Earliest Known Uses of Some of the Words of Mathematics

Earliest Known Uses of Some of the Words of Mathematics

p-ADIC INTEGER was coined by Kurt Hensel (1861-1941) (Katz, page 824).

P-VALUE is found in 1943 in Statistical Adjustment of Data by W. E. Deming (David, 1998).

PAIRWISE. An early use of this term is in Chowla, S.; Erdoes, Pal; Straus, E.G. On the maximal number of pairwise orthogonal latin squares of a given order, Canadian J. Math. 12, 204-208 (1960).

PANGEOMETRY is the term Nicholas Lobachevsky (1796-1856) gave to his non-Euclidea geometry (Schwartzman, p. 157).

PARABOLA was probably coined by Apollonius, who, according to Pappus, had terms for all three conic sections. Michael N. Fried says there are two known occasions where Archimedes used the terms "parabola" and "ellipse," but that "these are, most likely, later interpolations rather than Archimedes own terminology."

PARABOLIC GEOMETRY. See hyperbolic geometry.

PARACOMPACT. The term and the concept are due to J. Dieudonné (1906-1992), who introduced them in Une généralisation des espaces compacts, J. Math. Pures Appl., 23 (1944) pp. 65-76. A topological space X is paracompact if (i) X is a Hausdorff space, and (ii) every open cover of X has an open refinement that covers X and which is locally finite. The usefulness of the concept comes almost entirely from condition (ii), while the role of condition (i) has been somewhat controversial. Thus, in his book General Topology (1955), John Kelley (p. 156) replaces (i) by the condition that X be regular (and his definition of regularity does not include the Hausdorff separation axiom), while some other authors do not even mention (i) in defining paracompactness. In any case, however, it is possible to state this important fact (conjectured by Dieudonné in the paper above): every metric space is paracompact. This was proved by A. H. Stone in Paracompactness and product spaces, Bull. Amer. Math. Soc., 54 (1948) 977-982. [This entry was contributed by Carlos César de Araújo.]

PARACONSISTENT LOGIC. The first formal calculus of inconsistency-tolerant logic was constructed by the Polish logician Stanislaw Jaskowski, who published his paper "Propositional calculus for contradictory deductive systems" (in Polish) in Studia Societatis Scientiarum Torunensis, 55--77 in 1948. It was reprinted in English in Studia Logica 24, 143--157 (1969).

Newton Carneiro Affonso da Costa, one of the most prominent researchers in paraconsistent logic, referred to it as inconsistent formal systems in his 1964 thesis, which used that term as its title. [See the introduction of the work "Sistemas Formais Inconsistentes", Newton C. A. da Costa, Editora da UFPr, Curitiba, 1993, p. viii. This work is a reprint of the Prof. Newton's original 1964 thesis, the initial landmark of all studies in the matter.

The term paraconsistent logic was coined in 1976 by the Peruvian philosopher Francisco Miró Quesada, during the Terceiro Congresso Latino Americano.

[Manoel de Campos Almeida, Max Urchs]

PARALLEL appears in English in 1549 in Complaynt of Scotlande, vi. 47: "Cosmaghraphie ... sal delcair the eleuatione of the polis, and the lynis parallelis, and the meridian circlis" (OED2).

PARALLELEPIPED. According to Smith (vol. 2, page 292), "Although it is a word that would naturally be used by Greek writers, it is not found before the time of Euclid. It appears in the Elements (XI, 25) without definition, in the form of 'parallelepipedal solid,' the meaning being left to be inferred from that of the word 'parallelogrammic' as given in Book I."

Parallelipipedon appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

In the 1644 edition of his Cursus mathematicus (in Latin), Pierre Herigone used the spelling parallelepipedum.

The first citation in the OED2 with the shortened spelling parallelepiped is Walter Charleton (1619-1707), Chorea gigantum, or, The most famous antiquity of Great-Britain, vulgarly called Stone-heng : standing on Salisbury Plain, restored to the Danes, London : Printed for Henry Herringman, 1663.

Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon.

In Noah Webster's A compendious dictionary of the English language (1806) the word is spelled parallelopiped.

Mathematical Dictionary and Cyclopedia of Mathematical Science (1857) has parallelopipedon.

U. S. dictionaries show the pronunciation with the stress on the penult, but some also show a second pronunciation with the stress on the antepenult.

PARALLELOGRAM appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements (OED2).

In 1832 Elements of Geometry and Trigonometry by David Brewster, which is a translation of Legendre, has:

The word parallelogram, according to its etymology, signifies parallel lines; it no more suits the figure of four sides than it does that of six, of eight, &c. which have their opposite sides parallel. In like manner, the word parallelopipedon signifies parallel planes; it no more designates the solid with six faces, than the solid with eight, ten, &c. of which the opposite faces are parallel. The names parallelogram and parallelelopipedon*, have the additional inconvenience of being very long. Perhaps, therefore, it would be advantageous to banish them altogether from geometry; and to substitute in their stead, the names rhombus and rhomboid, retaining the term lozenge, for quadrilaterals whose sides are all equal.

*The word is misspelled this way in Brewster.

PARAMETER. Claude Mydorge used the word parameter with the meaning of "latus rectum" in 1631 according to two Italian scientific encyclopedias (Dizionario enciclopedico dei termini scientifici, Rizzoli, 1990; Dizionario biografico degli scienziati e dei tecnici, Zanichelli, 1999). This information was provided by Alessio Martini, who suggests the use occurred in his "Prodromi catoptricorum et dioptricorum, sive conicorum operis...", Paris, 1631).

According to Kline (page 340), parameter was introduced by Gottfried Wilhelm Leibniz (1646-1716). He used the term in 1692 in Acta Eruditorum 11 (Struik, page 272). Kline used the term in its modern sense.

PARAMETER (in statistics) is found in 1914 in E. Czuber, Wahrscheinlichkeitsrechnung, Vol. I (David, 1998).

Parameter is found in 1922 in R. A. Fisher, "On the Mathematical Foundations of Theoretical Statistics," Philosoophical Transactions of the Royal Society of London, Ser. A. 222, 309-368 (David, 1995).

The term was introduced by Fisher, according to Hald, p. 716.

PARAMETRIC EQUATION is found in 1894 in "On the Singularities of the Modular Equations and Curves" by John Stephen Smith in the Proceedings of the London Mathematical Society [University of Michigan Historical Math Collection].

PARTIAL DERIVATIVE and PARTIAL DIFFERENTIAL. Partial derivatives appear in the writings of Newton and Leibniz.

Partial differential equation was used in 1770 by Antoine-Nicolas Caritat, Marquis de Condorcet (1743-1794) in the title "Memoire sur les Equations aux différence partielles," which was published in Histoire de L'Academie Royale des Sciences (1773).

Partial differential is found in English in 1816 in the translation of Lacroix's Differential and Integral Calculus: "Usually expressed by saying that one is the partial differential relative to x, and the other the partial differential relative to y (OED2).

An early use of the term partial derivative in English is in an 1834 paper by Sir William Rowan Hamilton [James A. Landau].

Partial differential equation is found in English in 1836 in the second edition of Elements of the Differential Calculus by John Radford Young [James A. Landau].

