*Last revision: July 7, 2017*

**Q. E. D.** In the *Elements* Euclid concluded his proofs with
ὅπερ ἔδει δεῖξαι
“that which was to be shown”: see e.g. the end
of the proof of Proposition 4 on p. 11 of Fitzpatrick’s
Greek-English *Euclid*. Medieval
geometers translated the expression as *quod
erat demonstrandum* (“that which was to be proven”).

According to Veronika Oberparleiter, the earliest known use in print
of the phrase *quod erat demonstrandum* in a Euclid translation
appears in the translation by Bartholemew Zamberti published in Venice in 1505.

In *Dialogues Concerning Two New Sciences* (1638) in Latin
Galileo used
quod erat intentum,
quod erat demonstrandum,
quod erat probandum,
quod erat ostendendum,
quod erat faciendum,
quod erat determinandum, and
quod erat propositum.

In 1665 Benedictus de Spinoza (1632-1677) wrote a treatise on ethics,
*Ethica More Geometrico Demonstrata,* in which he proved various
moral propositions in a geometric manner. He wrote the abbreviation
Q. E. D., as a seal upon his proof of each ethical proposition.

Isaac Barrow used quod erat demonstrandum, quod erat faciendum (Q. E. F.), quod fieri nequit (Q. F. N.), and quod est absurdum (Q. E. A.).

Isaac Newton used the abbreviation Q. E. D.

[Martin Ostwald, Sam Kutler, Robin Hartshorne, David Reed]

See halmos (end of proof sign) on Earliest Uses of Symbols of Set Theory and Logic.

**QR ALGORITHM.** “The *QR* algorithm is currently the most popular
method for calculating the complete set of eigenvalues
of a full (i.e. small) matrix” writes D. S. Watkins “Understanding
the QR Algorithm,” *SIAM Review*, **24**, (1982), p. 427. Watkins attributes the
method to J. G. F. Francis “The QR transformation I,” *Comput. J.,* **4**,
(1961), 265-271 and V. N. Kublanovskaya “On Some Algorithms
for the Solution of the Complete Eignevalue Problem,” *USSR Comput. Math.
Phys*., **3**. 637-657.

**QUADRANGLE** is found in the geometry sense in English in about 1398 in
a translation of *Bartholomaeus Anglicus’ De Proprietatibus Rerum.* [OED]

The word was used by Shakespeare, in the sense of a rectangular space or courtyard.

**QUADRANT,** as one fourth of a circle, is found in 1571 in *Pantometria* by Digges:
"Biv, A Quadrant is the fourth part of a Circle, included with two Semidiameters" (OED).

In 1810 *Trigonometry, Plane and Spherical; with The Construction and
Application of Logarithms* by Thomas Simpson has:
"The difference of any arch from 90 degrees (or a quadrant) is called its
compliment; and its difference from 180 degrees (or a semicircle) its
supplement." [Ryan M. Castle]

In 1849, *Trigonometry and Double Algebra* by August de Morgan has:
"The Axes divide the plane into four quarters:
and as a line, revolving positively, passes from 0 to 90°,
from 90° to 180°, from 180° to 270° , and from 270° to 360°, it is
said to be in the first, second, third, and fourth quarters of space.
But these might equally well be designated as the ++, +-, --, and -+
quarters of space." [Pat Ballew]

In 1895, *Plane and
Spherical Trigonometry, Surveying, and Tables* by George A Wentworth has:
"Let, also, the four quadrants into which the circle is divided by the horizontal and
vertical diameters, A A’s, and B B’s, be numbered I, II, III, IV, in the
direction of motion." [Pat Ballew, Univ. of Mich. Digital Library]

**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.** The term *form* perhaps in this sense was used by Euler in
“Novae
demonstrations circa divisors numerorum formae xx _ nyy” which was read
Nov. 20, 1775 at the St. Petersburg Academy and published after his death in *Nova
acta acad. Petrop. I* [1783], 1747 47-74. This paper is cited in *Disquisitiones
Arithmeticae*, article 151, page 104.

pq(pp –qq), rt (rr – ss)

fiat quadratum , hic vero casus manifesto ad problema fatis notum deducitur , quo duo triangula rectangula in numeris quarruntur , quorum arese fint inter fe sequales ; quaeruntur igitur duo numeri , uterque formae xy (x x — y y) quorum productum fit quadratum , unde in aequenti Tabella simpliciores numeros huins formae exhibeamus per factores expressos, ubi quidem factores quadratos omittamus:

In hac sectione imprimis de functionibus duarum indeterminatarumx, y,huius formae

axx+ 2bxy+cyyubi

a, b, csunt integri dati, tractabimus, quasformas secundi gradussive simpliciterformasdicemus.

