Earliest Known Uses of Some of the Words of Mathematics (H)

See the entry BORSUK-ULAM THEOREM

HAMILTON and HAMILTONIAN are used to recall the work of William Rowan Hamilton (1805-1865). The terms in use today derive from his reformulation of mechanics and from the "Icosian game" which he invented and marketed. The game is now understood in terms of graph theory; see Mathworld. Hamilton’s long years of work on QUATERNIONS are recalled only in the name CAYLEY-HAMILTON THEOREM.

Hamilton’s two essays on a General Method in Dynamics in the Philosophical Transactions of the Royal Society of London (1834-5) gave rise to Hamilton's principle, the Hamiltonian form of the equations of motion and the Hamiltonian function.  All these terms appear in E. T. Whittaker Treatise on the Analytical Dynamics of Particles and Rigid Bodies (1904). The OED has a quotation from 1858, A. Cayley writing about forms of the equations of motion in Rep. Brit. Assoc. Adv. Sci. 1857 I. 15: "When there is no force function..the forms corresponding to the untransformed forms in T and U are as follows, viz. the Lagrangian form is dq/dt = q´, d(dT/dq´)/dt - dT/dq = Q, and the Hamiltonian form is dq/dt = dT/dp, dp/dt = - dT/dq + Q."

Hamiltonian mechanics involved the calculus of variations and so Hamiltonian terms appear in that field as well. A relatively recent example is the use of the letter H (for Hamiltonian) in The Mathematical Theory of Optimal Processes (1961) by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamrelidze & E. F. Mishchenko.

Hamiltonian Game appears in H. S. M. Coxeter's 1938 revision of Mathematical Recreations and Essays by W. W. Rouse Ball.

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).

HARD and SOFT MATHEMATICS.  "In mathematical slang, formal, abstract theorems are referred to as ‘soft’ and concrete theorems are called ‘hard.’  The words ‘hard’ and ‘soft’ are not to be taken in general as indications of difficulty or lack thereof; ‘hard’ analysis is not necessarily more (or less) difficult than any other kind of mathematics..." T. W. Gamelin & R. E. Greene Introduction to Topology, (1983, p. 74.)

These terms have been around for a long time but notoriously the history of slang is hard to document. In 1964 Jean Dieudonné was complaining of "traditional jokes" about hard and soft mathematics becoming "a little stale." ("Recent Developments in Mathematics," American Mathematical Monthly, 71, p. 247). JSTOR

HARDY CLASSES and SPACES.  According to the Encyclopedia of Mathematics article on Hardy Classes F. Riesz introduced the Hardy class of function in his "Ueber die Randwerte einer analytischen Funktion" Math. Z., 18 (1923) pp. 87-95. The name was chosen in honour of G. H. Hardy.  The higher dimensional analogues called Hardy spaces were introduced by E. Stein and G. Weiss, "On the theory of harmonic functions of several variables" Acta Math. , 103 (1960) pp. 25-62.

HARDY-LITTLEWOOD THEOREM. The 35-year collaboration between G. H. Hardy and J. E. Littlewood produced numerous theorems. Two are described in the article Hardy-Littlewood theorem. The Hardy-Littlewood Tauberian theorem was given in G.H. Hardy and J.E. Littlewood, "Tauberian theorems concerning power series and Dirichlet's series whose coefficients are positive," Proc. London. Math. Soc. (2), 13, (1914) pp. 174-191. The Hardy-Littlewood maximal theorem was given in "A maximal theorem with function-theoretic applications," Acta Math., 54 (1930) pp. 81-116.

See TAUBERIAN THEOREMS.

The HARDY-RAMANUJAN THEOREM was given by G. H. Hardy and S. Ramanujan "The normal number of prime factors of a number n, "Quart. J. Math. , 48, (1917) pp. 76-92. See the Encyclopedia of Mathematics article on the Hardy-Ramanujan theorem.

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, 54-59 (1929) [John Aldrich].