The term PARTIAL FRACTION occurs Traité élémentaire Calcul differéntiel et intégral (1797-1800) by Sylvestre Francois Lacroix.

La méthode générale pour intégrer les différentielles exprimées par des fractiones rationnelles consiste à les décomposer en d'autres dont les dénominateurs soient plus simples, qu'on désigne sous le norm de fractions partielles, et qu'on obtient comme il suit.

In English, the term is found in 1816 in Peacock and Herschel's translation of Lacroix: "The general method of integrating differentials of the above form, consists in decomposing them into others, whose denominators are more simple, which we designate by the name of partial fractions."

PARTIAL PRODUCT is found in an 1844 paper by Sir William Rowan Hamilton [James A. Landau].

PARTICULAR SOLUTION. The term particular case of the general integral is due to Lagrange (Kline, page 532).

Particular integral is found in English in 1814 in New Mathematical and Philosophical Dictionary by P. Barlow:

Particular Integral, in the Integral Calculus, is that which arises in the integration of any differential equation, by giving a particular value to the arbitrary quantity or quantities that enter into the general integral (OED2).

Particular solution is found in 1831 in Elements of the Integral Calculus (1839) by J. R. Young:

It must be here particularly remarked, that the value of c, as deduced from the equation (5), is not necessarily a function of the variables; for c may be connected with these variables in F (x, y, c) merely by way of addition or subtraction, in which case (5) will imply fc = 0, the roots of which equation will be particular constant values of c, which, substituted in the complete primitive, will furnish so many particular cases of that primitive; these, therefore, will be but particular solutions.

PASCAL'S TRIANGLE. Roger Cooke writes, "Although sources on Hindu mathematics tend to be unreliable, I'm fairly confident that 'Pascal's triangle' was discussed many hundreds of years earlier in Hindu writings, under the name Meru Prastara (Staircase of Mount Meru)."

Blaise Pascal used the term "arithmetical triangle" (triangle arithmetique) in his Traité du triangle arithmétique. He wrote, "I designate as an arithmetic triangle a figure whose construction is as follows..."

Arithmetical triangle of Pascal is found in Ed. Lucas, "Note sur le triangle arithmétique de Pascal et sur la série de Lamé, N. C. M. (1876).

Pascal's triangle appears in 1886 in Algebra by George Chrystal (1851-1911).

In Italy it is called Tartaglia's triangle and in China it is called Yang Hui's triangle.

The term PEANO-GOSPER CURVE was coined by Mandelbrot in 1977.

PEARLS OF SLUZE. Blaise Pascal (1623-1662) named the family of curves to honor Baron René François de Sluze, who studied the curves (Encyclopaedia Britannica article: "Geometry").

The term PEDAL CURVES is due to Olry Terquem (1782-1862) (Cajori 1919, page 228).

PELL'S EQUATION was so named by Leonhard Euler (1707-1783) in a paper of 1732-1733, even though Pell had only copied the equation from Fermat's letters (Burton, page 504) of 1657 and 1658.

The following is taken from Sir Thomas L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra, page 285-286:

Fermat rediscovered the problem and was the first to assert that the equation x2 - Ay2 = 1, where A is any integer not a square, always has an unlimited number of solutions in integers. His statement was made in a letter to Frénicle of February, 1657 (cf. Oeuvres de Fermat, II, pp. 333-4). Fermat asks Frénicle for a general rule for finding, when any number not a square is given, squares which, when they are respectively multiplied by the given number and unity is added to the product, give squares. If, says Fermat, Frénicle cannot give a general rule, will he give the smallest value of y which will satisfy the equations 61y2 + 1 = x2 and 109y2 + 1 = x2 ? ... The challenge was taken up in England by William, Viscount Brouncker, first President of the Royal Society, and Wallis. At first, owing apparently to some misunderstanding, they thought that only rational, and not necessarily integral solutions were wanted, and found of course no difficulty in solving this easy problem. Fermat was, naturally, not satisfied with this solution, and Brouncker, attacking the problem again, finally succeeded in solving it. The method is set out in letters of Wallis of 17th December, 1657, and 30th January, 1658, and in chapter XCVIII of Wallis' Algebra; Euler also explains it fully in his Algebra (Footnote 3: Part II, chap. VII), wrongly attributing it to Pell (Footnote 4: This was the origin of the erroneous description of our equation as the "Pellian" equation. Hankel (in Zur Geschichte der Math. im Alterthum und Mittlelalter, p. 203) supposed that the equation was so called because the solution was reproduced by Pell in an English translation (1668) by Thomas Brancker of Rahn's Algebra; but this is a misapprehension, as the so-called "Pellian" equation is not so much as mentioned in Pell's additions (Wertheim in Bibliotheca Mathematica, III, 1902, pp. 124-6); Konen, pp. 33-4 note). The attribution of the solution to Pell as a pure mistake of Euler's, probably due to a cursory reading by him of the second volume of Wallis' Opera where the solution of the equation ax2 + 1 = y2 is given as well as information as to Pell's work in indeterminate analysis. But Pell is not mentioned in connexion with the equation at all (Eneström in Bibliotheca Mathematica, III, 1902, p. 206).

The following is taken from Harold M. Edwards, Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, page 33:

This problem of Fermat is now known as "Pell's equation" as a result of a mistake on the part of Euler. In some way, perhaps from a confused recollection of Wallis's Algebra, Euler gained the mistaken impression that Wallis attributed the method of solving the problem not to Brouncker but to Pell, a contemporary of Wallis who is frequently mentioned in Wallis's works but who appears to have had nothing to do with the solution of Fermat's problem. Euler mentions this mistaken impression as early as 1730, when he was only 23 years old, and it is included in his definitive Introduction to Algebra written around 1770. Euler was the most widely read mathematical writer of his time, and the method from that time on has been associated with the name of Pell and the problem that it solved --- that of finding all integer solutions of y2 - Ax2 = 1 when A is a given number not a square --- has been known ever since as "Pell's equation", despite the fact that it was Fermat who first indicated the importance of the problem and despite the fact that Pell had nothing whatever to do with it.

These quotations were provided by Raul Nunes to a mathematics history mailing list.

The 1910 Encyclopaedia Britannica has: "Although Pell had nothing to do with the solution, posterity has termed the equation Pell's Equation" (OED2).

PENCIL OF LINES. Desargues coined the term ordonnance de lignes, which is translated an order of lines or a pencil of lines [James A. Landau].

PENTAGON. In 1551 in Pathway to Knowledge Robert Recorde used the obsolete word cinqueangle: "Defin., Figures of .v. sydes, other v. corners, which we may call cinkangles, whose sydes partlye are all equall as in A, and those are counted ruled cinkeangles" (OED2).

Pentagon appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

PENTAGRAM appears in English in 1833 in Fraser's Magazine (OED2).

The term PENTOMINO was coined by Solomon W. Golomb, who used the term in a 1953 talk to the Harvard Math Club. According to an Internet web page, the term was trademarked in 1975. (The first known pentomino problem is found in Canterbury Puzzles in 1907.)

PERCENTILE appears in 1885 in Francis Galton, "Some Results of the Anthropometric Laboratory," Journal of the Anthropological Institute, 14, 275-287: "The value which 50 per cent. exceeded, and 50 per cent. fell short of, is the Median Value, or the 50th per-centile, and this is practically the same as the Mean Value; its amount is 85 lbs." (OED2).

