The term **GALOIS CONNECTION** is
due to Oystein Ore, "Galois Connexions," *Trans. Amer. Math. Soc.*
55 (1944), 493-513:

The object of this paper is to discuss a general type of correspondence between structures which I have called Galois connexions. These correspondences occur in a great variety of mathematical theories and in several instances in the theory of relations. ... The name is taken from the ordinary Galois theory of equations where the correspondence between subgroups and subfields represents a special correspondence of this type.

The citation above was taken from a post by William C. Waterhouse. In another post, Phill Schultz writes:

The abstract notion of Galois Connection
appears in Garrett Birkhoff, "Lattice Theory," *Amer. Math. Soc.
Coll. Pub.,* Vol 25, 1940. I believe this is the first such occurrence,
since in later editions, Birkhoff refers to other publications, but they are
all later than 1940. The attribution 'Galois Connection' is simply because
classical Galois Theory, as developed by Artin in the 1930's, establishes a
correspondence between subfields of an algebraic number field and subgroups of
the group of automorphisms of that field which is a dual lattice isomorphism
between the lattice of normal subfields and the lattice of normal subgroups.
Birkhoff's idea is to replace the set of subfields and the set of subgroups by
arbitrary posets. The normal subfields and subgroups correspond to lattices of
'closed' elements of the posets. The Galois Connection is then an order
reversing correspondence between the posets which is a lattice dual isomorphism
between the posets of 'closed' elements.

**GALOIS FIELD.** See *field.*

**GALOIS GROUP.** *Galois' group* is found in J. De Perott,
"A construction of Galois' group of 660 elements," *Chicago Congr.
Papers* (1897).

*Galois group* is found in 1899 in the *Bulletin of the
American Mathematical Society* (OED).

*Galois group* is found in L. E. Dickson, "The *Galois*
group of a reciprocal quartic aquation," *Amer. Math. Monthly* 15.

**GALOIS THEORY.** *Galois equation theory* appears in Heinrich Weber, "Die allgemeinen Grundlagen der
Galois'schen Gleichungstheorie," *Mathematische Annalen,* 43 (1893)
[James A. Landau].

*Galois theory* is found in English in 1893 in the *Bulletin
of the New York Mathematical Society.*

**GAME THEORY.** See *theory of games.*

The term **GAMMA FUNCTION** was introduced by Legendre (Kline, page
424).

The term **GASKET** was coined by Benoit Mandelbrot. On page 131,
[Chapter 14] of "The Fractal Geometry of Nature", Benoit Mandelbrot
says:

*Sierpinski gasket* is the term I propose to denote the shape in
Plate 141.

And on page 142, Mandelbrot adds:

I call Sierpinski's curve a *gasket,*
because of an alternative construction that relies upon cutting out 'tremas', a
method used extensively in Chapter 8 and 31 to 35.

The citation above was provided by Julio González Cabillón.

The word **GAUGE** (in gauge theory) was introduced as the German
word *maßstab* by H. Weyl (1885-1955) in 1918 in *Sitzungsber. d.
Preuss. **Akad. d.
Wissensch.* 30 May 475 (OED2).

**GAUSS-JORDAN METHOD.** In *Matrix Analysis and Applied
Linear Algebra* (2000), Carl D. Meyer writes, "Although there has been
some confusion as to which Jordan should receive credit for this algorithm, it
now seems clear that the method was in fact introduced by a geodesist named
Wilhelm Jordan (1842-1899) and not by the more well known mathematician Marie
Ennemond Camille Jordan (1838-1922), whose name is often mistakenly associated
with the technique, but who is otherwise correctly credited with other
important topics in matrix analysis, the 'Jordan canonical form' being the most
notable."

The word **GAUSSIAN** was used (although not in a mathematical sense)
in a letter of Jan. 17, 1839, from William Whewell to Quételet: "Airy has
just put up his Gaussian apparatus..at Greenwich, including a Bifilar."

**GAUSSIAN CURVE** (normal curve) appears in a 1902 paper by Karl
Pearson [James A. Landau].

*Gaussian distribution* and *Gaussian law* were used
by Karl Pearson in 1905 in *Biometrika* I "Many of the other
remedies which have been proposed to supplement what I venture to call the
universally recognised inadequacy of the Gaussian law .. cannot .. effectively
describe the chief deviations from the Gaussian distribution" (OED2).

In an essay in the 1971 book *Reconsidering Marijuana,* Carl Sagan,
using the pseudonym "Mr. X," wrote, "I can remember one
occasion, taking a shower with my wife while high, in which I had an idea on
the origins and invalidities of racism in terms of gaussian distribution
curves. I wrote the curves in soap on the shower wall, and went to write the
idea down."

**GAUSSIAN INTEGER** is found in the title, "Sums of fourth
powers of Gaussian integers," by Ivan Niven (1915-1999), *Bull. Am.
Math. Soc.* 47, 923-926 (1941).

**GAUSSIAN LOGARITHM** appears in 1870 in *The portable transit
instrument in the vertical of the pole star,* a translation by Cleveland
Abbe of a memoir of William Döllen: "These auxiliary angles have, for the
computations of the present day---thanks to the increasing dissemination of the
Gaussian logarithms---lost, to a great extent, their former importance; they
afford a real relief in the computation generally, only when we have to do, not
with a single case but with many connected tegether, in which certain
quantities are common, as, for example, often in the computation of tables"
[University of Michigan Digital Library].

The name **GAUSS-MARKOV THEOREM** for the chief result on least
squares and best linear unbiassed estimation in the linear (regression) model
has a curious history. David (1998) refers to H. Scheffé's 1959 book *Analysis
of Variance* where the expression "Gauss-Markoff theorem" appears.
Before that the name "Markoff theorem" had been popularized by J.
Neyman, starting with his "On the Two Different Aspects of the
Representative Method" (*Journal of the Royal Statistical Society,*
97, 558-625). Neyman thought that this contribution from the Russian A. A.
Markov had been overlooked in the West. However in 1949 Plackett (*Biometrika,*
36, 149-157) showed that Markov had done no more than Gauss nearly a century
before in 1821/3. (In the nineteenth century the theorem was often referred to
as "Gauss's second proof of the method of least squares" - the
"first" being a Bayesian argument Gauss published in 1809). Following
Plackett, a few authors adopted the expression "Gauss theorem" but
"Markov" was well-entrenched and the compromise "Gauss-Markov
theorem" has become standard. [This entry was contributed by John
Aldrich.]

The term **GENERAL INTEGRAL** is due to Lagrange (Kline, page 532).

**GENERAL SOLUTION** is found in 1859 in George Boole, *Treat.
Differential Equations*: "The relation among the variables which
constitutes the general solution of a differential equation..is also termed its
complete primitive" (OED2).

**GENERAL TERM** is found in 1791 in "A new method of
investigating the sums of infinite series," by Rev. Samuel Vince, *Philos.
Trans. R. Soc.*: "To find the sum of the infinite series whose general
term is ..."

**GENETIC DEFINITION** was used by Christian Wolff (1679-1754) in *Philosophia
rat. sive logica* (1728, 3rd ed. 1740) [Bernd Buldt].

*Genetic definition* was also used by Immanuel Kant (1724-1804).

*Genetic definition* was used in English in 1837-38 by the Scottish
philosopher and logician William Hamilton (1788-1856) in *Logic* xxiv.
(1866) II. 13: "In Genetic Definitions the defined subject is considered
as in the progress to be, as becoming; the notion, therefore, has to be made,
and is the result of the definition, which is consequently synthetic"
(OED2).

The term **GENETIC METHOD** (as opposed to "axiomatic
method") was apparently introduced by David Hilbert (1862-1943), and its
first use may be its appearance in the 1900 essay "Ueber den
Zahlbegriff."

The term appears in English in Edward V. Huntington, "Complete Sets
of Postulates for the Theory of Real Quantities," *Transactions of the
American Mathematical Society,* July, 1903. Huntington popularized the use
of the term.

*Genetic method* was used earlier in a different sense by
Professor fuer hoehere Analysis und darstellende Geometrie Carl Reuschle
(1847-1909), son of the German mathematician Carl Gustav Reuschle (1812-1875),
in an article entitled "Constituententheorie, eine neue, principielle und
genetische Methode zur Invariantentheorie" (1897) [Julio González
Cabillón].

The term **GEODESIC** was introduced in 1850 by Liouville and was
taken from geodesy (Kline, page 886).

The term *geodesic curvature* is due to Pierre Ossian Bonnet
(1819-1892), according to the University of St. Andrews website.

However, according to Jesper Lützen in *The geometrization of
analytical mechanics: a pioneering contribution by Joseph Liouville (ca. 1850),*
"Liouville defined the 'geodesic curvature' (the name is due to
him)...."

**GEOID** was first used
in German (*geoide*) in 1872 by Johann Benedict Listing (1808-1882) in *Ueber
unsere jetzige Kenntniss der Gestalt u. Grösze der Erde* (OED2).

**GEOMETRIC MEAN.** The term *geometrical mean* is found in
the first edition of the *Encyclopaedia Britannica* (1768-1771) [James A.
Landau].

The term **GEOMETRIC PROGRESSION** was used by Michael Stifel in
1543: "Divisio in Arethmeticis progressionibus respondet extractionibus
radicum in progressionibus Geometricis" [James A. Landau].

*Geometrical progression* appears in English in 1557 in the *Whetstone
of Witte* by Robert Recorde: "You can haue no progression
Geometricalle, but it must be made either of square nombers, or els of like
flattes" (OED2).

*Geometric progression* appears in English in 1706 in *Syn.
Palmar. Matheseos* by William Jones: "The Curve describ'd by their
Intersection is called the Logarithmic Line... A Point from the Extremity
thereof, moving towards the Centre with a Velocity decreasing in a Geometric
Progression, will generate a Curve called the Logarithmic Spiral" (OED2).

**GEOMETRIC PROPORTION.** In 1551 Robert Recorde wrote in *Pathway
to Knowledge*: "Lycurgus .. is most praised for that he didde chaunge
the state of their common wealthe frome the proportion Arithmeticall to a
proportion geometricall" (OED2).