HARMONIC FUNCTION. According to Kline (p. 685) William Thomson (later called Lord Kelvin) introduced the term in 1850. The OED's earliest quotation is for spherical harmonic function as used in Thomson & P. G. Tait's Treatise on Natural Philosophy (1867, I. i. App. B): "The .. method .. commonly referred to by English writers as that of ‘Laplace's Co-efficients’..is here called spherical harmonic analysis.. A spherical harmonic function is defined as a homogeneous function, V, of x, y, z, which satisfies the equation ." In a post to the Historia Matematica mailing list Emili Bifet gives an earlier reference: Thomson’s "Dynamical Problems regarding Elastic Spheroidal Shells and Spheroids of Incompressible Liquid," Philosophical Transactions of the Royal Society, 153 (1863), p. 585.

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 harmonic mean 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 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).

HAT MATRIX in least squares analysis. The term appears in the title of David C. Hoaglin and Roy E. Welsch’s "The Hat Matrix in Regression and ANOVA," American Statistician, 32, (1978), 17-22. They write, "The term "hat matrix" is due to John W. Tukey who introduced us to the technique about ten years ago."

David (1998)

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

HAUSDORFF PARADOX refers to a result in Felix Hausdorff's (1868-1942) "Bemerkung über den Inhalt von Punktmengen," Mathematische Annalen, 75, (1914), 428-433: that one half of a sphere is congruent to one third of the same sphere. According to Gregory Moore Zermelo's Axiom of Choice: Its Origins, Development and Influence (1982, pp. 185-8) Hausdorff was demonstrating the non-existence of a certain kind of measure: if the measure existed the congruence proposition would follow. Émile Borel was the first to call the result a paradox: "The contradiction has its origins in the application of ... Zermelo's Axiom of Choice.... The paradox results from the fact that A is not defined, in the logical and precise sense of the word defined. If one scorns precision and logic one arrives at contradictions." (Moore's translation from Borel's Leçons sur la théorie des functions (2^nd edition 1914))

See AXIOM OF CHOICE, BANACH-TARSKI PARADOX and PARADOX

HAUSDORFF SPACE is found in E. W. Chittenden, "On General Topology and the Relation of the Properties of the Class of All Continuous Functions to the Properties of Space," Transactions of the American Mathematical Society, Vol. 31, No. 2 (Apr., 1929), pp. 290-321: "A topological space of the Hausdorff type will be referred to as a Hausdorff space." [JSTOR search]

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.

HAZARD RATE came into use in statistics in the 1960s as a general term for what is called the force of mortality in demography and the intensity function in extreme value theory. David (2001) finds "hazard rate" in R. E. Barlow; A. W. Marshall & F. Proschan "Properties of Probability Distributions with Monotone Hazard Rate," Annals of Mathematical Statistics, 34, (1963), 375-389. A JSTOR search found "death-hazard rate" in D. J. Davis "An Analysis of Some Failure Data," Journal of the American Statistical Association, 47, (1952), 113-150.

HEINE-BOREL THEOREM. Heine's name was connected to this theorem by Arthur Schoenflies in 1900, 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")

Grattan-Guinness (1997, pp. 671-2) refers to Borel's doctoral thesis of 1894 (his 23rd year), published in Ann. de l'École Normale, 12, (1895), 9-55. Borel "threw off a remark which was later recognized as an important pioneering example of a covering theorem: if every point on a line can be covered by a denumerable number of subintervals, then in fact a finite number could do the job .... [The covering technique of Heine in 1872] is the trivial one of dividing the interval into a finite number of subintervals."

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.

The HELMERT TRANSFORMATION is an orthogonal transformation that the geodesist F. R. Helmert used to obtain the distribution (in effect) of the sample variance for a normal population. (Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Fehlers directer Beobachtungen gleicher Genauigkeit, Astron. Nach., 88, (1876), 192-218) An extract from the paper is translated and annotated in H. A. David & A. W. F. Edwards (eds.) Annotated Readings in the History of Statistics (2001). Helmert was a well-known writer on the theory of errors and his work was described in German textbooks. However, Student (1908) and Fisher (1920) did not know it and re-derived the distribution. In 1931 Karl Pearson drew attention to Helmert in a "Historical Note on the Distribution of the Standard Deviations of Samples of any Size Drawn from an Indefinitely Large Normal Parent Population," Biometrika, 23, 416-418. Pearson's suggestion that the distribution be named after Helmert was not taken up but references to Helmert's transformation followed, e.g. in R. C. Geary's (1936), "The Distribution of "Student's" Ratio for Non-Normal Samples," Supplement to the Journal of the Royal Statistical Society, 3, 178-184.