According to Hald (p. 604), Galton introduced the term.

PERFECT NUMBER. According to Smith (vol. 2, page 21), the Pythagoreans used this term in another sense, because apparently 10 was considered by them to be a perfect number.

Proposition 36 of Book IX of Euclid's Elements is: "If as many numbers as we please beginning from a unit be set out continuously in double proportion, until the sum of all becomes a prime, and if the sum multiplied into the last make some number, the product will be perfect."

The Greek poet and grammarian Euphorion (born c. 275 BC?) used the phrase ". . . equal to his [or their] limbs, with the result that they are called perfect." This is an apparent reference to perfect numbers, according to J. L. Lightfoot, "An early reference to perfect numbers? Some notes on Euphorion, SH 417," Classical quarterly 48 (1998), 187-194.

The term was used by Nicomachus around A. D. 100 in Introductio Arithmetica (Burton, page 475). One translation is:

Among simple even numbers, some are superabundant, others are deficient: these two classes are as two extremes opposed to one another; as for those that occupy the middle position between the two, they are said to be perfect.

Nichomachus identified 6, 28, 496, and 8128 as perfect numbers.

St. Augustine of Hippo (354-430) wrote De senarii numeri perfectione ("Of the perfection of the number six") in De Civitate Dei. He wrote, in translation: "Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true. God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist."

Perfect number appears in English in 1570 in Sir Henry Billingsley's translation of Euclid.

In 1674, Samuel Jeake wrote in Arithmetic (1696) "Perfect Numbers are almost as rare as perfect Men" (OED2).

PERFECT SETS appears in Georg Cantor, "De la puissance des ensembles parfaits de points," Acta Mathematica 4 (1884) [James A. Landau].

PERIODOGRAM. A. Schuster (Terrestial Magnetism, 3, (1898), 13-41) introduced the term for a form of analysis he had been using since 1894 (David 2001).

PERMANENT (of a square matrix). In a paper written with M. Marcus ("Permanents", Amer. Math. Monthly, 1965, p. 577) Henryk Minc, one of the great authorities in permanents, wrote:

The name "permanent" seems to have originated in Cauchy's memoir of 1812 [B 3]. Cauchy's "fonctions symétriques permanentes" designate any symmetric function. Some of these, however, were permanents in the sense of the definition (1.1). (...) As far as we are aware the name "permanent" as defined in (1.1) was introduced by Muir [B 38].

The paper by T. Muir is "On a class of permanent symmetric functions", Proc. Roy. Soc. Edinburgh, 11 (1882) 409-418. [B3] is "Mémoire sur les fonctions Qui ne peuvent obtenir que deux valeurs égales et de signes contraires par suite des transpositions opérées entre les variables qu'elles renferment", J. de l'Éc. Polyt., 10 (1812) 29-112. According to J. H. van Lint in "The van der Waerden Conjecture: Two Proofs in One Year", The Mathematical Intelligencer:

In his book Permanents [9] H. Minc mentions that the name permanent is essentially due to Cauchy (1812) although the word as such was first used by Muir in 1882. Nevertheless a referee of one of Minc's earlier papers admonished him for inventing this ludicrous name!

[This entry was contributed by Carlos César de Araújo.]

PERMUTATION first appears in print with its present meaning in Ars Conjectandi by Jacques Bernoulli: "De Permutationibus. Permutationes rerum voco variationes..." (Smith vol. 2, page 528).

Earlier, Leibniz had used the term variationes and Wallis had adopted alternationes (Smith vol. 2, page 528).

The term PERMUTATION GROUP was coined by Galois (DSB, article: "Lagrange").

Permutation group appears in English in W. Burnside, "On the representation of a group of finite order as a permutation group, and on the composition of permutation groups," London M. S. Proc. 34.

PERPENDICULAR was used in English by Chaucer about 1391 in A Treatise on the Astrolabe. The term is used as a geometry term in 1570 in Sir Henry Billingsley's translation of Euclid's Elements.

The term PFAFFIAN was introduced by Arthur Cayley, who used the term in 1852: "The permutants of this class (from their connexion with the researches of Pfaff on differential equations) I shall term 'Pfaffians'." The term honors Johann Friedrich Pfaff (1765-1825).

PIECEWISE is found in 1933 in the phrase "vectors which are only piecewise differentiable" in Vector Analysis by H. B. Phillips (OED2).

PIE CHART is found in 1922 in A. C. Haskell, Graphic Charts in Business (OED2).

PIGEONHOLE PRINCIPLE. The principle itself is attributed to Dirichlet in 1834, although he apparently used the term Schubfachprinzip.

In Dirichlet's Vorlesungen über Zahlentheorie (Lectures on Number Theory, prepared for publication by Dedekind, first edition 1863), the argument is used in connection with Pell's equation but it bears no specific name [Peter Flor, Gunnar Berg].

In 1905 in Bachmann's "Zahlentheorie," part 5, the principle is stated as a "very simple fact" on which Dirichlet is said to have based his theory of units in number fields; no name is attached to the principle [Peter Flor].

In 1910 in Geometrie der Zahlen, Minkowski calls it "a famous method of Dirichlet" [Peter Flor].

According to Peter Flor, "the term Schubfachschluss, with or without a reference to Dirichlet, was used widely by German speaking number theorists at the universities of Vienna and Hamburg when I studied there in the 1950s. It occurs, among others, in the number theory books by Hasse and by Aigner."

In Swedish, the principle is called (in translation) "Dirichlets box principle" [Gunnar Berg]. The French term is "le principe des tiroirs de Dirichlet," which can be translated "the principle of the drawers of Dirichlet." In Portuguese, the term is "principio da casa dos pombos" (lit. principle of the house of the pigeons) or "das gavetas de Dirichlet" (lit. of the drawers of Dirichlet) [Julio González Cabillón].

Pigeonhole principle occurs in English in Raphael M. Robinson's paper "On the Simultaneous Approximation of Two Real Numbers," presented to the American Mathematical Society on November 23, 1940, and published in the Bulletin of the Society in 1941. Cf. volume 47, pp 512-513. In a footnote to this article, Robinson states:

The method used in this proof (Schubfachprinzip or "pigeonhole principle") was first used by Dirichlet in connection with a similar problem. We sketch the proof here in order to compare it with the proof of the theorem below, which also uses that method.

This citation was provided by Julio González Cabillón.

Paul Erdös referred to Dedekind's pigeon-hole principle in "Combinatorial Problems in Set Theory," an address he delivered in 1953 before the AMS [Julio González Cabillón].

Pigeon-hole principle occurs in English in Paul Erdös and R. Rado, "A partition calculus in set theory," Bull. Am. Math. Soc. 62 (Sept. 1956):

Dedekind's pigeon-hole principle, also known as the box argument or the chest of drawers argument (Schubfachprinzip) can be described, rather vaguely, as follows. If sufficiently many objects are distributed over not too many classes, then at least one class contains many of these objects.