A translation is: “In this section we shall treat particularly of functions in two unknowns *x, y* of the form
*axx* + 2*bxy* + *cyy*
where *a, b, c* are given integers. We will call these functions *forms of the second degree* or simply *forms.*”

Gauss also used Latin terms which are translated *binary, ternary,* and *quaternary forms.*

A Google print search finds *der binaren quadratischen Formen* in a letter dated July 9, 1831, which may have been written by Gauss.

Some other uses in German of *quadratic form* can be found using Google print search, but these seem to refer to the term
as it is used elsewhere in number theory or in chemistry.

*Quadratic form* is found in English 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 English in 1861 in
“Report on the Theory of Numbers—Part III” by H. J. Stephen Smith [Google print search].

[James A. Landau]

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, QUADRATURE OF THE CIRCLE,** and **MECHANICAL QUADRATURE.** The *OED* defines quadrature as the
“process of constructing geometrically a square equal in area to that of a
given figure, esp. a circle. More widely: the calculation of the area bounded
by, or lying under, a curve; the calculation of a definite integral, esp. by
numerical methods.”

The problem of squaring the circle was one of classical problems of Greek mathematics: see the MacTutor article
Squaring
the circle. The word *quadrature* appears in English in the 16^{th} century: the earliest quotation in
the *OED* is from 1569 in Sanford’s translation of the
contemporary Latin work by Heinrich Corelius Agrippa *Of the vanitie and uncertaintie of artes and sciences,*
“Yet no Geometrician hath founde out the true
Quadrature [L. *quadraturam*] of the Circle.” In classical Latin the word
meant a division of land into squares.

Agrippa’s book was not a mathematics book and the problem of the **quadrature of the circle** was
well-known outside specialist mathematical circles: the *OED*’s earliest quotation using the phrase to **square
the circle** is from a 1624 sermon by John Donne
(1572-1631): “Goe not Thou about to Square eyther circle [sc. God or thyself].”

Other quotations in the *OED* illustrate the quadrature of other curves and the
connection with CALCULUS: “A method for
the Quadrature of Parabola’s of all degrees” from an obituary of Fermat in *Philosophical
Transactions (1665-1678)*, **1**, (1665 - 1666), 15-16 and “Drawing Tangents to Curves, finding their Curvatures,
their Lengths, and Quadratures” from W. Emerson’s *The Doctrine of Fluxions* 1743.

The term **mechanical quadrature** is found in German in F. G. Mehler, “Bemerkungen zur Theorie
der mechanischen Quadraturen,” *J. Reine angew. Math* **63** (1864) 152-157.
By the early 20^{th} century the term was established
in the English mathematical literature. A *JSTOR* search found G. D. Birkhoff’s “General Mean Value and Remainder Theorems with
Applications to Mechanical Differentiation and Quadrature,” *Transactions of the American Mathematical
Society*, **7**, 1906), 107-136. The
methods of quadrature considered by Birkhoff went back to the early 18^{th} century—one was due to Roger Cotes.

Mechanical quadrature did not usually involve a machine of any kind and in the 20^{th} century the
term gave way to *numerical quadrature* and finally to **numerical integration**.
There is an early appearance of the latter term in the title *A Course in Integration and
Numerical Integration* (1915) by David Gibb. The term is used in the well-known
book of 1924, *The Calculus of Observations* by E. T. Whittaker & G.
Robinson. In the course of the 19^{th} century mechanical *integrating
machines* or **integrators** were
developed by physicists. One is described in James Thomson’s “On an Integrating Machine
Having a New Kinematic Principle,” *Proceedings of the
Royal Society of London*, **24**, (1875 - 1876), 262-265. The history
of these machines is related in “Integrators and Planimeters,” chapter 5 of H.
H. Goldstine *The Computer from Pascal to von Neumann*.

**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 **QUADRIVIUM** of the European Middle Ages consisted of the four mathematical sciences, arithmetic,
geometry, astronomy and music. (According to T. L. Heath *A History of Greek Mathematics*, vol. 1 p.
12, the Pythagoreans divided MATHEMATICS in the same way.) The term was used by the Roman scholar
Boethius
in his *Arithmetica* and, according
to the *DSB*, this is “probably the
first time the word was used.” Boethius had a profound influence on education
in medieval Europe. Preceding the quadrivium in the syllabus was the trivium,
for which see the entry LOGIC. The word *trivium* is echoed in the
modern word TRIVIAL.

