNCTION. According to the University of St. Andrews website, "the logical term 'disjunction' is certainly due to the Stoics and it is thought to have originated with" Chrysippus of Soli (280 BC - 206 BC).

DISME is an obsolete English word meaning "tenth." It occurs in 1608 in the title Disme: The Art of Tenths, or Decimall Arithmetike. This work is a translation by Robert Norman of La Thiende, by Simon Stevin (1548-1620), which was published in Flemish and in French in 1585.

Disme was used by Shakespeare in Troilus and Cressida (ii, 2, 15), which was first published in 1609. The use of this word is one of the pieces of evidence cited by defenders of the theory that Shakespeare's plays were actually written someone else, perhaps Francis Bacon.

DISPERSION (in statistics) is found in 1876 in Catalogue of the Special Loan Collection of Scientific Apparatus at the South Kensington Museum by Francis Galton (David, 1998).

DISTRIBUTED LAG. The term was used by Irving Fisher in his 1925 paper "Our Unstable Dollar and the so-called Business Cycle" (Journal of the American Economic Association, 20, 179-202) to describe a formulation he had used in "The Business Cycle Largely a 'Dance of the Dollar,'" Journal of the American Statistical Society, 18, (1923), 1024-1028 [John Aldrich].

The term DISTRIBUTION FUNCTION of a random variable is a translation of the Verteilungsfunktion of R. von Mises "Grundlagen der Wahrscheinlichkeitsrechnung," Math. Zeit. 5, (1919) 52-99.

The English term appears in J. L. Doob's "The Limiting Distributions of Certain Statistics," Annals of Mathematical Statistics, 6, (1935), 160-169.

The term cumulative distribution function was used by S. S. Wilks Mathematical Statistics (1943) (David 2001).

DISTRIBUTIVE. See commutative.

The term DIVERGENCE (of a vector field) was introduced by William Kingdon Clifford (1845-1879). Maxwell had earlier used the term convergence with a related meaning (Kline, page 785).

The DSB says that Maxwell introduced the term divergence in 1870; this seems to be incorrect.

DIVERGENT. See convergent.

DIVIDEND. Joannes de Muris (c. 1350) used dividendus (Smith vol. 2, page 131).

In English, the word is found in The Grovnd of Artes, by Robert Recorde, which was printed between 1540 and 1542: "Then begynne I at the hyghest lyne of the diuident, and seke how often I may haue the diuisor therin" (OED2).

The term DIVINE PROPORTION appears in 1509 in the title De Divina Proportione by Luca Pacioli (1445-1517). According to an Internet website, Pacioli coined the term.

Ramus wrote, "Christianis quibusdam divina quaedam proportio hic animadversa est..." in Scholarvm Mathematicarvm, Libri vnvs et triginta, Basel, 1569; ibid., 1578; Frankfort, 1599) (Smith vol. 2, page 291).

Kepler wrote, "Inter continuas proportiones unum singulare genus est proportionis divinae" (Frisch ed. of his Opera, I (1858). According to v. Baravalle (1948), Kepler used the term sectio divina.

DIVISION is found in English in "The crafte of nombrynge" (ca. 1300). The word is spelled dyuision (OED2).

Baker (1568) speaks of "Deuision or partition" and Digges (1572) says "To deuide or parte" (Smith vol. 2, page 129).

DIVISOR is found in English in "The crafte of nombrynge" (ca. 1300). The word is spelled dyvyser (OED2).

DODECAGON is found in English in a mathematical dictionary of 1658.

DOMAIN was used in 1886 by Arthur Cayley in "On Linear Differential Equations" in the Quarterly Journal of Pure and Applied Mathematics: "... for points x within the domain of the point a [University of Michigan Historic Math Collection].

Domain is found in 1896 in Bull. Amer. Math. Soc. Dec. 103: "The formation of an algebraic 'domain' and..the nature of the process of 'adjunction' introduced by Galois" (OED2).

Domain, referring to a power series, appears in 1898 in Introduction to the theory of analytic functions by J. Harkness and F. Morley.

Domain, in the sense of the values that an independent variable of a function can take, appears in the Encyclopaedia Britannica of 1902 (OED2).

DOPPELVERHÄLTNISS. Möbius introduced the term Doppelschnittverhältniss, meaning "ratio bisectionalis" or "double cut ratio," in his "Der barycentrische Calcul" (1827): gesammelte Werke, I (1885).

Jakob Steiner shortened the term to Doppelverhältniss (Smith vol. 2, page 334).

See also anharmonic ratio and cross-ratio.

DOT PRODUCT is found in 1901 in Vector Analysis by J. Willard Gibbs and Edwin Bidwell Wilson:

The direct product is denoted by writing the two vectors with a dot between them as

A·B

This is read A dot  B and therefore may often be called the dot product instead of the direct product.

[This citation was provided by Joanne M. Despres of Merriam-Webster Inc.]

DOUBLE INTEGRATION appears in 1831 in Elements of the Integral Calculus (1839) by J. R. Young.

DUALITY. The term "principle of duality" was introduced by Joseph Diaz Geronne (1771-1859) in "Considérations philosophiques sur les élémens de la science de l'étendue," Annales 16 (1825-1826) (DSB).

The term DUMMY VARIABLE is often used when describing the status of a variable like x in a definite integral. A. Church seems to be describing an established usage when he wrote in 1942, "A variable is free in a given expression ... if the expression can be considered as representing a function with that variable as an argument. In the contrary case the variable is called a bound (or apparent or dummy) variable." ("Differentials", American Mathematical Monthly, 49, 390.) [John Aldrich].

In regression analysis a DUMMY VARIABLE indicates the presence (value 1) or absence of an attribute (0).

A JSTOR search found "dummy variables" for social class and for region in H. S. Houthakker's "The Econometrics of Family Budgets" Journal of the Royal Statistical Society A, 115, (1952), 1-28.

A 1957 article by D. B. Suits, "Use of Dummy Variables in Regression Equations" Journal of the American Statistical Association, 52, 548-551, consolidated both the device and the name.

The International Statistical Institute's Dictionary of Statistical Terms objects to the name: the term is "used, rather laxly, to denote an artificial variable expressing qualitative characteristics .... [The] word 'dummy' should be avoided."

Apparently these variables were not dummy enough for Kendall & Buckland, for whom a dummy variable signifies "a quantity written in a mathematical expression in the form of a variable although it represents a constant", e.g. when the constant in the regression equation is represented as a coefficient times a variable that is always unity.

The indicator device, without the name "dummy variable" or any other, was also used by writers on experiments who put the analysis of variance into the format of the general linear hypothesis, e.g. O. Kempthorne in his Design and Analysis of Experiments (1952) [John Aldrich].

DUODECIMAL is dated 1663 in MWCD10.

Duodecimal appears in 1714 in the title A new and complete Treatise of the Doctrine of Fractions .. with an Epitome of Duodecimals by Samuel Cunn (OED2).

