Last revision: Sept. 2, 2016
ECCENTRICITY. Eccentric was a technical term in the astronomical system of Claudius Ptolemy. The English word comes via Latin from the Greek ekkentros for “out of the center” as opposed to concentric.
The earliest quotation in the OED is from 1398 but another quotation from 1561 better conveys the Ptolemaic meaning: “Eccentricke, is a circle which hath his center distant… from the center of the worlde.”
Eccentricity appears in English in “A direct and Geometrical method for finding the Aphelions, Eccentricities and proportions of the Orbs of the primary Planets, without supposing the Equality of the Angle of motion at the other Focus of the Planets Ellipsis, by Mr. Halley” in Philosophical Transactions of the Royal Society, number 128, September 25, 1676. The above text appears on page 679 and is the journal’s translation of the title of a Latin paper by Edmund Halley which appears on page 683. The Latin title, as best as I can transcribe it, is “Methodus directa & Geometrica, cujus ope investigantur Aphelia, Eccentricitates, Proportionesque orbium Planetarum primariorum, absque supposita aequalitate anguli motus, ad alterum Ellipseus focum, ab Astronomis hactenus usurpati.” [James A. Landau]
ECONOMETRICS. In the editorial in the first issue of the journal Econometrica, the organ of the Econometric Society, Ragnar Frisch, explained econometrics by referring to the aims of the Society: "Its main object shall be to promote studies that aim at a unification of the theoretical-quantitative and the empirical-quantitative approach to economic problems and that are penetrated by constructive and rigorous thinking similar to that which has come to dominate in the physical sciences. ... The mutual penetration of quantitative economic theory and statistical observation is of the essence of econometrics." Econometrica, 1, (1933), pp. 1-2. Frisch had already used the French version of the name, "l’économétrie," with the same meaning in a 1926 paper "Sur un problème d’économie pure." Earlier appearances have been found, e.g. in German "Oekonometrie" from 1910, but the word was not used in the sense that Frisch gave it. Equally, modern contributors to econometrics have not always shared Frisch’s vision of the subject.
EDGE (of a polyhedron). Euler used the Latin word acies. In a letter to Goldbach in 1750, he described "the junctures where two faces come together along their sides, which, for lack of an accepted term, I call acies." Acies is a Latin term which is comonly used for the sharp edge of a weapon, a beam of light, or an army lined up for battle. [Dave Richeson]
EDGEWORTH EXPANSION, SERIES, etc. derive from F. Y. Edgeworth "The Law of Error," Transactions of the Cambridge Philosophical Society, 20, (1905) 36-65, 113-141. The term "Edgeworth’s series" appears in E. P. Elderton’s review of Edgeworth’s paper (and some publications by C. V. Charlier), which appeared in Biometrika, 5, (1906), 206-210. In the 1920s H. Cramér developed a rigorous theory of "Edgeworth series" which he summarised in his Random Variables and Probability Distributions (1937). More recently the term "Edgeworth expansion" has come into circulation. A JSTOR search found this in S. T. David, M. G. Kendall & A. Stuart "Some Questions of Distribution in the Theory of Rank Correlation," Biometrika, 38, (1951), 131-140.
EFFECTS in the analysis of variance and experimental design. David (2001) traces the terms row effect, column effect and treatment effect to S. S. Wilks Mathematical Statistics (1943, p. 187). The term treatment effect was in use earlier, e.g. in J. O. Irwin "On the Independence of the Constituent Items in the Analysis of Variance," Supplement to the Journal of the Royal Statistical Society, Vol. 1, No. 2. (1934), pp. 236-251. The underlying concepts are implicit in Fisher’s earliest work--from 1923--on the analysis of variance.
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." (p. 310) 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." (p. 316) In the later literature Fisher’s concept would be described as asymptotic efficiency. Fisher did his first efficiency calculations in A Mathematical Examination of the Methods of Determining the Accuracy of an Observation by the Mean Error, and by the Mean Square Error (1920) unaware that Gauss had done similar calculations (Bestimmung der Genauigkeit der Beobachtungen (1816)) a century earlier. However the idea of efficiency in extracting information was novel. [This entry was contributed by John Aldrich, based on David (1995).]
See also ASYMPTOTIC, ESTIMATION and PETERS’ METHOD.
The EHRENFEST (URN) MODEL was introduced in a paper by Paul and Tatyana Ehrenfest "Über zwei bekannte Einwände gegen das Boltzmannsche H-Theorem," Physicalische Zeitschrift, 8, (1907), 311-314 to show that the irreversibility in Boltzmann’s thermodynamics did not contradict the recurrence properties of dynamic systems. The Ehrenfest model was discussed from a mathematical point of view by Mark Kac "Random Walk and the Theory of Brownian Motion," American Mathematical Monthly, 54, (1947), 369-391. It is treated as a Markov chain in the first edition (1950) of W. Feller’s An Introduction to Probability Theory and its Applications volume 1 while in the second (1957) it first appears in a new section on urn models. [John Aldrich]
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 physics.
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.))
An early (but not the first) use of secular is Jacobi’s famous paper in which he introduced his eigenvalue algorithm. The term secular (“continuing through long ages” OED) 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 xTAx subject to xTx = 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. Coll Math Papers, IV, 110
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.
Sylvester’s "On the Three Laws of Motion in the World of Universal Algebra" (Johns Hopkins University Circulars, 3, (1884)) contains 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." Coll Math Papers, IV, 146. Sylvester also used the term "latent equation" ("Lectures on the Principles of Universal Algebra" American J. of Math. VI (1884), 216) Coll Math Papers, IV, 208. "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 the first of Hilbert’s communications on integral equations “Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen”, Nachrichten von d. Königl. Ges. d. Wissensch. zu Göttingen (Math.-physik. Kl.) (1904) p. 49.-91. (The communications from 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.” Eigenvektor appears in the exposition of the finite-dimensional case in Courant & Hilbert Methoden der Mathematischen Physik (1924).
Hilbert’s original starting point was a non-homogeneous integral equation with a parameter λ for which the matrix counterpart is (I - λA)x = y. Hilbert called the values of λ 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. Courant & Hilbert (1924) uses κ for a “characteristiche Zahl” and λ = (1/κ) for an “Eigenwert”; Lichnerowicz (above) wrote the French equivalents. “Eigenwert” is used in a different way in J. von Neumann’s “Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren” Math. Ann. 102 (1929) 49-131: “Ein Eigenwert ist eine Zahl, zu der es eine Funktion f ≠ 0 mit Rf = λf ist dann Eigenfunktion.” This became the dominant usage so that by 1946 H. & B. Jeffreys (Methods of Mathematical Physics) were treating eigenvalue as synonymous with characteristic value and latent root.