*Geometrical proportion* appears in 1605 in Bacon, *Adv.
Learn.*: "Is there not a true coincidence between commutative and
distributive justice, and arithmetical and geometrical proportion?"
(OED2).

*Geometrical proportion* appears in 1656 in tr. *Hobbes's
Elem. Philos.*: "If four Magnitudes be in Geometrical Proportion, they
will also be Proportionals by Permutation, (that is, by transposing the Middle
Terms)" (OED2).

*Geometric proportion* appears in 1706 in *Synopsis
Palmariorum matheseos* by William Jones: "In any Geometric Proportion,
when the Antecedent is less than the Consequent, the Terms may be express'd by *a*
and *ar* (OED2).

**GEOMETRIC SERIES.** *Geometrical series* is found in the 1828
*Webster* dictionary.

*Geometical series* also appears in the 1830 American edition of
the 1828 second British edition of *Elements of Chemistry, Including the
Recent Discoveries and Doctrines of the Science* by Edward Turner:
"...the excess [caloric] remainng after each interval is, 9000/10,000,
8100/10,000, 7290/10000, 6560/10,000, 3905/10,000, 5316/10,000, &c. Is is
obvious that the numerators of these fractions constitute a geometrical series,
of which 1.111 is the ratio..." This quote might also appear in the 1827
first London edition of the book [James A. Landau].

*Geometric series* is found in English in 1837 in Whewell, *Hist.
Induct. Sci.* (1857): "The elasticity proceeds in a geometric
series" (OED2).

The term **GEOMETRY** was in use in the time of Plato and Aristotle,
and "doubtless goes back at least to Thales," according to Smith
(vol. 2, page 273).

Smith also writes (vol. 2, page 273) that "Plato, Xenophon, and Herodotus use the word in some of its forms, but always to indicate surveying."

However, Michael N. Fried points out that Smith may not be entirely correct:

In the *Epinomis* (whose Platonic
provenance is not completely clear), it is true that Plato refers to
mensuration or surveying as 'gewmetria' (990d), but elsewhere Plato is very
careful to distinguish between practical sciences concerning sensibles, such as
surveying, and theoretical sciences, such as geometry. For instance, in the *Philebus*
(of undisputed Platonic provenance), one has:

"SOCRATES: Then as between the calculating and measurement employed in
building or commerce and the geometry and calculation practiced in philosophy--
well, should we say there is one sort of each, or should we recognize two
sorts?

PROTARCHUS: On the strength of what has been said I should give my vote for
there being two" (57a).

This distinction reoccurs in Proclus'
neo-platonic commentary on Euclid's *Elements.* There Proclus writes:
"But others, like Geminus... think of one part [of mathematics] as
concerned with intelligibles only and of another as working with perceptibles
and in contact with them... Of the mathematics that deals with intelligibles
they posit arithmetic and geometry as the two primary and most authentic parts,
while the mathematics that attends to sensibles contains six sciences:
mechanics, astronomy, optics, geodesy, canonics, and calculation. Tactics they
do not think it proper to call a part of mathematics, as others do, though they
admit that it sometimes uses calculation... and sometimes geodesy, as in the
division and measurement of encampments" (Friedlein, p.38).

Even Herodotus does not identify geometry and
geodesy, but only claims that the origin of the former might have had it origin
in the later (*the Histories,* II.109).

Smith (vol. 2, page 273) writes, "Euclid did not call his treatise
a geometry, probably because the term still related to land measure, but spoke
of it merely as the *Elements.* Indeed, he did not employ the word
'geometry' at all, although it was in common use among Greek writers. When *Euclid*
was translated into Latin in the 12th century, the Greek title was changed to
the Latin form *Elementa,* but the word 'geometry' is often found in the
title-page, first page, or last page of the early printed editions" (Smith
vol. 2, page 273).

*Geometry* appears in English in 14th century
manuscripts. An anonymous 14th century manuscript begins, "Nowe sues here
a Tretis of Geometri wherby you may knowe the heghte, depnes, and the brede of
mostwhat erthely thynges" (Smith vol. I, page 237). The OED shows another
14th century use.

The term **GEOMETRY OF NUMBERS** was coined by Hermann Minkowski
(1864-1909) to describe the mathematics of packings and coverings. The term
appears in the title of his *Geometrie der Zahlen.*

**GÖDEL'S INCOMPLETENESS THEOREM.** *Entscheidungsproblem* (decision problem) appears in the title
"Beiträge zur Algebra der Logik, insbesondere zum
Entscheidungsproblem" in 1922 in *Mathematische Annalen* 86.

The term *Gödel's theorem* was used by Max Black in 1933 in *The
Nature of Mathematics* (OED2).

In 1955 K. R. Popper in P. A. Schilpp Philos. of R. Carnap (1963) refers to his "two famous incompleteness theorems" (OED2).

**GOLDBACH'S CONJECTURE.** *Théorème de Goldbach* is found in G. Eneström, "Sur un théorème
de Goldbach (Lettre à Boncompagni)," *Bonc. Bull.* (1886).

*Théorème de Goldbach*
is found in G. Cantor, *Vérification jusqu'à 1000 du théorème de Goldbach,*
Association Française pour l'Avancement des Sciences, Congrès de Caen (1894).

*Goldbach's theorem* is
found in 1896 in M.-P. Stackel, "Über Goldbach's empirisches
Theorem," *Gött. **Nachrichten,*
1896.

*Goldbach-Euler theorem* appears in the title of an article
"On the Goldbach-Euler theorem regarding prime numbers" by James
Joseph Sylvester, which appeared in *Nature* in 1896/7.

*Goldbach's problem* is found in English in 1902 in Mary Winton
Newson's translation of Hilbert's 1900 address in the *Bulletin of the
American Mathematical Society.*

*Goldbach's theorem* occurs in English in the *Century Dictionary*
(1889-1897).

*Goldbach's hypothesis* is found in J. G. van der Corput,
"Sur l'hypothese de Goldbach pour presque tous les nombres pairs" *Acta
Arith.* 2, 266-290 (1937).

*Goldbach's conjecture* is found in 1919 in Dickson:
"No complete proof has been found for Goldbach's conjecture in 1742 that
every even integer is a sum of two primes."

*Goldbach's conjecture* appears in the title of the novel *Uncle
Petros and Goldbach's Conjecture* by Apostolos Doxiadis, published on March
20, 2000, by Faber and Faber.

**GOLDEN SECTION.** According to *Greek Mathematical Works I -
Thales to Euclid,* "This ratio is never called the Golden Section in
Greek mathematics." According to an Internet web page, Euclid used *Reliqua
Sectio.*

Leonardo da Vinci used *sectio aurea* (the golden section),
according to H. V. Baravalle in "The geometry of the pentagon and the
golden section," *Mathematics Teacher,* 41, 22-31 (1948).

The OED2 has: "This celebrated proportion has been known since the 4th century b.c., and occurs in Euclid (ii. 11, vi. 30). Of the several names it has received, golden section (or its equivalent in other languages) is now the usual one, but it seems not to have been used before the 19th century."

*Goldenen Schnitt* appears in print for the first time in 1835 in
the second edition of *Die reine Elementar-Mathematik* by Martin Ohm:

Diese Zertheilung einer beliebigen Linie *r* in 2 solche Theile, nennt
man wohl auch den goldenen Schnitt.

In the
earlier 1826 edition, the term does not occur, but instead *stetige
Proportion* is used.

Roger Herz-Fischler in *A Mathematical History of Division in Extreme
and Mean Ratio* (Wilfred Laurier University Press, 1987, reprinted as *A
Mathematical History of the Golden Number,* Dover, 1998) concludes
"that Ohm was not making up the name on the spot and that it had gained at
least some, and perhaps a great deal of currency, by 1835" [Underwood
Dudley].

The term appears in 1844 in J. Helmes in *Arch. **Math. und Physik* IV. 15 in the heading "Eine..Auflösung der
sectio aurea."

The term appears in 1849 in *Der allgemeine goldene Schnitt und sein
Zusammenhang mit der harmonischen Teilung* by A. Wigand.

According to David Fowler, it was the publications of Adolf Zeising's *Neue
Lehre von den Proportionen des menschlischen Körpers* (1854), *Äesthetische
Forschungen* (1855), and *Der goldne Schnitt* (1884) that did the most
to widely popularize the name.

*Golden section* is found in English in 1872 in *The science
of aesthetics; or, The nature, kinds, laws, and uses of beauty* by Henry
Noble Day: "The rule of the 'golden section' has been one of the fruits of
these researches. This principle is the same as the geometrical section into
extreme and mean ratio. A line is said to be so cut when the square on the
larger of the two parts is equal to the rectangle of the whole line and the
less part; or when the whole bears the same ratio to the greater part that this
part bears to the less" [University of Michigan Digital Library].

*Golden mean* appears in English in 1917 in *On Growth and
Form* by Sir D'Arcy Wentworth Thompson (1860-1948): "This celebrated
series, which..is closely connected with the *Sectio aurea* or Golden
Mean, is commonly called the Fibonacci series" (OED2).

The term **GOODNESS OF FIT** is found in the sentence, "The
'percentage error' in ordinate is, of course, only a rough test of the goodness
of fit, but I have used it in default of a better." This citation is a
footnote in "Contributions to the Mathematical Theory of Evolution II Skew
Variation in Homogeneous Material," which was in *Philosophical
Transactions of the Royal Society of London* (1895) Series A, vol 186, pp
343-414 [James A. Landau].

**GOOGOL** and **GOOGOLPLEX** are apparently found in
Edward Kasner, "New Names in Mathematics," *Scripta Mathematica.*
5: 5-14, January 1938.

*Googol* and *googolplex* are found in March 1938
in *The Mathematics Teacher*: "The following examples are of
mathematical terms coined by Prof. Kasner himself: turbine, polygenic
functions, parhexagon, hyper-radical or ultra-radical, googol and googolplex. A
googol is defined as 10^{100}. A googolplex is 10^{googol},
which is 10^{10100}." [This quotation is part of a review of the
January 1938 article above.]