DYAD. The Greek word dyad is found in Euclid in Proposition 36 of Book IX and in Nichomachus' Introduction to Arithmetic (Book I, Chapter VII) [Mary Townsend].

In the sense of a mathematical operator, dyad was used in 1884 by Josiah Willard Gibbs (1839-1903) and is found in his Vector-Analysis of 1901 and his Collected Works of 1928 (OED2).

 

EFFICIENCY. The terms efficiency and efficient applied to estimation were introduced by R. A. Fisher in "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922). He described the criterion of efficiency as "satisfied by those statistics which, when derived from large samples, tend to a normal distribution with the least possible standard deviation." He also wrote: "To calculate the efficiency of any given method, we must therefore know the probable error of the statistic calculated by that method, and that of the most efficient statistic which could be used. The square of the ratio of these two quantities then measures the efficiency." Fisher seems not to have known that such calculations had been done by Gauss a century earlier. However the idea of efficiency in extracting information was novel. [This entry was contributed by John Aldrich, based on David (1995).]

EIGENVALUE, EIGENFUNCTION, EIGENVECTOR and related terms. "Almost every combination of the adjectives proper, latent, characteristic, eigen and secular, with the nouns root, number and value, has been used in the literature for what we call a proper value" (P. R. Halmos Finite Dimensional Vector Spaces (1958, 102)). To add to the confusion, both the values and their reciprocals have been important: in A. Lichnerowicz's Algèbre et analyse linéaires (1947), valeur charactéristique and valeur propre are reciprocals of one another, but the English (1967) translation has eigenvalue and proper value, which derive from the same German word, Eigenwert. The adjectives also combine with other sorts of nouns, including equations, solutions, functions and vectors. The story of the terminology ranges across algebra, analysis and mechanics, classical and quantum.

Modern expositions of spectral theory often begin with a matrix A and introduce value and vector x together in the value/vector-equation Ax = x : any value for which this equation is satisfied for a non-null x is a value and the associated x is a vector. The existence of a non-null solution x for (A - I)x = 0 requires the determinant of (A - I) to be zero, i.e. that the roots of a polynomial are significant. This finite-dimensional case is used to motivate the treatment of differential equations and integral equations which involve infinite-dimensional spaces where the vector is now a function. The historical order of development was more or less the reverse. The polynomial equation generated from the differential equations of celestial mechanics came first, ca. 1780, then the equation was expressed using determinants ca. 1830, then the equation was associated with matrices ca. 1880, then integral equations were studied ca. 1900 until finally the modern order of topics starting from the value/vector-equation became established ca. 1940. (Based on Kline ch. 29, 33, 45 & 46 and Hawkins (1975 and -7.))

The term secular ("continuing through long ages" OED2) recalls that one of the origins of spectral theory was in the problem of the long-run behaviour of the solar system investigated by Laplace and Lagrange. See Hawkins (1975). The 1829 paper in which Cauchy established that the roots of a symmetric determinant are real has the title, "Sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des mouvements des planétes"; this signifed only that Cauchy recognized that his problem, of choosing x to maximise x^TAx subject to x^Tx = 1 (to use modern notation), led to an equation like that studied in celestial mechanics. Sylvester's title "On the Equation to the Secular Inequalities in the Planetary Theory" (see below) was even more misleading as to content. In this tradition the "Säkulärgleichung" of Courant & Hilbert Methoden der Mathematischen Physik (1924) and the "secular equation" of E. T. Browne's "On the Separation Property of the Roots of the Secular Equation" American Journal of Mathematics, 52, (1930), 843-850 refer to the characteristic equation of a symmetric matrix. The term "secular equation" appears in the modern numerical linear algebra literature.

The characteristic terms derive from Augustin Louis Cauchy (1789-1857), who introduced the term l'equation caractéristique and investigated its roots in his "Mémoire sur l'integration des équations linéaires," Exercises d'analyse et de physique mathématique, 1, 1840, 53 = Oeuvres, (2), 11, 76 (Kline, page 801). Frobenius referred to this memoir when he introduced the phrase "die charakteristische Determinante" in his fundamental paper on matrices, "Über lineare Substitutionen und bilineare Formen," Jrnl. für die reine und angewandte Math. (1874), 84, 1-63. In Les méthodes nouvelles de la mécanique céleste (1892) Poincaré wrote about exposants (exponents) caractéristiques.

Sightings in JSTOR show the further expansion of the characteristic family and its spread into English: characteristic value in G. D. Birkhoff, "Boundary Value and Expansion Problems of Ordinary Linear Differential Equations," Trans. American Mathematical Society, 9, (1908), 373-395, characteristic root in H. Hilton "Properties of Certain Homogeneous Linear Substitutions," Annals of Mathematics, 2nd Ser., 15, (1913-1914) 195-201, characteristic solution in W. D. A. Westfall "Existence of the Generalized Green's Function" Annals of Mathematics, 2nd Ser., 10, (1909), 177-180, characteristic vector in J. W. Alexander "On the Class of a Covariant Tensor" Annals of Mathematics, 2nd Ser., 28, (1926-1927), 245-250 and F. D. Murnaghan & A. Wintner "A Canonical Form for Real Matrices under Orthogonal Transformations" Proceedings of the National Academy of Sciences of the United States of America, 17, (1931), 417-420. C. C. MacDuffee's standard work Theory of Matrices (1933) used characteristic root, function and equation but found no use for characteristic vector.

The latent terminology was introduced by James Joseph Sylvester (1814-1897) in the 1883 paper "On the Equation to the Secular Inequalities in the Planetary Theory" Phil. Mag. 16, 267.

It will be convenient to introduce here a notion (which plays a conspicuous part in my new theory of multiple algebra), viz. that of the latent roots of a matrix -- latent in a somewhat similar sense as vapour may be said to be latent in water or smoke in a tobacco-leaf.

One of Sylvester's three "laws of motion in the world of universal algebra" was the "law of congruity ... which affirms that the latent roots of a matrix follow the march of any functional operation formed on a matrix, not involving the action of any foreign matrix" (Johns Hopkins University Circulars, 3, (1884)). Sylvester also used the term "latent equation" ("Lectures on the Principles of Universal Algebra" American J. of Math. VI (1884), 216). "Latent vector" came later, perhaps as late as 1937 with A. C. Aitken's "Studies in Practical Mathematics II. The Evaluation of the Latent Roots and Latent Vectors of a Matrix," Proc. Royal Soc. Edinburgh, 57, 269-304. Previously H. Turnbull & Aitken (Theory of Canonical Matrices, 1932) used the term "latent point" which they attributed to Sylvester.