The development of MATRX MECHANICS in the 1920s promoted the use of the eigen terminology, for the new theory was written in the language of Hilbert and his school. See chapter 3 of M. Born, W. Heisenberg, and P. Jordan “Zur Quantenmechanik II,” Zeitschrift für Physik, 35, (1926), 557-615. [English translation with commentary in B. L. van der Waerden, (editor) Sources of Quantum Mechanics (Dover Publications, 1968.)]
The passage of the eigen terms into English physics 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) (OED). Eigenvalue appears 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" (OED). Eigenvector appears in R. Brauer & H. Weyl’s “Spinors in n Dimensions,” Amer. J. Math., 57, (1935) 425-449 (JSTOR). While eigenvalue, eigenfunction and eigenvector were translations of German originals, the eigenstate of Dirac’s Principles of Quantum Mechanics (1930, p. 35) was a new construction and it marks the arrival of eigen as a fully English particle. Eigenstate was translated into German as Eigenzustand.
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.
Paul Halmos (see above) followed von Neumann’s English writings and used “proper value” in his widely-used Finite Dimensional Vector Spaces (1958). However Halmos admitted defeat in his A Hilbert Space Problem Book (1967, p. x):
For many years I have battled for proper values, and against the one and a half times translated German-English hybrid that is often used to refer to them. I have now become convinced that the war is over, and eigenvalues have won it; in this book I use them.
The above information was contributed by John Aldrich.
In 2016, Jan Peter Schäfermeyer wrote:
Some time ago I researched the history of the power and related methods to compute eigenvectors and found out that these methods were used first for differential and integral equations, before they were used for the matrix eigenvalue problem. Here are the sources, mostly in German.
An interesting aside, that might be of interest for your webpage, concerns the nomenclature of eigenvalues and eigenfunctions, terms that were quickly adopted in the German literature after Hilbert had proposed them in 1904. Before that, I haven't seen any particular names for these entities. Schwarz simply speaks of a function w and a number c.
The French were more inventive: Picard in two papers, Comptes Rendus 117 (1893), and C.R. 118 (1894), speaks of “points singuliers” and “valeurs singulieres,” but also has no particular name for eigenfunctions:
Poincare proposes the names “nombres caracteristiques” and “fonctions harmoniques” in an 1894 paper, and in 1896 “fonctions fondamentales.”
And in the English literature I have found the terms “characteristic numbers” and “characteristic functions” used by Bocher in his 1909 book on integral equations.
EINSTEIN SUMMATION CONVENTION in TENSOR ANALYSIS. This simplified way of writing tensor expressions was introduced by Einstein in his great paper on general relativity, “Die Grundlage der allgemeinen Relativitätstheorie” Annalen der Physik, 49 (1916). “Later he said in jest to a friend, ‘I have made a great discovery in mathematics; I have suppressed the summation sign every time that the summation must be made over an index which occurs twice..’” Abraham Pais ‘Subtle is the Lord...’: The Science and Life of Albert Einstein (1982, p. 216.)
ELEMENT in set theory. The entry SET and SET THEORY describes how the German word Menge used by Cantor came to be translated as set. The history of the companion word element is much simpler for the term was adopted from an early date and it is essentially the same in German, French and English: Element, élément and element.
Cantor first used the German word Element in his “Ueber unendliche, lineare Punktmannichfaltigkeiten,” Mathematische Annalen (1882) XX p. 114. The French word élément appeared in his “Une contribution á la théorie des ensembles,” Acta Mathematica, 2, (1883), p. 311. The English word appears in Bertrand Russell’s Principles of Mathematics (1903).
Element was not a new word: derivatives of the Latin word elementum had been in English, French and German for centuries and had been used in various mathematical contexts. Thus Euclid’s work was translated into English in 1570 as The Elements of Geometry while in 1867 C. L. Dodgson was referring to the elements of a block (i.e. a MATRIX). In German von Staudt used the term Elemente (elements) in his Geometrie der Lage (2nd ed., 1856): “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 ... ”
[John Aldrich and Ken Pledger contributed to this entry.]
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 April 1893 in “On the Determination of Groups Whose Order is a Power of a Prime” by J. W. A. Young in American Journal of Mathematics Vol. 15, No. 2, pp. 124-178. [Google print search by James A. Landau]
ELEMENTARY OPERATIONS (TRANSFORMATIONS) on (of) a matrix. According to Wedderburn Lectures on Matrices (1934, p. 169) “the notion of an elementary transformation seems to be due in the main” to Grassmann Ausdehnungslehre 1862. Elementary transformations of a matrix are defined by M. Bôcher Introduction to Higher Algebra (1909, p. 55). More recently the term elementary operation has been preferred. For an early instance see W.J. Duncan, R.A. Fraser, A. R. Collar Elementary Matrices and Some Applications to Dynamics and Differential Equations (1938).
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.”
Eliminant is found in 1859 in “Lessons introductory to the Modern Higher Algebra” by George Salmon. Page 18 has “The result of eliminating n variables between n homogeneous equations of any degree is called the eliminant of these equations.”
Eliminant is found in 1881 in The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms by William Snow Burnside and Arthur William Panton: "Being given a system of n equations, homogeneous between n variables, or non-homogeneous between n - 1 variables, if we combine these equations in such a manner as to eliminate the variables, and obtain an equation R = 0, containing only the coefficients of the equations; the quantity R is, when expressed in a rational and integral form, called their Resultant or Eliminant." [Google print search by James A. Landau]
ELIMINATE is found in English in 1818 in Elements of Algebra by S. F. Lacroix, Translated from the French For the use of the students of the University at Cambridge, New England. [Google print search by James A. Landau]
ELIMINATION is found in 1764 in Bézout’s “Recherches sur le degré des Équations résultantes de l'évanouissement des inconnues, & sur les moyens qu'il convient d'employer pour trouver ces Équations,” Histoire de l'Académie royale des sciences Ann. 1764.
Elimination is found in English in 1801 in a review of La Croix’s Elémens d’Algèbre in The Monthly Review; or Literary Journal. [Google print search by James A. Landau].
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. According to the 1900 translation by Beman and Smith of Geschichte der Elementar-Mathematik by Karl Fink: "Instead of Euler’s names: 'elliptoid, elliptic-hyperbolic, hyperbolic-hyerbolic, elliptic-parabolic, parabolic-hyperbolic surface,' the terms now in use, 'ellipsoid, hyperboloid, paraboloid,' were naturalized by Biot and Lacroix."
Ellipsoid appears in a letter written in 1672 by Sir Isaac Newton [James A. Landau].
ELLIPTIC CURVE. In 1901 “A Problem and its Solutions” by Euplio Conoscente in Proceedings of the London Mathematical Society has: “It ought to be so for the coordinates of an elliptic curve are expressed by elliptic functions of the Abelian integral of the first kind relative to its equation.” [Google print search by James A. Landau]
In 1908, volume three of Lehrbuch der Algebra by Heinrich Weber has an initial section called "Die elliptische Integrale" with subsection 6 called "Elliptische Kurven." That subsection includes this statement:
Wir wollen hier alle Kurven, deren Differentiale und Integrale auf elliptische reduzierbar sind, e l l i p t i s c h e K u r v e n nennen. ... Wir werden sehen, dass alle Kurven dritten grades ohne Doppel- oder Rueckkehrpunkt zu den elliptischen gehoeren. Kurven hoeheren Grades koennen nur dann dazu gehoeren, wenn sie eine gewisse Anzahl singulaerer Punckte haben.