*Googol* and *googolplex* were coined by Milton
Sirotta, nephew of American mathematician Edward Kasner (1878-1955), according
to *Mathematics and the Imagination* (1940) by Kasner and James R. Newman:

Words of wisdom are spoken by children at least as often as by scientists. The name "googol" was invented by a child (Dr. Kasner's nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it. He was very certain that this number was not infinite, and therefore equally certain that it had to have a name. At the same time that he suggested "googol" he gave a name for a still larger number: "Googolplex." A googolplex is much larger than a googol, but is still finite, as the inventor of the name was quick to point out.

This
quotation was taken from the article "New Names for Old" found in *The
World of Mathematics* (1956) by Newman. The article is identified as an
excerpt from *Mathematics and the Imagination.*

**GRAD** or **GRADE** originally meant one ninetieth
of a right angle, but the term is now used primarily to refer to one hundredth
of a right angle.

*Gradus* is a Latin word equivalent to
"degree."

Nicole Oresme called the difference between two successive *latitudines*
a *gradus* (Smith vol. 2, page 319).

The OED2 shows a use of *grade* in English in about 1511, referring
to one-ninetieth of a right angle.

The OED2 shows a use of *grade,* meaning one-hundredth of a right
angle, in 1801 in Dupré *Neolog. Fr. Dict.* 127: "*Grade* .. the
grade, or decimal degree of the meridian."

The term may have been used in the modern sense in the unpublished
French *Cadastre* tables of 1801.

In 1857, *Mathematical Dictionary and Cyclopedia of Mathematical Science*
has: "The French have proposed to divide the right angle into 100 equal
parts, called *grades,* but the suggestion has not been extensively
adopted."

The calculator that is part of Microsoft Windows 98, in the scientific view, allows the user to choose between degrees, radians, and (erroneously) gradients.

**GRADIENT** was introduced by Horace Lamb (1849-1934) in *An
Elementary Course of Infinitesimal Calculus* (Cambridge: Cambridge
University Press, 1897):

It is convenient to have a name for the property of a curve which is measured by the derived function. We shall use the term "gradient" in this sense.

Sylvester used the term in a different sense in 1887 (OED2).

The *DSB* says that Maxwell introduced the term in 1870; this seems
to be incorrect.

**GRAHAM'S NUMBER.** The term "Graham-Spencer number"
appears in N. D. Nenov and N. G. Khadzhiivanov, "On the Graham-Spencer
number," *C. R. Acad. Bulg. Sci.* 32 (1979).

The term "Graham's number" appears the 1985 *Guinness Book
of World Records,* and it may appear in earlier editions of that book.

The number is discussed in M. Gardner, "Mathematical Games," *Sci.
Amer.* 237, Nov. 1977.

**GRAPH (older sense, noun)** is due to Sylvester, according to
the OED2, which states that he shortened the word *graphic* and applied it
to mathematics. The OED2 shows a use of the term by Sylvester in 1878 in *American
Journal of Mathematics* I. 65.

The phrase *graph of a function* was used by Chrystal in 1886 in *Algebra*
I. 307: "This curve we may call the graph of the function" (OED2).

**GRAPH (verb)** is found in 1898 in Perry, *Applied
Mechanics* 21: "Students will do well to graph on squared paper some
curves like the following" (OED2).

**GRAPH (in graph theory)** "appears to have been coined
by A. Cayley," according to an Internet web page.

However, Martin Gardner wrote in *Scientific American* in April
1964, "In the 1930s, the German mathematician Dénes König made the first
systematic study of all such patterns, giving them the generic name
'graphs.'" König
published *Theorie der endlichen und unendlichen Graphen* in Leipzig in
1936.

*Graph theory* appears in English in W. T. Tutte, "A
ring in graph theory," *Proc. **Camb. Philos. Soc.* 43, 26-40 (1947).

**GREAT CIRCLE** is found in English in 1594 in the title, *The
Sea-mans Secrets .. wherein is taught the 3 kindes of Sailing, Horizontall,
Paradoxall, and Sayling vpon a great Circle,* by John Davis. Davis wrote,
"Navigation consiseth of three parts, ... The third is a great Circle
Navigation, which teacheth bow upon a great Circle, drawn between any two
places assigned (being the only shortest way between place and place) the Ship
may be conducted and to performed by the skilful application of Horizontal and
Paraboral Navigation."

**GREATEST COMMON DIVISOR** in Latin books was usually written
as *maximus communis divisor.*

Cataneo in 1546 used *il maggior
commune ripiego* in Italian.

*Greatest common measure* is found in English in 1570 in
Billingsley, *Elem. Geom.*: "It is required of these three magnitudes
to finde out the greatest common measure" (OED2).

Cataldi in 1606 wrote *massima
comune misura* in Italian.

*Highest common divisor* is found in 1858 in Isaac
Todhunter, *Algebra*: "The term greatest common measure is not very
appropriate in Algebra..It would be better to speak of the highest common
divisor or of the highest common measure" (OED2).

*Greatest common divisor* is found in English in 1811 in *An
Elementary Investigation in the Theory of Numbers* [James A. Landau].

In 1881 G. A. Wentworth uses the phrase "highest common
factor" in *Elements of Algebra,* although the phrase "G. C. M.
of *a* and *b*" is found, where the context shows he is
referring to the greatest common divisor [James A. Landau].

Olaus Henrici (1840-1918), in a Presidential address to the London Mathematical Society in 1883, said, "Then there are processes, like the finding of the G. C. M., which most boys never have any opportunity of using, except perhaps in the examination room."

**GREEN'S THEOREM** appears in P. G. Tait, "On Green's and
other allied theorems," *Trans. of Edinb.* (1870).

The theorem bears the name of Mikhail Ostrogradski (1801-1861) in Russia.

**GREGORY'S SERIES** appears in 1859(?) in *Plane Trigonometry*
by the Right Rev. John William Colenso (1814-1883) [University of Michigan
Historical Math Collection].

*Madhava-Gregory series* is found in 1973 in R. C. Gupta,
"The Madhava-Gregory series," *Math. Education* 7 (1973),
B67-B70 [James A. Landau].

A web page by Antreas Hatzipolakis dated Dec. 12, 1998, says the series "is now called the Madhava-Gregory-Leibniz series."

Another web page dated March 5, 2000, calls the series the "Leibniz-Gregory-Madhava series."

The series is said to have been discovered by Madhava, who lived around 1400.

**GROEBNER BASES.** Bruno Buchberger introduced Groebner bases in
1965 and named them for W. Gröbner (1899-1980), his thesis adviser, according
to *Ideals, Varieties, and Algorithms* by Cox, Little, and O'Shea [Paul
Pollack].

The term **GROUP** was coined (as *groupe* in French) by
Evariste Galois (1811-1832). According to Cajori (vol. 2, page 83), the word
group was first used in a technical sense by Galois in 1830. The modern
definition of a group is somewhat different from that of Galois, for whom the
term denoted a subgroup of the group of permutations of the roots of a given
polynomial.

Klein and Lie used the term *closed system* in their earliest
writing on the subject of groups.

The term **GROUP OF AN EQUATION** was used by Galois (Kramer).

**GROUP THEORY.** *Theory of groups* is found in Arthur
Cayley, "On the theory of groups, as depending on the symbolic equation
[theta]^{n}=1," *Philosophical Magazine,* 1854, vol. 7, pp.
40-47. Reprinted in Collected Works as no. 125, pp. 123-130 [Dirk Schlimm].

*Group theory* is found in English in 1898 in *Proc. Calf.
Acad. Science* (OED2).

**GRUNDLAGENKRISIS** (foundational crisis). Walter Felscher writes,
"As far as I am aware, 'Grundlagenkrisis' was a term invented during the
Hilbert-Weyl discussion between 1919 and 1922, occurring e.g. in Weyl's *Über
die neue Grundlagenkrise der Mathematik,* Math.Z. 10 (1921) 39-79."

The term **GUDERMANNIAN** was introduced by Arthur Cayley
(1821-1895), according to Chrystal in *Algebra,* vol. II. The term appears
in an 1862 article by him in the *Philosophical Magazine* [University of
Michigan Historical Math Collection].

**GYROID,** as the name of a minimal surface, was coined by
Alan H. Schoen. (The discovery of this intriguing surface is also due to him.)
On October 31, 2000 Schoen wrote (private correspondence):

My records don't show exactly when I thought of the name "gyroid", but I do find in my files a copy of a letter to Bob Osserman on March 3, 1969 in which I wrote as follows:

The gyroid. This is my latest choice of a name
for this surface, which is the only surface associate to the two
intersection-free adjoint Schwarz surfaces ("P" and "D")
that is free of self-intersections. (*Webster's 3d International Dictionary*
defines gyroidal as "spiral or gyratory in arrangement -- used esp. of the
planes of crystals".)

When Bob wrote back shortly afterward, he mentioned that he approved of the name. I suppose it was at least in part my having studied Latin and Greek in highschool and college that impelled me to search for a classical-sounding name for this surface. As soon as I stumbled on a name that shared its 'oid' ending with the helicoid and catenoid, I decided to look no further!

This entry was contributed by Carlos César de Araújo.

**HAMILTONIAN CIRCUIT.** *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).

**HARMONIC ANALYSIS.** According to Grattan-Guinness
(679), the phrase is due to W. Thomson (later Lord Kelvin). In an obituary of
Archibald Smith (*Proc. Royal Soc.* **22.** (1873 - 1874) p. vi)
Thomson wrote "One of Smith's earliest contributions to the compass problem
was the application of Fourier's grand and fertile theory of the expansion of a
periodic function in series of sines and cosines of the argument and its
multiples, now commonly called the harmonic analysis of a periodic
function." Thomson invented the harmonic analyser; in 1879 the Royal
Society allocated him £50 for "completing a Tidal Harmonic Analyser"
(*Proc. Royal Soc.,* **29,** 442.)

The phrase "harmonic
analysis" was prominent in N. Wiener's writings of the 1920s, see e.g.
"The Harmonic Analysis of Irregular Motion (Second Paper)," *J.
Math. and Phys.* **5** (1926) 158-189. These writings culminated in the
"generalized harmonic analysis" of 1930 (*Acta Mathematica,* 55,
117-258).

