*Last revision: June 28, 2017*

**HAAR MEASURE.** Alfréd Haar
described the an invariant measure on locally compact groups in his “Der Massbegriff in
der Theorie der kontinuierlichen Gruppen” *Annals of Mathematcs*, **34**,
(1933), 147–169. The expression *Haar measure* was soon in circulation: a *JSTOR* search found it in
S. Bochner’s “Average Distribution of Arbitrary Masses under Group Translations,”
*Proceedings of the National Academy of Sciences*, **20**, (1934), 206-210. See also
*Enyclopedia of Mathematics*.

**HADAMARD** or **SCHUR PRODUCT** of matrices. There is nothing puzzling about the association
of Schur
with the entry-wise product for he wrote the seminal paper on the subject:
Bemerkungen zur Theorie der beschränkten
Bilinearformen mit unendlich vielen Veränderlichen,
*Journal für die reine und angewandte Mathematik*,
**141**, (1911), 1-28.
Hadamard,
on the other hand, never wrote about this type of product. The popularity of the expression *Hadamard product* appears
to derive from its appearance in Halmos’s *Finite-Dimensional
Vector Spaces*
(1948). Halmos later explained that von Neumann used
the expression in lectures and it has been suggested that von Neumann
was
alluding to Hadamard’s well-known work on products of power
series (“Théorème sur les séries
entières,”
*Acta Mathematica*, **22**, (1899), 55-63) and that von Neumann was
associating Hadamard’s name with term-by-term products of all kinds.
(Based on the historical remarks in R. A. Horn’s chapter “The Hadamard Product” in C. R. Johnson (ed.)
*Matrix Theory and Applications*, AMS (1990).) [John Aldrich]

**HAHN-BANACH THEOREM** in functional analysis. The result is due to
Hans Hahn
“Ueber lineare Gleichungsysteme in linearen Räume,”
*J. Reine Angew. Math.,* **157**, (1927), 214–229 and
Stefan Banach
“Sur les fonctionelles linéaires,”
*Studia Math.*, **1,** (1929), 211–216 and
“Sur les fonctionelles linéaires II,”
*Studia Math.*, **1** (1929), 223–239.
See also Enyclopedia of Mathematics.

**HAIRY BALL THEOREM.** James A. Landau recalls that in Freshman Orientation at Michigan State in 1965, a math professor gave a talk
about topology. The professor said, “I know it’s hard to believe [it was very windy that
day] but there is a point on the earth at which the wind does not blow.” Landau also recalls that the professor
then went on to say this theorem meant “you can't comb a hairy billiard ball
without leaving a cowlick.”

According to M. W. Hirsch *Differential Topology*
(1976) Every vector field on *S*^{2n} is zero somewhere; more
picturesquely a hairy ball cannot be combed.” (p.125) Hirsch refers to this
result as the “so-called ‘hairy ball theorem’” and the name appears to have
become popular in the 1970s. However the theorem goes back to L. E. J. Brouwer
“Über Abbildung von Mannigfaltigkeiten,”
*Mathematisches
Annalen*, (1912), 97-115 and beyond that to Ch XIII pp. 203-8 of
Poincaré’s “Sur les courbes définies par les équations différentielles,”
*Journal de mathématiques pures et appliquées*
(1885), 167-244.

**HALMOS**. See the entry on Earliest Uses of Symbols of Set Theory and Logic.

**HAM SANDWICH THEOREM.**
Hugo Steinhaus
posed the following problem in the famous Lvov
*Scottish Book*
"Given are three sets *A*_{1}, *A*_{2}, *A*_{3}
located in the 3-dimensional Euclidean space and
with finite Lebesgue measure. Does there exist a plane cutting each of the three
sets *A*_{1}, *A*_{2},
*A*_{3} into two parts of equal measure? The same for *n* sets
in the *n*-dimensional space." A solution was published in a note, "A Note
on the Ham Sandwich Theorem," in the journal *Mathesis
Polska* in 1938. The note was by Steinhaus and others but the proof they presented was due to
Banach;
this is based on the Borsuk-Ulam Theorem. The note indicates that the problem
can be formulated as follows: "Can we place a piece of ham under a meat cutter
so that meat, bone and fat are cut in halves?" (Based on WA Beyer, A Zardecki
"The early history of the ham sandwich theorem," *American Mathematical Monthly*, January
2004, pp. 58-61, which contains a translation of the original note.)