The eigen terms are associated with David Hilbert (1862-1943), though he may have been following such constructions as Eigentöne in acoustics (cf. H. L. F. Helmholtz Lehre von den tonempfindungen). Eigenfunktion and Eigenwert appear in Hilbert's work on integral equations (the original papers from the Gött. Nachr. 1904-1910 were collected as Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen 1912). In Whittaker & Watson's Course of Modern Analysis "Eigenfunktion" is translated as "autofunction." Hilbert's starting point was a non-homogeneous integral equation with a parameter for which the matrix counterpart is (I - A)x = y. Hilbert called the values of that generate a non-null solution to the homogeneous version of these equations Eigenwerte; they are the reciprocals of the characteristic/latent roots of A. The influential Courant & Hilbert volume Methoden der Mathematischen Physik (1924) uses for a "characteristiche Zahl" and = (1/ ) for an "Eigenwert"; Lichnerowicz (above) wrote the French equivalents. "Eigenvektor" appears in Courant & Hilbert's exposition of the finite-dimensional case.

J. von Neumann's "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren" Math. Ann. 102 (1929) 49-131 used "Eigenwert" in a different way: "Ein Eigenwert ist eine Zahl, zu der es eine Funktion f 0 mit Rf = f gibt; f ist dann Eigenfunktion." This became the dominant usage and by 1946 H. & B. Jeffreys (Methods of Mathematical Physics) were treating eigenvalue as synonymous with characteristic value and latent root.

The anglicising of the eigen terms can be followed through the OED and JSTOR. In 1926 P. A. M. Dirac was writing "a set of independent solutions which may be called eigenfunctions" ("On the Theory of Quantum Mechanics," Proc. Royal Soc. A, 112, 661-677) (OED2). Eigenvalue appears, perhaps with humorous intent, in a letter to Nature (July 23, 1927) from A. S. Eddington beginning "Among those ... trying to acquire a general acquaintance with Schrödinger's wave mechanics there must be many who find their mathematical equipment insufficient to follow his first great problem -- to determine the eigenvalues and eigenfunctions for the hydrogen atom" (OED2). Eigenvector appears in R. Brauer & H. Weyl's "Spinors in n Dimensions," Amer. J. Math., 57, (1935) 425-449 (JSTOR).

Proper has been a standard English rendering of "eigen" -- thus in the 19th century Helmholtz's Eigentöne became "proper tones." "Proper values" and "proper functions" appear in von Neumann's English writings, e.g. in his and S. Bochner's "On Compact Solutions of Operational-Differential Equations. I," Annals of Mathematics, 2nd Ser., 36, (1935), 255-291 although "eigenvalue" is used in the English translation (1949) of his Mathematische Grundlagen der Quantenmechanik (1932). Dirac (Principles of Quantum Mechanics) argued against the proper terminology (and for the eigen) on the ground that "proper" had other meanings in physics.

See also spectrum.

[This entry was contributed by John Aldrich.]

ELEMENT. The term Elemente (elements) is found in Geometrie der Lage (2nd ed., 1856) by Carl Georg Christian von Staudt: "Wenn man die Menge aller in einem und demselben reellen einfoermigen Gebilde enthaltenen reellen Elemente durch n + 1 bezeichnet und mit diesem Ausdrucke, welcher dieselbe Bedeutung auch in den acht folgenden Nummern hat, wie mit einer endlichen Zahl verfaehrt, so ..." [Ken Pledger].

Cantor also used the German Element in Math. Ann. (1882) XX. 114.

The term ELEMENTARY DIVISOR was first used in German by Weierstrass, according to Maxime Bôcher in Introduction to Higher Algebra.

Elementary divisor is found in English in H. T. Burgess, "A practical method for determining elementary divisors, American M. S. Bull. (1916).

ELIMINANT. According to George Salmon in Modern Higher Algebra (1885), "The name 'eliminant' was introduced I think by Professor De Morgan. ... The older name 'resultant' was employed by Bezout, Histoire de l'Académie de Paris, 1764."

ELIMINATE is found in 1845 in the Penny Cyclopedia: "If by means of one of these we eliminate p from the rest, the process ... would allow of our eliminating both x and y by one equation only (OED2).

ELIMINATION is found in 1845 in the Penny Cyclopedia: "As to equations which are not purely algebraical ... we cannot ... say that there is any organized method of elimination existing, except that of solution" (OED2).

ELLIPSE was probably coined by Apollonius, who, according to Pappus, had terms for all three conic sections. Michael N. Fried says there are two known occasions where Archimedes used the terms "parabola" and "ellipse," but that "these are, most likely, later interpolations rather than Archimedes own terminology."

James A. Landau writes that the curve we call the "ellipse" was generally called an ellipsis in the seventeenth century, although the word Elleipse appears in a letter written by Robert Hooke in 1679. Ellipse appears in a letter written by Gilbert Clerke in 1687.

ELLIPSOID appears in a letter written in 1672 by Sir Isaac Newton [James A. Landau].

ELLIPTIC CURVE is found in 1727 in Thomson, To Mem. Sir J. Newton: "He, first of Men, with awful Wing pursu'd The Comet thro' the long Elliptic Curve" (OED2).

Elliptic curve is found in Webster's New International Dictionary (1909).

The term ELLIPTIC FUNCTION was used by Adrien Marie Legendre (1752-1833) in 1825 in volume 1 of Traité des Fonctions Elliptiques and may appear in 1811 in volume 1 of his Exercises du Calcul Intégral. He used the term for what is now called an elliptic integral.

The term appears in the title Recherches sur les fonctions elliptiques by Niels Henrik Abel (1802-1829), which was published in Crelle's Journal in September 1827.

Elliptic function appears in 1845 in the Penny Cyclopedia 1st Supp.: "A large class of integrals closely related to and containing among them the expression for the arc of an ellipse have received the name of Elliptic functions" (OED2).

Elliptic function appears in 1876 in Arthur Cayley, Elliptic Functions: "sn u is a sort of sine function, and cn u, dn u are sorts of cosine-functions of u; these are called Elliptic Functions" (OED2).

Elliptic function appears in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "A one-valued doubly periodic function, whose only essential singular point is at , is called an elliptic function.

ELLIPTIC GEOMETRY. See hyperbolic geometry.

ELLIPTIC INTEGRAL. According to the DSB, "Giulio Carlo Fagnano dei Toschi (1682-1766) gave the name 'elliptic integrals' to integrals of the form int f(x₁ sqrt P[x]) dx where P(x) is a polynomial of the third or fourth degree."

According to Elliptic Functions and Elliptic Integrals by V. Prasolov and Y. Solovyev, AMS, 1997 (Translations of Mathematical Monographs, v. 170):

The most remarkable properties of the lemniscate were discovered by an Italian mathematician Count Fagnano (1682-1766). By the way, it was Fagnano who coined the term elliptic integrals. Fagnano discovered that the arc length of the lemniscate can be expressed in terms of an elliptic integral of the first kind. He obtained an addition theorem for this integral and, therefore, demonstrated that the division of arcs of the lemniscate into n equal parts is an algebraic problem.