E. C. Milner and R. Rado, "The pigeon-hole principle for ordinal numbers," Proc. Lond. Math. Soc., III. Ser. 15 (Oct., 1965) begins similarly:

Dirichlet's pigeon-hole principle (chest-of-drawers principle, Schubfachprinzip) asserts, roughly, that if a large number of objects is distributed in any way over not too many classes, then one of these classes contains many of these objects.

PLACE VALUE appears in 1911 in The Hindu-Arabic Numerals by David Eugene Smith and Louis Charles Karpinski: "It was therefore impossible to have any place value, and the numbers like twenty, thirty, and other multiples of ten, one hundred, and so on, required separate symbols except where they were written out in words."

The word PLAGIOGRAPH was coined by James Joseph Sylvester (DSB).

PLANE GEOMETRY appears in English in a letter from John Collins to Oldenburg for Tschirnhaus written in May 1676: "...Mechanicall tentative Constructions performed by Plaine Geometry are much to be preferred..." [James A. Landau].

PLATONISM. In the specific sense now widely used in discussions on the foundations of mathematics, this term was introduced by Paul Bernays (1888-1977) in Sur lê platonisme dans les mathematiques, Einseignement Math., 34 (1935-1936), 52-69. We quote the relevant passage:

If we compare Hilbert's axiom system to Euclid's (...), we notice that Euclid speaks of figures to be constructed, whereas, for Hilbert, systems of points, straight lines, and planes exist from the outset. (...) This example shows already that the tendency (...) consists in viewing the objects as cut off from all links with the reflecting subject. Since this tendency asserted itself especially in the philosophy of Plato, allow me to call it "platonism".

(The translation from the French is by Charles Parsons. This entry was contributed by Carlos César de Araújo.)

PLUQUATERNION was coined by Thomas Kirkman (1806-1895), as he attempted to extend further the notion of quaternions.

PLUS and MINUS. From the OED2:

The quasi-prepositional use (sense I), from which all the other English uses have been developed, did not exist in Latin of any period. It probably originated in the commercial langauge of the Middle Ages. In Germany, and perhaps in other countries, the Latin words plus and minus were used by merchants to mark an excess or deficiency in weight or measure, the amount of which was appended in figures. The earliest known examples of the modern sense of minus are German, of about the same date as our oldest quotation. ... In a somewhat different sense, plus and minus had been employed in 1202 by Leonardo of Pisa for the excess and deficiency in the results of the two suppositions in the Rule of Double Position; and an Italian writer of the 14th century used meno to indicate the subtraction of a number to which it was prefixed.

PLUS OR MINUS. The expression "plus or minus" is very old, having been in common use by the Romans to indicate simply "more or less" (Smith vol. 2, page 402).

PLUS OR MINUS SIGN. This symbol (±) is called the ambiguous sign in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

PLUS SIGN. Positive sign is found in 1704 in Lexicon Technicum.

Plus sign is found in 1841 in J. R. Young, Mathematical Dissertations: "The ordinary convention ... as to the disposal of the plus sign" (OED2).

The 1857 Mathematical Dictionary and Cyclopedia of Mathematical Science has affirmative sign.


The term POINT-SERIES GEOMETRY was coined by E. A. Weiss [DSB, article: "Reye"].

The term POINT-SET TOPOLOGY was coined by Robert Lee Moore (1882-1974), according to the University of St. Andrews website.

POINT-SLOPE FORM. Slope-point form is found in 1904 in Elements of the Differential and Integral Calculus by William Anthony Granville [James A. Landau].

Point-slope form is found in 1924 in Analytic Geometry by Arthur M. Harding and George W. Mullins: "This equation is known as the point-slope form of the equation of the straight line."

POISSON DISTRIBUTION. Poisson's exponential binomial limit appears in 1914 in the title "Tables of Poisson's Exponential Limit" by Herbert Edward Soper in Biometrika, 10, 25-35 (David, 1995).

Poisson distribution appears in 1922 in Ann. Appl. Biol. IX. 331: "When the statistical examination of these data was commenced it was not anticipated that any clear relationship with the Poisson distribution would be obtained" (OED2).

The term POLAR was introduced by Joseph-Diez Gergonne (1771-1859) in its modern geometric sense in 1810 (Smith vol. I).

POLAR COORDINATES. According to Daniel L. Klaasen in Historical Topics for the Mathematical Classroom:

Isaac Newton was the first to think of using polar coordinates. In a treatise Method of Fluxions (written about 1671), which dealt with curves defined analytically, Newton showed ten types of coordinate systems that could be used; one of these ten was the system of polar coordinates. However, this work by Newton was not published until 1736; in 1691 Jakob Bernoulli derived and made public the concept of polar coordinates in the Acta eruditorum. The polar system used for reference a point on a line rather than two intersecting lines. The line was called the "polar axis," and the point on the line was called the "pole." The position of any point in a plane was then described first by the length of a vector from the pole to the point and second by the angle the vector made with the polar axis.

According to Smith (vol. 2, page 324), "The idea of polar coordinates seems due to Gregorio Fontana (1735-1803), and the name was used by various Italian writers of the 18th century."

Polar co-ordinates is found in English in 1816 in a translation of Lacroix's Differential and Integral Calculus: "The variables in this equation are what Geometers have called polar co-ordinates" (OED2).

POLE. The term pôle (in projective geometry) was introduced by François Joseph Servois (1768-1847) in 1811 (Smith vol. 2, page 334). It was introduced in his first contribution to Gergonne's Annales de mathématiques pures et appliquées (DSB).

POLISH SPACE is defined in Nicolas Bourbaki, Topologie Generale [Stacy Langton].

POLYGON was used in classical Greek. Euclid, however, preferred "polypleuron," designating many sides rather than many vertices.

Polygon appears in English in 1570 in Sir Henry Billingsley's translation of Euclid, folio 125. In an addition after Euclid IV.16, which Billingsley ascribes to Flussates (François de Foix, Bishop of Aire), he mentions "Poligonon figures;" and in a marginal note explains "A Poligonon figure is a figure consisting of many sides." [Ken Pledger]

In 1571 in A Geometricall Practise, named Pantometria, Thomas Digges (d. 1595) wrote, "Polygona are such Figures as haue moe than foure sides" (OED2).

Multangle is found in 1674 in Samuel Jeake, Arith. (1696): "If 3 [angles] then called a Triangle, if 4 a Quadrangle, if more a Multangle or Polygone" (OED2).

In 1768-1771 the first edition of the Encyclopaedia Britannica has: "Every other right lined figure, that has more sides than four, is in general called a polygon."

In the 1828 Webster dictionary, the definition of polygon is: "In geometry, a figure of many angles and sides, and whose perimeter consists at least of more than four sides." In this dictionary, the word polygon appears in the definition of the enneagon (nine sides) and the dodecagon, but not in the definitions of figures consisting of fewer than nine sides.

In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has: "Regular polygons may have any number of sides: the equilateral triangle is one of three sides; the square is one of four."

POLYGONAL NUMBER and FIGURATE NUMBER. Pythagoras was acquainted at least with the triangular numbers, and very probably with square numbers, and the other polygonal numbers were treated by later members of his school (Burton, page 102).

According to Diophantus, Hypsicles (c. 190 BC-120 BC) defined polygonal numbers.