This entry was contributed by John Aldrich, based on David & Edwards. There is also a discussion in Hald (1998).

The term HEMIHEDRON was coined by Cayley according to Arthur Cayley: Mathematician Laureate of the Victorian Age by Tony Crilly.

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).

HERMITE POLYNOMIAL (also known as Hermite-Chebyshev polynomial) is named after Charles Hermite who gave the construction in his "Sur un nouveau développement en série de functions," Compt. Rend. Acad. Sci. Paris, 58, (1864), 93-100 and 266-273, reprinted in Oeuvres. 2, pp. 293-308. The construction had been described a few years earlier by P. L. Chebyshev, "Sur le développement des fonctions à une seule variable," Bull. ph.-math., Acad. Imp. Sc. St. Pétersbourg 1,  (1859), 193-200, reprinted in Oeuvres, 1, pp. 501-508.  Hald (1998, p. 341) reports that Laplace had used polynomials proportional to Hermite’s in his analysis of a diffusion problem in 1811.

Regarding the use of these polynomials in English language statistics, they appear to have become prominent in the 1920s. A JSTOR search found "Hermite’s polynomial" in Alf. Guldberg "Expansions useful in the theory of frequency distributions," Journal of the Royal Statistical Society, 83, (1920), 127-130. In his Advanced Theory of Statistics, I of 1943 M. G.  Kendall noted Chebyshev’s priority and accordingly used the term "Tchebycheff-Hermite polynomial," However this ordering is rare and even "Hermite-Chebyshev" is much less common than just "Hermite."

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. C. C. MacDuffee, The Theory of Matrices, Springer (1933, p. 62) refers to C. Hermite "Remarques sur un théorème de M. Cauchy," C. R. Acad Sci., 41 (1855), p. 181. The phrase "Hermitian matrix" appears in 1866 in Arthur Cayley, "A Supplementary Memoir on the Theory of Matrices," Philosophical Transactions of the Royal Society of London Papers 4, p. 438 But this was in reference to a different matrix and to a different Hermite publication.

The term HESSIAN was coined by James Joseph Sylvester (1814-1897), named for Ludwig Otto Hesse. Hesse introduced the concept in a paper published in 1844 and reproduced in Werke pp. 89-122.

Hessian appears in Sylvester’s “Sketch of a Memoir on Elimination, Transformation, and Canonical Forms,” Cambr. & Dublin Math. Jrnl., 6, 186-200, reprinted in Math. Papers J. S. S., 1:184-197: "The Hessian, or as it ought to be termed, the first Boolian Determinant" (OED).

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.

The HILBERT terms recall the work of David Hilbert (1862-1943). Hilbert contributed to many branches of mathematics, including the theory of invariants (HILBERT’S FINITE BASIS THEOREM), algebraic number theory and geometry. By 1900 Hilbert’s standing was such that he could pose problems for the 20th century: see HILBERT’S PROBLEMS. In the first decade of the century Hilbert made a deep study of integral equations from which his followers developed HILBERT SPACE theory. In the 1920s Hilbert became preoccupied with the foundations of mathematics: see HILBERT’S PROGRAM.

For some non-eponymous terms with Hilbert connections, see the entries BETWEENNESS, EIGENVALUE, GENETIC METHOD, KERNEL, METAMATHEMATICS, PREDICATE CALCULUS.

The HILBERT CURVE was given by Hilbert. "Über die stetige Abbildung einer Linie auf ein Flachenstück." Math. Ann. 38, 459-460, 1891.  See Mathworld.

See PEANO CURVE.

HILBERT SPACE is named after David Hilbert (1862-1943) but it was not introduced by him in the sense that BANACH SPACE was introduced by Banach. Hilbert did not give a set of postulates and indeed he "never refers explicitly to 'spaces' or 'function spaces' in any of his work. [His] chief aim is to get on with the job of solving analytic problems" wrote M. Bernkopf (p. 10) in "The Development of Function Spaces with Particular Reference to their Origins in Integral Equation Theory" Arch. Hist. Exact Sci. 3, (1966), 1-96.