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*(*d*T/*dq´*)/*dt* - *d*T/*dq*
= Q, and the Hamiltonian form is *dq*/*dt* = *d*T/*dp, dp*/*dt*
= - *d*T/*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).

**HANDWAVING.** This
slang term first appears in print in the 1960s. A *JSTOR* search found these early examples: “By
keeping the handwaving and the use of the villainous “it is clear that” to a
minimum, the author has imposed discipline on and revealed the beauty of a
subject that has attracted hitherto the more than its share of mathematical
cavaliers” from Wimp *SIAM Review*, **7**, (1965), p. 577 and “The
presentation is ... for the most parts rigorous. Unfortunately there is a great
deal of handwaving in the geometric parts of the chapter on conformal mapping…”
from Leibowitz in *American Mathematical
Monthly*, **74**, (1967), p. 1154.

**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*

It seems likely that the terms originated as extensions of the older terms **hard analysis** and **soft analysis**. *JSTOR*
sightings of these terms date only from the late 1950s
although it is clear that the terms were already well-established. Typical
subjects for hard analysis included the SPECIAL FUNCTIONS, while soft analysis was often another term for
FUNCTIONAL ANALYSIS.

**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*.)

A Google print search finds the term *harmonic analysis* in “Committee for the purpose of promoting the extension, improvement, and harmonic analysis of Tidal Observations,” a report by Sir W. Thomson dated December 1867.
It appears in the Report of the Annual Meeting, Volume 38, Part 1868 by the
British Association for the Advancement of Science. [James A. Landau]

In statistics *harmonic analysis* is found in R. A. Fisher, "
Tests of Significance in Harmonic Analysis.," *Proc. Roy. Soc. A,* 125, 54-59 (1929) [John Aldrich].

**HARMONIC ANALYZER.** A Google print search finds this term in the Proceedings of the Royal Society of London in the title of the paper
“Harmonic Analyzer,” Shown and explained by
Sir William Thomson, F.R.S., Professor of Natural Philosophy in the University
of Glasgow. Received May 9, 1878. [James A. Landau]

**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 theIntroduction 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.

Harmozeinjust 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 theIntroductionNicomachus 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" [OED].

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" [OED].

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)

The **HAUPTSATZ.** In German this label
for “main theorem” is attached to several propositions–see
Wikipedia–but in English mathematics/mathematical
logic it is now attached to only one of these, *der Gentzensche Hauptsatz*.
The proposition appears in
Gerhard
Gentzen’s “Untersuchungen über das logische Schließen,”
*Mathematische
Zeitschrift*, **39**, (1934-5), 405-431. In
English “Gentzen’s Hauptsatz” is mentioned in the *Journal of Symbolic Logic*
of 1938 and for decades it was referred to by this name or as “the Gentzen
Hauptsatz.” Eventually it became possible to drop the reference to Gentzen without
risk of confusion, as in the title of D. Prawitz’s “Hauptsatz for Higher Order
Logic,” *Journal of Symbolic Logic*, **33**, (1968), 452-457. In English
the theorem is also called the “cut-elimination theorem”; see
Wikipedia. [John Aldrich]

The **HAUPTVERMUTUNG** refers to a
conjecture originally formulated by Steinitz and Tietze in 1908. In their *Topologie*
Alexandroff & Hopf (1935) call it “die sog. Hauptvermutung der
kombinatorischen Topologie” (the so-called main conjecture of combinatorial topology.
In English the short form “Hauptvermutung” appears in quotes in J. H. C.
Whitehead’s “On the Homotopy Type of Manifolds,” *Annals of Mathematics*,
(1940) but from around 1950 the word has appeared without quotes. Thus a description
in German of general application has become the English name for a particular
conjecture. See Wikipedia
and the Edinburgh Hauptvermutung
page which makes available many of the main contributions. [John Aldrich]