[From a post in sci.math by David Cantrell]

EMPTY SET is found in Walter J. Bruns, "The Introduction of Negative Numbers," The Mathematics Teacher, October 1940: "For our purposes we still need a symbol for an 'empty' set, that means for a multitude containing no element."

Dorothy Geddes and Sally I. Lipsey, "The Hazards of Sets," The Mathematics Teacher, October 1969 has: "The fact that mathematicians refer to the empty set emphasizes the rather unique nature of this set."

An older term is null set, q. v.

The term EPICYCLE was employed by Ptolemy, according to the Mathematical Dictionary and Cyclopedia of Mathematical Science (1857).

EQUAL (in area and volume). In 1832 Elements of Geometry and Trigonometry by David Brewster (which is a translation of Legendre) has:

It is customary with Euclid, and various geometrical writers, to give the name equal triangles, to triangles which are equal only in surface; and of equal solids, to solids which are equal only in solidity. We have thought it more suitable to call such triangles or solids equivalent; reserving the denomination equal triangles, or solids, for such as coincide when applied to each other.

EQUATION. Equatio appears in the ordinary high school sense of the word in Fibonacci's Liber Abbaci, Ch. 15, section 3 [Barnabas Hughes].

Equatio was used by Medieval writers.

Ramus used aequatio in his arithmetic (1567).

Equation appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "Many rules...of Algebra, with the equations therein vsed." It also appears in the preface to the translation, by John Dee: "That great Arithmeticall Arte of Aequation: commonly called...Algebra."

Viete defines the term equation in chapter 8 of In artem analyticem isagoge (1591), according to the DSB.

EQUIANGULAR. Equiangle appears in English in 1570 in Sir Henry Billingsley's translation of Euclid: "To describe a triangle equiangle vnto a triangle geuen" (OED2).

Equiangular is found in English in 1660 in Barrow, Euclid: "An Equiangular or equal-angled figure is that whereof all the angles are equal" (OED2).

Isogon (or isagon) is found in a 1696 dictionary.

EQUILATERAL. Aequilaterum appears in English in 1551 in Pathway to Knowledge by Robert Recorde: "That the Greekes doo call Isopleuron, and Latine men aequilaterum: and in english it may be called a threlike triangle" (OED2).

Equilateral triangle is found in English in 1570 in Sir Henry Billingsley's translation of Euclid: "How to describe an equilaterall triangle redily and mechanically" (OED2).

EQUIPROBABLE was used in 1921 by John Maynard Keynes in A Treatise on Probability: "A set of exclusive and exhaustive equiprobable alternatives" (OED2).

ERGODIC. Ludwig Boltzmann (1844-1906) coined the term Ergode (from the Greek words for work + way) for what Gibbs later called a "micro-canonical ensemble"; Ergode appears in the 1884 article in Wien. Ber. 90, 231. Later P. & T. Ehrenfest (1911) "Begriffiche Grundlagen der statistischen Auffassung in der Mechanik" (Encyklopädie der mathematischen Wissenschaften, vol. 4, Part 32) discussed "ergodische mechanischer Systeme" the existence of which they saw as underlying the gas theory of Boltzmann and Maxwell. (Based on a note on p. 297 of Lectures on Gas Theory, S. G. Brush's translation of Boltzmann's Vorlesungen über Gastheorie.)

After the impossibility of an ergodic mechanical system was demonstrated, various related hypotheses were investigated. "Ergodic" and "quasi-ergodic" theorems were proved in the 1930s, by, amongst others, G. D. Birkhoff in Proc. Nat. Acad. Sci. (1931) 17, 651 -- "I propose ... to establish a general recurrence theorem and thence the 'ergodic theorem'" -- and J. von Neumann Proc. Nat. Acad. Sci. (1932) 18, 70-82.

Ergodic theorems originated in classical mechanics but in the theory of stochastic processes they appear as versions of the law of large numbers, see e.g. J. L. Doob's Stochastic Processes (1954). [This entry was contributed by John Aldrich.]

ESCRIBED CIRCLE is found in 1855 in A treatise on plane and spherical trigonometry by William Chauvenet: "Let PO' = D', Fig. 26, O' being the center of the escribed circle lying within the angle A" (University of Michigan Digital Library).

ESTIMATION. Long before the terminology stabilized around estimation the activity was called calculation, determination or fitting.

The terms estimation and estimate were introduced in R. A. Fisher's "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922). He writes (none too helpfully!): "Problems of estimation are those in which it is required to estimate the value of one or more of the population parameters from a random sample of the population." Fisher uses estimate as a substantive sparingly in the paper.

The phrase unbiassed estimate appears in Fisher's Statistical Methods for Research Workers (1925, p. 54) although the idea is much older.

The expression best linear unbiased estimate appears in 1938 in F. N. David and J. Neyman, "Extension of the Markoff Theorem on Least Squares," Statistical Research Memoirs, 2, 105-116. Previously in his "On the Two Different Aspects of the Representative Method" (Journal of the Royal Statistical Society, 97, 558-625) Neyman had used mathematical expectation estimate for unbiased estimate and best linear estimate for best linear unbiased estimate (David, 1995).

The term estimator was introduced in 1939 in E. J. G. Pitman, "The Estimation of the Location and Scale Parameters of a Continuous Population of any Given Form," Biometrika, 30, 391-421. Pitman (pp. 398 & 403) used the term in a specialised sense: his estimators are estimators of location and scale with natural invariance properties. Now estimator is used in a much wider sense so that Neyman's best linear unbiased estimate would be called a best linear unbiased estimator (David, 1995). [This entry was contributed by John Aldrich.]

The term ETHNOMATHEMATICS was coined by Ubiratan D'Ambrosio. In 1997 he wrote the following to Julio González Cabillón (who provided the translation from Spanish to English):

In 1977, the AAAS hosted a conference on Native American Sciences, where I presented a paper about "Science in Native Cultures," and where I called attention to the need for extending the methodology of Botany (ethnobotany was already in use) to the scientific knowledge as a whole, and to mathematics, doing "something like ethnoscience and ethnomathematics." That paper was never published. Five years later, in a meeting in Suriname in 1982, the concept was mentioned more explicitly. In the following years, I began using ethnomathematics. But it was not until ICME 5 that the term was "officially" recognized. My book Socio-cultural Bases for Mathematics Education brings the first study about Ethnomathematics.

In the above, ICME 5 refers to the Fifth International Congress on Mathematics Education, held in Adelaide, Australia, in August 1984. The AAAS is the American Association for the Advancement of Science, of which Ubiratan D'Ambrosio is a Fellow.

EUCLIDEAN was used in English in 1660 by Isaac Barrow (1630-1677) in the preface of an edition of the Elements (OED2).

EUCLIDEAN GEOMETRY appears in English about 1865 in The Circle of the Sciences, edited by James Wylde.

