Earliest Known Uses of Some of the Words of Mathematics

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 10¹⁰⁰. A googolplex is 10^googol, which is 10¹⁰¹⁰⁰." [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], C₈ (or the hyper-cube) is the simplest, because, though with more bounding solids than C₅, it is right-angled throughout" (OED2).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Aⁿ = 0, and Aⁿ = A; but with reference to idempotent expressions, it will always be assumed that they are of the form

A² = 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 E² = 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 s₁ = identity, s₂, 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_le such that, for every element a, i_la = a" (OED2).

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, x³ - 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 x³ - 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 -a² 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 ÌX) (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 u₂ 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 ax⁴ + 4bx³y + 6cx²y² + 4dxy³ + ey⁴." 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."