**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
the Metrization problem and Related Problems in the Theory of Abstract Sets,”
*Bull. Amer. Math. Soc*. (1927) **33**, pp. 13-34. Referring to Hausdorff’s
*Grundzüge der Mengenlehre,* Chittenden
writes, “3. *Hausdorff Spaces.* A remarkable and important class of topological spaces has been
defined by F. Hausdorff ….” (p. 15)

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")

A Google print search shows *Heine-Borel’schen Theorems* in “Bericht uber die Mengenlehre”
by Arthur Schoenflies in
*Jahresbericht der Deutschen Mathematiker-Vereinigung,* (1899). [James A. Landau]

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. Helmert used it in his paper on the accuracy of PETERS’ METHOD,
“Die Genauigkeit der Formel von Peters zur Berechnung des
wahrscheinlichen Fehlers directer Beobachtungen gleicher Genauigkeit,”
*Astronomische
Nachrichten*, **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). See also CHI SQUARE.

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

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."

*Hermite polynomial* is found in 1939 in *Orthogonal Polynomials* by Gabor Szegö [Google print search,
James A. Landau].

**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.

**HESSENBERG MATRIX** is found in 1954 in
Neue Verfahren zur direkten Lösung des allgemeinen Matrizeneigenwertproblemes
by Sigurd Falk. [Jan Peter Schäfermeyer]

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.
See the paper
“When
Did We Begin to Spell ‘Heteros*edasticity’ Correctly?”
by Alfredo R. Paloyo.
[This entry was
contributed by John Aldrich, referring to OED2 and David, 1995.]

**HETEROLOGICAL.** See AUTOLOGICAL and
HETEROLOGICAL.

**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). A Google print search suggests uses in 1949 and 1950, but
only snippet view is available.

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.** Hexagons are discussed in Book IV of Euclid’s
Elements.
The word “hexagon” appears
in English in 1570 in Sir Henry Billingsley’s translation of the *Elements* (*OED*).
Earlier in 1551 in *Pathway to Knowledge* Robert Recorde used the word *siseangle*: “Def., Likewyse shall you iudge of
siseangles, which haue sixe corners” (*OED*).
When writing English Recorde preferred to create new English words rather than borrow
classical words. See EQUILATERAL and PENTAGON. [John Aldrich]

**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.

**HILBERT’S ANSATZ.** See ANSATZ.

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’S SATZ 90.** The theorem is the 90th in David Hilbert’s *Die Theorie der algebraischen
Zahlkörper*, the famous *Zahlbericht* of 1897: p. 149 in
*Hilbert’s
Gesammelte Abhandlungen Erster Band*. See the *Wikipedia*
entry.

**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 (*l ^{2}*) set-up. See
e.g. E. R. Hedrick "On Properties of a Domain for Which Any Derived Set
is Closed."

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 NULLSTELLENSATZ** (theorem of the location of zeros) in algebraic geometry;
see Wikipedia article.
The theorem appears in Hilbert’s
"Ueber die Theorie der algebraischen Formen," *Mathematische Annalen*, **36**,
(1890) 473-534.
The theorem came to be called *der
Hibertsche Nullstellensatz*, as in J. L. Rabinowitsch "Zum Hibertschen
Nullstellensatz," *Mathematische Annalen*, **102**,
(1929) 520.
In English the expression became *Hilbert’s Nullstellensatz* as in O. Zariski “A
New Proof of Hilbert’s Nullstellensatz,” *Bulletin
of the American Mathematical Society*, **53**, (1947), 362-368. That expression, or *the Hilbert Nullstellensatz*,
has remained the standard one in English. Contrast with the French
practice of referring to “le théorème des zéros de
Hilbert” or with the fate of other German expressions in English, such as *das Entscheidungsproblem* which
became DECISION PROBLEM.

**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.”

An address by Eliakim Hastings Moore in *Science,* March 13, 1903, has:

One has then the feeling that the carrying out in an absolute sense of the program of the abstract mathematicians will be found impossible. At the same time, one recognizes the importance attaching to the effort to do precisely this thing. The requirement of rigor tends toward essential simplicity of procedure, as Hilbert has insisted in his Paris address, and the remark applies to this question of mathematical logic and its abstract expression.

