**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 x^{2} - Ay^{2} = 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 61y^{2} +
1 = x^{2} and 109y^{2} + 1 = x^{2} ? ... 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
ax^{2} + 1 = y^{2} 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 y^{2} - Ax^{2} = 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.*

**POINT OF ACCUMULATION.** See *limit point.*

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 2^{4q+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.

and

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.,

**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 = ax^{2} + bxy + cy^{2}
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).