In statistics the term is
found in R. A. Fisher, "Tests of significance in harmonic analysis," *Proc.
Roy. Soc. A,* 125, page 54 (1929) [John Aldrich].

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

The term was also used by Aristotle.

According to the *Catholic
Encyclopedia,* the word *harmonic* first appears in a work on conics by
Philippe de la Hire (1640-1718) published in 1685.

*Harmonical mean* is found in English in the 1828 *Webster*
dictionary:

Harmonical mean, in arithmetic and algebra, a term used to express certain relations of numbers and quantities, which are supposed to bear an analogy to musical consonances.

*Harmonic
mean* is found in
1851 in *Problems in illustration of the principles of plane coordinate
geometry* by William Walton [University of Michigan Digital Library].

*Harmonic mean* is also found in 1851 in *The principles of
the solution of the Senate-house 'riders,' exemplified by the solution of those
proposed in the earlier parts of the examinations of the years 1848-1851* by
Francis James Jameson: "Prove that the discount on a sum of money is half
the harmonic mean between the principal and the interest" [University of
Michigan Digital Library].

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

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

In a letter dated Feb. 15, 1671, James Gregory wrote to Collins, "As to yours, dated 24 Dec., I can hardly beleev, till I see it, that there is any general, compendious & geometrical method for adding an harmonical progression...."

**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).

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

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

**HARMONIC SERIES** appears in 1727-51 in *Chambers Cyclopedia*:
"Harmonical series is a series of many numbers in continual harmonical
proportion" (OED2).

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

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

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

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

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

In June 1907 in the *Bulletin des Sciences mathématiques,* Lebesgue
denied any paternity of the theorem and wrote that in his opinion the name of
the theorem should bear only the name of Borel [Udai Venedem].

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

**HEPTAGON.** In 1551 in *Pathway to Knowledge* Robert
Recorde used *septangle.*

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

**HERMITIAN FORM** is found in "On Quadratic, Hermitian and
Bilinear Forms" by Leonard Eugene Dickson, *Transactions of the American
Mathematical Society,* 7 (Apr., 1906).

**HERMITIAN MATRIX** appears in 1866 in Arthur Cayley, "A
Supplementary Memoir on the Theory of Matrices," *Philosophical
Transactions of the Royal Society of London*: "I consider from a
different point of view the theory of a matrix ... or, as we may call it, a
Hermitian matrix" [University of Michigan Historical Math Collection].

The term **HESSIAN** was coined by James Joseph Sylvester
(1814-1897), named for Otto Hesse, who had used the term *functional
determinants.*

*Hessian* appears in his "Sketch of a Memoir on
Elimination, Transformation, and Canonical Forms," *Math. Papers J. S.
S.,* 1:184-197.

*Hessian* appears in 1851 in *Cambr. & Dublin
Math. Jrnl.* 6: "The Hessian, or as it ought to be termed, the first
Boolian Determinant" (OED2).

**HETERO-** and **HOMOSCEDASTICITY.** The terms *heteroscedasticity*
and *homoscedasticity* were introduced in 1905 by Karl Pearson in "On
the general theory of skew correlation and non-linear regression," *Drapers'
Company Res. Mem.* (Biometric Ser.) II. Pearson wrote, "If ... all
arrays are *equally scattered* about their means, I shall speak of the
system as a *homoscedastic* system, otherwise it is a *heteroscedastic*
system." The words derive from the Greek *skedastos* (capable of
being scattered).

Many authors prefer the spelling *heteroskedasticity.* J. Huston
McCulloch (*Econometrica* 1985) discusses the linguistic aspects and
decides for the *k*-spelling. Pearson recalled that when he set up *Biometrika*
in 1901 Edgeworth had insisted the name be spelled with a *k.* By 1932
when *Econometrica* was founded standards had fallen or tastes had
changed. [This entry was contributed by John Aldrich, referring to OED2 and
David, 1995.]

**HEXADECIMAL.** *Sexadecimal* appears in 1891 in the *Century*
dictionary.

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

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

**HEXAGON.** In 1551 in *Pathway to Knowledge* Robert
Recorde used the obsolete word *siseangle*: "Def., Likewyse shall you
iudge of siseangles, which haue sixe corners" (OED2).

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

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

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

**HILBERT SPACE** is found in E. W. Chittenden, "On the
relation between the Hilbert space and the calcul fonctionnel of Frechet,"
*Palermo Rend.* (1921).

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

*Arabic numeral* appears in 1799 in T. Green, *Lover of Lit.*
(1810): "Writing, he deduces, from pictural representations, through
hieroglyphics ... to arbitrary marks ... like the Chinese characters and Arabic
numerals.

*Hindu numerals* is found in 1872 in *Chambers's
encyclopaedia*: "After the introduction of the decimal system and the
Arabic or Hindu numerals about the 11th c., Arithmetic began to assume a new
form..." [University of Michigan Digital Library].

*Indo-Arabic system* appears in 1884 in the *Encyclopaedia
Britannica*: "In Europe, before the introduction of the algorithm or
full Indo-Arabic system with the zero" (OED2).

*Indo-Arabic numeral* appears in 1902 in the second edition of *The
Number-System of Algebra* by Henry B. Fine: "At all events, it is
certain that the Indo-Arabic numerals, 1, 2, ..., 9 (not 0), appeared in
Christian Europe more than a century before the complete positional system and *algorithm.*"
The term may occur in the 1890 edition also.

*Hindu notation* appears in 1906 in *A History of Mathematics*
by Florian Cajori: "Generally we speak of our notation as the 'Arabic'
notation, but it should be called the 'Hindoo' notation, for the Arabs borrowed
it from the Hindoos. ... These Singhalesian signs, like the old Hindoo
numerals, are supposed originally to have been the initial letters of the
corresponding numerical adjectives." Presumably the terms appear in the
earlier 1893 edition of Cajori.

*Hindu-Arabic numeral* appears in 1911 in the title *The
Hindu-Arabic Numerals* by David Eugene Smith and Louis Charles Karpinski
[Julio González Cabillón].

**HISTOGRAM.** The term *histogram* was coined by Karl
Pearson.

In *Philos. Trans. R. Soc. A.* CLXXXVI, (1895) 399 Pearson
explained that term was "introduced by the writer in his lectures on
statistics as a term for a common form of graphical representation, i.e., by
columns marking as areas the frequency corresponding to the range of their
base."

S. M. Stigler writes in his *History of Statistics* that Pearson
used the term in his 1892 lectures on the geometry of statistics.

The earliest citation in the OED2 is in 1891 in E. S. Pearson *Karl
Pearson* (1938).

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

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

Halphen proposed that the terms be replaced by "integral" and "fractional."

**HOMOGENEOUS EQUATIONS** is found in 1815 in the second
edition of Hutton's mathematics dictionary: "Homogeneous Equations ... in
which the sum of the dimensions of *x* and *y*... rise to the same
degree in all the terms" (OED2).

The term **HOMOGRAPHIC** is due to Michel Chasles (1793-1880) (Smith,
1906).

**HOMOLOGOUS** is found in English in 1660 in Barrow's
translation of Euclid: "B and D are homologous or magnitudes of a like
ratio" (OED2).

The term is found in a modern sense in 1879 in *Conic Sections* by
George Salmon (1819-1904): "Two triangles are said to be homologous, when
the intersections of the corresponding sides lie on the same right line called
the axis of homology; prove that the lines joining corresponding vertices meet
in a point" (OED2).

**HOMOLOGY** is found in 1879 in *Conic Sections* by
George Salmon: "Two triangles are said to be homologous, when the
intersections of the corresponding sides lie on the same right line called the
axis of homology; prove that the lines joining corresponding vertices meet in a
point" (OED2).

*Homology* is found in 1885 in Charles Leudesdorf's
translation of Cremona's *Elements of Projective Geometry* "Two
corresponding straight lines therefore always intersect on a fixed straight
line, which we may call s; thus the given figures are in homology, O being the
centre, and s the axis, of homology" (OED2).

*Homology* is found in a more modern usage, originally in
algebraic topology but now more widespread (as in homological algebra) in 1895
in H. Poincare, *Analysis situs* [Joseph Rotman].

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

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

**HORNER'S METHOD** appears in 1842 in the *Penny Cyclopedia*:
"The use of Horner's method is very much more easy than that of
Newton" (OED2).

*The method of Horner* appears in the second edition of *Theory
and Solution of Algebraical Equations* (1843) by J. R. Young.

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

*Hyperbola* was used in English in 1668 by Barrow in
correspondence: "The rules I sent you concerning the hyperbola, I cannot
well exemplify."

*Hyperbola* also appears in English in 1668 in the *Philosophical
Transactions of the Royal Society* (OED2).

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

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

**HYPERBOLIC LOGARITHM.** Because of the relation between
natural logarithms and the areas of hyperbolic sectors, natural logarithms came
to be called *hyperbolic logarithms.* The connection between natural
logarithms and sectors was discovered by Gregory St. Vincent (1584-1667) in
1647, according to Daniel A. Murray in *Differential and Integral Calculus*
(1908).

Abraham DeMoivre (1667-1754) used *Hyperbolic Logarithm* in English
in his own English translation of a paper presented to some friends on Nov. 12,
1733. His translation appears in the second edition (1738) of *The Doctrine
of Chances.*

*Hyperbolic logarithm* appears in 1743 in Emerson, *Fluxions*:
"The Fluxion of any Quantity divided by that Quantity is the Fluxion of
the Hyperbolic Logarithm of that Quantity" (OED2).

Euler called these logarithms "natural or hyperbolic" in 1748
in his *Introductio,* according to Dunham (page 26), who provides a
reference to Vol. I, page 97, of the *Introductio.*

**HYPERBOLIC PARABOLOID** appears in 1836 in the second
edition of *Elements of the Differential Calculus* by John Radford Young
[James A. Landau].

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

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

**HYPERCUBE** is found in *Scientific American* of July
1909: "Of these [regular hyper-solids], C_{8} (or the hyper-cube)
is the simplest, because, though with more bounding solids than C_{5},
it is right-angled throughout" (OED2).

