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

Last revision: Mar. 24, 1999

EIGENVALUE, EIGENFUNCTION, EIGENVECTOR. According to a post in sci.math.num-analysis by G. W. Stewart, eigenvalue derives from the German Eigenwert, which was introduced by David Hilbert (1862-1943) in 1904 to denote for integral equations the reciprocal of the corresponding quantities for matrices. At some point Hilbert's Eigenwerte inverted themselves and became attached to matrices. The reference is: Hilbert, 1912, Grundz\uge einer allgemeinen Theorie der linearen Integralgleicungen, note: A collection of papers that appeared in G\ott. Nachr. 1904--1910.

The OED shows a use of eigenvalue and eigenfunction from 1927 Nature 23 July 117/1:

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
Eigenvector is dated 1941 in MWCD10.

Paul R. Halmos wrote in Finite Dimensional Vector Spaces (1942, 1958), "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." Proper value is from the French valeur propre. An eigenfunction is also called an autofunction.

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.

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].

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.

The term was also used 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 English in its modern sense in 1876 in the title Elliptic Functions by Cayley.

ELLIPTIC INTEGRAL is found in 1851 in the title Theory of Elliptic Integrals by James Booth [James A. Landau].

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.

EQUILATERAL appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

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

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.

EUCLID'S ALGORITHM appears in the 1907 edition of Introduction to Higher Algebra by Maxime Bôcher [James A. Landau].

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

EULER'S CONSTANT (for 0.577...) appears in 1886 in Integral Calculus by Isaac Todhunter (OED2). He writes, "The quantity C is called Euler's constant." Eulerian constant, referring to the same number, 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). Webster's New International Dictionary of 1909 refers to the constant as the Eulerian constant.

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).

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].

EXPECTATION. The word expectatio first appears in van Schooten's translation of a tract by Huygens (Burton, page 461).