The term

The term was also used by Aristotle.

For its use in English, *harmonic mean* is dated 1856 in MWCD10.

**HARMONIC NUMBER.** A treatise on trigonometry by Levi ben Gerson
(1288-1344) was translated into Latin under the title *De numeris harmonicis.*

**HARMONIC PROGRESSION.** Sir Isaac Newton used the phrase "harmonical
progression" in a letter of 1671 (New Style) [James A. Landau].

**HARMONIC PROPORTION** appears in English in 1706 in *Synopsis
palmariorum matheseos* by William Jones (OED2).

The term **HARMONIC RANGE** developed from the Greek "harmonic mean."
Collinear points A, B, C, D form a harmonic range when the length AC is
the harmonic mean of AB and AD, i.e. 2/AC = 1/AB + 1/AD. It's then easy
to deduce the more modern condition that the cross ratio (AC,BD) = -1.

In "A Treatise of Algebra," 1748, Appendix, p. 20, Maclaurin says "atque
hae quatuor rectae, Cl. D. *De la Hire,* Harmonicales dicuntur." In
"Nouvelle methode en geometrie pour les sections des superficies coniques
et cylindriques ...," 1673, by Philippe de la Hire, p.1, his first words
are: "Definition. J'appelle une ligne droitte AD couppée en 3 parties harmoniquement
quand le rectangle contenu sous la toutte AD & la partie du milieu
BC est égal au rectangle contenu sous les deux parties extremes AB, CD
...." This statement AD.BC = AB.CD is another variant of the conditions
given above, disregarding signs. [Ken Pledger]

**HARMONIC SERIES** is defined in an 1866 dictionary.

The term **HARMONIC TRIANGLE** was coined by Leibniz (Julio González
Cabillón).

**HAUSDORFF MEASURE** occurs in E. Best, "A theorem on Hausdorff
measure," *Quart. J. Math.,* Oxford Ser. 11, 243-248 (1940).

**HAUSDORFF SPACE** is found in Lawrence M. Graves, "On the completing
of a Hausdorff space," *Ann. of Math.,* II. Ser. 38, 61-64 (1937).

The term **HAVERSINE** was introduced by James Inman (1776-1859)
in 1835 in the third edition of *Navigation and Nautical Astronomy for
the use of British Seamen.*

**HEINE-BOREL THEOREM.** Heine's name was connected to this theorem
by A. Schoenfliess, although he later omitted Heine's name. The validity
of the name has been challenged in that the covering property had not been
formulated and proved before Borel. (DSB, article: "Heine").

The term **HELIX** is due to Archimedes, "to a spiral already studied
by his friend Conon" (Smith vol. 2, page 329). It is now known as the spiral
of Archimedes.

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

**HERMITIAN MATRIX** is found in G. Richard Trott, "On the canonical
form of a non-singular pencil of Hermitian matrices," *Amer. J. Math.*
56, 359-371 (1934).

The term **HESSIAN** was coined by James Joseph Sylvester (1814-1897)
(Cajori 1919, page 345).

**HEXADECIMAL** is found in Carl-Erik Froeberg, *Hexadecimal conversion
tables,* Lund: CWK Gleerup 20 S. (1952).

In 1955, R. K. Richards wrote in *Arithmetic Operations in Digital
Computers*: "Octonary, duodecimal, and sexadecimal are the accepted
terms applying to radix eight, twelve, and sixteen, respectively" [James
A. Landau].

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

**HEXAHEDRON.** The word "hexahedron" was used by Heron to refer
to a cube; he used "cube" for any right parallelepiped (Smith vol. 2, page
292).

The term **HIGHER-DIMENSIONAL ALGEBRA** was coined by Ronald Brown,
according to an Internet web page.

**HINDU-ARABIC NUMERAL.** In his *Liber abaci* (1202), Fibonacci
used the term *Indian figures*: "The nine Indian figures are: 9 8
7 6 5 4 3 2 1. With these nine figures and with the sign 0 ... any number
may be written, as is demonstrated below."

*Arabic numeral* appears in an 1847 *Webster* dictionary.

*Hindu-Arabic numeral* appears in the title *The Hindu-Arabic
Numerals* by David Eugene Smith and Louis Charles Karpinski, Boston
and London: Ginn and Company Publishers, 1911 [Julio González Cabillón].