[This citation was provided by William C. Waterhouse.]
Elliptic curve is found in Webster’s New International Dictionary (1909), defined as “one of the genus 1.”
ELLIPTIC FUNCTION has had two meanings: 1) what we now call an “elliptic integral” 2) the doubly periodic functions which are the inverse of the elliptic integral.
Fonction elliptique was used with meaning 1) by Legendre in his 1792 paper “Mémoire Sur Les Transcendantes Elliptiques.” Elliptic function appeared in English in 1809 in the translation of this paper in Thomas Leybourn, ed The Mathematical Repository, New Series, Volume III (1809).
Fonctions elliptiques with meaning 2) appeared in a letter by Jacobi, “Suite des notices sur les fonctions elliptiques,” which was published in volume III (1828) of Crelle.
In 1844 elliptic function appears in the title A Treatise on the Integral calculus, containing the Integration of Explicit Functions of One Variable; Together With The Theory of Definite Integrals and of Elliptic Functions by J. Hymers, DD.
In 1847 in Report of the Sixteenth Meeting of the British Association for the Advancement of Science in the article “On the Recent Progress of Analysis (Theory of the Comparison of Transcendentals)” by R. L. Ellis writes, “Following M. Jacobi, we shall henceforth give the name of elliptic functions to those which are analogous to circular functions. It is on this account better to call Legendre’s functions elliptic integrals than, as he has done, elliptic functions.”
[This entry was contributed by James A. Landau.]
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]
EM ALGORITHM. The term and the method were introduced by A. P. Dempster; N. M. Laird & D. B. Rubin "Maximum Likelihood from Incomplete Data via the EM Algorithm," Journal of the Royal Statistical Society, B, 39, (1977), 1-38. "This paper presents a genereal approach to iterative computation of maximum-likelihood estimates when the observations can be viewed as incomplete data. Since each iteration of the algorithm consists of an expectation step followed by a maximization step we call it the EM algorithm." The paper gives many references to related earlier work, the oldest to a paper from 1926 by A. G. McKendrick. (David (2001)).
EMPTY SET. A JSTOR search found the term used--without explanation--in J. E. McAtee "Modular Invariants of a Quadratic Form for a Prime Power Modulus," American Journal of Mathematics, 41, (1919), p. 237. The term began to become common in the 1930s, although in these early days it was nothing like as common as null set.
ENDOGENOUS/EXOGENOUS VARIABLE. In the SIMULTANEOUS EQUATIONS MODEL of ECONOMETRICS variables are either endogenous or exogenous. The terms were coined around 1945 by Tjalling C. Koopmans. He explained them as follows, "[Exogenous variables] are variables that influence the endogenous variables but are not themselves influenced by the endogenous variables". [The] exogenous variables are treated as if they are given functions of time, the values of which remain the same in repeated samples." p. 56 of Koopmans & H. Rubin "Measuring the Equation Systems of Dynamic Economics," chapter II of Koopmans (ed.) Statistical Inference in Dynamic Economic Models (1950). The terms endogenous and exogenous were taken from business cycle theory. In his work on the simultaneous equations model Haavelmo had used the terms "dependent" and "autonomous." The Probability Approach in Econometrics," Econometrica, 12, Supplement. (1944), pp. 86-7.
In econometrics the term exogenous variable is often used where in other branches of statistics independent variable or regressor or covariate might be used.
See DEPENDENT/INDEPENDENT VARIABLE, REGRESSION, COVARIATE.
ENTROPY. The concept originated in thermodynamics, was extended by analogy to information theory and then by analogy again to dynamical systems.
The concept was introduced into physics by Rudolf Clausius (1822-1888) "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie," Annalen der Physik und Chemie, 125, (1865) 353. Clausius gave a full account of how he chose the name:
If we wish to designate S by a proper name we can say of it that it is the transformation content of the body, in the same way that we say of the quantity U that it is the heat and work content of the body. However, since I think it is better to take the names of such quantities as these, which are important for science, from the ancient languages, so that they can be introduced without change into all the modern languages, I proposed to name the magnitude S the entropy of the body, from the Greek word , a transformation. I have intentionally formed the word entropy so as to be as similar as possible to the word energy, since both these quantities, which are to be known by these names, are so nearly related to each other in their physical significance that a certain similarity in their names seemed to me advantageous.Ludwig Boltzmann (1844-1906) gave a probabilistic interpretation of entropy and expressed entropy in terms of the integral of f log f in "Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen" Sitzungsberichte. Akad. Wiss. Wien, part II, 66, (1872), 275-370. There is an English translation in S. G. Brush, Kinetic Theory, Vol. II, pp. 88-175, (1965).
In A Mathematical Theory of Communication (1948, p.11) C. E. Shannon (1916-2001) set down a "reasonable" set of requirements for a measure of information and showed that the only expression satisfying them has a form that "will be recognized as that of entropy as defined in certain formulations of statistical mechanics." So he called the measure "entropy." See INFORMATION THEORY.
A. N. Kolmogorov applied information theory to the ergodic theory of dynamical systems in "On the Entropy per Time Unit as a Metric Invariant of Automorphisms," Doklady akademi nauk, 124, (1959), 754-5.
An article by J. Willard Gibbs, “Graphical Methods in the Thermodynamics of Fluids” has a footnote: “The term entropy, it will be observed, is here used in accordance with the original suggestion of Clausius, and not in the sense in which it has been employed by Professor Tait and others after his suggestion. The same quantity has been called by Professor Rankine the Thermodynamic function. ...”
This entry was contributed by John Aldrich, with the footnote added by James A. Landau.
ENTSCHEIDUNGSPROBLEM. See DECISION PROBLEM.
The term EPICYCLE was used in the astronomical system of Claudius Ptolemy to signify a small circle, having its centre on the circumference of a bigger circle. The English word came via Latin from the Greek upon + circle. The earliest quotation in the OED is from Chaucer’s Treatise on the Astrolabe (about 1391): “The Moone Moeuyth the contrarie from othere planetes as in hire Episicle.”
The term EPICYCLOID was used by Philippe de La Hire in Traité des épicycloïdes et leur usage en mécanique, which he read to the Academy of Sciences in 1694. A footnote in the Collected Works of Jacob Bernoulli says the term is apparently due to La Hire. [Information provided by François Ziegler.]
Epicycloid is found in English in 1695 in “The Quadrature of a Portion of the Epicycloid” by a Mr. Caswel or Caswell (both spellings appear) in Philosophical Transactions For the Month of October, 1695. [Google print search by James A. Landau]
EPIMOMORPHISM. See the entry MONOMORPHISM.