[Google print search, James A. Landau]

It appears from a Google print search with snippet view that in 1930 *On the Logic of Measurement* by Ernest Nagel
has:

Consequently, it is Hilbert’s program to reduce the whole of mathematics to a formal,meaninglessplay with marks, so that the pursuit of mathematics becomes a glorified game of chess. Therulesof the game, designated as “metamathematics,” have a meaning, but only in the terms of the meaningless operations with signs to which “mathematics” has been. . . .

A *JSTOR* search found *Hilbert program*
in 1934 in Alonzo Church’s “The Richard Paradox,” *American Mathematical Monthly*, **41**, p.
360.

**HINDU-ARABIC NUMERAL.** According to Katz, the earliest available arithmetic text to deal
with Hindu numbers is the Arabic work *Book on Addition and Subtraction after the Method of the Indians* by
Muhammad
ibn Musa al-Khwarizmi. This system of notation was introduced into Europe by
Fibonacci.

In his *Liber abaci* (1202) Fibonacci referred to *Indian figures*:

Novem figure indorum he sunt 9 8 7 6 5 4 3 2 1. [The nine Indian figures are 9 8 7 6 5 4 3 2 1.]

For some time the Hindu origins of the notation were forgotten so that the term *Arabic characters* appears in *Chambers Cyclopedia*:
“The Arabic characters stand contradistinguished to the Roman" (1727-51).
*Arabic numerals* appears in in T. Green, *Extracts from the diary of a lover of literature*:
“Writing, he deduces, from pictural representations, through hieroglyphics ...
to arbitrary marks ... like the Chinese characters and Arabic numerals.” (1810).

The 1872 edition of *Chambers Encyclopaedia* uses the term *Hindu numerals*: “After the
introduction of the decimal system and the Arabic or Hindu numerals about the
11th c., Arithmetic began to assume a new form.” The term *Indo-Arabic system* appears in the 1884 edition of the *Encyclopaedia Britannica*: “In Europe, before the introduction of the algorithm or full Indo-Arabic system with the zero.”

The historians of mathematics wanted to set the record straight. In *A History of Mathematics* Florian Cajori
wrote:

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.

(This is from the 1906 edition but presumably the passage appears in the earlier 1893 edition).

David Eugene Smith and Louis Charles Karpinski called their book (1911)
*The
Hindu-Arabic Numerals.*

For more information see the MacTutor articles Indian numerals and Arabic numerals.

**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. See the entry RADIOGRAM.

**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.’

A Google print search finds *holomorphic* in English in “Solution of the Equation of the Fifth Degree,”
*The Analyst,* Nov. 1878. The article is a translation of the above work by Briot and Bouquet. [James A. Landau]

A Google print search finds *meromorphic* in English in 1885 in “Extrait d’une letter de M. Hermite,”
*American Journal of Mathematics*: “the function *f*(*z*)/*z*^{(n-1)} being meromorphic over the whole sphere must be a rational fraction....” [James A. Landau]

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 sonthoméomorphes, 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. However,
as G. H. Moore points out in his “The evolution of the concept of homeomorphism,” *Historia Mathematica*,
**34**, (2007), 333-343, Poincaré meant by
homeomorphism “something quite different from and more restricted than” what is
meant by the term today: in particular, Poincaré’s notion of homeomorphism had
strong requirement of differentiability and smoothness that “have nothing
directly to do with topology.” Moore identifies a string of contributors to the
modern notion of homeomorphism which was established around 1930. The modern notion is found in
Lefschetz *Topology* (1930),
Kuratowski
*Topologie I* (1933) and
Alexandroff &
Hopf *Topologie* (1935).

This entry was contributed by John Aldrich. See also 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" [OED].

**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.** Writing in Latin, Cavalieri used *homologe* in 1643 in *Geometria indivisibilibus continuorum: noua quadam ratione promota.* [Google print search, James A. Landau]

*Homologous* is found in English in 1656 in *The History of Philosophy, in Eight Parts*
by Thomas Stanley: “Of equiangle triangles, the sides that are about equall angles are
proportionall, and the sides that subtend the equall angles are homolgous.” [Google print search, James A. Landau]

*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 1749 in James Stirling, *The Differential Method, or a Treatise Concerning Summation and Interpolation of
Infinite Series, Translated into English, with the Author’s Approbation, by Francis Holliday*:
“Consequently by gathering the homologous terms into one, the series will be
reduced to its first form, and will be ομολογια, homology, or
likeness, of the terms no where depends on the co-efficients A, B, C, D, &c. but
altogether upon the indeterminate quantity *z* ....” [Google print search, 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 lesaxes de symptosedes deux coniques données, et que les deux angles du quadrilatère circonscrit, situés sur la droite qui passe par les deux centres de contact, sont lescentres d’homologiedes 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 appeleronsaxes 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" [OED].