**HISTOGRAM** is found in 1891 in E. S. Pearson *Karl Pearson*
(1938) (OED2).

The terms **HOLOMORPHIC FUNCTION** and **MEROMORPHIC FUNCTION**
were introduced by Charles A. A. Briot (1817-1882) and Jean-Claude Bouquet
(1819-1885).

The earlier terms *monotypique, monodrome, monogen,* and *synetique*
were introduced by Cauchy (Kline, page 642).

**HOMOMORPHIC** is found in English in 1935 in the *Proceedings
of the National Academy of Science* (OED2).

**HOMOMORPHISM** is found in English in 1935 in the *Duke Mathematical
Journal* (OED2).

**HYPERBOLA** was probably coined by Apollonius, who, according to
Pappus, had terms for all three conic sections. The word appears in English
in 1668 in the *Philosophical Transactions of the Royal Society* (OED2).

The term **HYPERBOLIC FUNCTION** was introduced by Lambert in 1768
[Ken Pledger].

The terms **HYPERBOLIC GEOMETRY, ELLIPTIC GEOMETRY,** and **PARABOLIC
GEOMETRY** were introduced by Felix Klein (1849-1925) in 1871 in "Über
die sogenannte Nicht-Euklidische Geometrie" (On so-called non-Euclidean
geometry), reprinted in his Gesammelte mathematische Abhandlungen I (1921)
p. 246 (Ken Pledger and Smart, p. 301).

**HYPERBOLIC LOGARITHM** is an early term for *natural logarithm.*
It was used by Euler and others.

**HYPERBOLIC SINE** and **HYPERBOLIC COSINE.** Vincenzo Riccati
(1707-1775) introduced hyperbolic functions in volume I of his *Opuscula
ad Res Physicas et Mathematicas pertinentia* of 1757. Presumably he
used these terms, since he used the notation Sh *x* and Ch *x.*

**HYPERBOLOID.** Boyer (page 419) implies this term was introduced
by Christopher Wren (1632-1723).

**HYPERCOMPLEX** is dated ca. 1889 in MWCD10.

**HYPERDETERMINANT** was Cayley's term for independent invariants
(DSB). He coined the term around 1845.

According to Eric Weisstein's Internet web page, "Cayley (1845) originally coined the term, but subsequently used it to refer to an Algebraic Invariant of a multilinear form."

The term **HYPERELLIPTICAL FUNCTION** (*ultra-elliptiques*)
was coined by Legendre, according to an article by Jacobi in *Crelle's
Journal* in which Jacobi went on to propose instead the term Abelian
transcendental function (*Abelsche Transcendenten*) (DSB).

The term **HYPERGEOMETRIC** (to describe a particular differential
equation) is due to Johann Friedrich Pfaff (1765-1825) (Kline, page 489).

**HYPERGEOMETRIC DISTRIBUTION** occurs in H. T. Gonin, "The use of
factorial moments in the treatment of the hypergeometric distribution and
in tests for regression," *Philos. Mag.,* VII. Ser. 21, 215-226 (1936).

The term **HYPERGEOMETRIC SERIES** was introduced by John Wallis
(1616-1703), according to Cajori (1919, page 185).

However, the term *hypergeometric series* is due to Pfaff, according
to Smith (vol. 2, page 507).

**HYPERPLANE** appears in a paper by James Joseph Sylvester published
in 1863. He also used the words *hyperplanar, hyperpyramid,* and *hypergeometry*
[James A. Landau].

**HYPERSET.** This term is due to Jon Barwise and appeared for the
first time in the expository article *Hypersets* (Mathematical Intelligencer
13 (1991), 31-41) by him and Larry Moss. It is a new name for "non-well-founded
set", a concept which was banished from set theory by Dimitry Mirimanoff
(1861-1945) in two papers of 1917, and later by von Neumann (1925) and
Zermelo (1930). Such "exceptional sets" begun to attract attention in the
1980s mainly through the work of Peter Aczel, which prompted Barwise and
John Etchemendy to apply them to the mathematical modeling of circular
phenomena. Barwise used the term "hyperset" having in mind an analogy with
the hyperreals of non-standard analysis and intending to avoid the "negative
connotations" of the previous name. [Carlos César de Araújo]

**HYPOTENUSE** was used by Pythagoras (c. 540 BC). It is found in
English in 1571 in *A geometrical practise named Pantometria* by Thomas
Digges (1546?-1595): "Ye squares of the two contayning sides ioyned together,
are equall to the square of ye Hypothenusa" (OED2).