EPONYMY and LAWS OF EPONYMY. Names in mathematics often make reference to a person. These pages contain hundreds of examples, from ABELIAN EQUATION to ZORN’S LEMMA, or, in time, from the PYTHAGOREAN THEOREM to GRAHAM’S NUMBER. There is no uniform relation between "thing" and person: ZORN’S LEMMA is a name that contemporaries of Zorn gave to a proposition that can be found (more or less) in a work by him; CARTESIAN PRODUCT arrived nearly 300 years after Descartes in a branch of mathematics that did not exist in his time and there was never any question of looking it up in his work.
Instances of eponymy prompt obvious questions, e.g. of BAYES it has been asked, "Who discovered Bayes’s theorem?" and "Was Bayes a Bayesian?" Sometimes a ‘wrong’ answer inspires an attempt to alter usage: see MACLAURIN’s SERIES for an unsuccessful effort and GAUSS-MARKOV THEOREM for a half-successful one. The story of another half-successful attempt, the RAO-BLACKWELL THEOREM, throws some light on what makes an eponym successful. Sometimes an incorrect attribution just stands with no attempt to change it, e.g. VANDEMONDE DETERMINANT.
The general practice of eponymy has attracted attention, especially the widespread phenomenon of misattribution. Stigler (see below) quotes a cynical remark from an unnamed historian of science, "Every scientific discovery is named after the last individual too ungenerous to give due credit to his predecessors." Misattribution has also inspired laws of eponymy--eponymous laws of eponymy, naturally.
Boyer’s law, that "mathematical formulas and theorems are usually not named after their original discoverers," was proposed by H. C. Kennedy ("Who Discovered Boyer’s Law?" Amer. Math. Monthly, 79:1 (1972), 66-67) on the basis of the many instances described in Carl B. Boyer’s History of Mathematics (1968). Boyer gave his opinion of eponymy in the exclamation, "Clio, the muse of history, often is fickle in attaching names to theorems!"
Stigler’s law of eponymy, that "no scientific discovery is ever named after its original discoverer," was formulated by Stephen Stigler ("Stigler’s Law of Eponymy" (1980), reprinted in Stigler (1999)). Unlike Boyer, Stigler saw patterns in naming and attempted to explain them using the ideas of the sociologist Robert K. Merton. Stigler tried out his hypotheses in a case study of the GAUSSIAN (NORMAL) distribution.
Another law complicating eponymy is Whitehead’s law: "Everything of importance has been said before by someone who did not discover it." (The remark was popularised by Merton but Michael Berry calls it a law.) See the entry RAO-BLACKWELL for an illustration of the difficulty of applying this law, especially to the work of a living discoverer who argues back. Whitehead’s remark is from the essay Organisation of Thought published in a collection with the same title (1917). He was discussing the relation between traditional logic and modern logic and the sentence before read, "To come near to a true theory and to grasp its precise application are two very different things, as the history of science teaches." Berry, incidentally, calls the Boyer-Stigler law Arnold’s law after V. I. Arnol’d who complains of Western neglect of Russian contributions: see the entry CAUCHY-SCHWARZ for an instance.
[This entry was contributed by John Aldrich.]
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. Diophantus used the words isotas and isosis in his Arithmetica. This was translated ca. 860 CE by Qusta ibn Luqa with the Arabic word . The Arabic term was used by many later algebraists in the Islamic world. The Latin equatio is probably a translation from the Arabic. [Jeffrey A. Oaks]
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. 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" (OED). Billingsley simply anglicised the Latin term.
A few years before (in 1551) Robert Recorde had proposed an English expression in his Pathway to Knowledge: "That the Greekes doo call Isopleuron, and Latine men aequilaterum: and in english it may be called a threlike triangle" Recorde’s English terms (there was a family of them) did not take root as he acknowledged in his last work. In this—it is written in the form of a dialogue—the “Master” states, “if I should give them [mathematical terms] newe names, many would make a quarrelle against, for obscuring the old Arte with newe names; as some all redy have done.” In German local terms did take root and gleichseitig (equal-sided) became the standard term. (Based on G. Howson A History of Mathematics Education in England 1982.)
See also the entry ISOSCELES.
EQUIPROBABLE. In 1853 a review of the book Essays on Life Assurance by Edward Sang has: “Now Mr. Sang ought to have known, not merely that M. Legendre did not attempt to demonstrate the method of minimum squares, but that Gauss and La Place did attempt to do so, the former aided by the consideration of equi-probable errors, and the latter, with complete success, by that of equi-possible combinations.” The review is found in The Assurance Magazine, and Journal of the Institute of Actuaries. [Google print search by James A. Landau]
EQUIVALENT (MATRIX) was introduced by K. Hensel (as “ähnlich”) in “Ueber lineare Substitutionen und bilineare Formen,” J. reine angew. Math. Vol. 84 (1879) p. 21. This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933).
Equivalent is found in 1884 in an article in The Messenger of Mathematics. [Google print search by James A. Landau]
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.]
ERLANGEN PROGRAM (in German, Erlanger Programm.) In 1872 the 24 year-old Felix Klein became a professor at the University of Erlangen and "by tradition he was required to circulate a written programme of work plans." Grattan-Guinness (p. 547) Klein’s program was entitled "Vergleichende Betrachtungen über neuere geometrische Forschungen" (A Comparative Review of Recent Researches in Geometry.) The paper was widely translated and Klein reissued it in Mathematische Annalen, 43, (1893),63-100.
The phrase "Erlangen program" has referred variously to the paper, to the injunction which it contains, "Given a manifold and a group of transformations of the manifold, to study the manifold configurations with respect to those features which are not altered by the transformations of the group." (p. 67, translated in Roberto Torretti Nineteenth Century Geometry), to the work that followed the injunction and to the underlying interpretation of geometry. [John Aldrich]
ERROR, LAW OF ERROR, THEORY OF ERRORS, etc. Between 1750 and 1830 mathematicians and astronomers (who were usually the same people) created the first elaborate theory of statistical inference, the theory of errors. In the 20th century the theory was extended to new areas including regression and the analysis of variance. Many special terms were introduced in the 19th century as the theory was expounded and consolidated. In the 20th century more new terms appeared and some of the old ones were retired.
Thomas Simpson’s On the Advantage of Taking the Mean of a Number of Observations (1755) was an early attempt to subject to probabilistic analysis the "method practiced by Astronomers" of taking the mean of several observations "in order to diminish the errors arising from the imperfections of instruments and the organs of sense." By the early 19th century some far reaching theory had been developed with Gauss and Laplace the leading contributors. See CENTRAL LIMIT THEOREM and METHOD OF LEAST SQUARES
Astronomers recognised different kinds of errors, e.g. W. Chauvenet A Manual of Spherical and Practical Astronomy ... with an Appendix on the Method of Least Squares (4th edition, 1871, p. 470) distinguished between irregular and constant errors, subdividing the latter into 3 categories. The theory of errors was about eliminating the effects of irregular errors.
