早期數學字彙的歷史 (H)

Last revision: Aug. 2, 1999


The term HARMONIC MEAN is due to Archytas of Tarentum, according to the University of St. Andrews website, which also states that it had been called sub-contrary in earlier times.

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 coupp嶪 en 3 parties harmoniquement quand le rectangle contenu sous la toutte AD & la partie du milieu BC est 嶲al 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 Gonz嫮ez Cabill鏮).

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 Gonz嫮ez Cabill鏮].

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 "鈁er 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 C廥ar de Arajo]

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).


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