Hamiltonian circuit is found in W. T. Tutte, "On Hamiltonian circuits," J. London Math. Soc. 21, 98-101 (1946).
Hamiltonian path is found in V. Mierlea, "An algorithm for finding the minimal length Hamiltonian path in a graph," Econom. Comput. econom. Cybernetics Studies Res. 1973, No. 2, 77-89 (1973).
HARMONIC ANALYSIS. According to Grattan-Guinness (679), the phrase is due to W. Thomson (later Lord Kelvin). In an obituary of Archibald Smith (Proc. Royal Soc. 22. (1873 - 1874) p. vi) Thomson wrote "One of Smith's earliest contributions to the compass problem was the application of Fourier's grand and fertile theory of the expansion of a periodic function in series of sines and cosines of the argument and its multiples, now commonly called the harmonic analysis of a periodic function." Thomson invented the harmonic analyser; in 1879 the Royal Society allocated him £50 for "completing a Tidal Harmonic Analyser" (Proc. Royal Soc., 29, 442.)
The phrase "harmonic analysis" was prominent in N. Wiener's writings of the 1920s, see e.g. "The Harmonic Analysis of Irregular Motion (Second Paper)," J. Math. and Phys. 5 (1926) 158-189. These writings culminated in the "generalized harmonic analysis" of 1930 (Acta Mathematica, 55, 117-258).
In statistics the term is found in R. A. Fisher, "Tests of significance in harmonic analysis," Proc. Roy. Soc. A, 125, page 54 (1929) [John Aldrich].
HARMONIC MEAN, HARMONIC PROPORTION. A surviving fragment of the work of Archytas of Tarentum (ca. 350 BC) states, "There are three means in music: one is the arithmetic, the second is the geometric, and the third is the subcontrary, which they call harmonic."
The term was also used by Aristotle.
According to one theory, the term subcontrary may refer to the fact that a tone based on this mean reverses the order of the two fundamental musical intervals in a scale.
The term harmonic may have been used because it produced a division of the octave in which the middle tone stood in the most harmonious relationship to the tonic and the octave.
In a message posted to a history of mathematics mailing list in 2003, Michael N. Fried writes:
In the Introduction to Arithmetic (Arithmetike eisagoge), Nicomachus gives his account of the name harmonic proportion (which defines the harmonic mean) in contradistinction to the arithmetic and geometric proportions (which define the arithmetic and geometric means, respectively). He writes as follows:The harmonic proportion was so called because the arithmetic proportion was distinguished by quantity, showing an equality in this respect with the intervals from one term to another [for if m is the mean between a and b, then a-m is precisely the same as m-b], and the geometric by quality, giving similar qualitative relations between one term and another [for a-m:m-b is precisely the same ration as a:m and m:b], but this form, with reference to relativity, appears now in one form, now in another, nether in its terms exclusively nor in its differences exclusively, but partly in the terms and partly in the differences [here, a-m:m-b is the same ratio only as a:b]; for as the greatest term is to the smallest, so also is the difference between the greatest and the next greatest, or middle, term to the difference between the least term and the middle term, and vice versa" (II.25, trans. by Martin Luther D'Ooge)I am not quoting this passage to contradict the musical explanations for the word "harmonic" that have been already given over the past few days. Indeed, immediately following this passage Nicomachus reminds us of the connection between the relative, the harmonic, and the musical. But it is important to remember that both music and harmony had, for Nicomachus, among others in antiquity, a wider intention than merely aural phenomena--and here is where one must tread with care.Harmozein just means to join or to fit together--for example, in Elem. IV.1, when Euclid requires a segment of a given length to be fit in a given circle, he uses the related word, enarmozein. Consonance is a fitting together of sounds and is just an example of a more general fitting together of different parts of the world. It is in this spirit that earlier in the Introduction Nicomachus gives own definition, if you like, of the word harmonic:
Everything that is harmoniously constituted is knit together out of opposites and, of course, out of real things; for neither can non-existent things be set in harmony, nor can things that exist, but are like one another, nor yet things that are different, but have no relation one to another. It remains, accordingly, that those things out of which a harmony is made are both real, different, and things with some relation to one another" (I.6, trans. again by M. L. D'Ooge)Incidentally, Archytas, for one, refers not only to the harmonic mean as musical, but equally so to the geometric and arithmetic means!