Error was associated not only with the individual observations but with the inferences based upon them. Thus the title of Gauss’s Theoria combinationis observationum erroribus minimis obnoxiae (1821/3) translates as the Theory of the combination of observations least subject to error and the main business of this work was to show that the method of least squares produces the required combination. The 20th century terms sampling error (see Fisher Statistical Methods for Research Workers (1925, §26)) and Type I error are concerned with errors of inference in estimation or testing and have no necessary connection with errors of observation. See the entries GAUSS-MARKOV THEOREM and TYPE I ERROR.
Bessel’s probable error (wahrscheinliche Fehler) was a measure of (un)reliability that could apply both to an individual observation and to a combination of observations like the mean. Today there are two distinct terms, the standard deviation and the standard error. For details see the entries PROBABLE ERROR, STANDARD ERROR and STANDARD DEVIATION.
Law of error A. De Morgan Essay on Probabilities (1838) asked, "what do we mean by a law of error?" and went on to describe "the standard law of facility of error" (p. 143). This law had been used by Gauss in his first (1809) theory of least squares and is called the normal or Gaussian distribution today. There were many variations on the De Morgan terminology but the boldest and simplest was the law of error. There is a chapter on the law in W. S. Jevons’s Principles of Science (1874). See the entries NORMAL and GAUSSIAN and also Symbols Associated with the Normal Distribution on the Symbols in Probability and Statistics page.
Theory of errors. According to Sheynin Theory of Probability: A Historical Essay p. 57, the term “theory of errors” was not used by Gauss or Laplace and is due to Bessel who first used it in 1820 in his “Beschreibung des auf des Königsberger Sternwarte,” Astron. Jahrb. (Berlin) für 1823, 161–168. The English expression appears in the title of G. B Airy’s 1861 book, On the Algebraical and Numerical Theory of Errors of Observation and the Combination of Observations. This title contains another popular name for the subject, viz. the combination of observations.
Error function. In the course of the19th century the function from the theory of errors appeared in several contexts unrelated to probability, e.g. refraction and heat conduction. In 1871 J. W. Glaisher wrote that "Erf x may fairly claim at present to rank in importance next to the trigonometrical and logarithmic functions." Glaisher introduced the symbol Erf and the name error function for a particular form of the law as follows:
As it is necessary that the function should have a name, and as I do not know that any has been suggested, I propose to call it the Error-function ... and to write("On a Class of Definite Integrals, Philosophical Magazine, 42, 1871, p. 296) Glaisher also introduced the error-function-complement with the symbol Erfc. The names and symbols continue to be used though often the definitions are slightly different from Glaisher’s.
When R. A. Fisher wrote about regression and the analysis of variance he was reviving the theory of errors. He and his followers reinterpreted old terms and produced new ones. They considered their extensions so radical that they were no longer working in the theory of errors; that was now part of history: see Fisher’s Statistical Methods for Research Workers (1925, §1).
Experimental error. In "The Arrangement of Field Experiments", Journal of the Ministry of Agriculture of Great Britain, 33, (1926), 503-513 Fisher extended the notion of error when he explained that "experiments of the uniformity trial type have demonstrated the magnitude and ubiquity of that class of error which cannot be ascribed to carelessness in measuring the land or weighing the produce, and which is consequently described as due to ’soil heterogeneity.'" In the analysis of variance Fisher had a category "variance due to experimental error": see Statistical Methods for Research Workers (1925, §48, Table 58). See the entry VARIANCE.
Error term. In his formulation of REGRESSION Fisher wrote the regression equation as a relation between the mean value of the dependent variable and the values of the independent variables: see Statistical Methods for Research Workers (1925, §26). In the econometric literature the equations of the SIMULTANEOUS EQUATIONS MODEL were written with "random shifts" as in T. Haavelmo "The Probability Approach in Econometrics," Econometrica, 12, Supplement. (1944), p. 99 or with "disturbances" as in Koopmans & H. Rubin "Measuring the Equation Systems of Dynamic Economics" p. 59. However these expressions were soon displaced by "error." The expression error term appears in the title of D. Cochrane & G. H. Orcutt’s "Application of Least Squares Regression to Relationships Containing Auto-Correlated Error Terms," Journal of the American Statistical Association, 44, (1949), 32-61.
Fisher’s regression was given a new look by J. Durbin and G. S. Watson ("Testing for Serial Correlation in Least Squares Regression: I," Biometrika, 37, (1950), p. 410) when they wrote
in "which y the dependent variable and x, the dependent variable, are observed, the errors ε being unobserved." They (p. 411) also gave the matrix version
Although the error formulation of the regression equation was introduced by writers on time series analysis/econometrics, who were used to including random shocks or disturbances in their models, the formulation was equally natural in experiments and the analysis of variance and was adopted in such works as O. Kempthorne’s Design and Analysis of Experiments (1952) and H. Scheffé’s Analysis of Variance (1959). See the entries REGRESSION, RESIDUAL and VARIANCE.
Errors in variables and errors in equations. These expressions came into use around 1950 to indicate two ways of modelling the relationship between variables. They appear, for instance, in M. G. Kendall’s "Regression, Structure and Functional Relationship Part I," Biometrika, 38, (1951), p. 11. The errors in equation formulation figures in the Fisher regression model and in the SIMULTANEOUS EQUATIONS MODEL and has just been described under the heading error term. In the errors in variables formulation there is an exact relationship between unobserved true values of the variables but the observations are contaminated by measurement error. Interest in this formulation went back to the late 19th century. D. V. Lindley began his literature review with the work of C. H. Kummel (1879) in his paper "Regression Lines and the Linear Functional Relationship," Supplement to the Journal of the Royal Statistical Society, 9, (1947), 218-244.
This entry was contributed by John Aldrich.
ESCRIBED CIRCLE is found in 1847 in The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in Plane Co-ordinate Geometry by Thomas Gaskin: "The lines joining the angles of a triangle with the points in which the escribed circles touch the opposite sides, meet in a point. Shew also that if the base and the sum of the other sides is constant, the locus of the centre of the escribed circle touching the base is an ellipse." [Google print search]
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." (p. 310) Fisher uses estimate as a substantive sparingly in the paper. The same paper introduced three criteria of estimation: consistency, efficiency and sufficiency.
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 (see entry on Gauss-Markov theorem). In his "On the Two Different Aspects of the Representative Method" (Journal of the Royal Statistical Society, 97, (1934), 558-625) Neyman had used mathematical expectation estimate for unbiased estimate and best linear estimate for best linear unbiased estimate.
