Earliest Known Uses of Some of the Words of Mathematics (P)

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.

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).

The term PARAMETER was introduced by Gottfried Wilhelm Leibniz (1646-1716) (Kline, page 340). He used the term in 1692 in Acta Eruditorum 11 (Struik, page 272).

PARAMETRIC EQUATION. "Mono-parametric equation" occurs in "On the Geometry of Planes in a Parabolic Space of Four Dimensions," Irving Stringham, Transactions of the American Mathematical Society, Vol. 2, No. 2 (Apr., 1901).

Parametric equation is found in "Covariants of Systems of Linear Differential Equations and Applications to the Theory of Ruled Surfaces," E. J. Wilczynski, Transactions of the American Mathematical Society, Vol. 3, No. 4. (Oct., 1902).

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

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

Jacobi used the term differentialia partialia in 1841 in "De determinantibus functionalibus" (Cajori vol. 2, page 236)

The term 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).

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).

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 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).

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 F. Galton, Jrnl. Anthrop. Inst. Feb. 276: "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).

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 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.

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].

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].

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!

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").

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.

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 XIV (OED2).

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

The French term is "le principe des tiroirs de Dirichlet," which can be translated "the principle of the drawers of Dirichlet."

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.

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.

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".

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).

POINT OF ACCUMULATION appears in 1927 in Bull. Amer. Math. Soc. XXXIII. 14: "The Metrization Problem. The problem is to state in terms of the concepts point, and point of accumulation the conditions that a topological space be metric" (OED2).

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.

The term POISSON DISTRIBUTION was coined in 1914 by H. E. Oper, according to an Internet web page.

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 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."

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).

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]

POLYGONAL NUMBERS were defined by Hypsicles (DSB).

The term 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.

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].

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."

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 (for a four dimensional convex solid) was introduced by Alicia Boole Stott (1860-1940), 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.

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 dated 1941 in MWCD10.

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

Cardano (1545) called positive numbers numeri ueri or ueri 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.

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).

The term POTENTIAL is due to Gauss (1840), according to Smith (1906).

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 (Smith, 1906; Encyclopaedia Britannica, article: "Green").

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 SERIES is dated 1893 in MWCD10.

PRECALCULUS (an adjective) is dated 1964 in MWCD10.

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).

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 L. E. Dickson, Hist. Theory Numbers: "A test for the primality of 2n +/- 1" (OED2).

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. For example, Primary Elements of Algebra for Common Schools and Academies (1866) 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.

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.

The term PRIMITIVE ROOT was introduced by Leonhard Euler (1707-1783), according to Dickson, 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].

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

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 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].

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.

Pascal did not use the term (DSB).

PROBABILITY DENSITY FUNCTION. 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].

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].

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 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).

Linear programming appears in 1953 in Cooper & Henderson, W. W. Cooper et al., Introd. Linear Programming "Linear programming is concerned with the problem of planning a complex of interdependent activities in the best possible (optimal) fashion" (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 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).

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).

PURE IMAGINARY is found in 1881 in Elements of Algebra by G. A. Wentworth: "When a is zero, the root is a pure imaginary." The term may appear in the 1876 edition of An Elementary Treatise on Elliptic Functions by Arthur Cayley, which has not been consulted; it does, however, appear in the 1961 Dover "corrected republication" of the 1895 edition of that work [James A. Landau].

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).

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.

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

[Randy K. Schwartz contributed to this entry.]