[Martin Ostwald, Sam Kutler, Robin Hartshorne, David Reed]
QUADRANGLE is found in English in the fifteenth century.
The word was later used later by Shakespeare.
QUADRATIC is derived from the Latin quadratus, meaning "square." In English, quadratic was used in 1668 by John Wilkins (1614-1672) in An essay towards a real character, and a philosophical language [London: Printed for Sa. Gellibrand, and for John Martyn, 1668]. He wrote: "Those Algebraical notions of Absolute, Lineary, Quadratic, Cubic" (OED2)
In his Liber abbaci, Fibonacci referred to problems involving quadratic equations as questiones secundum modum algebre.
QUADRATIC FORM is dated 1859 in MWCD10.
The term QUADRATIC RESIDUE was introduced by Euler in a paper of 1754-55 (Kline, page 611). The term non-residue is found in a paper by Euler of 1758-59, but may occur earlier.
QUADRATRIX. The quadratrix of Hippias was probably invented by Hippias but it became known as a quadratrix when Dinostratus used it for the quadrature of a circle (DSB, article: "Dinostratus"; Webster's New International Dictionary, 1909).
The term QUADRATRIX OF HIPPIAS was used by Proclus (DSB, article: "Dinostratus").
The quadratrix of Hippias is the first named curve other than circle and line, according to Xah Lee's Visual Dictionary of Special Plane Curves website.
QUADRILATERAL appears in English in 1650 in Thomas Rudd's translation of Euclid.
See also quadrangle.
The term QUADRIVIUM was used by Anicius Manlius Severinus Boethius (ca. 480 - 524/525) in his Arithmetica. According to the DSB, this is "probably the first time the word was used."
The term QUANTICS was used by Arthur Cayley (1821-1895).
QUARTILE is found in D. McAlister, Proc. R. Soc. XXIX: "As these two measures, with the mean, divide the curve of facility into four equal parts, I propose to call them the 'higher quartile' and the 'lower quartile' respectively. It will be seen that they correspond to the ill-named 'probable errors' of the ordinary theory" (OED2).
The term QUASI-PERIODIC FUNCTION was introduced by Ernest Esclangon (1876-1954) (DSB, article: Bohl).
The term QUATERNION was introduced by William Rowan Hamilton (1805-1865). He used the word in a paper of 1843.
QUEUEING. The OED2 shows a use of "a queueing system" and "a complex queueing problem" in 1951 in the Journal of the Royal Statistical Society, and a use of "queueing theory" in 1954 in Science News. [An interesting fact about the word queueing is that it contains five consecutive vowels, the longest string of vowels in any English word, except for a few obscure words not generally found in dictionaries.]
QUINTIC was used in English as an adjective in 1853 by Sylvester in Philosophical Magazine: "May, To express the number of distinct Quintic and Sextic invariants." The word was used as a noun in 1856 by Cayley: "In the case of a quantic of the fifth order or quintic" (from his Works, 1889) (OED2).
QUINTILE is found in 1922 in "The Accuracy of the Plating Method of Estimating the Density of Bacterial Populations," Annals of Applied Biology by R. A. Fisher, H. G. Thronton, and W. A. Mackenzie: "Since the 3-plate sets are relatively scanty, we can best test their agreement with theory by dividing the theoretical distribution of 43 values at its quintiles, so that the expectation is the same in each group." There are much earlier uses of this term in astrology [James A. Landau].
QUOTIENT. Joannes de Muris (c. 1350) used numerus quociens.
In the Rollandus Manuscript (1424) quotiens is used (Smith vol. 2, page 131).
Pellos (1492) used quocient.
QUOTIENT GROUP (for factor group) appears in 1893 in the Bulletin of the New York Mathematical Society (OED2).
QUOTIENT RING is found in D. G. Northcott, "Some properties of analytically irreducible geometric quotient rings," Proc. Camb. Philos. Soc. 47, 662-667 (1951).