Following Neyman’s (ibid.) introduction of confidence intervals (q.v.) the expression interval estimation came into use. It appears in E. S. Pearson "Some Aspects of the Problem of Randomization" Biometrika, 29, (1937), pp. 53-64, though Neyman himself (Phil. Trans. R. Soc. A 236, (1937), p. 346) wrote of "estimation by unique estimate" and "estimation by interval". Point estimation and interval estimation are used as contrasting terms in Henry Scheffé "Statistical Inference in the Non-Parametric Case" Annals of Mathematical Statistics, 14, (1943), pp. 305-332.
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.
This entry was contributed by John Aldrich. See also CONFIDENCE INTERVAL, CRAMÉR-RAO INEQUALITY, CRITERIA OF ESTIMATION, DECISION THEORY, GAUSS-MARKOV THEOREM, MAXIMUM LIKELIHOOD, METHOD OF LEAST SQUARES, MINIMUM χ2, RAO-BLACKWELL THOREM, STANDARD ERROR.
ETHNOMATHEMATICS. In a 1959 Italian paper, “Per una maggiore attenzione all’etnologia matematica,” Rivista di Antropologia, O. Falsirol, appraising the work of Ewald Fettweis, wrote:
Professore all’Accademia Pedagogica di Aachen, dove tenne lezioni di didattica, di storia delle matematiche e di etnomatematica fino al 1954, egli dedicò e viene dedicando parte considerevole della sua attività scientifica alla matematica e all’astronomia dei popoli considetti primitivi.
[Professor at the Paedagogische Akademie in Aachen, he lectured on mathematical education, history of mathematics, and ethnomathematics until 1954. He focused his research on the study of the mathematics and astronomy of primitive people.]
This information, including the English translation, is taken from “Ethnomathematics in the 1930s — The Contribution of Ewald Fettweis to the History of Ethnomathematics,” by Andrea V. Rohrer and Gert Schubring, For the Learning of Mathematics 31, 2 (July, 2011). The article states that the term first appeared in print in 1959.
In 1997 Ubiratan D'Ambrosio stated that he had coined the term, writing 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 ALGORITHM. Gauss used the term continued fraction algorithm.
Der Euclidische Algorithmus is found in 1902 in Nidere Zahlentheorie. Erster Teil by Dr. Paul Bachmann.
Euclidean algorithm is found in Englsih in July 1903 in a review of the above book in the Bulletin of the American Mathematical Society.
[Google print searches by James A. Landau.]
EUCLIDEAN GEOMETRY is found in English in 1723 in the title A System of the Mathematics, Containing The Euclidean Geometry, Plane and Spherical Trigonometry, the Projection of the Sphere, both Orthographic and Stereographic, Astronomy, the Use of the globes and Navigation by James Hodgson. [Google print search by James A. Landau]
EULER. Leonhard Euler (1707-83) is generally considered the most prolific mathematician of all time. Karl Pearson wrote, "It does not matter what field you take up in pure or applied mathematics you will find Euler’s name attached to one or more theorems, or one or more theorems to which it ought to be attached." Lectures on the History of Statistics in the 17th and 18th Centuries (p. 245). Besides theorems, there are Euler conjectures, diagrams, equations, formulas, etc. The terms appear in analysis, topology, number theory, combinatorics, mechanics, ...
MathWorld has entries for over 80 Euler terms. The entry for Euler’s theorem in the Century Dictionary (1889-1897) began lettering them but then seemed to give up:
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 = nrf(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 Leonhard Euler (1707-83).
Euler introduced many other terms and symbols besides those that bear his name. These can be found by using the Search on the front page. On the Symbols pages see especially Earliest Uses of Symbols of Number Theory and Earliest Uses of Function Symbols. For information on Euler’s writings, see the Euler Archive.
EULER CHARACTERISTIC or EULER-POINCARÉ CHARACTERISTIC. These terms in algebraic topology and graph theory are associated with Poincaré’s generalisation of Euler’s convex polyhedra formula in §16 of his "Analysis Situs," Journal de l’École Polytechnique, 1, (1892) 1-121 See Mathworld entry.
See the entries EULER-DESCARTES RELATION and TOPLOGY.
EULER’s CIRCLES or DIAGRAMS. In about 1690 Leibniz used circle diagrams to illustrate all of the syllogisms of the Aristotelian logic. The source is: Leibniz, De Formae Logicae comprobatione per linearum ductus, ed. in: L. Couturat, Opuscules et fragmentes inedits de Leibniz, Paris 1903, p. 292-320. There is a German translation in: Fragmente zur Logik. Ausgewählt, übersetzt und erläutert von Franz Schmidt, Berlin 1960, p.373ff.
In his 1768 Lettres a une Princesse d'Allemagne (see letter CIII) Euler used circles to illustrate some basic propositions in logic. The device was used in many nineteenth century textbooks on logic. The phrase Euler’s scheme of notation is found in 1858 in Elements of logic by Henry Coppée (1821-1895): "Euler’s scheme of notation is altogether the one best suited to our purpose, and we shall limit ourselves to the explanation of that. It is essentially an arrangement of three circles, to represent the three terms of a syllogism, and, by their combination, the three propositions" [University of Michigan Digital Library]. Euler’s diagram appears in 1884 in Elementary Lessons in Logic by W. Stanley Jevons: "Euler’s diagram for this proposition may be constructed in the same manner as for the proposition I as follows:..." Euler’s circles appears in 1893 in Logic by William Minto (1845-1893): "The relations between the terms in the four forms are represented by simple diagrams known as Euler’s circles."
Wilfried Neumaier and John Aldrich contributed to this entry. See also VENN DIAGRAM.
EULER’s EQUATION in the calculus of variations. L. Bolza Lectures on the Calculus of Variations (1904, p. 22) writes that "Euler’s (differential) equation" was given in 1744 in ch. 2 article 21 of "Methodus inveniendo lineas curvas maximi minimore propietas." In a footnote Bolza notes that Hilbert had referred to it as "Lagrange’s equation" but that Lagrange himself attributed it to Euler: "cette equation est celle qu'Euler a trouvé le premier." Ouevres, X, p. 397.
In 1848 Cayley used the term Euler’s equation: “By a method founded on that of Lagrange for the solution of Euler’s equation.” This is from “Notes on the Abelian Integrals.—Jacobi’s System of Differential Equations,” The Cambridge and Dublin Mathematical Journal, vol. III. (1848), pp. 51-54. [James A. Landau]
See the entry CALCULUS OF VARIATIONS.
EULER’s EQUATION, FORMULA or IDENTITY eix = cos x + i sin x. In A Concise History of Mathematics (p. 120) D. J. Struik writes that Euler presented this equation in the Introductio in analysin infinitorum of 1748 although it was "already discovered by Johann Bernoulli and others in different forms". Struik reproduces pp. 103-4 of the Introductio which contain the relation. James Thomson calls it Euler’s formula in his An Introduction to the Differential and Integral Calculus, 2nd ed. (1849).