EUCLIDEAN ALGORITHM. Euclid's algorithm appears in "On the Group Defined for any Given Field by the Multiplication Table of Any Given Finite Group," Leonard Eugene Dickson, Transactions of the American Mathematical Society 3 (Jul., 1902).

Euklidischer Algorithmus was used by Paul Bachmann in 1902 in Niedere Zahlentheorie [Heinz Lueneburg].

Euclidean algorithm is found in L. L. Dines, "Independant postulates for a generalized euclidean algorithm," Bulletin A. M. S. (1929).

The EULERIAN INTEGRAL was named by Adrien Marie Legendre (1752-1833) (Cajori 1919; DSB). He used Eulerian integral of the first kind and second kind for the beta and gamma functions. Eulerian integral appears in 1825-26 in the his Traité des Fonctions elliptiques et des Intégrales Eulériennes [James A. Landau].

EULER LEHMER PSEUDOPRIME. Euler pseudoprime first appears in Solved and Unsolved Problems in Number Theory, 2nd ed. by Daniel Shanks (Chelsea, N. Y., 1979).

Euler Lehmer pseudoprime and strong Lehmer pseudoprime are found in A. Rotkiewicz, "On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with parameters L,Q in arithmetic progressions," Prepr., Inst. Math., Pol. Acad. Sci. 220 (1980).

EULER-MASCHERONI CONSTANT. Euler's constant appears in 1868 in the title "Second and Third Supplementary Paper on the Calculation of the Numerical Value of Euler's Constant," Proc. of Lond. XVI. 154 and 299-300.

In 1872, "On the History of Euler's Constant" by J. W. L. Glaisher in The Messenger of Mathematics has: "It has sometimes (as in Crelle, t. 57, p. 128) been quoted as Mascheroni's constant, but it is evident that Euler's labours have abundantly justified his claim to its being named after him."

Eulerian constant appears in the Century Dictionary (1889-1897).

In his famous address in 1900, David Hilbert (1862-1943) used the term Euler-Mascheroni constant (and the symbol C).

EULER'S THEOREM and EULER'S FORMULA. Euler's theorem is found in 1847 in Phil. Mag. 3rd Ser. XXX. 424: "Recent researches.., in reference to the new analytical theory of imaginary quantities, have revived attention to Euler's theorem, that the sum of four squares multiplied by the sum of four squares produces the sum of four squares" (OED2).

Euler's formula appears in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson, referring to to e^ix = cos x + i sin x.

Euler's formula, referring to a formula for representing the relation between the load and the moving power in machines, is found in 1853 in A dictionary of science, literature & art by William Thomas Brande [University of Michigan Digital Library].

Euler's formula of verification is found in 1853 in Elements of trigonometry, plane and spherical, with its application to navigation and surveying, nautical and practical astronomy and geodesy, with logarithmic, trigonometrical, and nautical tables by Rev. C. W. Hackley [University of Michigan Digital Library].

The Century Dictionary (1889-1897) has:

Euler's theorem. (a) The proposition that at every point of a surface the radius of curvature [rho] of a normal section inclined at an angle [theta] to one of the principal sections is determined by the equation [....].; so that in a synclastic surface [rho1] and [rho2] are the maximum and minimum radii of curvature, but in an anticlastic surface, where they have opposite signs, they are the two minima radii. (b) The proposition that in every polyhedron (but it is not true for one which enwraps the center more than once) the number of edges increased by two equals the number of faces and of summits. (c) One of a variety of theorems sometimes referred to as Euler's, with or without further specification: as, the theorem that (xd/dx + yd/dy)^r f(x, y)ⁿ = n^rf(x, y)ⁿ; the theorem, relating to the circle, called by Euler and others Fermat's geometrical theorem; the theorem on the law of formation of approximations to a continued fraction; the theorem of the 2, 4, 8, and 16 squares; the theorem relating to the decomposition of a number into four positive cubes. All the above (except that of Fermat) are due to Leonahrd Euler (1707-83).

Euler's equation appears in October 1904 in Gilbert Ames Bliss, "Sufficient Condition for a Minimum With Respect to One-Sided Variations," Transactions of the American Mathematical Society. It refers to the differential equation.

In 1909, Webster's New International Dictionary has Euler's formula (an engineering formula) and Euler's theorem (in differential geometry).

In 1934 Webster's New International Dictionary, 2nd ed., has Euler's formula (an engineering formula) and Euler's equation (e^ix = cos x + i sin x).

Euler's formula is found in 1947 in Courant & Robbins, What is Mathematics?: "On the basis of Euler's formula it is easy to show that there are no more than five regular polyhedra" (OED2).

In 1949, Mathematics Dictionary has Euler's equation (a differential equation), the equation of Euler (in differential geometry), Euler's theorem on polyhedrons, and Euler's theorem on homogeneous functions.

Euler-Descartes relation is found in March 1961 in New Scientist: "The result V + F - E = 2 was originally derived for polyhedra: this is the Euler-Descartes relation, known to Descartes but first explicitly proved by Euler" (OED2).

In 1961, Webster's Third New International Dictionary has Euler's formula (an engineering formula) and Euler's equation defined as e^ix = cos x + i sin x or as "any of several differential equations of dynamics."

In 1990 The Mathematical Intelligencer (vol. 12, no. 3) reported that readers of the magazine had selected V + F = E + 2 as the second-most-beautiful theorem in mathematics. The article referred to the theorem as Euler's formula for a polyhedron.

EULER'S METHOD appears in 1851 in Bonnycastle's introduction to algebra by John Bonnycastle (1750?-1821): "Several new rules are introduced, those of principal note are the following: ... the Solution of Biquadratics by Simpson's and Euler's methods..." [University of Michigan Digital Library].

EULER'S NUMBERS (for the coefficients of a series for the secant function) were so named by H. F. Scherk in 1825 in Vier mathematische Abhandlungen (Cajori vol. 2, page 44).

EVEN FUNCTION. Functiones pares is found in 1727 in "Problematis traiectoriarum reciprocarum solutio," presented to the Petersburg Academy in July 1727 by Leonhard Euler:

Primo loco notandae sunt functiones, quas pares appello, quarum haec est proprietas, ut immutatae maneant, etsi loco x ponatur -x. [In the first place are noted functions, which I call even, of which there is this property, that they remain unchanged if in place of x is put -x.]

This citation was found by Ed Sandifer.

Even function is found in 1849 in Trigonometry and Double Algebra by Augustus De Morgan [University of Michigan Historical Math Collection].

EVENT has been in probability in English from the beginning. A. De Moivre's The Doctrine of Chances (1718) begins "The Probability of an Event is greater or less, according to the number of chances by which it may happen, compared with the whole number of chances by which it may either happen or fail."