*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* [OED].

**HOMOMORPHISM** is found in English in 1935 in the *Duke
Mathematical Journal* [OED].

**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.** “Mr. Horner’s new method of solving numerical equations
of all orders, by continuous approximations” appears in “A New Method of Solving Numerical Equations of All Orders, By Continuous Approximation,” by W. G. Hoener, Esq. Communicated by Davies Gilbert, Esq. F.R.S.
Read July 1, 1819, *Philosophical Transactions of the Royal Society of London for the year MDCCCXIX.* [Google print search,
James A. Landau]

Theophilus Holdred wrote in a letter, “The method I have given at the end of my Supplement, Mr. Nicholson calls Mr.
Horner’s method; and says I have been anticipated by himself in point of
publication: but it resembles Mr. Horner’s in no respect....” The letter appears in
*The Philosophical Magazine and Journal
Volume LVI For July, August, September, October, November, and December, 1820.*

*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” [OED].

**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* [OED].

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"
[OED].

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.*

*Hyperbolic sine* and *hyperbolic cosine* are found in English in
“Report of the Third Meeting of the British Association for the Advancement of
Science, Held at Cambridge in 1833.” [Google print search, James A. Landau]

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

**HYPERCUBE** is found in Joseph Delboeuf, “L'ancienne et les nouvelles
géométries. II,” *Revue Philosophique de la France
et de l'étranger* **37** (1894), 353-383 [Google print search].

*Hyper-cube* is found in English in
1909 in *Scientific American* 6/3: “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.” [OED]

**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'" [OED].

*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" [OED].

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" [OED].

**HYPERSET** is found in 1987 in *The Liar: An Essay on Truth and Circularity* by
John Barwise and John Etchemendy: “the sets we get on Aczl’s conception include all those in the
traditional, wellfounded universe. But in addition to these, we get a rich
class of nonwellfounded sets, or as we will sometimes call them,
*hyper*sets.” [Google print search, James A. Landau]

*Hyperset* is also found in 1991 in the article “Hypersets,” by Barwise and Larry Moss in *Mathematical Intelligencer.*

**HYPERSPHERE.** Ludwig Schlaefli [Schläfli], *On the
multiple integral ∫ ^{n} dx dy ... dz, whose limits are*

We do not dwell further on this general subject, but we proceed to one of its most particular cases, the general spheric equation ; the origin here assumed may be calledcentreand the constant aradius.For the curved continuum itself, represented by this equation, I propose the termpolysphereorn-sphere; thus, a pair of points in a straight line will be amonosphere,the circle adisphere,and the spherical surface properly so called atrisphere.These terms are very harsh, but I do not know how to avoid this.

George Salmon, *On the relation which
connects the mutual distances of five points in space,*
Quarterly Journal
of Pure and Applied Mathematics **3** (1860), 282-288, has on p. 285:

It must have occurred to any person who has observed the analogy between the processes of the analytic geometry of three and of two dimensions, that as far as algebraic processes are concerned there is nothing to limit the methods employed to the case of three dimensions. If, for example, there were any beings who could conceive space of four dimensions, we could with our present faculties write as good a book of analytic geometry for their use as if we could conceive space of four dimensions ourselves. We should take fourhyper-planes at right angle to each other, and define the position of any point by four coordinates expressing its four distances from these; and so proceed in every manner exactly as we proceed in the analytic geometry of three dimensions. [...] Now the process followed in Art. (2) may be extended word for word to space of four dimensions and gives the following relation between the distances of five points on ahyper-sphere.

[This entry was contributed by Dave L. Renfro.]

**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" [OED].

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 *H _{0}*. Neyman did not like the term "null
hypothesis," arguing (

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.