EULER’s GRAECO-LATIN SQUARES CONJECTURE. In his "Recherches sur une Nouvelle Espèces de Quarrés Magique" (1782) conjectured that there exist no Graeco-Latin (or Graeco-Roman) squares of order n= 4+k for k = 1, 2, .... The conjecture was shown to be false by Bose, Shrikhande, and Parker in 1959. In 1921 MacNeish used the term Euler square for a generalisation of the Graeco-Latin square but the word is now often used to refer to the Graeco-Latin square itself. See Mathworld
See the entry LATIN SQUARE.
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 his Traité des Fonctions elliptiques et des Intégrales Eulériennes [James A. Landau].
See the entry BETA and GAMMA FUNCTIONS.
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). See Mathworld for this and other senses of Euler numbers.
EULER’s THEOREM. The OED2’s earliest quotation for Euler’s theorem is from 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"
EULER’s THEOREM on homogeneous functions was given in his Institutiones Calculi Differentialis (1755). (F. Cajori, A History of Mathematics, p. 239.)
EULER-DESCARTES RELATION V + F - E = 2 for polyhedra. The history of this theorem is used as a case study by Imre Lakatos Proofs and Refutations (1976). According to Lakatos (p. 6), the relation was conjectured by Descartes around 1639 but first published by Euler in "Elementa Doctrinae Solidorum," Comment. Acad. Sci. U. Petrop. 4, (1758), 109-140, (Read in November 1750).
In his "Recherches sur les Polyèdres" (1813) Oeuvres II, 1, p. 16 Cauchy refers to "la théorème d’Euler."
The recognition of Descartes’ priority is fairly recent. OED2 finds Euler-Descartes relation 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"
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.
See the entry TOPOLOGY.
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-MACLAURIN SUMMATION FORMULA. The formula was published by Euler in 1738 in Comm. Acad. Sci. Imp. Petrop. 6, p. 68 and by Colin Maclaurin in 1742 in Treatise of Fluxions p. 672. (According to E. T. Whittaker and G. Robinson The Calculus of Observations (1924, p. 135.)) Both men had discovered the result somewhat earlier; see MacTutor’s Leonhard Euler for an account.
EULER-MASCHERONI CONSTANT. See the entry on the Euler-Mascheroni constant on the Earliest Uses of Symbols for Constants for references to the writings of Euler and Mascheroni.
Euler’s constant appears in 1868 in the title "On the Calculation of the Numerical Value of Euler’s Constant, Which Professor Price, of Oxford, Calls E," Proc. of R. Soc Lond. XV. 429-432. (JSTOR) Eulerian constant appears in the Century Dictionary (1889-1897).
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."
In his famous address in 1900, David Hilbert (1862-1943) used the term Euler-Mascheroni constant (and the symbol C).
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 English in 1809 in a translation of Legendre’s 1792 paper “Mémoire Sur Les Transcendantes Elliptiques.” [James A. Landau]
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].
Evolute and anti-evolute are found in 1717 in English in The Lives of the French, Italian and German Philosophers. [Google print search by James A. Landau]
EXACT DIFFERENTIAL EQUATION. Exact differential is found in 1824 in First Principles of the Differential and Integral Calculus, or the Doctrine of Fluxions, Intended as an Introduction to the Physico-Mathematical Sciences; Taken Chiefly from the Mathematics of Bézout, And translated from the French For the Use of the Students of the University at Cambridge, New England: “But when the numerator is the differential of the denominator, multiplied or divided by a constant number, the proposed differential must be decomposed into two factors, of which one shall be a fraction having for its numerator the exact differential of the denominator, and the other a constant number.”
Exact differential equation is found in 1858 in A Treatise on Differential Equations and on the Calculus of Finite Differences, 2nd ed., by J. Hymers.
[Google print searches by James A. Landau]
EXCENTER, EXCIRCLE. See the entry CIRCUMCENTER, CIRCUMCIRCLE, EXCENTER, EXCIRCLE, INCENTER, INCIRCLE, MIDCIRCLE.
Interest in EXCHANGEABLE random variables dates from the 1920s with Bruno de Finetti (1906 - 1985) the most influential contributor. However "exchangeable" emerged only slowly as the standard term. De Finetti used "équivalent" in his most widely read work, "La prévision: ses lois logiques, ses sources subjectives," Annales de l'Institute Henri Poincaré, 7, (1937) 1-68. L. J. Savage The Foundations of Statistics (1954) used the term "symmetric." Polyá suggested the term "échangeable" and it appeared in a 1943 book by Fréchét. De Finetti took up the term and "exchangeable" was adopted in the 1964 English translation of his 1937 work. David (2001) cites M. Loève’s Probability Theory (1955) for the first occurrence of "exchangeable" in the English literature.
[This entry was contributed by John Aldrich, based on J. von Plato Creating Modern Probability (1994).]
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).
EXHAUSTION, METHOD OF. See METHOD OF EXHAUSTION.
The expression EXHAUSTIVE ESTIMATION was used by R. A. Fisher in his 1936 address Uncertain Inference Proceedings of the American Academy of Arts and Science, 71: 245-258. Fisher borrowed the phrase from the book Méthodes d'Estimation by G. Darmois where exhaustif was used to translate Fisher’s term sufficient. Fisher wrote to Darmois, “I notice you use the term ‘exhaustive estimation’ for what I speak of as ‘sufficient statistics’. I should like, if you approve, to use the term ‘exhaustive estimation’ in a somewhat wider sense, including not only the cases in which sufficient statistics, but also those cases … in which the whole of the ancillary information may be utilised by taking account of characteristics of the sample, which, by themselves, yield no information respecting the parameter sought.” Correspondence p. 65.
See the entries SUFFICIENCY and ANCILLARY.
EXPECTATION. The tract by Christiaan Huygens, in Latin De Ratiociniis in Ludo Aleae (1657), is about calculating expectations and the concept of expectation has been attributed to him, although A. W. F. Edwards (Pascal’s Arithmetical Triangle) argues that Pascal had already developed it.
The term "expectation" was used neither by Pascal nor by Huygens. The latter’s Dutch text has no term that translates as "expectation;" it speaks rather of the "value" of a game. Expectatio appears in the Latin translation by van Schooten and expectation appears in the English translation of the Latin. The translators, however, do not use these terms in the modern way. Huygens’s Proposition I is rendered, "If I expect a or b, and have an equal chance of gaining either of them, my Expectation is worth (a + b)/2." The "worth or "value" of the expectation is the modern concept of expectation. The same phraseology is used in De Moivre’s Doctrine of Chances (1718), although "the expectation is worth" is often shortened to "the expectation is."
The expression "l'espérance mathématique," or mathematical expectation, with its modern meaning is found in a letter by Gabriel Cramer dated 21st May 1728: see letter 8 in Correspondence of Nicholas Bernoulli concerning the St Petersburg game with Montmort, Daniel Bernoulli and Cramer (translation by Richard J. Pulskamp.) Laplace adopted "l'espérance mathématique" along with Cramer’s complementary term, "l'espérance morale": see his Théorie Analytique des Probabilités, livre II, chapitre I, p. 189. The English "mathematical expectation" appears in A. de Morgan’s Essay on Probabilities (1838, p. 97), "The balance is the average required, and is known by the name of the mathematical expectation." (OED).