Event took on a technical existence when Kolmogorov in the Grundbegriffe der Wahrscheinlichkeitsrechnung (1933) identified "elementary events" ("elementare Ereignisse") with the elements of a collection E (now called the "sample space") and "random events" ("zufällige Ereignisse") with the elements of a set of subsets of E [John Aldrich].

The term EVOLUTE was defined by Christiaan Huygens (1629-1695) in 1673 in Horologium oscillatorium. He used the Latin evoluta. He also described the involute, but used the phrase descripta ex evolutione [James A. Landau].

EXACT DIFFERENTIAL EQUATION. Exact differential is found in English in 1825 in D. Lardner, Elementary Treatise on Differential and Integral Calculus: "As there are may differentials of two variables which are not exact differentials, so also there are many differential equations which are not the immediate differentials of any primitve equation" (OED2).

Exact differential equation is found in W. W. Johnson, "Symbolic treatment of exact linear differential equations," American J. (1887).

EXCENTER is found in 1893 in A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples by John Casey. It is spelled excentre [University of Michigan Historic Math Collection].

EXCIRCLE was used in 1883 by W. H. H. Hudson in Nature XXVIII. 7: "I beg leave to suggest the following names: circumcircle, incircle, excircle, and midcircle" (OED2).

EXPECTATION. According to A. W. F. Edwards, expectatio occurs in 1657 in Huygens's De Ratiociniis in Ludo Alae (David 1995).

According to Burton (p. 461), the word expectatio first appears in van Schooten's translation of a tract by Huygens.

Expectation appears in English in Browne's 1714 translation of Huygens's De Ratiociniis in Ludo Alae (David 1995).

See also mathematical expectation.

EXPLEMENT appears in the Century Dictionary (1889-1897): "In geom., the amount by which an angle falls short of four right angles."

See also conjugate angle.

EXPLICIT FUNCTION is found in 1814 New Mathematical and Philosophical Dictionary: "Having given the methods ... of obtaining the derived functions, of functions of one or more quantities, whether those functions be explicit or implicit, ... we will now show how this theory may be applied" (OED2).

The term EXPONENT was introduced by Michael Stifel (1487-1567) in 1544 in Arithmetica integra. He wrote, "Est autem 3 exponens ipsius octonarij, & 5 est exponents 32 & 8 est exponens numeri 256" (Smith vol. 2, page 521).

In the Logarithm article in the 1771 edition of the Encyclopaedia Britannica, the word is spelled differently: "Dr. Halley, in the philosophical transactions, ... says, they are the exponements of the ratios of unity to numbers" [James A. Landau].

EXPONENTIAL CURVE is found in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris (OED2).

EXPONENTIAL FUNCTION. This term was used by Jakob Bernoulli, according to an Internet web page.

Lacroix used fonctions exponentielles in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).

Exponential function appears in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "Thus, a^x, a log x, sin x, &c., are transcendental functions: the first is an exponential function, the second a logarithmic function, and the third a circular function" [James A. Landau]

EXTERIOR ANGLE is found in English in 1756 in Robert Simson's translation of Euclid.

EXTRANEOUS ROOT is found in 1861 in Arthur Cayley, "Note on Mr. Jerrard's Researches on the Equation of the Fifth Order," Philosophical Magazine: "The principle which furnishes what in a foregoing foot-note is called the à priori demonstration of Lagrange's theorem is that an equation need never contain extraneous roots..." [University of Michigan Historic Math Collection].

The term EXTREMAL (for a resolution curve) was introduced by Adolf Kneser (1862-1930), who also introduced these other terms in the calculus of variations: field (for a family of extremals), transversal, strong and weak extremum (DSB).

EXTREME VALUE appears in E. J. Gumbel, "Les valeurs extrêmes des distributions statistiques," Ann. Inst. H. Poincaré, 5 (1934).

See also L. H. C. Tippett, "On the extreme individuals and the range of samples taken from a normal population," Biometrika 17 (1925) [James A. Landau].

EXTREMUM was first used as a mathematical term (in German) by Paul Du Bois-Reymond (1831-1889) in German in 1878 in Math. Ann. XV. 564, according to the OED2.

Extremum was used in English in 1904 by Oskar Bolza (1857-1942) in Lectures on the Calculus of Variations: "The word 'extremum' will be used for maximum and minimum alike, when it is not necessary to distingish between them" (OED2).

 

F DISTRIBUTION. The F distribution was tabulated - and the letter introduced - by G. W. Snedecor Calculation and Interpretation of Analysis of Variance and Covariance (1934). (David, 1995). The letter was chosen to honor Fisher.

The term 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).

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

Factor appears in English in 1673 in Elements of Algebra by John 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).

The term FACTOR ANALYSIS was introduced by Louis L. Thurstone (1887-1955) in 1931 in "Multiple Factor Analysis," Psychological Review, 38, 406-427: "It is the purpose of this paper to describe a more generally applicable method of factor analysis which has no restrictions as regards group factors and which does not restrict the number of general factors that are operative in producing the correlations" (OED2).

FACTOR GROUP. See quotient group.

FACTORIAL. The earlier term faculty was introduced around 1798 by Christian Kramp (1760-1826).

Factorial was coined (in French as factorielle) by Louis François Antoine Arbogast (1759-1803).

Kramp withdrew his term in favor of Arbogast's term. In the Preface, pp. xi-xii, of his "Éléments d'arithmétique universelle," Hansen, Cologne (1808), Kramp remarks:

...je leur avais donné le nom de facultés. Arbogast lui avait substitué la nomination plus nette et plus française de factorielles; j'ai reconnu l'avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage à la mémoire de mon ami. [...I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend.]

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].

FERMAT'S LAST THEOREM. Fermat's General Theorem (referring to this theorem) appears in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

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 Rapport sur un memoire de M. Lamé, Cauchy remarks:

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.

Fermat's Undemonstrated Theorem appears in 1845 in Phil. Mag. XXVII. 286 (OED2).

James Joseph Sylvester concluded an 1847 paper as follows: "I venture to flatter myself that as opening out a new field in connexion with Fermat's renowned Last Theorem, and as breaking new ground in the solution of equations of the third degree, these results will be generally allowed to constitute an important and substantial accession to our knowledge of the theory of numbers."

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.]

FERMAT'S LITTLE THEOREM is found in 1913 in Zahlentheorie by Kurt Hensel: "Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist." [There is a fundamental theorem holding in every finite group, usually called Fermat's little Theorem because Fermat was the first to have proved a very special part of it.]

This citation was provided by Peter Flor to a math history mailing list.

Fermat's "little theorem" is found in English in Irving Kaplansky, "Lucas's Tests for Mersenne Numbers," American Mathematical Monthly, 52 (Apr., 1945) [W. Edwin Clark].

FIBONACCI (as a name for Leonardo of Pisa). There is no evidence that the name Fibonacci was ever used by Leonardo or his contemporaries.