Harmonic proportion appears in English in 1660 in R. Coke, Justice Vind., Arts & Sc.: "Harmonical proportion increases neither equally nor proportionally: nor do the extremes added or multiplied produce the like number with the mean" (OED2).
According to the Catholic Encyclopedia, the word harmonic first appears in a work on conics by Philippe de la Hire (1640-1718) published in 1685.
Harmonical mean is found in English in the 1828 Webster dictionary:
Harmonical mean, in arithmetic and algebra, a term used to express certain relations of numbers and quantities, which are supposed to bear an analogy to musical consonances.Harmonic mean is found in 1851 in Problems in illustration of the principles of plane coordinate geometry by William Walton [University of Michigan Digital Library].
Harmonic mean is also found in 1851 in The principles of the solution of the Senate-house 'riders,' exemplified by the solution of those proposed in the earlier parts of the examinations of the years 1848-1851 by Francis James Jameson: "Prove that the discount on a sum of money is half the harmonic mean between the principal and the interest" [University of Michigan Digital Library].
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].
In a letter dated Feb. 15, 1671, James Gregory wrote to Collins, "As to yours, dated 24 Dec., I can hardly beleev, till I see it, that there is any general, compendious & geometrical method for adding an harmonical progression...."
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 appears in 1727-51 in Chambers Cyclopedia: "Harmonical series is a series of many numbers in continual harmonical proportion" (OED2).
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 Arthur Schoenflies, 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").
In June 1907 in the Bulletin des Sciences mathématiques, Lebesgue denied any paternity of the theorem and wrote that in his opinion the name of the theorem should bear only the name of Borel [Udai Venedem].
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. In 1551 in Pathway to Knowledge Robert Recorde used septangle.
Heptagon appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.
HEPTAKAIDECAGON (17-sided polygon) appears in Webster's Third New International Dictionary (1961).
HERMITIAN FORM is found in "On Quadratic, Hermitian and Bilinear Forms" by Leonard Eugene Dickson, Transactions of the American Mathematical Society, 7 (Apr., 1906).
HERMITIAN MATRIX appears in 1866 in Arthur Cayley, "A Supplementary Memoir on the Theory of Matrices," Philosophical Transactions of the Royal Society of London: "I consider from a different point of view the theory of a matrix ... or, as we may call it, a Hermitian matrix" [University of Michigan Historical Math Collection].
The term HESSIAN was coined by James Joseph Sylvester (1814-1897), named for Otto Hesse, who had used the term functional determinants.
Hessian appears in his "Sketch of a Memoir on Elimination, Transformation, and Canonical Forms," Math. Papers J. S. S., 1:184-197.
Hessian appears in 1851 in Cambr. & Dublin Math. Jrnl. 6: "The Hessian, or as it ought to be termed, the first Boolian Determinant" (OED2).
HETERO- and HOMOSCEDASTICITY. The terms heteroscedasticity and homoscedasticity were introduced in 1905 by Karl Pearson in "On the general theory of skew correlation and non-linear regression," Drapers' Company Res. Mem. (Biometric Ser.) II. Pearson wrote, "If ... all arrays are equally scattered about their means, I shall speak of the system as a homoscedastic system, otherwise it is a heteroscedastic system." The words derive from the Greek skedastos (capable of being scattered).
Many authors prefer the spelling heteroskedasticity. J. Huston McCulloch (Econometrica 1985) discusses the linguistic aspects and decides for the k-spelling. Pearson recalled that when he set up Biometrika in 1901 Edgeworth had insisted the name be spelled with a k. By 1932 when Econometrica was founded standards had fallen or tastes had changed. [This entry was contributed by John Aldrich, referring to OED2 and David, 1995.]