In the course of the 19th century the qualification "mathematical" became optional, perhaps because "moral expectation" fell out of use. The term "expected value" came into use in the 20th century. The OED gives a quotation from Arne Fisher’s Mathematical Theory of Probabilities (1915) p. 97, "Russian and Scandinavian writers and the followers of the Lexis statistical school of Germany have preferred to make another quantity known as the ‘probable' or ‘expected value', the nucleus of their investigations.." Fisher uses expected value as the general term, reserving expectation for situations where a sum of money is involved.
See also MORAL EXPECTATION and the discussion of EXPECTATION on the Symbols in Probability and Statistics page.
[This entry was contributed by John Aldrich, based on Edwards (op. cit.) and Hald (1986)]
The term EXPERIMENT has been used in expositions of probability theory since W. Feller adopted it in his Introduction to Probability Theory and its Applications, volume one (1950): "The mathematical theory of probability gains practical value and an intuitive meaning in connection with real or conceptual experiments or phenomena such as [a long list of examples follows]." (p. 8)
See SAMPLE SPACE.
EXPERIMENTS, DESIGN OF in Statistics. See DESIGN OF EXPERIMENTS.
EXPLEMENT is found in 1827 in Mathematical and astronomical tables by William Galbraith: "...the explement, or difference from four right angles" [Google print search].
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).
EXPLORATORY DATA ANALYSIS is the title of J. W. Tukey’s 1971 book Exploratory Data Analysis volume 1. Exploratory data analysis is "about looking at data to see what it seems to say," while confirmatory data analysis is about "assessing the precision with which our inference from sample to population is made."
David (2001). See DATA ANALYSIS and DESCRIPTIVE STATISTICS.
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).
Exponent is found in English in 1745 in Sir Isaac Newton’s Two Treatises of the Quadrature of Curves, and Analysis by Equations of an infinite Number of Terms, explained by John Stewart: “Wherefore universally the Rule for finding the Fluxion of any Power of a flowing Quantity, be it’s Exponent positive or negative, integral or fractional, and whether the Quantity affected by the Exponent be simple or compound, is this, multiply the flowing Quantity by the Exponent of the Power, diminish the Exponent by a Unity, and multiply further by the Fluxion of the Quantity or Root.” [Google print search by James A. Landau]
In the 1771 edition of the Encyclopaedia Britannica, exponent is spelled differently (in the Logarithm article): "Dr. Halley, in the philosophical transactions, ... says, they are the exponements of the ratios of unity to numbers" [James A. Landau].
EXPONENTIAL is found in English in 1704 in John Harris, Lexicon Technicum: "Exponential curves are such as partake both of the nature of Algebraick and Transcendent ones" (OED2).
In Statistics the adjective EXPONENTIAL has been attached to several objects--including at times the normal distribution. Today the most common constructions are:
Exponential distribution. David (2001) cites Karl Pearson’s reference to the "negative exponential curve" on p. 345 of his "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material," Philosophical Transactions of the Royal Society of London. A, 186. (1895), 343-414.
Double exponential distribution. David (2001) cites R. A. Fisher A Mathematical Examination of the Methods of Determining the Accuracy of an Observation by the Mean Error, and by the Mean Square Error. (1920, p. 770) for the double exponential frequency curve. Laplace discussed this function in 1774 and it has been known as "the first law of Laplace" (with the normal distribution as the second.) Other names include the bilateral exponential and two-tailed exponential.
Exponential family (of distributions). In 1934/6 several authors including R. A. Fisher, G. Darmois, E. J. Pitman and B. O. Koopman discussed the type of distribution for which a sufficient statistic exists: in Koopman’s words, such distributions are of a "very special exponential type." The papers are Fisher Two new properties of mathematical likelihood, Proc. Roy. Soc., A, 144 (1934), 285-307, Darmois ’sur les lois de probabilité à estimation exhaustive', C.R. Acad. Sci. 200, (1935), 1265-1266, Pitman "Sufficient statistics and intrinsic accuracy," Proceedings of the Cambridge Philosophical Society, 32, (1936), 567-579 and Koopman "On Distributions admitting a sufficient statistic" Transactions of the American Mathematical Society, 39, (1936), 399-409. Since around 1950 it has been common to refer to the exponential family of distributions. David (2001) cites M. A. Girshick & L. J. Savage "Bayes and minimax estimates for quadratic loss functions." Proceedings of the 2nd Berkely Symposium, (1951) 43-73. Also used are names on the pattern "Fisher-Pitman-Koopman-Darmois family," though usually without all of the names.
[This entry was contributed by John Aldrich]
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, ax, 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 1658 in John Newton Trigonometria Britanica: Or The Doctrine of Triangles, in Two Books: “Therefore any two angles of a triangle are lesse then two right. And any side being continued, the exterior angle is equal to the two interior opposite ones.” [Google print search by James A. Landau]
EXTRANEOUS ROOT is found in 1843 in Theory and Solution of Algebraical Equations of the Higher Orders, 2nd ed., by John Radford Young: “In the researches of Fourier, so fully discussed in Chapters VIII and IX, we have seen that the object first to be accomplished, in the analysis of a doubtful interval, is to reduce the last three of the indices to the values 0, 1, 2. Till we get three such consecutive indices, either for the three final functions, or for some preceding three, Fourier’s peculiar method of proceeding does not come into operation. In bringing about this [page 301] arrangement of indices, the proposed interval must be narrowed, with a view to the detection of the situations of those roots of the derived equations which require to be passed over, or separated, before the desired arrangement can be present. In the search after these extraneous roots, in the proposed interval, we are not supplied by Fourier, with any guide. ...” [Google print search by James A. Landau]
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). The term apparently is in his Lehrbuch der Variationsrechnung (1900).
Extremal is found in English in 1901 in “Sufficient Conditions in the Calculus of Variations” by W. F. Osgood in Annals of Mathematics: “Any function y of x (or curve representing such a function) which, at all interior points of an interval (x’, x”) throughout which it is considered satisfies Lagrange’s equation (2) is called an extremal.” [Google print search by James A. Landau]
EXTREME VALUE appears in E. J. Gumbel, "Les valeurs extrêmes des distributions statistiques," Ann. Inst. H. Poincaré, 5 (1935), 115-138.
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].
See also WEIBULL DISTRIBUTION.
EXTREMUM was first used as a mathematical term (in German) by Paul Du Bois-Reymond (1831-1889) in German in 1879 in "Fortsetzung der Erläuterungen zu den Anfangsgründen der Variationsrechnung," 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 p. 10: "The word 'extremum' will be used for maximum and minimum alike, when it is not necessary to distinguish between them" (OED2).