In Baldassarre Boncompagni, Della vita e delle opere di Leonardo Pisano matematico del secolo decimoterzo (1852), Baldassarre Boncompagni listed the writers who used the name "Fibonacci":

John Leslie 1820
P. D. Pietro Cossali 1797-99
Giovanni Gabriello Grimaldi 1790-1792
Guillaume Libri 1838-1841
Chasles 1837
Nicollet 1811-1818
S. Ersch & I. G. Gruber 1818 and subsequent years
August de Morgan 1847 (he also used Bonacci)

He also listed writers who explaine "Fibinacci = filio Bonacci"

Flaminio dal Borgo 1765
Tiraboschi 1822-1828
Ranieri Tempesti 1787
Giovanni Andres 1808-1817
Grimaldi 1790-1792
Libri 1838-1841

Then his arguments for Fibonacci = de filiis Bonacci follow.

According to Boncompagni, Cossali wrote in Origine, trasporto in Italia, primi progressi in essa dell' algebra (2 vols., Parma 1797-1799) the following, on the last page of volume II:

Nel corso dell'Opera ho chiamato il benemerito Leonardo di Pisa, Leonardo Bonacci, laddove da altri fu detto Leonardo Fibonacci, accozzando la prima sillaba Fi di filius al paterno nome Bonacci. Io ho stimato di volger questo a cognome, come assai volte si è fatto. A taluno sarebbe forse piu\ piacciuto (sic) il dire Leonardo di Bonacci.

However, according to Menso Folkerts, Cossali gives on page 1, volume 1, Leonardo's name as "Leonardo Bonacci di Pisa." Later on he only has "Leonardo". In the summary, printed at the beginning of the book, Cossali writes "Leonardo Pisano".

Leonardo Fibonacci appears on pages 2 and 109 in Scritti inediti del P. D. Pietro Cossali. Edited by B. Boncompagni. Roma 1857.

On page 20 of volume two of "Histoire des sciences mathematiques en Italie" (1838) by the historian of mathematics Guillaume Libri (1803-1869) a footnote begins:

Fibonacci est une contraction de filius Bonacci, contraction dont on trouve de nombreux exemples dans la formation des noms des familles toscanes.

According to Victor Katz in A History of Mathematics (p. 283), "Leonardo [of Pisa], often known today by the name Fibonacci (son of Bonacci) given to him by Baldassarre Boncompagni, the nineteenth century editor of his works, was born around 1170."

[Most of this entry was taken from a post to the historia matematica mailing list by Heinz Lueneburg.]

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.

FIDUCIAL PROBABILITY and FIDUCIAL DISTRIBUTION first appeared in R. A. Fisher's 1930 paper "Inverse Probability," Proceedings of the Cambridge Philosophical Society, 26, 528-535 (David (2001)).

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 Zahlkörper (body of numbers) is due to Richard Dedekind (1831-1916) (Kline, page 1146). Dedekind used the term in his lectures of 1858 but the term did not come into general use until the early 1890s. Until then, the expression used was "rationally known quantities," which means either the field of rational numbers or some finite extension of it, depending on the context.

Zahlenkörper appears in Stetigkeit und Irrationale Zahlen (Continuity & Irrational 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'.

Dedekind did not allow for finite fields; for him, the smallest field was the field of rational numbers. According to a post in sci.math by Steve Wildstrom, "Dedekind's 'Koerper' is actually what we would call a division ring rather than a field as it does not require that multiplication be commutative."

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 = 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, µ₁, µ₂ ... µ_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 Körper;. KRONECKER's term Rationalitätsbereich, 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 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.

First derivative, used attributively, is found in English in 1850 in The calculus of operations by John Paterson (1801-1883). He also uses the terms second derivative and third derivative, although they are used attributively [University of Michigan Digital Library].

First derivative is found in English in 1881 in A Treatise on Electricity and Magnetism by James Clerk Maxwell: "The first derivatives of a continuous function may be discontinuous" (OED2).

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

FLOATING-POINT is found in 1948 in Math. Tables & Other Aids to Computation III. 318: "Floating-point operation greatly reduces the need for scale factors, but complicates the operations of addition and subtraction" (OED).

The terms FLUXION and FLUENT are associated with Isaac Newton, but Richard Suiseth (also known as Calculator; fl. 1350) used the words fluxus and fluens in his Liber calculationum, according to Carl B. Boyer in The History of the Calculus and its Conceptual Development.

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).

Newton composed the treatise Method of Fluxions in Latin in 1671. [It was first published in 1736, translated into English.] Newton wrote (in translation): "Now those quantities which I consider as gradually and indefinitely increasing, I shall hereafter call fluents, or flowing quantities, and shall represent them by the final letters of the alphabet, v, x, y, and z; ... and the velocities by which every fluent is increased by its generating motion (which I may call fluxions, or simply velocities, or celerities), I shall represent by the same letters pointed...."

FOCUS (of a conic section). Carl Boyer writes in A History of Mathematics, "As the curves are now introduced in textbooks, the foci play a prominent role, yet Apollonius had no names for these points, and he referred to them only indirectly."

The term focus (of an ellipse) was introduced by Johannes Kepler (1571-1630) in a treatment of the conic sections in his Ad Vitellionem paralipomena, quibus Astronomiae pars optica traditur (1604).

Umbilic point is an entry in 1700 in Joseph Moxon's dictionary of mathematics: "Umbilique Points, or the 2 Focus or Centre-Points in an Elipsis."

FOIL (standing for "first, outer, inner, last," a method of multiplying two binomials) is found in Charles P. McKeague, Beginning Algebra: A Text/Workbook Second Edition San Dieg Academic Press (imprint of Harcourt Brace Jovanovich), 1985, ISBN 0-15-505230-6: "Rule: To multiply any two polynomials, multiply each term in the first with each term in the second. There are two wasy we can put this rule to work. FOIL Method."

FOIL is found in 1986 in the third edition of Intermediate Algebra by Charles P. McKeague.

FOIL is also found in 1988 in the third edition of Basic Mathematics for Calculus by Dennis G. Zill, Jacqueline M. Dewar, and Warren S. Wright: "Formula (1), illustrated in Figure 0.10, is sometimes called the FOIL method after the first letter in each of the boldface words." The term is not found in the 1983 second edition of this textbook.

These uses of FOIL were found by James A. Landau. The are almost certainly not the earliest uses; if any readers of this page come across any earlier usages, I would appreciate hearing from you.

FOLIUM OF DESCARTES. According to Eves (page 302), Barrow called this curve la galande.

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.

FOLK THEOREM. This term became very popular in the English literature by the middle of the twentieth century. It is just what the name implies: a result (usually not very deep) which belongs to the "folklore." As far as I know, the first to offer an explicit definition was E. J. McShane in his Retiring Presidential Address ("Maintaining Communication," Amer. Math. Monthly, 1957, 309-317):

One hears of "folk theorems", established (presumably) by some expert, communicated verbally or more likely mentioned in an off-hand way during some conversation with another expert or two, and thereafter unpublishable forevermore because no one would want to publish a "known theorem."