Hilbert treated square summable sequences in the fourth of his communications on integral ("Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. Vierte Mitteilung" Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1906), pp. 157-227. His student Erhard Schmidt gave the treatment a geometric interpretation in his "Über die Auflösung linearer Gleichungen mit unendlich vielen Unbekannten," Rendiconti di Matematico di Palermo, 25, (1908), 53-77. The expression "the Hilbert space" was used first for this (l²) set-up. See e.g. E. R. Hedrick "On Properties of a Domain for Which Any Derived Set is Closed." Transactions of the American Mathematical Society, 12, (1911), 285-294 (JSTOR search).

Hilbert's work inspired many mathematicians and his methods were applied to other set-ups. In 1929 J. von Neumann presented axioms for abstract Hilbert space: see p. 55 of his "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren" Math. Ann. 102, 49-131. In M. H. Stone's Linear Transformations in Hilbert Space (1932) "ordinary Hilbert space" is the space of square summable sequences and "abstract Hilbert space" the von Neumann space. Eventually "Hilbert space" without qualification came to mean the abstract space.

This entry was contributed by John Aldrich, based on Bernkopf and Kline chapter 46. See also EIGENVALUE.

HILBERT'S FINITE BASIS THEOREM. This theorem was given by David Hilbert in his "Ueber die Theorie der algebraischen Formen," Mathematische Annalen, 36, (1890) 473-534.

It was with reference to this theorem that Paul Gordan is supposed to have said, "This is theology, not mathematics" ("Das ist keine Mathematik, das ist Theologie!") Max Noether's obituary of Gordan in Mathematische Annalen 75, (1914) 1-41, page 18. (This topic was discussed in the Historia Matematica forum in 2001 and 1999.)

HILBERT'S PROBLEMS. In a famous address delivered in 1900 David Hilbert (1862-1943) posed 23 problems for the 20th century. Among them were the CONTINUUM HYPOTHESIS, FERMAT’S LAST THEOREM and the RIEMANN HYPOTHESIS. The problems had a great influence on the development of mathematics in the 20th century but the phrase "Hilbert’s problems" seems to have become automatic only relatively recently: a JSTOR search found it in 1950 in the American Mathematical Monthly, the "so-called ‘Hilbert problems’".

HILBERT'S PROGRAM, named for David Hilbert (1862-1943), refers to the program in the foundations of mathematics he formulated in the 1920s. According to Richard Zach’s Hilbert’s program, the program “calls for a formalization of all of mathematics in axiomatic form, together with a proof that this axiomatization of mathematics is consistent.“ A JSTOR search found "Hilbert program" in 1934 in Alonzo Church's "The Richard Paradox," American Mathematical Monthly, 41, p. 360.

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 his Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material, Philosophical Transactions of the Royal Society A, 186, (1895) Pearson explained in a footnote (p. 399) that the 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.”

The term histogram appears in a lecture of November 1891 in the series of lectures on the “Geometry of Statistics” that Pearson gave at Gresham College in the academic year 1891-2. The lectures are described by S. M. Stigler History of Statistics pp. 326-7 and T. M. Porter Karl Pearson: The Scientific Life in a Statistical Age, p. 236.

[Pekka Pere contributed to this entry.]

HÖLDER MEAN and HÖLDER’S INEQUALITY both derive from Otto Hölder 1889 paper "Über ein Mittelwertsatz," Nachrichten von der Königl. Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen, 38-47. Hölder’s inequality is discussed in Inequalities by G. H. Hardy, J. E. Littlewood and G. Polya (1934).

Hölder mean has also been used to refer to a construction in the theory of divergent series. This concept comes from Hölder’s "Grenzwerthe von Reihen an der Convergenzgrenze," Mathematische Annalen, 20, (1882), 535-49. A JSTOR search found Hölder summability in W. A. Hurwitz & L. L. Silverman "On the Consistency and Equivalence of Certain Definitions of Summability," Transactions of the American Mathematical Society, 18, (1917), 1-20. See Kline (pp. 1112-3).