HEXADECIMAL. Sexadecimal appears in 1891 in the Century dictionary.
Hexadecimal is found in Carl-Erik Froeberg, Hexadecimal conversion tables, Lund: CWK Gleerup 20 S. (1952).
In 1955, R. K. Richards used sexadecimal 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. In 1551 in Pathway to Knowledge Robert Recorde used the obsolete word siseangle: "Def., Likewyse shall you iudge of siseangles, which haue sixe corners" (OED2).
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.
HILBERT SPACE is found in E. W. Chittenden, "On the relation between the Hilbert space and the calcul fonctionnel of Frechet," Palermo Rend. (1921).
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 characters appears in 1727-51 in Chambers Cyclopedia: "The Arabic characters stand contradistinguished to the Roman" [OED2].
Arabic numerals appears in 1799 in T. Green, Lover of Lit. (1810): "Writing, he deduces, from pictural representations, through hieroglyphics ... to arbitrary marks ... like the Chinese characters and Arabic numerals.
Hindu numerals is found in 1872 in Chambers's encyclopaedia: "After the introduction of the decimal system and the Arabic or Hindu numerals about the 11th c., Arithmetic began to assume a new form..." [University of Michigan Digital Library].
Indo-Arabic system appears in 1884 in the Encyclopaedia Britannica: "In Europe, before the introduction of the algorithm or full Indo-Arabic system with the zero" (OED2).
Indo-Arabic numeral appears in 1902 in the second edition of The Number-System of Algebra by Henry B. Fine: "At all events, it is certain that the Indo-Arabic numerals, 1, 2, ..., 9 (not 0), appeared in Christian Europe more than a century before the complete positional system and algorithm." The term may occur in the 1890 edition also.
Hindu notation appears in 1906 in A History of Mathematics by Florian Cajori: "Generally we speak of our notation as the 'Arabic' notation, but it should be called the 'Hindoo' notation, for the Arabs borrowed it from the Hindoos. ... These Singhalesian signs, like the old Hindoo numerals, are supposed originally to have been the initial letters of the corresponding numerical adjectives." Presumably the terms appear in the earlier 1893 edition of Cajori.
Hindu-Arabic numeral appears in 1911 in the title The Hindu-Arabic Numerals by David Eugene Smith and Louis Charles Karpinski [Julio González Cabillón].
HISTOGRAM. The term histogram was coined by Karl Pearson.
In Philos. Trans. R. Soc. A. CLXXXVI, (1895) 399 Pearson explained that term was "introduced by the writer in his lectures on statistics as a term for a common form of graphical representation, i.e., by columns marking as areas the frequency corresponding to the range of their base."
S. M. Stigler writes in his History of Statistics that Pearson used the term in his 1892 lectures on the geometry of statistics.
The earliest citation in the OED2 is in 1891 in E. S. Pearson Karl Pearson (1938).
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).
Halphen proposed that the terms be replaced by "integral" and "fractional."
HOMOGENEOUS EQUATIONS is found in 1815 in the second edition of Hutton's mathematics dictionary: "Homogeneous Equations ... in which the sum of the dimensions of x and y... rise to the same degree in all the terms" (OED2).
The term HOMOGRAPHIC is due to Michel Chasles (1793-1880) (Smith, 1906).
HOMOLOGOUS is found in English in 1660 in Barrow's translation of Euclid: "B and D are homologous or magnitudes of a like ratio" (OED2).
The term is found in a modern sense in 1879 in Conic Sections by George Salmon (1819-1904): "Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology; prove that the lines joining corresponding vertices meet in a point" (OED2).
HOMOLOGY is found in 1879 in Conic Sections by George Salmon: "Two triangles are said to be homologous, when the intersections of the corresponding sides lie on the same right line called the axis of homology; prove that the lines joining corresponding vertices meet in a point" (OED2).