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

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

*Hyperdeterminant* was used by Cayley in 1845 in *Camb. Math.
Jrnl.* IV. 195: "The function u whose properties we proceed to
investigate may be conveniently named a 'Hyperdeterminant'" (OED2).

*Hyperdeterminant* was used by Cayley about 1846 in *Camb.
& Dublin Math. Jrnl.* I. 104: "The question may be proposed 'To
find all the derivatives of any number of functions, which have the property of
preserving their form unaltered after any linear transformations of the
variables'... I give the name of Hyperdeterminant Derivative, or simply of
Hyperdeterminant, to those derivatives which have the property just
enunciated" (OED2).

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

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

The term **HYPERGEOMETRIC CURVE** is found in the title "De
curva hypergeometrica hac aequatione expressa y=1*2*3*...*x" by Leonhard
Euler. The paper was presented in 1768 and published in 1769 in *Novi
Commentarii academiae scientiarum Petropolitanae.*

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

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

However, the term *hypergeometric series* is due to Pfaff,
according to Smith (vol. 2, page 507) and Smith (1906).

The 1816 translation of Lacroix's *Differential and Integral Calculus*
has: "These series, in which the number of factors increases from term to
term, have been designated by Euler ... hypergeometrical series" (OED2).

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

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

**HYPOTENUSE** was used by Pythagoras (c. 540 BC).

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

In English, the word has also been spelled *hypothenusa, hypotenusa,*
and *hypothenuse.*

**HYPOTHESIS** was used in English in a mathematical context
in 1660 by Barrow in his translation of *Euclid* i. xxvii. (1714) 23:
"Which being supposed, the outward angle AEF will be greater than the
inward angle DFE, to which it was equal by Hypothesis" (OED2).

**HYPOTHESIS TESTING.** *Test of hypothesis* is found in 1928 in
J. Neyman and E. S. Pearson, "On the use and Interpretation of Certain
Test Criteria for Purposes of Statistical Inference. Part I," *Biometrika,*
20 A, 175-240 (David, 1995).

**ICOSAHEDRON** is found in English in Sir Henry
Billingsley's 1570 translation of Euclid's *Elements* (OED2).

**IDEAL (point or line)** was introduced as *idéal* by
J. V. Poncelet in *Traité des Propriétés proj. des Figures* (1822).

**IDEAL (number theory)** was introduced by Richard Dedekind
(1831-1916) in P. G. L. Dirichlet *Vorles. über Zahlentheorie* (ed. 2,
1871) Suppl. x. 452 (OED2).

**IDEAL NUMBER.** Ernst Eduard Kummer (1810-1893)
introduced the term *ideale zahl* in 1846 in *Ber. über die zur
Bekanntmachung geeigneten Verh. d. K. Preuss. Akad. d. Wiss. zu Berlin* 87
(OED2).

**IDEMPOTENT** and **NILPOTENT** were used by
Benjamin Peirce (1809-1880) in 1870:

When an expression raised to the square or any
higher power vanishes, it may be called *nilpotent;* but when, raised to a
square or higher power, it gives itself as the result, it may be called *idempotent.*

The defining equation of nilpotent and
idempotent expressions are respectively *A ^{n}* = 0, and

*A*^{2} = *A,*

unless it be otherwise distinctly stated.

This
citation is excerpted from "Linear Associative Algebra," a memoir
read by Benjamin Peirce before the National Academy of Sciences in Washington,
1870, and published by him as a lithograph in 1870. In 1881, Peirce's son,
Charles S. Peirce, reprinted it in the *American Journal of Mathematics.*
[Julio González Cabillón]

The OED2 shows a 1937 citation with a simplified definition of *idempotent*
in *Modern Higher Algebra* (1938) iii 88 by A. A. Albert: "A matrix *E*
is called idempotent if *E*^{2} = *E.* [Older dictionaries
pronounce idempotent with the only stress on the second syllable, but newer
ones show a primary stress on the first syllable and a secondary stress on the
penult.]

**IDENTITY (type of equation)** is found in 1831 in the second
edition of *Elements of the Differential Calculus* (1836) by John Radford
Young: "This is obvious, for this first term is what the whole development
reduces to when *h* = 0, but we must in this case have the identity *f*(*x*)
= *f*(*x*); hence *f*(*x*) is the first term" [James
A. Landau].

Young also uses the term *identical equations* in the same work.

**IDENTITY (element)** is found in 1894 in the *Bulletin of the
American Mathematical Society* I: "Given an (abstract) group G_{n}
... with elements s_{1} = identity, s_{2}, s_{n}
(OED2).

*Identity element* is found in 1902 in *Transactions of the
American Mathematical Society* III. 486: "There exists a left-hand
identity element, that is, an element *i _{l}e* such that, for
every element

**IDENTITY MATRIX** is found in "Representations of the
General Symmetric Group as Linear Groups in Finite and Infinite Fields,"
Leonard Eugene Dickson, *Transactions of the American Mathematical Society,*
Vol. 9, No. 2. (Apr., 1908).

The term is also found in "Concerning Linear Substitutions of
Finite Period with Rational Coefficients," Arthur Ranum, *Transactions
of the American Mathematical Society,* Vol. 9, No. 2. (Apr., 1908).

**IFF.** On the last page of his autobiography, Paul R.
Halmos (1916- ) writes:

My most nearly immortal contributions are an
abbreviation and a typographical symbol. I invented "iff", for
"if and only if" -- but I could never believe that I was really its
first inventor. I am quite prepared to believe that it existed before me, but I
don't *know* that it did, and my invention (re-invention?) of it is what
spread it thorugh the mathematical world. The symbol is definitely not my
invention -- it appeared in popular magazines (not mathematical ones) before I
adopted it, but, once again, I seem to have introduced it into mathematics. It
is the symbol that sometimes looks like [an empty square], and is used to
indicate an end, usually the end of a proof. It is most frequently called the
"tombstone", but at least one generous author referred to it as the
"halmos".

This quote
is from *I Want to Be a Mathematician: An Automathography,* by Paul R.
Halmos, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985, page 403.

The earliest citation of "iff" in the OED2 is 1955 in *General
Topology* by John L. Kelley:

F is equicontinuous at *x* iff there is a
neighborhood of *x* whose image under every member of F is small.

Kelley credited the term to Halmos.

The terms **IMAGINARY** and **REAL** were introduced in French by
Rene Descartes (1596-1650) in "La Geometrie" (1637):

...tant les vrayes racines que les fausses ne sont pas tousiours réelles;
mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien toujiours
en imaginer autant que aiy dit en chàsque Equation; mais qu'il n'y a
quelquefois aucune quantité, qui corresponde à celles qu'on imagine. comme
encore qu'on en puisse imaginer trois en celle cy, x^{3} - 6xx + 13x -
10 = 0, il n'y en a toutefois qu'une réelle, qui est 2, & pour les deux
autres, quois qu'on les augmente, ou diminué, ou multiplié en la façon que ie
viens d'éxpliquer, on ne sçauroit les rendre autres qu'imaginaires. [...neither the true roots nor the
false are always real; sometimes they are, however, imaginary; namely, whereas
we can always imagine as many roots for each equation as I have predicted,
there is still not always a quantity which corresponds to each root so
imagined. Thus, while we may think of the equation x^{3} - 6xx + 13x -
10 = 0 as having three roots, yet there is just one real root, which is 2, and
the other two, however, increased, diminished, or multiplied them as we just
laid down, remain always imaginary.] (page 380)

An early
appearance of the word *imaginary* in English is in "A treatise of
algebra, both historical and practical" (1685) by John Wallis (1616-1703):

We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative)... These *Imaginary* Quantities (as they are commonly called) arising from *Supposed* Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.

The
quotation above is from Chapter LXVI (p. 264), *Of NEGATIVE SQUARES, and
their IMAGINARY ROOTS in Algebra.* This work is a translation of "De
Algebra Tractatus; Historicus & Practicus" written in Latin in 1673.
For the Latin edition of the latter consult "Opera mathematica", vol.
II, Oxoniae, 1693. [Julio González Cabillón]

As a way of removing the stigma of the name, the American mathematician
Arnold Dresden (1882-1954) suggested that imaginary numbers be called *normal*
numbers, because the term "normal" is synonymous with perpendicular,
and the y-axis is perpendicular to the x-axis (Kramer, p. 73). The suggestion
appears in 1936 in his *An Invitation to Mathematics.*

Some other terms that have been used to refer to imaginary numbers include "sophistic" (Cardan), "nonsense" (Napier), "inexplicable" (Girard), "incomprehensible" (Huygens), and "impossible" (many authors).

The first edition of the *Encyclopaedia Britannica* (1768-1771)
has: "Thus the square root of -a^{2} cannot be assigned, and is
what we call an *impossible* or *imaginary* quantity."

There are two modern meanings of the term *imaginary number.* In *Merriam-Webster's
Collegiate Dictionary,* 10th ed., an imaginary number is a number of the
form *a* + *bi* where *b* is not equal to 0. In *Calculus and
Analytic Geomtry* (1992) by Stein and Barcellos, "a complex number that
lies on the *y* axis is called **imaginary.**"

The term **IMAGINARY GEOMETRY** was used by Lobachevsky, who in 1835
published a long article, "Voobrazhaemaya geometriya" (Imaginary
Geometry).

The term **IMAGINARY PART** appears in 1836 in the second edition of *Elements
of the Differential Calculus* by John Radford Young [James A. Landau].

The term **IMAGINARY UNIT** was used (and apparently introduced) by
by Sir William Rowan Hamilton in "On a new Species of Imaginary Quantities
connected with a theory of Quaternions," *Proceedings of the Royal Irish
Academy,* Nov. 13, 1843: "...the extended expression...which may be
called an imaginary unit, because its modulus is = 1, and its square is
negative unity."

