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 González Cabillón.
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 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).
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 GÖDEL'S THEOREM is used by Max Black in 1933 in The Nature of Mathematics (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 Schwartzman (page 100) golden section was apparently first used in print in 1835 by Georg Simon Ohm.
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) 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.
The term GRAPH in mathematics 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).
The term graph in the context of graph theory "appears to have been coined by A. Cayley," according to an Internet web page.
GRAPH THEORY appears in 1953 in the title Graph Theory as a Mathematical Model in Social Science by Harary and Norman (OED2).
GREATEST COMMON DIVISOR is dated ca. 1924 in MWCD10. 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].
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. 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 (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 Über die neue Grundlagenkrise der Mathematik, Math.Z. 10 (1921) 39-79 .