GALOIS GROUP is found in 1899 in the Bulletin of the American Mathematical Society (OED).
GALOIS THEORY is found in 1893 in the Bulletin of the New York Mathematical Society.
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 Gonz嫮ez Cabill鏮.
The word GAUGE (in gauge theory) was coined by Hermann Weyl (1885-1955).
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 (OED2).
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 1874 in Rep. Brit. Assoc. (1873) (OED2).
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) [University of St. Andrews website].
GEOMETRIC MEAN. The term geometrical mean is found in the 1771 edition of the Encyclopaedia Britannica [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].
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 is found in English in 1837 (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: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).
"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).
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.
G淗EL'S INCOMPLETENESS THEOREM. The term G鐰el'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).
GOLDEN SECTION. According to Greek Mathematical Works I - Thales to Euclid (which is Loeb 335): "This ratio is never called the Golden Section in Greek mathematics. The name appeared in print for the first time, as the goldene Schnitt, in Die reine Elementar-Mathematik by Martin Ohm (1835)." This citation is from a footnote on page 510 [John Conway].
According to an Internet web page, Euclid used Reliqua Sectio and Leonardo used Sectio Aurea.
According to Schwartzman (page 100) golden section was apparently first used in print in 1835 by Georg Simon Ohm.
According to Smith (vol. 2, page 291), "This term seems to have come into general use in the 19th century. It is found in the Archiv der Math. und Physik (IV, 15-22) as early as 1844."
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 both dated 1938 in MWCD10. Both terms 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 (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).
Grade, defined as a hundredth of a right angle, is found in 1898 in Houston Elec. Dict., in which both spellings are given. [Joanne M. Despres of Merriam-Webster Inc.] The term may have been used in the unpublished French Cadastre tables of 1801.
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 D幯es K霵ig made the first systematic study of all such patterns, giving them the generic name 'graphs.'" K霵ig 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 was used by John Davis in Seamans Secrets, which was written in 1594: "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.
Cataldi in 1606 wrote massima comune misura in Italian.
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].
Greatest common divisor is found in English in "A New Definition of the General Abelian Linear Group," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 1, No. 1. (Jan., 1900). The term may also occur much earlier.
GREEN'S THEOREM appears in the 1902 Encyclopaedia Britannica [James A. Landau].
GROEBNER BASES. Bruno Buchberger introduced Groebner bases in 1965 and named them for W. Gr鐽ner (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 (Hans Wussing, "Die Genesis des abstrakten Gruppenbegriffes," Berlin 1969; translated as "The Genesis of the Abstract Group Concept," M.I.T. Press 1984.). [Ken Pledger]
The term GROUP OF AN EQUATION was used by Galois (Kramer).
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 鈁er die neue Grundlagenkrise der Mathematik, Math.Z. 10 (1921) 39-79."