**IMPLICIT DEFINITION.** In the literature of mathematics,
this term was introduced by Joseph-Diaz Gergonne (1771-1859) in *Essai sur la
théorie des définitions*, Annales de Mathématique Pure et Appliquée (1818)
1-35, p. 23. (The *Annales* begun to be published by Gergonne himself in
1810.) He also emphasized the contrast between this kind of definition and the
other "ordinary" ones which, according to him, should be called
"explicit definitions". According to his own example, given the words
"triangle" and "quadrilateral" we can define (implicitly)
the word "diagonal" (of a quadrilateral) in a satisfactory way just
by means of a *property* that individualizes it (namely, that of dividing
the quadrilateral in two equal triangles). Gergonne’s observations are now
viewed by many as an anticipation of the "modern" idea of
"definition by axioms" which was so fruitfully explored by Dedekind,
Peano and Hilbert in the second half of the nineteenth century. In fact, still
today the axioms of a theory are treated in many textbooks as "implicit
definitions" of the primitive concepts involved. We can also view
Gergonne’s ideas as anticipating, to a certain extent, the use of
"contextual definitions" in Russell’s theory of descriptions (1905). [Carlos César de Araújo]

**IMPLICIT DIFFERENTIATION** is dated ca. 1889 in MWCD10.

**IMPLICIT FUNCTION** is found in 1814 *New Mathematical and
Philosophical Dictionary*: "Having given the methods ... of obtaining
the derived functions, of functions of one or more quantities, whether those
functions be explicit or implicit, ... we will now show how this theory may be
applied" (OED2).

**IMPROPER FRACTION** was used in English in 1542 by Robert Recorde
in *The ground of artes, teachyng the worke and practise of arithmetike*:
"An Improper Fraction...that is to saye, a fraction in forme, which in
dede is greater than a Unit."

**IMPROPER DEFINITE INTEGRAL** occurs in "Concerning
Harnack's Theory of Improper Definite Integrals" by Eliakim Hastings
Moore, *Trans. Amer. Math. Soc.,* July 1901.

*Improper integral* appears in the same paper.

**INCENTER** is dated ca. 1890 in MWCD10.

**INCIRCLE** was used in 1883 by W. H. H. Hudson in *Nature*
XXVIII. 7: "I beg leave to suggest the following names: circumcircle,
incircle, excircle, and midcircle" (OED2).

**INCLUDED** (angle or side) appears in 1806 in Hutton, *Course
Math.*: "If two Triangles have Two Sides and the Included Angle in the
one, equal to Two Sides and the Included Angle in the other, the Triangles will
be Identical, or equal in all respects" (OED2).

In 1828, *Elements of Geometry and Trigonometry* (1832) by David
Brewster (a translation of Legendre) has: "...Two triangles are equal when
they have two angles and an interjacent side in each equal."

**INCOMMENSURABLE.** *Incommensurability* is found in Latin in
the 1350s in the title *De commensurabilitate sive incommensurabilitate
motuum celi* (the commensurability or incommensurability of celestial
motions) by Nicole Oresme.

The term **INDEFINITE INTEGRAL** is defined by Sylvestre-François
Lacroix (1765-1843) in *Traité du calcul différentiel et integral* (Cajori
1919, page 272).

*Indefinite integral* also appears in 1831 in *Elements of the
Integral Calculus* (1839) by J. R. Young:

In the practical applications of the calculus,
it is not the general, or, as it is usually called, the *indefinite,*
integral that is ultimately required, because here the constant which completes
the integral is indeterminate, whereas, in every particular inquiry this
constant has a corresponding particular value, thus rendering the integral *definite.*

*Indefinite
integral* also
appears in 1835 in "On the determination of the attractions of ellipsoids
of variable densities" by George Green [University of Michigan Historical
Math Collection].

**INDEPENDENT EVENT** and **DEPENDENT EVENT** are found in 1738
in *The Doctrine of Chances* by De Moivre: "Two Events are
independent, when they have no connexion one with the other, and that the
happening of one neither forwards nor obstructs the happening of the other. Two
events are dependent, when they are so connected together as that the
Probability of either's happening is alter'd by the happening of the
other."

**INDEPENDENT VARIABLE** is is found in the 1816 translation
of *Differential and Integral Calculus* by Lacroix: "Treating the
subordinate variables as implicit functions of the independent ones"
(OED2).

**INDETERMINATE FORM** is found in *An Elementary Treatise on
Curves, Functions and Forces* (1846) by Benjamin Peirce (1809-1880).

Forms such as 0/0 are called *singular values* and *singular
forms* in in 1849 in *An Introduction to the Differential and Integral
Calculus,* 2nd ed., by James Thomson.

In *Primary Elements of Algebra for Common Schools and Academies*
(1866) by Joseph Ray, 0/0 is called "the symbol of indetermination."

**INDEX.** Schoner, writing his commentary on the work of
Ramus, in 1586, used the word "index" where Stifel had used
"exponent" (Smith vol. 2).

**INDICATOR.** See *totient.*

**INDICATOR FUNCTION** and **INDICATOR RANDOM VARIABLE.** The term
*indicator* of a set appears in M. Loève's *Probability Theory*
(1955) and, according to W. Feller (*An Introduction to Probability Theory
and its Applications volume II*), Loève was responsible for the term.
Loève's *Probability Theory* did not use term *indicator random variable*
but this soon appeared, see e.g. H. D. Brunk's "On an Extension of the
Concept Conditional Expectation" *Proceedings of the American
Mathematical Society,* **14**, (1963), pp. 298-304. (See *characteristic
function* of a set) [John Aldrich]

The term **INDUCTION** was first used in the phrase *per modum
inductionis* by John Wallis in 1656 in *Arithmetica Infinitorum.*
Wallis was the first person to designate a name for this process; Maurolico and
Pascal used no term for it (Burton, page 440). [See also *mathematical
induction, complete induction, successive induction.* ]

**INDUCTIVE (PARTIALLY) ORDERED SET.** The adjective "inductive"
used in this context was introduced by Bourbaki in *Élements de mathématique.
**I. Théorie des
ensembles. Fascicule de résultats,* Actualités Scientifiques et Industrielles, no. 846, Hermann, Paris, 1939. Bourbaki's original term and
definition is now standard among mathematicians: a poset (*X,* £) is inductive if every totally ordered subset of it has a supremum,
that is:

(1) ("*A* Ì *X*) (*A* is a chain Þ *A has a supremum*).

A notion of "completeness" is usually associated with conditions of this kind. Thus, if

(2) (" *A* Ì *X*) (*A has a supremum*),

then (*X,* £) becomes a "complete
lattice." Similarly, (*X,* £) is said to be
"order-complete" (or "Dedekind-complete") if

(3) (" *A* Ì*X*) (*A* Æ and *A has
an upper bound* Þ *A has a supremum*).

This may explain why some computer scientists prefer the term "complete poset" instead of "inductive poset." (However, "complete poset" is also used by many of them in a related but different sense.)

[Carlos César de Araújo]

**INFINITE DESCENT.**
Pierre de Fermat (1607?-1665) used the term *method of infinite descent*
(Burton, page 488; DSB).

A paper by Fermat is titled "La méthode de la 'descente infinie ou indéfinie.'" Fermat stated that he named the method.

The term **INFINITELY SMALL** was used by Christian Huygens
(1629-1695) (*DSB*).

The term **INFINITESIMAL ANALYSIS** was used in 1748 by Leonhard
Euler in *Introductio in analysin infinitorum* (Kline, page 324).

**INFIX (notation)** is found in D. Wood, "A proof of
Hamblin's algorithm for translation of arithmetic expressions from infix to
postfix form," *BIT, Nordisk Tidskr. Inform.-Behandl.* 9 (1969).

**INFLECTION POINT** appears in a 1684 paper by Leibniz, according
to Katz (page 528), who has a footnote referring to Struik, Source Book, page
275.

*Point of inflexion* appears in 1743 in *Fluxions* by Emerson:
"The Point of Inflexion or contrary Flexure is that Point which separates
the convex from the concave Part of the Curve" (OED2).

In *An Elementary Treatise on Curves, Functions and Forces* (1846),
Benjamin Peirce writes, "When a curve is continuous at a point, but
changes its direction so as to turn its curvature the opposite way at this
point, the point is called *a point of contrary flexure,* or *a point of
inflexion.*"

**INFORMATION, AMOUNT OF, QUANTITY OF** in the theory of statistical
estimation. R. A. Fisher first wrote about "the whole of the information
which a sample provides" in 1920 (*Mon. Not. Roy. Ast. Soc.,* 80,
769). In 1922-5 he developed the idea that information could be given
quantitative expression as minus the expected value of the second derivative of
the log-likelihood. The formula for "the amount of information in a single
observation" appears in the 1925 "Theory of Statistical
Estimation," *Proc. **Cambr. Philos. Soc.*
22. 700-725. In the
modern literature the qualification Fisher's information is common,
distinguishing Fisher's measure from others originating in the theory of
communication as well as in statistics. [John Aldrich and David (1995)].

**INFORMATION THEORY.** The OED2 shows a number of citations for this
term from 1950.

**INJECTION** was used in 1950 by S. MacLane in the *Bulletin
of the American Mathematical Society* (OED2).

**INJECTIVE** was used in 1952 by Eilenberg and Steenrod in *Foundations
of Algebraic Topology* (OED2).

The term **INNER PRODUCT** was coined (in German as *inneres
produkt*) by Hermann Günther Grassman (1809-1877) in *Die lineale
Ausdehnungslehre* (1844).

According to the OED2 it is "so named because an inner product of two vectors is zero unless one has a component 'within' the other, i.e. in its direction."

According to Schwartzman (p. 155):

When the German Sanskrit scholar Hermann
Günther Grassman (1809-1877) developed the general algebra of hypercomplex
numbers, he realized that more than one type of multiplication is possible. To
two of the many possible types he gave the names *inner* and *outer.*
The names seem to have been chosen because they are antonyms rather than for
any intrinsic meaning.

In English,
*inner product* is found in a 1909 *Webster* dictionary, although
Cajori (1928-29) uses the terms *internal* and *external product.*

The term **INNUMERACY** was popularized as the title of a recent book
by John Allen Paulos. The word is found in 1959 in *Rep. Cent. Advisory
Council for Educ. (Eng.)* (Ministry of Educ.): "If his numeracy has
stopped short at the usual Fifth Form level, he is in danger of relapsing into
innumeracy" (OED2).