The terms HOLOMORPHIC FUNCTION and MEROMORPHIC FUNCTION were introduced by Charles A. A. Briot (1817-1882) and Jean-Claude Bouquet (1819-1885) in their Théorie des fonctions elliptiques (1859). According to the OED and to Schwartzman, mero- comes from the Hellenistic Greek μερο and signifies ‘part’ or ‘partial’ while holo- is based on the Greek and signifies ‘whole.’ The terms were adopted in the English treatises Theory of Functions of a Complex Variable by A. R. Forsyth (1893) and A Treatise on the Theory of Functions by J. Harkness & F. Morley (1893). (OED and Google Print Search)

Cauchy had used the terms monotypique, monodrome, monogen, and synetique in his articles of 1851 Comptes rendus, 32 pp. 68-75 and 162-4. (Kline, p. 642).

HOMEOMORPHISM. The terms homéomorphisme and homéomorphe appear in J. H. Poincaré’s "Analysis Situs," Journal de l’École Polytechnique, 1, (1892) 1-121: "nous dirons que les deux variétés V et V’ sont equivalents au point de vue de l’ Analysis situs, ou pour abrégrer le langage, qu’elles sont homéomorphes, c’est-à-dire de forme pareille." A JSTOR search found the English terms homeomorphic and homeomorphism in respectively S. Lefschetz Proceedings of the National Academy of Sciences of the United States of America, 5, (1919), 103-106 and H. R. Brahana "Systems of Circuits on Two-Dimensional Manifolds," Annals of Mathematics, 23, (1921), pp. 144-168.

For Analysis Situs see the entry TOPOLOGY.

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).

HOMOGENEOUS DIFFERENTIAL EQUATION is found in 1868 in A Treatise on the Differential and Integral Calculus: And on the Calculus of Variations by Edward Henry Courtenay: "Prop. To determine the factor necessary to render a homogenous differential equation exact." [Google print search]

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).

Homologous is found in Arthur Cayley, “On the Triple Tangent Planes of Surfaces of the Third Order,” The Cambridge and Dublin Mathematical Journal, vol. IV (1849), pp. 118-132: “Considering two lines in the same triple tangent plane, the remaining triple tangent planes through these two lines respectively are homologous systems.” [James A. Landau]

HOMOLOGY is found in Arthur Cayley, “Sur le Problème des Contacts,” Journal für die reine and angewandte Mathematik (Crelle), tom. XXXIX (1850), pp. 4-13:

...nous dirons que les deux côtés du quadrangle inscrit qui se coupent dans le point d’intersection des deux chordes de contact, sont les axes de symptose des deux coniques données, et que les deux angles du quadrilatère circonscrit, situés sur la droite qui passe par les deux centres de contact, sont les centres d’homologie des deux coniques données.

Soient maintenant inscrites trois coniques à la même conique.  En combinant deux à deux ces trois coniques, les six axes de symptose se couperont trois à trois en quatre points que nous nommerons centres de symptose, et les six centres d’homologie seront situés trois à trois sur quatre droites que nous appelerons axes d’homologie.

...ces equations ... correspondent à un centre d’homologie des deux coniques  ... on obtient facilement, pour un des axes d’homologie des trois coniques ... et celles des trois autres axes d’homologie ...

Cherchons le pôle de l’axe d’homologie dont nous venons de trouver l’équation

[James A. Landau]

Homology is found in English 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 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).

HOMOTOPY. The OED refers to O. Veblen Analysis Situs (1922) (Cambridge Colloq. Lect., Vol. 5, Pt. 2) v. 126 "The term homotopy will be used to designate a deformation in the general sense..and two generalized complexes..will be said to be homotopic if one can be carried into the other by means of a homotopy."

According to Lefschetz Topology (1930, p. 77), the term was introduced in Dehn & Heegaard’s Analysis situs in the Encyklopädie der mathematischen Wissenschaften (1907, pp. 153ff.)

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.

HOUSEHOLDER TRANSFORMATION. This matrix was used by Alston S. Householder in his “Unitary Triangularization of a Nonsymmetric Matrix,” Journal of the ACM (JACM), 5, Issue 4 (October 1958), 339 – 342. Householder did not give the matrix a name but in The Theory of Matrices in Numerical Analysis (1964, p. 4) he calls it an “elementary Hermitian matrix.” The term “elementary reflector” is also used. A JSTOR search found the term Householder transformation in use by the mid-1960s. [John Aldrich]

HYPERBOLA was probably coined by Apollonius, who, according to Pappus, had terms for all three conic sections.