Nicomachus discussed polygonal numbers in the Introductio.

A tract on polygonal numbers attributed to Diophantus exists in fragmentary form.

Boethius defined figurate numbers as numbers "qui circa figuras geometricas et earum spatia demensionesque versantur" (Smith vol. 2, page 24).

In 1646 Vieta (1540-1603) referred to triangular and pyramidal numbers: "In prima adfectione per unitatis crementum, in secunda per numeros triangulos, in tertia per numeros pyramidales, in quarta per numeros triangulo-triangulos, in quinta per numeros triangulo-pyramidales."

In 1665 Pascal wrote his Treatise on Figurative Numbers.

Pentagonal number appears in English in 1670 in Collins in Rigaud Corr. Sci. Men (1841): "It is likewise a pentagonal number, or composed of two, three, four, or five pentagonal numbers" (OED2).

Pyramidal number appears in English in 1674 in Samuel Jeake's Arithmetic: "Six is called the first Pyramidal Number; for the Units therein may be so placed, as to represent a Pyramis" (OED2).

Polygonal number is found in English in 1704 in Lexicon Technicum: "Polygonal Numbers, are such as are the Sums or Aggregates of Series of Numbers in Arithmetical Progression, beginning with Unity; and so placed, that they represent the Form of a Polygon" (OED2).

Figurate number and triangular (as a noun) appear in English in 1706 in William Jones, Synopsis palmariorum matheseos: "The Sums of Numbers in a Continued Arithmetic Proportion from Unity are call'd Figurate ... Numbers. ... In a Rank of Triangulars their Sums are called Triangulars or Figurates of the 3d Order" (OED2).

Triangular number appears in English in 1796 in Hutton's Math. Dict.: "The triangular numbers 1, 3, 6, 10, 15, &c" (OED2).

In 1811 Peter Barlow used multangular numbers in An Elementary Investigation of the Theory of Numbers [James A. Landau].

POLYHEDRON. According to Ken Pledger, polyhedron was used by Euclid without a proper definition, just as he used "parallelogram." In I.33 he constructs a parallelogram without naming it; and in I.34 he first refers to a "parallelogrammic (parallel-lined) area," then in the proof shortens it to "parallelogram." In a similar way, XII.17 uses "polyhedron" as a descriptive expression for a solid with many faces, then more or less adopts it as a technical term.

However, according to Smith (vol. 2, page 295), "The word 'polyhedron' is not found in the Elements of Euclid; he uses 'solid,' 'octahedron,' and 'dodecahedron,' but does not mention the general solid bounded by planes."

In English, polyhedron is found in 1570 in Sir Henry Billingsley's translation of Euclid XII.17. Early in the proof (folio 377) Billingsley amplifies it to "...a Polyhedron, or a solide of many sides,..." [Ken Pledger].

In English, in the 17th through 19th centuries, the word is often spelled polyedron.

POLYNOMIAL was used by François Viéta (1540-1603) (Cajori 1919, page 139).

The word is found in English in 1674 in Arithmetic by Samuel Jeake (1623-1690): "Those knit together by both Signs are called...by some Multinomials, or Polynomials, that is, many named" (OED2). [According to An Etymological Dictionary of the English Language (1879-1882), by Rev. Walter Skeat, polynomial is "an ill-formed word, due to the use of binomial. It should rather have been polynominal, and even then would be a hybrid word."]

The term POLYOMINO was coined by Solomon W. Golomb in 1954 (Schwartzman, p. 169).

The term POLYSTAR was coined by Richard L. Francis in 1988 (Schwartzman, p. 169).

The word POLYTOPE was introduced by Alicia Boole Stott (1860-1940) to mean a four dimensional convex solid, according to the University of St. Andrews website.

PONS ASINORUM usually refers to Proposition 5 of Book I of Euclid. From Smith vol. 2, page 284:

The proposition represented substantially the limit of instruction in many courses in the Middle Ages. It formed a bridge across which fools could not hope to pass, and was therefore known as the pons asinorum, or bridge of fools. It has also been suggested that the figure given by Euclid resembles the simplest form of a truss bridge, one that even a fool could make. The name seems to be medieval.

The proposition was also called elefuga, a term which Roger Bacon (c. 1250) explains as meaning the flight of the miserable ones, because at this point they usually abandoned geometry (Smith vol. 2, page 284).

Pons asinorum is found in English in 1751 in Smollett, Per. Pic.: "Peregrine..began to read Euclid..but he had scarce advanced beyond the Pons Asinorum, when his ardor abated" (OED2).

According to Smith, pons asinorum has also been used to refer to the Pythagorean theorem.

POPULATION. See sample.

POSET, an abbreviation of "partially ordered set", is due to Garret Birkhoff (1911-1996), as said by himself in the second edition (1948, p. 1) of his book Lattice Theory. The term is now firmly established [Carlos César de Araújo].

POSITIONAL NOTATION is found in "Our Symbol for Zero" by George Bruce Halsted in American Mathematical Monthly, Vol. 10, No. 4. (Apr., 1903), pp. 89-90 [JSTOR].

POSITIVE. In the 15th century the names "positive" and "affirmative" were used to indicate positive numbers (Smith vol. 2, page 259).

In 1544 in Arithmetica integra Stifel called positive numbers numeri veri (Smith vol. 2, page 260).

Cardano (1545) called positive numbers numeri veri or veri numeri (Smith vol. 2, page 259).

Napier (c. 1600) used the adjective abundantes to designate positive numbers (Smith vol. 2, page 260).

Positive is found in English in the phrase "the Affirmative or Positive Sign +" in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris.

POSITIVE DEFINITE appears in 1905 in volume I of The Theory of Functions of Real Variables by James Pierpont [James A. Landau].

POSTERIOR PROBABILITY and PRIOR PROBABILITY. These contractions of "probability a priori" and "probability a posteriori" were introduced by Wrinch and Jeffreys ("On Certain Fundamental Principles of Scientific Inquiry," Philosophical Magazine, 42, (1921), 369-390). The longer forms were used by Lubbock & Drinkwater-Bethune (On Probability, 1830?) presumably following Laplace (Théorie Analytique des Probabilités (1812)) who wrote of "la probabilité de l'évenement observé, déterminée à priori" though Laplace did not use the à posteriori form [John Aldrich, using David (2001) and Hald (1998, p. 162)].

POSTFIX NOTATION is found in R. M. Graham, "Bounded Context Translation," Proceedings of the Eastern Joint Computer Conference, AFIPS, 25 (1964) [James A. Landau].

POSTULATE appears in the early translations of Euclid and was commonly used by the medieval Latin writers (Smith vol. 2, page 280).

In English, postulate is found in 1646 in Pseudodoxia epidemica or enquiries into very many eceived tenents by Sir Thomas Browne in the phrase "the postulate of Euclide" (OED2).

POTENTIAL FUNCTION. This term was used by Daniel Bernoulli in 1738 in Hydrodynamica (Kline, page 524).

