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

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].

**ELLIPTIC CURVE** is found in *Webster's New International Dictionary*
(1909).

The term **ELLIPTIC FUNCTION** was used by Adrien Marie Legendre
(1752-1833) in 1825 in volume 1 of *Traité des Fonctions Elliptiques*
and may appear in 1811 in volume 1 of his *Exercises du Calcul Intégral.*

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 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(x_{1} sqrt P[x]) dx where P(x) is a polynomial
of the third or fourth degree."

**EMPTY SET** is found in W. Peremans, "Free algebras with an empty
set of generators," *Nederl. Akad. Wet., Proc.,* Ser. A 59 (1956).

An older term is *null set,* q. v.

The word **EQUATION** 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.

**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).

**ESCRIBED CIRCLE** is found in 1871 in *Geometry* by William
Chauvenet (1820-1870) (OED1).

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 usingIn 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.ethnomathematics.But it was not until ICME 5 that the term was "officially" recognized. My bookSocio-cultural Bases for Mathematics Educationbrings the first study about Ethnomathematics.

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

**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).

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

In his *Adnotationes ad calculum integrale Euleri* (1790), Mascheroni
calculated the consant to 32 decimal places; the figure was corrected from
the 20th decimal place by Johann von Soldner in 1809.

**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).

See also *mathematical expectation.*

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

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

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

**EXPONENTIAL FUNCTION** appears in an 1843 paper by Sir William
Rowan Hamilton [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).

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