However, G. J. Toomer believes the names parabola and hyperbola are older than Apollonius, based on an Arabic translation of Diocles' On burning mirrors.

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. The page can be viewed here (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 appears in the title of H. T. Gonin, "The use of factorial moments in the treatment of the hypergeometric distribution and in tests for regression," Philosophical Magazine, 7, Ser. 21, 215-226 (1936).

The name is relatively recent but the distribution first appears as the solution to Problem IV of Huygens’s De Ratiociniis in Ludo Aleae (1657, p. 12). Several people, besides Huygens, solved the problem and James Bernoulli and de Moivre gave solutions for the general case. See Hald (1990, pp. 201-2). At the end of the 19^th century Karl Pearson wrote a paper in which he considered fitting the distribution (given by the "hypergeometrical series") to data: "On Certain Properties of the Hypergeometrical Series, and on the Fitting of such Series to Observation Polygons in the Theory of Chance," Philosophical Magazine, 47, (1899), 236-246. [John Aldrich]

HYPERGEOMETRIC SERIES. According to Geschichte der Elementar-Mathematik by Karl Fink, Wallis and Euler used this term for the series in which the quotient of any term divided by the preceding is an integral linear function of the index, and J. F. Pfaff proposed the term for the general series in which the quotient of any term divided by the preceding is a function of the index.

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 and HYPOTHESIS TESTING in Statistics. Although the use of probability in testing hypotheses is almost as old as the study of probability, the modern terminology was largely created by R. A. Fisher and the team of J. Neyman and E. S. Pearson in the 1920s and -30s. These parties disagreed about the theory of testing but their terminologies are blended in modern Statistics. (For the earliest test see the entry SIGNIFICANCE.)

Hypothesis and test only became standard terms in the 1920s: they were not used in such classics as Karl Pearson (1900) on goodness of fit or Student (1908) on tests on the normal mean. The term was very prominent in R. A. Fisher’s Statistical Methods for Research Workers (1925). See the entries on CHI-SQUARE, P-VALUE and STUDENT'S t-DISTRIBUTION

Fisher's favoured term was test of significance while 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. (This paper introduced the likelihood ratio test.) See the entries on SIGNIFICANCE and LIKELIHOOD RATIO.

Neyman and Pearson (1928) speak of accepting and rejecting hypotheses. Fisher had not used the term rejecting but he had no objection to it. However he did object to accepting (see below).

Alternative hypothesis is used several times in the 1928 paper but it was canonised in Neyman and Pearson’s most important paper, "On the Problem of the Most Efficient Tests of Statistical Hypotheses," Philosophical Transactions of the Royal Society of London. Series A, 231. (1933), pp. 289-337. The role they gave to the alternative hypothesis also drew criticism from Fisher. See POWER.

The 1933 paper introduced the terms simple hypothesis and composite hypothesis. The paper contained many other new terms and new results. See the entries on CRITICAL REGION, NEYMAN-PEARSON LEMMA, SAMPLE SPACE, SIMILAR REGION, SIZE, TYPE I ERROR

Null hypothesis appears in 1935 in Fisher's The Design of Experiments. He writes, "[W]e may speak of this hypothesis as the 'null hypothesis,' and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation." (p. 19)

The "null hypothesis" is often identified with the "hypothesis tested" of the Neyman-Pearson 1933 paper and represented by their symbol H₀. Neyman did not like the term "null hypothesis," arguing (First Course in Probability and Statistics, 1950, p. 259) that "the original term 'hypothesis tested' seems more descriptive." It is not clear, however, that "hypothesis tested" was ever floated as a technical term.

This entry was contributed by John Aldrich. See also ASYMPTOTIC RELATIVE EFFICIENCY, CHI-SQUARE, CRITICAL REGION, NEYMAN-PEARSON (FUNDAMENTAL) LEMMA, NUISANCE PARAMETER, P-VALUE, POWER, SIGNIFICANCE, SIMILAR REGION, SIZE, TYPE I ERROR, STUDENT'S t-DISTRIBUTION.