According to Smith (1906) and the Encyclopaedia Britannica (article: "Green"), the term potential function was introduced by George Green (1793-1841) in 1828 in Essay on the Application of Mathematical Analysis to the Theory of Electricity and Magnetism: "Nearly all the attractive and repulsive forces..in nature are such, that if we consider any material point p, the effect, in a given direction, of all the forces acting upon that point, arising from any system of bodies S under consideration, will be expressed by a partial differential of a certain function of the co ordinates which serve to define the point's position in space. The consideration of this function is of great importance in many inquiries... We shall often have occasion to speak of this function, and will therefore, for abridgement, call it the potential function arising from the system S."

POTENTIAL as the name of a function was introduced by Gauss in 1840, according to G. F. Becker in Amer. Jrnl. Sci. 1893, Feb. 97. [Cf. Gauss Allgem. Lehrsätze d. Quadrats d. Entfernung Wks. 1877 V. 200: "Zur bequemern Handhabung..werden wir uns erlauben dieses V mit einer besonderen Benennung zu belegen, und die Grösse das Potential der Massen, worauf sie sich bezieht, nennen."]

POWER appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "The power of a line, is the square of the same line."

POWER (in set theory) was coined by Georg Cantor (1845-1918) (Katz, page 734). He used the German word Machtigkeit.

The expression POWER OF A POINT WITH RESPECT TO A CIRCLE was coined (in German) by Jacob Steiner [Julio González Cabillón].

POWER (of a test) is found in 1933 in J. Neyman and E. S. Pearson, "The Testing of Statistical Hypotheses in Relation to Probabilities A Priori," Proceedings of the Cambridge Philosophical Society, 24, 492-510 (David (2001)).

POWER SERIES is found in English in 1893 in Theory of Functions of Complex Variable by A. R. Forsyth: "Any one of the continuations of a uniform function, represented by a power-series, can be derived from any other" (OED2).

PRECALCULUS is found in 1947 in Mary Draper Boeker, The Status of the Beginning Calculus Students in Pre-Calculus College Mathematics, Bureau of Publications, Teachers College, Columbia University.

Precalculus is found as a noun in December 1969 in "Homomorphism: A Unifying Concept" by Eugene F. Krause in The Mathematics Teacher in the heading "Precalculus and early calculus."

PREDICATE CALCULUS occurs in G. Kreisel, "Note on arithmetic models for consistent formulae of the predicate calculus," Fundam. Math. 37 (1950).

Predicate calculus is also found in 1950 in tr. Hilbert & Ackermann’s Princ. Math. Logic: "The terminology has been adapted to that of the Grundlagen der Mathematik by Hilbert and Bernays. For example, the term 'functional calculus' has been everywhere replaced by 'predicate calculus'. ... We will now proceed, just as we did for the sentential calculus, to set up for the predicate calculus a system of axioms from which the remaining true sentences of the predicate calculus may be obtained by means of certain rules" (OED2).

PREFIX (notation) is found in S. Gorn, "An axiomatic approach to prefix languages," Symbol. Languages in Data Processing, Proc. Sympos., March. 26-31, 1962, 1-21 (1962).

PRESENT VALUE appears in Edmund Halley, "An Estimate of the Degrees of the Mortality of Mankind," Philosophical Transactions of the Royal Society, XVII (1693) [James A. Landau].

PRE-WHITENING occurs in G. Hext, "A note on pre-whitening and recolouring," Stanford Univ. Dept. Statist. Tech. Rep no. 13 (1964) [James A. Landau].

PRIMALITY is is found in 1919 in Dickson: "T. E. Mason described a mechanical device for applying Lucas' method for testing the primality of 24q+3 - 1."

The term PRIME NUMBER was apparently used by Pythagoras.

Iamblichus writes that Thymaridas called a prime number rectilinear since it can only be represented one-dimensionally.

In English prime number is found in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

Some older textbooks include 1 as a prime number.

In his Algebra (1770), Euler did not consider 1 a prime [William C. Waterhouse].

In 1859, Lebesgue stated explicitly that 1 is prime in Exercices d'analyse numérique [Udai Venedem].

In 1866, Primary Elements of Algebra for Common Schools and Academies by Joseph Ray has:

All numbers are either prime or composite; and every composite number is the product of two or more prime numbers. The prime numbers are 1, 2, 3, 5, 7, 11, 13, 17, etc. The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, etc.

In 1873, The New Normal Mental Arithmetic by Edward Brooks has on page 58:

Numbers which cannot be produced by multiplying together two or more numbers, each of which is greater than a unit, are called prime numbers.

In 1892, Standard Arithmetic by William J. Milne has on page 92:

A number that has no exact divisor except itself and 1 is called a Prime Number. Thus, 1, 3, 5, 7, 11, 13, etc. are prime numbers.

A list of primes to 10,006,721 published in 1914 by D. N. Lehmer includes 1.

[James A. Landau provided some of the above citations.]

PRIME NUMBER THEOREM. Edmund Georg Herman Landau (1877-1938) used the term Primzahlsatz (Cajori 1919, page 439).

PRIMITIVE (in group theory). The German word primitiv appears in Sophus Lie, Theorie der Transformationsgruppen (1888).

Primitive appears in 1888 in Amer. Jrnl. Math. X: "A group in the plane is primitive when with each ordinary point which we hold, no invariant direction is connected" (OED2).

PRIMITIVE FUNCTION. Lacroix used fonction primitive in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).

Primitive (noun) and primitive function appear in 1831 in Elements of the Integral Calculus (1839) by J. R. Young.

PRIMITIVE RECURSIVE FUNCTION was coined by Rózsa Péter (1905-1977) in "Über den Zusammenhang der verschiedenen Begriffe der rekursiven Funktion," Mathematische Annalen, 110:612-632, 1934 [Cesc Rossello, Dirk Schlimm].

The term PRIMITIVE ROOT was introduced by Leonhard Euler (1707-1783), according to Dickson, vol. I, page 181.

In "Demonstrationes circa residua ex divisione potestatum per numeros primos resultantia," Novi commentarii academiae scientiarum Petropolitanae 18 (1773), Euler wrote: "Huiusmodi radices progressionis geometricae, quae series residuorum completas producunt, primitivas appellabo" [Heinz Lueneburg].

Primitive root is found in English in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

The term PRINCIPAL GROUP was introduced by Felix Klein (1849-1925) (Katz, page 791).

PRINCIPAL SQUARE ROOT appears in 1898 in Text-Book of Algebra by G. E. Fisher and I. J. Schwatt, according to Manning (1970).

The term PRINCIPLE OF CONTINUITY was coined by Poncelet (Kline, page 843).

The term PRINCIPLE OF THE PERMANENCE OF EQUIVALENT FORMS was introduced by George Peacock (1791-1858) (Eves, page 377).

PRISM is found in English in Sir Henry Billingsley's 1570 translation of Euclid's Elements (OED2).

PRISMATOID (as a geometric figure) occurs in the title Das Prismatoid, by Th. Wittstein (Hannover, 1860) [Tom Foregger].

Prismatoid is found in English in 1881 in Metrical geometry. An elementary treatise on mensuration by George Bruce Halsted: "XXXIV. A prismatoid is a polyhedron whose bases are any two polygons in parallel planes, and whose lateral faces are determined by so joining the vertices of these bases that each line in order forms a triangle with the preceding line and one side of either base. REMARK. This definition is more general than XXXIII., and allows dihedral angles to be concave or convex, though neither base contain a reentrant angle. Thus, BB' might have been joined instead of A'C" [University of Michigan Digital Library].