**INTEGER** and **WHOLE NUMBER.** Writing in Latin,
Fibonacci used *numerus sanus.*

According to Heinz Lueneburg, the term *numero sano* "was used
extensively by Luca Pacioli in his *Summa.* Before Pacioli, it was already
used by Piero della Francesca in his *Trattato d'abaco.* I also find it in
the second edition of Pietro Cataneo's *Le pratiche delle due prime
matematiche* of 1567. I haven't seen the first edition. Counting also
Fibonacci's Latin *numerus sanus,* the word *sano* was used for at
least 350 years to denote an integral (untouched, virginal) number. Besides the
words *sanus, sano,* the words *integer, intero, intiero* were also
used during that time."

The first citation for *whole number* in the OED2 is from about
1430 in *Art of Nombryng* ix. EETS 1922:

Of nombres one is lyneal, ano(th)er superficialle, ano(th)er quadrat, ano(th)cubike or hoole.

In the
above quotation (th) represents a thorn. In this use, *whole number* has
the obsolete definition of "a number composed of three prime
factors," according to the OED2.

*Whole number* is found in its modern sense in the title of
one of the earliest and most popular arithmetics in the English language, which
appeared in 1537 at St. Albans. The work is anonymous, and its long title runs
as follows: "An Introduction for to lerne to reken with the Pen and with
the Counters, after the true cast of arismetyke or awgrym in hole numbers, and
also in broken" (Julio González Cabillón).

Oresme used *intégral.*

*Integer* was used as a noun in English in 1571 by
Thomas Digges (1546?-1595) in *A geometrical practise named Pantometria*:
"The containing circles Semidimetient being very nighe 11 19/21 for
exactly nether by integer nor fraction it can be expressed" (OED2).

*Integral number* appears in 1658 in Phillips: "In
Arithmetick integral numbers are opposed to fraction[s]" (OED2).

*Whole number* is most frequently defined as Z+, although it
is sometimes defined as Z. In *Elements of the Integral Calculus* (1839)
by J. R. Young, the author refers to "a whole number or 0" but later
refers to "a positive whole number."

**INTEGRABLE** is found in English in 1727-41 in Chambers' *Cyclopaedia*
(OED2).

The word **INTEGRAL** first appeared in print by Jacob (or James or
Jacques I) Bernoulli (1654-1705) in May 1690 in *Acta eruditorum,* page
218. He wrote, "Ergo et horum Integralia aequantur" (Cajori vol. 2,
page 182; Ball). According to the DSB this represents the first use of *integral*
"in its present mathematical sense."

However, Jean I (or Johann or John) Bernoulli (1667-1748) also claimed to have introduced the term. According to Smith (vol. I, page 430), "the use of the term 'integral' in its technical sense in the calculus" is due to him.

The the following terms to classify solutions of nonlinear first order
equations are due to Lagrange: *complete solution* or *complete
integral, general integral, particular case* of the general integral, and *singular
integral* (Kline, page 532).

**INTEGRAL CALCULUS.** Leibniz originally used the term *calculus
summatorius* (the calculus of summation) in 1684 and 1686.

Johann Bernoulli introduced the term *integral calculus.*

Cajori (vol. 2, p. 181-182) says:

At one time Leibniz and Johann Bernoulli
discussed in their letters both the name and the principal symbol of the
integral calculus. Leibniz favored the name *calculus summatorius* and the
long letter [long S symbol] as the symbol. Bernoulli favored the name *calculus
integralis* and the capital letter *I* as the sign of integration. ...
Leibniz and Johann Bernoulli finally reached a happy compromise, adopting
Bernoulli's name "integral calculus," and Leibniz' symbol of
integration.

According
to Smith (vol. 2, page 696), Leibniz in 1696 adopted the term *calculus
integralis,* already suggested by Jacques Bernoulli in 1690.

According to Stein and Barcellos (page 311), the term *integral
calculus* is due to Leibniz.

The term "integral calculus" was used by Leo Tolstoy in *Anna
Karenina,* in which a character says, "If they'd told me at college
that other people would have understood the integral calculus, and I didn't,
then ambition would have come in."

**INTEGRAL DOMAIN** is found E. J. Finan, "A determination of
the integral domains of the complete rational matric algebra of order 4," *Bulletin
A. M. S.* (1930).

**INTEGRAL EQUATION (calculus sense).** According to Kline (page 1052) and
Cajori 1919 (page 393), the term *integral equation* is due to Paul du
Bois-Reymond (1831-1889), *Jour. für Math.,* 103, 1888, 288. However,
Euler used a phrase which is translated *integral equation* in the paper
"De integratione aequationis differentialis," *Novi Commentarii
Academiae Scientarum Petropolitanae 6, 1756-57* (1761) [James A. Landau].

*Integral equation* is found in English in 1802 in Woodhouse, *Phil.
Trans.* XCII. 95: "Expressions deduced from the true integral
equations" (OED2).

The term **INTEGRAL GEOMETRY** is due to Wilhelm Blaschke
(1885-1962), according to the University of St. Andrews website.

**INTEGRAND.** Sir William Hamilton of Scotland used this word
in logic. It appears in his *Lectures on metaphysics and logic*
(1859-1863): "This inference of Subcontrariety I would call Integration,
because the mind here tends to determine all the parts of a whole, whereof a
part only has been given. The two propositions together might be called the
integral or integrant (propositiones integrales vel integrantes). The given
proposition would be styled the integrand (propositio integranda); and the
product, the integrate (propositio integrata)" [University of Michigan
Digital Library].

*Integrand* appears in the calculus sense in 1893 in *A
Treatise on the Theory of Functions* by James Harkness and Frank Morley:
"When *c* moves to *t,* the integrand of *u*_{2}
remains finite and continuous."

**INTEGRATING FACTOR** is found in May 1845 in a paper by Sir George
Gabriel Stokes published in the *Cambridge Mathematical Journal*
[University of Michigan Historical Math Collection].

**INTEGRATION BY PARTS** appears in 1831 in *Elements of
the Integral Calculus* (1839) by J. R. Young: "...a formula which
reduces the integration of *udv* to that of *vdu,* and which is known
by the name of *integration by parts.*"

The term appears in a paper by George Green under the heading "General Preliminary Results."

The method was invented by Brook Taylor and discussed in *Methodus
incrementorum directa et inversa* (1715).

**INTEGRATION BY SUBSTITUTION** is found in about 1870 in *Practical
treatise on the differential and integral calculus, with some of its
applications to mechanics and astronomy* by William Guy Peck: "Integration
by Substitution, and Rationalization. 67. An irrational differential may
sometimes be made rational, by substituting for the variable some function of
an auxiliary variable; when this can be done, the integration may be effected
by the methods of Articles 65 and 66. When the differential cannot be
rationalized in terms of an auxiliary variable, it may sometimes be reduced to
one of the elementary forms, and then integrated" [University of Michigan
Digital Library].

**INTERIOR ANGLE** is found in English in 1756 in Robert Simson's
translation of Euclid: "The three interior angles of any triangle are
equal to two right angles" (OED2).

**INTERMEDIATE VALUE THEOREM** appears in 1937 in *Differential
and Integral Calculus,* 2nd. ed. by R. Courant [James A. Landau].

The term **INTERPOLATION** was introduced into mathematics by John
Wallis (DSB; Kline, page 440).

The word appears in the English translation of Wallis' algebra
(translated by Wallis and published in 1685), although the use that has been
found in the excerpt in Smith's *Source Book in Mathematics* appears not
to be his earliest use of the term.

**INTERQUARTILE RANGE** is found in 1882 in Francis Galton,
"Report of the Anthropometric Committee," *Report of the 51st
Meeting of the British Association for the Advancement of Science, 1881,*
pp. 245-260: "This gave the upper and lower 'quartile' values, and
consequently the 'interquartile' range (which is equal to twice the 'probable
error') (OED2).

**INTERSECTION** (in set theory) is found in *Webster's New
International Dictionary* of 1909.

**INTRINSICALLY CONVERGENT SEQUENCE** is the term used by Courant for
"Cauchy sequence" in *Differential and Integral Calculus,* 2nd.
ed. (1937) [James A. Landau].

The term **INTRINSIC EQUATION** was introduced in 1849 by William
Whewell (1704-1886) (Cajori 1919, page 324).

**INVARIANT** appears in 1851 in James Joseph Sylvester,
"On A Remarkable Discovery in the Theory of Canonical Forms and of
Hyperdeterminants," *Philosophical Magazine,* 4th Ser., 2, 391-410:
"The remaining coefficients are the two well-known hyperdeterminants, or,
as I propose henceforth to call them, the two Invariants of the form ax^{4}
+ 4bx^{3}y + 6cx^{2}y^{2} + 4dxy^{3} + ey^{4}."
In the same article he wrote, "If I (a, b,..l) = I (a', b',..l'), then I
is defined to be an invariant of f."

The term is due to Sylvester (1814-1897), according to Cajori (1919,
page 345) and Kline (page 927), who supplies the reference *Coll. Math.
Papers,* I, 273. Sylvester coined the term in 1851, according to Karen
Hunger Parshall in "Toward a History of Nineteenth-Century Invariant
Theory."

See also *normal subgroup.*

**INVERSE (element producing identity element)** appears in 1900 in *Ann. Math.*
(OED2).

**INVERSE (in logic)** appears in 1896 in Welton, *Manual of Logic*:

Inversion is the inferring, from a given proposition, another proposition whose subject is the contradictory of the subject of the original proposition. The given proposition is called the Invertend, that which is inferred from it is termed the Inverse... The rule for Inversion is: Convert either the Obverted Converse or the Obverted Contrapositive.

[OED2]

**INVERSE FUNCTION** appears in in English in 1816 in the
translation of Lacroix's *Differential and Integral Calculus*: "*e ^{x}*
and log

*Inverse function* also appears in 1849 in *An Introduction to
the Differential and Integral Calculus,* 2nd ed., by James Thomson: "A
very convenient notation for expressing these and other *inverse*
functions, as they have been called, has been proposed by Sir John Herschel."