**QUALITY CONTROL.** In 1929 W. A. Shewhart gave an address to a meeting of the American
Association for the Advancement of Science on "Quality control of the manufactured
article." In 1930 Bell Telephone Labs. issued Shewhart’s "Economic quality control
of manufactured product". This 26 page pamphlet was followed the next year by
a 501 page book *Economic Control of Quality of Manufactured Product.* (*JSTOR* search)

The term **QUANTIC** for a homogeneous function of two or more variables with constant
coefficients was introduced by Arthur Cayley in “An Introductory Memoir on Quantics,”
*Philosophical Transactions of the Royal Society of London,* **144** (1854). See
*Collected Papers II* p. 221.

**QUANTIFIER** (logic.) According to A. Church *Introduction
to Mathematical Logic* (1956, p. 288), "The notion of propositional function
and the use of quantifiers, originated with Frege in his *Begriffsschrift*
of 1879... The terms ‘quantifier’ and ‘quantification’ are Peirce’s." In 1885
C. S. Peirce wrote, "If the quantifying part, or Quantifier, contains Σ_{x}
and we wish to replace the *x* by a new index *i*, not already in
the Quantifier, and such that every *x* is an *i*, we can do so at
once by simply multiplying every letter of the Boolian having *x* as an
index by *x _{i}*."

The *terms* **universal quantifier** and **existential
quantifier** became established in the 1930s. The OED gives a reference to
Łukasiewicz and Tarski writing in Polish in 1930. A *JSTOR* search
found *universal quantifier* in H. B. Curry "The Universal Quantifier in Combinatory Logic,"
*Annals of Mathematics*, **32**, (1931), 154-180 and
*existential quantifier* in H. B. Curry "Apparent Variables From the Standpoint of Combinatory
Logic," *Annals of Mathematics*, **34**, (1933), 381-404.

For the symbols and for references to the work of Peano and Russell, see the
*Earliest Use of Symbols* page.

**QUANTILE** as a general term
covering quartiles, percentiles, etc. arrived about 60 years after the terms
it was generalising.

David (2001) gives M. G. Kendall’s "Note
on the Distribution of Quantiles for Large Samples," *Supplement to the Journal
of the Royal Statistical Society*, **7**, (1940 - 1941), 83-85.

See QUARTILE.

**QUARTILE**. *Higher* and *lower quartile* are found in 1879 in Donald McAlister,
The Law
of the Geometric Mean, *Proc. R. Soc.* XXIX, p. 374:
"As these two measures, with the mean, divide the
curve of facility into four equal parts, I propose to call them the ’shigher
quartile’s and the ’slower quartile’s respectively. It will be seen that they correspond
to the ill-named ’sprobable errors’s of the ordinary theory" (OED2). McAlister
defined octiles on a similar principle. The topic of McAlister’s paper was suggested
by Francis Galton (see LOGNORMAL) and Galton may have had a hand in the terms.
However, McAlister’s words seems clear enough.

*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

(From David (1995) and Jeffrey K. Aronson "Francis Galton
and the Invention of Terms for Quantiles," *Journal of Clinical Epidemiology*,
**54**, (2001) 1191-1194.)

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."

William Rowan Hamilton
(1805-1865) used the word to refer to a 4-dimensional object consisting of a real component and three
imaginary ones. He introduced quaternions in a paper he presented in 1843,
On a new Species of Imaginary Quantities connected with
a theory of Quaternions. Hamilton devoted the last 20 years of his
life to the development of quaternions; his magnum opus was the
*Lectures
on Quaternions* (1853).

See also HAMILTON, VECTOR and VECTOR ANALYSIS. For quaternion terms that have not survived (in their original meaning at least) see TENSOR and VERSOR.

**QUEUEING.** The OED2 shows a use of "a queueing system" and "a
complex queueing problem" in 1951 in David G. Kendall’s
"Some Problems in the Theory of Queues," *Journal of the Royal Statistical Society,*
*Series B*, **13**, 151-185 and a use of "queueing theory" in 1954 in *Science News.*
Kendall does not give a history of the "Theory of Queues"
but he says that, "the first major study of congestion problems was undertaken by
A. K. Erlang
in 1908 ... under the auspices of the Copenhagen Telephone Company."

[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. Thornton, 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 1946 in *Foundations of Algebraic Geometry*
by Andre Weil: "Furthermore, every quotient-ring derived from this ring and a prime ideal in it
will be Noetherian."
[Google print search]