PRISMOID is found in 1704 in Lexicon Technicum: "I, Prismoid, is a solid Figure, contained under several Planes whose Bases are rectangular Parallelograms, parallel and alike situate" [OED2].

PROBABILISTIC is found in Tosio Kitagawa, Sigeru Huruya, and Takesi Yazima, The probabilistic analysis of the time-series of rare event, Mem. Fac. Sci. Kyusyu Univ., Ser. A 2 (1942).

The term PROBABILITY may appear in Latin in De Ratiociniis in Ludo Aleae (1657) by Christiaan Huygens, since the 1714 English translation has:

As, if any one shou'd lay that he wou'd throw the Number 6 with a single die the first throw, it is indeed uncertain whether he will win or lose; but how much more probability there is that he shou'd lose than win, is easily determin'd, and easily calculated.


TO resolve which, we must observe, First, That there are six several Throws upon one Die, which all have an equal probability of coming up.

The opening sentence of De Mensura Sortis (1712) by Abraham de Moivre (1667-1754) is translated:

If p is the number of chances by which a certain event may happen, and q is the number of chances by which it may fail; the happenings as much as the failings have their degree of probability: But if all the chances by which the event may happen or fail were equally easy; the probability of happening will be to the probability of failing as p to q.

The first citation for probability in the OED2 is in 1718 in the title The Doctrine of Chances: or, a Method of Calculating the Probability of Events in Play by De Moivre.

Pascal did not use the term (DSB).

PROBABILITY DENSITY FUNCTION. Probability function appears in J. E. Hilgard, "On the verification of the probability function," Rep. Brit. Ass. (1872).

Wahrscheinlichkeitsdichte appears in 1912 in Wahrscheinlichkeitsrechnung by A. A. Markoff (David, 1998).

In J. V. Uspensky, Introduction to Mathematical Probability (1937), page 264 reads "The case of continuous F(t), having a continuous derivative f(t) (save for a finite set of points of discontinuity), corresponds to a continuous variable distributed with the density f(t), since F(t) = integral from -infinity to t f(x)dx" [James A. Landau].

Probability density appears in 1939 in H. Jeffreys, Theory of Probability: "We shall usually write this briefly P(dx|p) = f'(x)dx, dx on the left meaning the proposition that x lies in a particular range dx. f'(x) is called the probability density" (OED2).

Probability density function appears in 1946 in an English translation of Mathematical Methods of Statistics by Harald Cramér. The original appeared in Swedish in 1945 [James A. Landau].

PROBABILITY DISTRIBUTION appears in a paper published by Sir Ronald Aylmer Fisher in 1920 [James A. Landau].

PROBABLE ERROR appears in 1812 in Phil. Mag.: "All that can be gained is, that the errors are as trifling as possible--that they are equally distributed--and that none of them exceed the probable errors of the observation" (OED2).

According to Hald (p. 360), Friedrich Wilhelm Bessel (1784-1846) introduced the term probable error (wahrscheinliche Fehler) without detailed explanation in 1815 in "Ueber den Ort des Polarsterns" in Astronomische Jahrbuch für das Jahr 1818, and in 1816 defined the term in "Untersuchungen über die Bahn des Olbersschen Kometen" in Abh. Math. Kl. Kgl. Akad. Wiss., Berlin. Bessel used the term for the 50% interval around the least-squares estimate.

In 1872 Elem. Nat. Philos. by Thomson & Tait has: "The probable error of the sum or difference of two quantities, affected by independent errors, is the square root of the sum of the squares of their separate probable errors" (OED2).

In 1889 in Natural Inheritance, Galton criticized the term probable error, saying the term was "absurd" and "quite misleading" because it does not refer to what it seems to, the most probable error, which would be zero. He suggested the term Probability Deviation be substituted, opening the way for Pearson to introduce the term standard deviation (Tankard, p. 48).

The term PROBABLE PRIME TO BASE a was suggested by John Brillhart [Carl Pomerance et al., Mathematics of Computation, vol. 35, number 151, July 1980, page 1021].

PRODUCT (in multiplication). According to the OED2, Albertus Magnus (1193-1280) used productum in his Metaphysicorum.

Fibonacci (1202) used factus ex multiplicatione and also the phrase "contemptum sub duobus numeris" (Smith, vol. 1).

Art of Nombryng (about 1430) uses both product and sum for the result in multiplication: "In multiplicacioun 2 nombres pryncipally ben necessary,..the nombre multiplying and the nombre to be multipliede... Also..the 3 nombre, the whiche is clepide product or pervenient. [...] Multiplie .3. by hym-selfe, and þe some of alle wolle be 9" (OED2).

Sum was used for the result in multiplication by Pacioli (1494), Ortega (1512; 1515), and Recorde (c. 1542; 1558) (Smith, vol. 1).

In 1542 Robert Recorde in Ground of Artes (1575) used the obsolete term offcome: "The ofcome or product" (OED2). Offcome also appears in English in 1570 in Billingsley's translation of Euclid.

According to Smith (vol. 1), Licht (1500) used simply productus, dropping the numerus from numerus productus. Clichtoveus (1503) used both numerus productus and tota summa. Fine (1530) used sum as well as numerus productus. Glareanus (1538) used summa producta. Ramus (1569) used factus.

PROGRAMMING (solving an optimization problem) appears in 1949 in the title "The Programming of Interdependent Activities: General discussion" by Marshall K. Wood and George B. Dantzig in Econometrica 17, July-October, 1949 [James A. Landau].

Linear programming. Programming in a Linear Structure is the title of a work by George B. Dantzig published in 1948.

According to Linear Programming and Network Flows by Mokhtar S. Bazaraa, John J. Jarvis, and Hanif D. Sherali, 2nd Edition, 1990), the term linear programming was coined by the economist and mathematician Tjalling Charles Koopmans (1910-1985) in the summer of 1948 while he and George B. Dantzig strolled near the beach in Santa Monica, California.

In an interview of Merrill Flood conducted by Albert Tucker on May 14, 1984, Flood indicated that he and John Tukey coined the term linear programming:

Flood: One of the friendly arguments that Tjallings Koopmans and I had concerned an appropriate name for what is now known as linear programming theory. When I was responsible for organizing the December meeting of the Allied Social Science Associations in Cleveland, probably 1947, I wanted to include a session of what was then commonly referred to as input-output analysis, after the work of Wassily Leontief. Tjallings agreed to organize such a session for the meeting, and we met in California to discuss the arrangements just prior to Neyman's Second Berkeley Symposium. Actually we discussed this while enroute from Stanford to Berkeley in a car whose other passengers were John Tukey, Francis Dresch, and a Stanford mathematician (Spencer?) who was driving. I knew a bit about Leontief's work because of the work under Marshall Wood, by George Dantzig and others, that had been pushed and encouraged by Duane Evans, who was then at the Bureau of Labor Statistics - because of my position as Chief Civilian Scientist on the War Department General Staff, with some minor responsibility for the effort in the Air Force under Marshall Wood. When Tjallings and I were trying to decide what to call the session in Cleveland I was unhappy with the input-output analysis title and wanted something that was broader and peppier, partly because of the related Air Force work. Tjallings proposed "activity analysis" as a name for the session, with some support from the economist Dresch, but Tukey and I were not satisfied. As you know, John Tukey is very good at creating good names for things, and between us John and I soon settled upon "linear programming" as an excellent name for the session. As I recall vaguely now Tjallings did not call the Cleveland session "linear programming" but his own 1948 Chicago conference went by that name even though he used 'activity analysis' in the title of his published proceedings. I forget just how Tukey and I arrived at the name 'linear programming', but it has certainly stuck. I doubt that Tukey even remembers the California incident now.