The term **INVERSE GAUSSIAN DISTRIBUTION** was coined in 1948 by M.
C. K. Tweedie, according to Gerard Letac.

**INVERSE VARIATION.** *Inverse ratio* and *inversely* are
found in English in 1660 in Barrow's translation of Euclid.

*Inverse proportion* is found in 1793 in Beddoes, *Math. Evid.*:
"A balance of which one arm should be ten inches, and the other one inch
long, and each arm should be loaded in an inverse proportion to its
length" (OED2).

*Inversely proportional* is found in Thomas Graham, "On
the Law of the Diffusion of Gases," *Philosophical Magazine* (1833).
The paper was read before the Royal Society in Edinburgh on Dec. 19, 1831:
"Which volumes are not necessarily of equal magnitude, being, in the case
of each gas, inversely proportional to the square root of the density of that
gas." [James A. Landau]

*Varies inversely* is found in 1834 in M. Somerville, *Connex.
Phys. Sc.* xxv. (1849): "The elasticity or tension of steam..varies
inversely as its volume" (OED2).

*Inverse variation* is found in 1856 in *Ray's higher
arithmetic. The principles of arithmetic, analyzed and practically applied*
by Joseph Ray (1807-1855):

Variation is a general method of expressing proportion often used, and is either direct or inverse. Direct variation exists between two quantities when they increase togeether, or decrease together. Thus the distance a ship goes at a uniform rate, varies directly as the time it sails; which means that the ratio of any two distances is equal to the ratio of the corresponding times taken in the same order. Inverse variation exists between two quantities when one increases as the other decreases. Thus, the time in which a piece of work will be done, varies inversely as the number of men employed; which means that the ratio of any two times is equal to the ratio of the numbers of men employed for these times, taken in reverse order.

This citation was taken from the University of Michigan Digital Library [James A. Landau].

**INVERTIBLE** is found in the phrase "invertible
elements of a monoid *A*" in 1956 in *Fundamental Concepts of
Algebra* ii. 27 by C. Chevelley (OED2).

The term **INVOLUTION** is due to Gérard Desargues (1593-1662)
(Kline, page 292).

**IRRATIONAL.** Cajori (1919, page 68) writes, "It is
worthy of note that Cassiodorius was the first writer to use the terms
'rational' and 'irrational' in the sense now current in arithmetic and
algebra."

*Irrational* is used in English by Robert Recorde in 1551
in *The Pathwaie to Knowledge*: "Numbres and quantitees surde or
irrationall."

The term **IRREDUCIBLE INVARIANT** was used by Arthur Cayley
(1821-1895).

**ISOGRAPHIC** is the word used by Ernest Jean Philippe
Fauquede Jonquiéres (1820-1901) to describe the transformations he had
discovered, later called *birational transformations* (DSB).

**ISOMETRIC.** *Isometrical* is found in 1838 in the
title *Treatise on Isometrical Drawing* by T. Sopwith (OED2).

*Isometric* is found in the *Penny Cyclopaedia* in
1840: "This specific application of projection was termed isometric by the
late Professor Farish, who pointed out its practical utility, and the facility
of its application to the delineation of engines, etc. ... A scale for
determining the lengths of the axes of the isometric projection of a
circle" (OED2).

*Isometrische Abbildung* (isometric mapping) is found in the
1944 edition of Hausdorff's *Grundzuge der Mengenlehre* and may occur in
the first 1914 edition [Gerald A. Edgar].

**ISOMETRY.** Aristotle used the word *isometria.*

*Isometry* is found in English in *Appletons'
Cyclopaedia of Drawing* edited by W. E. Worthen, which is dated 1857 but
appears to be cited in a catalog printed in 1853 [University of Michigan
Digital Library].

In its modern sense, *isometry* occurs in English in 1941 in *Survey
of Modern Algebra* by MacLane and Birkhoff: "An obvious example is
furnished by the symmetries of the cube. Geometrically speaking, these are the
one-one transformations which preserve distances on the cube. They are known as
'isometries,' and are 48 in number" (OED).

**ISOMORPHISM** was used
by Walter Dyck (1856-1934) in 1882 in *Gruppentheoretische Studien* (Katz,
page 675).

**ISOSCELES** was used in English 1551 by Robert Recorde in *The
Pathwaie to Knowledge*: "There is also an other distinction of the name
of triangles, according to their sides, whiche other be all equal...other els
two sydes bee equall and the thyrd vnequall, which the Greekes call *Isosceles,*
the Latine men *aequicurio,* and in english tweyleke may they be
called."

In English, an isosceles triangle was called an *equicrure* in 1644
and an *equicrural triangle* in 1650 (OED2). These are the earliest uses
for the alternate term of Latin origin in the OED2.

The term **ITERATED FUNCTION SYSTEM** was coined by Michael Barnsley,
according to an Internet website.

**J-SHAPED** is found in 1911 in *An Introduction to the
Theory of Statistics* by G. U. Yule (David, 1995).

The term **JACOBIAN**
was coined by James J. Sylvester (1814-1897), who used the term in 1852 in *The
Cambridge & Dublin Mathematical Journal.*

Sylvester also used the
word in 1853 *Philosophical Transactions of the Royal Society of London,*
CXLIII, Part III, pp. 407-548: "In Arts. 65, 66, I consider the relation
of the Bezoutiant to the differential determinant, so called by Jacobi, but
which for greater brevity I call the Jacobian."

**JERK** was used by J. S. Beggs in 1955 in *Mechanism*
iv. 122: "Since the forces to produce accelerations must arise from
strains in the materials of the system, the rate of change of acceleration, or
jerk, is important" (OED2).

**JORDAN CURVE** appears in W. F. Osgood, "On
the Existence of the Green's Function for the Most General Simply Connected
Plane Region," *Transactions of the American Mathematical Society,*
Vol. 1, No. 3. (July 1900): "By a *Jordan curve* is meant a curve of
the general class of continuous curves without multiple points, considered by
Jordan, *Cours d'Analyse,* vol. I, 2d edition, 1893..." (OED2).

**JORDAN CURVE THEOREM** is dated 1915-20 in RHUD2.

*Jordan curve-theorem* is found in D. W. Woodard, "On
two-dimensional analysis situs with special reference to the Jordan
curve-theorem," *Fundamenta* (1929).

*Jordan curve theorem* is also found in L. Zippin,
"Continuous curves and the Jordan curve theorem," *Bulletin A. M.
S.* (1929).

** **

**k-STATISTICS.** k-statistics are sample cumulants
and were introduced with them by R. A. Fisher in 1929. The *term*
"k-statistic" appears in the 1932 edition of his *Statistical
Methods for Research Workers* [John Aldrich].

**KERNEL (an integrand).** David Hilbert used the German word *kern*
in *Nachrichten von d. Königl. Ges. d. Wissensch. zu Göttingen*
(Math.-physik. Kl.)
(1904) 49 (OED2)

*Kernel* occurs in English in 1909 in *Introd.
Study Integral Equations* by M. Bôcher: "K is called the kernel of
these equations" (OED2).

**KERNEL (elements mapped
into identity element)** is found in English in 1946 in E. Lehmer's translation of *Pontrjagin's
Topological Groups* (OED2).

**KITE.** *Deltoid* appears in 1879 in *Dictionary
of Scientific Terms*: "*Deltoid,* a four-sided figure formed of
two unequal isosceles triangles on opposite sides of a common base"
(OED2).

*Kite* appears as a geometric term in the
1893 *Funk and Wagnalls Standard Dictionary.*

**KLEIN BOTTLE** occurs in C. Tompkins, "A flat
Klein bottle isometrically embedded euclidean 4-space," *Bull. Am. Math.
Soc.* 47, 508 (1941).

**KLEIN FOUR GROUP.** *Vierergruppe* is found in 1884 in *Vorlesungen uber das
Ikosaeder und die Aufloesung der Gleichungen vom funften Grade* by Felix
Klein:

Offenbar umfasst unsere neue Gruppe von der Identitaet abgesehen nur
Operationen von der Periode 2, und es ist zufaellig, das wir eine dieser
Operationen an die Hauptaxe der Figur, die beiden anderen an die Nebenaxe
geknupft haben. Dementsprechend will ich die Gruppe mit einem besonderen Namen
belegen, der nicht mehr and die Dieder- configuration erinnert, und sie als *Vierergruppe*
benennen.

The above citation was provided by Gunnar Berg.

The term "(Kleinsche) Vierergruppe" was used by Bartel
Leendert van der Waerden (1903-1996) in 1930 in his influential textbook *Moderne
Algebra.* It denotes the permutation group generated by (12)(34) and
(13)(24), rather than the abstract product of two 2-cyclic groups. The term
does not occur in the older algebra books by Weber and by Perron [Peter Flor].

The term **KLEINIAN GROUP** was used by Henri Poincaré.

**KNOT.** The first mathematical paper which mentions
knots is "Remarques sur les problemes de situation" (1771) by Alexandre-Theophile
Vandermonde (1735-1796).

**KNOT THEORY** appears in 1932 in the title *Knotentheorie*
by Kurt Werner Friedrick Reidemeister (1893-1971).

**KOLMOGOROV-SMIRNOV TEST** appears in F. J. Massey Jr.,
"The Kolmogorov-Smirnov test of goodness of fit," *J. Amer.
Statist. Ass.* 46 (1951).

See also W. Feller, "On the Kolmogorow-Smirnov limit theorems for
empirical distributions," *Ann. Math. Statist.* 19 (1948) [James A.
Landau].

The term **KRONECKER DELTA** is found in 1926 in *Riemannian
Geometry* by Luther Pfahler Eisenhard: "These are called the Kronecker
deltas and are used frequently throughout this work." [Joanne M. Despres
of Merriam-Webster Inc.]

**KURTOSIS** was used by
Karl Pearson in 1905 in "Das Fehlergesetz und seine Verallgemeinerungen
durch Fechner und Pearson. A Rejoinder," *Biometrika,* 4, 169-212, in the phrase
"the degree of kurtosis." He states therein that he has used the term
previously (OED2).