Homology is found in 1885 in Charles Leudesdorf's translation of Cremona's Elements of Projective Geometry "Two corresponding straight lines therefore always intersect on a fixed straight line, which we may call s; thus the given figures are in homology, O being the centre, and s the axis, of homology" (OED2).
Homology is found in a more modern usage, originally in algebraic topology but now more widespread (as in homological algebra) in 1895 in H. Poincare, Analysis situs [Joseph Rotman].
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).
HORNER'S METHOD appears in 1842 in the Penny Cyclopedia: "The use of Horner's method is very much more easy than that of Newton" (OED2).
The method of Horner appears in the second edition of Theory and Solution of Algebraical Equations (1843) by J. R. Young.
HYPERBOLA was probably coined by Apollonius, who, according to Pappus, had terms for all three conic sections.
Hyperbola was used in English in 1668 by Barrow in correspondence: "The rules I sent you concerning the hyperbola, I cannot well exemplify."
Hyperbola also 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. Because of the relation between natural logarithms and the areas of hyperbolic sectors, natural logarithms came to be called hyperbolic logarithms. The connection between natural logarithms and sectors was discovered by Gregory St. Vincent (1584-1667) in 1647, according to Daniel A. Murray in Differential and Integral Calculus (1908).
Abraham DeMoivre (1667-1754) used Hyperbolic Logarithm in English in his own English translation of a paper presented to some friends on Nov. 12, 1733. His translation appears in the second edition (1738) of The Doctrine of Chances.
Hyperbolic logarithm appears in 1743 in Emerson, Fluxions: "The Fluxion of any Quantity divided by that Quantity is the Fluxion of the Hyperbolic Logarithm of that Quantity" (OED2).
Euler called these logarithms "natural or hyperbolic" in 1748 in his Introductio, according to Dunham (page 26), who provides a reference to Vol. I, page 97, of the Introductio.
HYPERBOLIC PARABOLOID appears in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young [James A. Landau].
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.
HYPERCOMPLEX is dated ca. 1889 in MWCD10.
HYPERCUBE is found in Scientific American of July 1909: "Of these [regular hyper-solids], C8 (or the hyper-cube) is the simplest, because, though with more bounding solids than C5, it is right-angled throughout" (OED2).
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."
Hyperdeterminant was used by Cayley in 1845 in Camb. Math. Jrnl. IV. 195: "The function u whose properties we proceed to investigate may be conveniently named a 'Hyperdeterminant'" (OED2).
Hyperdeterminant was used by Cayley about 1846 in Camb. & Dublin Math. Jrnl. I. 104: "The question may be proposed 'To find all the derivatives of any number of functions, which have the property of preserving their form unaltered after any linear transformations of the variables'... I give the name of Hyperdeterminant Derivative, or simply of Hyperdeterminant, to those derivatives which have the property just enunciated" (OED2).
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).
The term HYPERGEOMETRIC CURVE is found in the title "De curva hypergeometrica hac aequatione expressa y=1*2*3*...*x" by Leonhard Euler. The paper was presented in 1768 and published in 1769 in Novi Commentarii academiae scientiarum Petropolitanae.
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) and Smith (1906).
The 1816 translation of Lacroix's Differential and Integral Calculus has: "These series, in which the number of factors increases from term to term, have been designated by Euler ... hypergeometrical series" (OED2).
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).
In English, the word has also been spelled hypothenusa, hypotenusa, and hypothenuse.
HYPOTHESIS was used in English in a mathematical context in 1660 by Barrow in his translation of Euclid i. xxvii. (1714) 23: "Which being supposed, the outward angle AEF will be greater than the inward angle DFE, to which it was equal by Hypothesis" (OED2).
HYPOTHESIS TESTING. Test of hypothesis is found in 1928 in J. Neyman and E. S. Pearson, "On the use and Interpretation of Certain Test Criteria for Purposes of Statistical Inference. Part I," Biometrika, 20 A, 175-240 (David, 1995).