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

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 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. Théorie de Galois appears as a section heading in Camille Jordan's in Traité des substitutions et des équations algébriques (1870). Here the term refers to higher degree congruences [Margherita Barile].

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.

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 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 IV: "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.

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.

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.

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 is due to James Joseph 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.

In a note "Chemistry and Algebra" in Nature 17 (1877-1878), Sylvester wrote: "Every invariant and covariant thus becomes expressible by a graph precisely identical with a Kekulean diagram or chemicograph" [Len Smiley, Adalbert Kerber].

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 (of a function). 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 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.

Group appears in English in Arthur Cayley, "On the theory of groups, as depending on the symbolic equation θⁿ = 1," Philosophical Magazine, 1854, vol. 7, pp. 40-47: "A set of symbols, 1, α, β, ..., all of them different, and such that the product of any two of them (no matter what order), or the product of any one of them into itself, belongs to the set, is said to be a group."

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 θⁿ = 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!