[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. In 1853 Arthur Cayley referred to "...the transformation of a quadratic form of four indeterminates into itself" in "On the homographic transformation of a surface of the second order into itself" in the Philosophical Magazine [University of Michigan Historical Math Collection].
Quadratic form is found in 1859 in G. Salmon, Less. Mod. Higher Alg.: "A quadratic form can be reduced in an infinity of ways to a sum of squares, yet the number of positive and negative squares in this sum is fixed" (OED2).
Binary quadratic form is found in 1929 in L. E. Dickson, Introd. Theory Numbers: "The function q = ax2 + bxy + cy2 is called a binary quadratic form" (OED2).
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.
QUADRATURE OF THE CIRCLE is found in English in 1596 in a pamphlet Have with You to Saffron Walden by Thomas Nashe (1567-1601): "As much time..as a man might haue found out the quadrature of the circle in (OED2).
Square the circle appears in English in 1624 a sermon of John Donne (1572-1631): "Goe not Thou about to Square eyther circle [sc. God or thyself]" (OED2).
QUADRILATERAL appears in English in 1650 in Thomas Rudd's translation of Euclid.
See also quadrangle.
QUADRIVARIATE is found in J. A. McFadden, "An approximation for the symmetric, quadrivariate normal integral," Biometrika 43, 206-207 (1956).
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."
QUANTICS appears in Arthur Cayley, "An Introductory Memoir on Quantics," Philosophical Transactions of the Royal Society of London, 144 (1854).
The term QUARTILE was introduced by Francis Galton (Hald, p. 604).
Higher and lower quartile are found in 1879 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).
Upper and lower quartile appear in 1882 in F. Galton, "Report of the Anthropometric Committee," Report of the 51st Meeting of the British Association for the Advancement of Science, 1881, p. 245-260 (David, 1995).
The term QUASI-PERIODIC FUNCTION was introduced by Ernest Esclangon (1876-1954) (DSB, article: Bohl).
QUATERNION (a group of four things) dates to the 14th century in English.
The word appears in the King James Bible (Acts 12:4), which refers to "four quaternions of soldiers."
The word was introduced in mathematics by William Rowan Hamilton (1805-1865), who 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.]
QUINDECAGON is found in English in 1570 in Henry Billingsley's translation of Euclid: "In a circle geuen to describe a quindecagon or figure of fiftene angles" (OED2).
The OED2 shows one citation, from 1645, for pendecagon.
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."
Quintic 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 theory). This term was introduced by Hölder in 1889, according to a paper by Young in 1893.
Quotient appears in English in 1893 in a paper by Cayley, "Note on the so-called quotient G/H in the theory of groups."
QUOTIENT GROUP. Otto Hölder (1859-1937) coined the term factorgruppe. He used the term in 1889.
Quotient group is found in 1893 in Bull. N.Y. Math. Soc. III. 74: "The quotient-group of any two consecutive groups in the series of composition of any group is a simple group" (OED2).
Factor-group appears in English in G. L. Brown, "Note on Hölder's theorem Concerning the constancy of factor-groups," American M. S. Bull. (1895).
QUOTIENT RING is found in D. G. Northcott, "Some properties of analytically irreducible geometric quotient rings," Proc. Camb. Philos. Soc. 47, 662-667 (1951).