Tucker: The first paper by George Dantzig and Marshall Wood was called "Programming in a Linear Structure".

Flood: Well, it is possible that is where we got the idea.

Tucker: And the two words, interchanged, were pulled out of that. I think that's the official story.

Flood: When was that paper? Was that before that, do you think? It may be the other way around.

Tucker: Well, that paper appears in the Activity Analysis volume. But it appeared earlier. The Activity Analysis volume was published in 1951. 1 think that the paper had appeared about two years before that.

Flood: That would be later. I suspect that it came the other way around. I'm remembering vaguely the conversation Tukey and I had, and I don't remember any awareness of any such terminology by Dantzig and company.

Linear programming was used in 1949 by George B. Dantzig (1914- ) in Econometrica 17: "It is our purpose now to discuss the kinds of restrictions that fit naturally into linear programming" (OED2).

Nonlinear programming appears in the title "Nonlinear Programming" in Jerzy Neyman (ed.), Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability (1950) [James A. Landau].

Mathematical programming occurs in the title "Mathematical Programming," by A. Henderson and R. Schlaifer, Harvard Business Review 32, May-June 1954 [James A. Landau].

Dynamic programming is found in Richard Bellman, Dynamic Programming of Continuous Processes, The RAND Corporation, Report R-271, July 1954 [James A. Landau].

Quadratic programming is found in 1958 the title On Quadratic Programming by E. W. Barankin and R. Dorfman [James A. Landau].

PROGRESSION. Boethius (c. 510), like the other Latin writers, used the word progressio (Smith vol. 2, page 496).

PROJECTIVE GEOMETRY is found in 1873 in E. d'Ovidio, "Studio sulla geometria projettiva," Brioschi Ann.

Projective geometry is found in English in 1885 in Charles Leudesdorf's translation of Cremona's Elements of Projective Geometry (OED2).

PROPER FRACTION appears in English in 1674 in Samuel Jeake Arithmetic (1701): "Proper Fractions always have the Numerator less than the Denominator, for then the parts signified are less than a Unit or Integer" (OED2).

PROPER VALUE and VECTOR. See Eigenvalue.

PROPORTION. See ratio and proportion.

PROPOSITIONAL CALCULUS occurs in 1903 in Principia Mathematica by Bertrand Russell (OED2).

PSEUDO-PARALLEL was apparently coined by Eduard Study (1862-1930) in 1906 Ueber Nicht-Euklidische und Linien Geometrie.

The term PSEUDOPRIME appears in Paul Erdös, "On pseudoprimes and Carmichael numbers," Publ. Math., 4, 201-206 (1956).

The term was also used by Ivan Niven (1915-1999) in "The Concept of Number," in Insights into Modern Mathematics, 23rd Yearbook, NCTM, Washington (1957), according to Kramer (p. 500). She seems to imply Niven coined the term.

PSEUDOSPHERE. Kramer (p. 53) and the DSB imply this term was coined by Eugenio Beltrami (1835-1900).

PSYCHOMATHEMATICS. Henry Blumberg coined this term and used it in the article "On the technique of generalization" (AMM, 1940, pp. 451-462). He wrote: "By Psychomathematics, I understand, namely, that union of mathematics and psychology - using the latter in a broad, non technicals sense - whose function it is to explain how mathematical ideas arise, and to formulate heuristically helpful principles in mathematical exploration" [Carlos César de Araújo].

PURE IMAGINARY is found in 1857 in "A Memoir Upon Caustics" by Arthur Cayley in the Philosophical Transactions of the Royal Society of London: "... the radius may be either a real or a pure imaginary distance ..." [University of Michigan Historical Math Collection].

PURE MATHEMATICS was used by Francis Bacon: "In mathematics I can report no deficiency, except it be that men do not sufficiently understand the excellent use of the Pure Mathematics" (Boyer, page 339).

Leonahrd Euler used the term in 1761 in the title "Specimen de usu observationum in mathesi pura."

The first edition of the Encyclopaedia Britannica (1768-1771) has: "Pure mathematics have one peculiar advantage, that they occasion no disputes among wrangling disputants, as in other branches of knowledge; and the reason is, because the definitions of the terms are premised, and every body that reads a proposition has the same idea of every part of it."

PYRAMID. According to Smith (vol. 2, page 292), "the Greeks probably obtained the word 'pyramid' from the Egyptian. It appears, for example, in the Ahmes Papyrus (c. 1550 B. C.). Because of the pyramidal form of a flame the word was thought by medieval and Renaissance writers to come from the Greek word for fire, and so a pyramid was occasionally called a 'fire-shaped body.'"

PYTHAGOREAN THEOREM. Apollodorus, Cicero, Proclus, Plutarch, Athenaeus, and other writers referred to this proposition as a discovery of Pythagoras, according to Heath's edition of Euclid's Elements.

The term Pythagorean theorem appears in English in 1743 in A New Mathematical Dictionary, 2nd ed., by Edmund Stone.

Pythagorean theorem, is the 47th Prop. of the first Book of Euclid.

This citation was provided by John G. Fauvel, who suggests the term may also be contained in the first edition of 1726, but he does not have a copy of that edition.

Pythagorean axiom appears in 1912 in G. Kapp, Electr.: "The well-known Pythagorean axiom that the sum of the squares of the kathetes in a rectangular triangle is equal to the square of the hypotenuse" (OED2).

Some early twentieth-century U. S. dictionaries have Pythagorean proposition, rather than Pythagorean theorem.

[Randy K. Schwartz contributed to this entry.]

PYTHAGOREAN TRIPLE. Pythagorean triad appears in 1909 in Webster's New International Dictionary.

Pythagorean number triplet appears in 1916 in Historical Introduction to Mathematical Literature by George Abram Miller: "Pythagorean number triplets appear also in some of the Hindu writings which antedate the lifetime of Pythagoras."

Pythagorean triple is found in March 1937 in The Mathematics Teacher: "Later in the book the quest for primitive Pythagorean triples, a beautiful illustration, by the way, of the methods of mathematical reasoning, leads just as naturally to a consideration of 'Fermat's last theorem' and other topics in the theory of numbers." The term may be much older, however.


Q. E. D. Euclid (about 300 B. C.) concluded his proofs with hoper edei deiksai, which Medieval geometers translated as quod erat demonstrandum ("that which was to be proven"). 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. The Q. E. D. abbreviation was also used by Isaac Newton in the Principia, by Galileo in a Latin text, and by Isaac Barrow, who additionally used quod erat faciendum (Q. E. F.), quod fieri nequit (Q. F. N.), and quod est absurdum (Q. E. A.).

[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).