[This entry was contributed by Carlos César de Araújo.]

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

FOUR-COLOR PROBLEM. The problem itself dates to 1852, when Francis Guthrie, while trying to color the map of counties of England, noticed that four colors sufficed.

The first printed reference is due to Cayley in 1878, "On the colourings of maps.," Proc. Royal Geog. Soc. 1 (1879), 259-261.

Problem of the four colors appears in A. B. Kempe, "On the geographical problem of the four colors," Amer. J. Math., 2 (1879), 193-200.

"Map-Colour Theorem" is the title of a paper by Percy John Heawood (1861-1955) which appeared in the Quarterly Journal of Pure and Applied Mathematics in 1890.

Four-color map theorem is found in R. P. Baker, "The four-color map theorem," American M. S. Bull. (1916).

Four color problem is found in Ph. Franklin, "The four color problem," American J. (1922).

Four colour theorem is found in H. S. M. Coxeter's 1933 revision of Mathematical Recreations and Essays by W. W. Rouse Ball.

Four color conjecture is found in 1969 in F. Harary, Proof Techniques in Graph Theory: "By far the most celebrated problem concerning graphs is the Four Color Conjecture."

FOURIER SERIES appears in Björling, "Fourierska serierna," Öfv. af. Förh. Stockh. (1868).

Fourier's series appears in English in 1872 in Spectrum analysis in its application to terrestrial substances, and the physical constitution of the heavenly bodies by Dr. Heinrich Schellen, translated from a German edition: "Other expansions similar to Fourier's series can be conceived, in which the terms, instead of representing pendulous vibrations, would represent vibrations of any other prescribed form..." [University of Michigan Digital Library].

Fourier series appears in English in 1879 in the article "Function" by Arthur Cayley in the Encyclopaedia Britannica.

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

Isaac Todhunter in the third edition of An Elementary Treatise on the Theory of Equations (1875) refers to "a theorem which English writers usually call Fourier's theorem, and which French writers connect with the name of Budan as well as with that of Fourier."

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. According to Franceschetti (p. 357), "In the winter of 1975, while he was preparing the manuscript of his first book, Mandelbrot thought about a name for his shapes. Looking into his son's Latin dictionary, he came across the adjective fractus, from the verb frangere, meaning 'to break.' He decided to name his shapes 'fractals.'"

Fractal appears in 1975 in Les Objets fractals: Forme, hasard, et dimension by Benoit Mandelbrot (1924- ). The title was translated as Fractals: Form, Chance, and Dimension (1977).

Fractal appears (as an adjective in "the idea of the fractal dimension") in the Nov. 1975 Scientific American (OED2).

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).

Broken number is found in 1542 in Ground of Artes (1575) by Robert Recorde: "A Fraction in deede is a broken number" (OED2).

Fragment is found in English in in 1579 in Stratioticos by Thomas Digges: "The Numerator of the last Fragment to be reduced."

FREQUENCY DISTRIBUTION is found in 1895 in Karl Pearson, Phil. Trans. R. Soc. A. CLXXXVI. 412: "A method is given of expressing any frequency distribution by a series of differences of inverse factorials with arbitrary constants" (OED2).

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 or the Quincuncial Lozenge, or Net-work Plantations of the Ancients ... Considered by Sir Thomas Brown: "In the parts thereof [plants] we finde..frustums of Archimedes" (OED2).

This word is commonly misspelled as "frustrum" in, for example, Samuel Johnson's abridged 1843 Edition of his dictionary. The word is spelled correctly in the "Frustum" entry and the "Hydrography" entry in the 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science, but it is misspelled in the entry "Altitude of a Frustrum." The word is misspelled in the 1962 Crescent Dictionary of Mathematics and remains misspelled in the 1989 Webster's New World Dictionary of Mathematics, which is a revision of the Crescent dictionary. The word is also misspelled in at least three places in The History of Mathematics: An Introduction (1988) by David M. Burton.

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 September 1694, Johann Bernoulli wrote in a letter to Leibniz, "quantitatem quomodocunque formatam ex indeterminatis et constantibus," although there is no explicit reference to the Latin term functio. The letter appears in Mathematische Schriften.

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.

Function is found in English in 1779 in Chambers' Cyclopedia: "The term function is used in algebra, for an analytical expression any way compounded of a variable quantity, and of numbers, or constant quantities" (OED2).

(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 A History of Algebraic and Differential Topology 1900-1960 Jean Dieudonné 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 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).

Fundamental theorem of algebra is found in English in 1891 in a translation by George Lambert Cathcart of the German An introduction to the study of the elements of the differential and integral calculus by Axel Harnack [University of Michigan Historical Math Collection].

FUZZY was coined in 1962 by Lotfi A. Zadeh (1921- ), according to an Internet source. The term appears in 1963 in his paper, "A computational approach to fuzzy quantifiers in natural languages," Computers & Mathematics With Applications 9(1), 149-184.

The OED2 shows this 1964 citation: L. A. Zadeh et al., Memorandum (Rand Corporation) RM-4307-PR 1: "The notion of a 'fuzzy' set..extends the concept of membership in a set to situations in which there are many, possibly a continuum of, grades of membership."

In an interview with Betty Blair, Zadeh said:

I coined the word "fuzzy" because I felt it most accurately described what was going on in the theory. I could have chosen another term that would have been more "respectable" with less pejorative connotations. I had thought about "soft," but that really didn't describe accurately what I had in mind. Nor did "unsharp," "blurred," or "elastic." In the end, I couldn't think of anything more accurate so I settled on "fuzzy".

Fuzzy logic appears in 1969 in IEEE Trans. Computers XVIII. 348/2: "In the digital field, pattern recognitions and classification are .. potential users of fuzzy logic" (OED2).

Fuzzy appears in 1976 in Numbers and Games by J. H. Conway: "We say that G and H are confused or that G is fuzzy against H."

Fuzzy mathematics is found in 1963 in the title Journal of Fuzzy Mathematics.

On May 26, 1997, a column by in U. S. News and World Report by John Leo has: "Now 'Deep Thoughts' are available on greeting cards, including one that pokes fun at the fuzzy new math in the schools."

Fuzzy math appears on June 11, 1997, in the Wall Street Journal in the headline "President Clinton's Mandate for Fuzzy Math." In the article, Lynne V. Cheney wrote, "Sometimes called 'whole math' or 'fuzzy math,' this latest project of the nation's colleges of education has some formidable opponents."

In a Presidential debate on Oct. 3, 2000, George W. Bush said, "Look this is the man who's got great numbers. He talks about numbers. I'm beginning to think not only did he invent the Internet, but he invented the calculator. It's fuzzy math."

G-K