Earliest Known Uses of Some of the Words of Mathematics (E)

Last revision: June 01, 2003

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 lambda and vector x together in the value/vector-equation Ax = lambda x : any value lambda 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 - lambda I)x = 0 requires the determinant of (A - lambda 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 lambda for which the matrix counterpart is (I - lambda A)x = y. Hilbert called the values of lambda 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 kappa for a "characteristiche Zahl" and lambda = (1/ kappa ) 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 not equal 0 mit Rf = lambda 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 A Poem sacred to the Memory of Sir Isaac Newton by James Thomson: "He, first of Men, with awful Wing pursu'd The Comet thro' the long Elliptic Curve" (OED2).

In 1908, volume three of Lehrbuch der Algebra by Heinrich Weber has an inital 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."

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 infinity symbol , 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 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.]

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).