# Earliest Known Uses of Some of the Words of Mathematics (A)

**ABBE-HELMERT CRITERION**for randomness is named for Ernst Abbe and F. R. Helmert. The criterion is based on the first-order serial correlation coefficient and, according to H. O. A. Wold & L. Juréen

*Demand Analysis: A Study in Econometrics*p. 332 , "was first proposed by ˜ (1863) and a large-sample theory was developed by Helmert (1905)." The articles referred to are Abbe (1863), "über die Gesetzmässigkeit in der Vertheilung der Fehler bei Beobachtungsreihen" in

*Gesammelte Abhandlungen*,

**2**, 55 – 81. and Helmert (1905) "über die Genauigkeit der Kriterien des Zufalls bei Beobachtungsreihen,"

*Sitzungsberichte der Kgl. Preussischen Akad. der Wissenschaften*1905, 594-512. ˜'s work is particularly remarkable, coming so many years before the era of modern time series analysis. Compare the entries SERIAL CORRELATION and CORRELOGRAM.

**ABELIAN EQUATION**is named for a kind of equation treated by Niels Henrik Abel (1802-1829) in his "Mémoire sur une classe particulière d'équations résoluble algébriquement" (1829)

*Oeuvres Complètes,*, 478-514.

**1**Leopold Kronecker (1823-1891) introduced the term

*Abelsche Gleichung*in an 1853 paper on algebraically soluble equations, "Über die algebraisch auflösbren Gleichungen, erste Abhandlung" in Werke IV, p. 27ff. Kronecker used the term to describe an equation which in modern terms would be described as having cyclic Galois group. According to B. L. van der Waerden

*A History of Algebra*(1985, p. 149) Kronecker was the first German mathematician to fully recognise the importance of the investigations of Abel and Galois on the solubility of algebraic equations.

*Abelian equation*is found in English in 1861 in

*Rep. Brit. Assoc. Advancem. Sci. 1860*: "As we can always express the roots of an Abelian equation by radicals.., so also the solution of an Abelian congruence depends ultimately on the solution of binomial congruences." [John Aldrich, Peter M. Neumann, OED]

**ABELIAN FUNCTION.**C. G. J. Jacobi (1804-1851) proposed the term

*Abelsche Transcendenten*(Abelian transcendental functions) in

*Crelle's Journal*8 (1832)

*Werke III*, p. 481. (DSB).

*Abelian (function)*is found in English in 1846 in

*Examples of the Processes of the Differential and Integral Calculus*(ed. 2) ii by D. F. Gregory: "There are also many memoirs in Crelle's Journal, both on elliptic functions, and on those which are called hyper-elliptic or Abelian." [OED]

*Abelian function*appears in the title "Zur Theorie der Abelschen Functionen" by Karl Weierstrass (1815-1897) in Crelle's

*Journal,*47 (1854) reprinted in

*Werke I*, p. 132.

Weierstrass' first publications on Abelian functions appeared in the Braunsberg school prospectus (1848-1849) reprinted in

*Werke I,*p. 111.

**ABELIAN GROUP.**Camille Jordan (1838-1922) wrote

*groupe abélien*in 1870 in

*Traité des Substitutions et des Equations Algébraiques.*However, Jordan does not mean a commutative group as we do now, but instead means the symplectic group over a finite field (that is to say, the group of those linear transformations of a vector space that preserve a non-singular alternating bilinear form). In fact, Jordan uses both the terms "groupe abélien" and "équation abélienne." The former means the symplectic group; the latter is a natural modification of Kronecker's terminology and means an equation of which (in modern terms) the Galois group is commutative.

An early use of "Abelian" to refer to commutative groups is H. Weber, "Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist,"

*Mathematische Annalen*, 20 (1882), 301--329. The term is used in the first paragraph of the paper without definition; it is given an explicit definition in the middle of p. 304.

*Abelian group*is found in English in 1892 in F. N. Cole tr.

*E. Netto Theory Substitutions Contents*: "Groups analogous to the Abelian groups." [Peter M. Neumann, Julia Tompson, OED]

The term

**ABELIAN INTEGRAL**is found in a letter of Sept. 8, 1844, from William Henry Fox Talbot: "What is the definition of an Abelian Integral? for it appears to me that most integrals possess the Abelian property." The letter was addressed to John Frederick William Herschel, who, in his reply of Sept. 13, 1844, wrote: "I suppose the most general definition of an Abelian Integral might be taken to be this that between +(

*x*) and +(φ(

*x*)) there shall subsist an algebraical relation between several such functions." [The Talbot letters are available at http://foxtalbot.dmu.ac.uk/.]

The term

**ABELIAN THEOREM**was introduced by C. G. J. Jacobi (1804-1851). He proposed the term in

*Crelle's Journal*8 (1832), writing that it would be "very appropriate" (DSB).

**ABSCISSA.**According to Cajori (1919, page 175), "The words 'abscissa' and 'ordinate' were not used by Descartes. ... The technical use of 'abscissa' is observed in the eighteenth century by C. Wolf and others. In the more general sense of a 'distance' it was used earlier by B. Cavalieri in his Indivisibles, by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome, and by others."

*Abscissa*is found in Latin in 1656 in

*Exercitationum Mathematicarum*by Frans van Schooten. See page 285. [Bill Stockich]

According to Cajori (1906, page 185), "The term

*abscissa*occurs for the first time in a Latin work of 1659, written by Stefano degli Angeli (1623-1697), a professor of mathematics in Rome." A footnote attributes this information to Moritz Cantor.

*Abscissa*was used in Latin by Gottfried Wilhelm Leibniz (1646-1716) in "De linea ex lineis numero infinitis ordinatim ductis inter se concurrentibus formata....,"

*Acta Eruditorum 11*1692, 168-171 (Leibniz,

*Mathematische Schriften,*Abth. 2, Band I (1858), 266-269).

According to Struik (page 272), "This term, which was not new in Leibniz's day, was made by him into a standard term, as were so many other technical terms."

For the word in English the

*OED*cites a passage from 1706: H. Ditton

*An Institution of Fluxions*p. 31 "'Tis required to find the relation of the Fluxion of the Ordinate to the Fluxion of the Abscisse."

See the entry ORDINATE.

**ABSOLUTE CONTINUITY.**The concept was introduced in 1884 by E. Harnack "Die allgemeinen Sätze über den Zusammenhang der Functionen einer reellen Variabelen mit ihren Ableitungen. II. Theil."

*Math. Ann.*

**21**, (1884), 217-252. The term was introduced in 1905 by G. Vitali. In the meantime several mathematicians had used the concept. See T. Hawkins

*Lebesgue's Theory of Integration*.

**ABSOLUTE CONVERGENCE**was defined by Cauchy (DSB).

*Absolute convergence*and

*unconditional convergence*appear in 1893 in

*A Treatise on the Theory of Functions*by James Harkness and Frank Morley: "A series is said to converge

*absolutely*when it still converges after all negative signs have been changed to positive. ... A convergent series which is subject to the commutative law is said to be

*unconditionally*convergent; otherwise it is said to be

*conditionally convergent.*... Absolute convergence implies unconditional convergence."

**ABSOLUTE DIFFERENTIAL CALCULUS**or

*il calcolo differenziale assoluto*was the name given by the Italian geometer Gregorio Ricci-Curbastro to a form of differential calulus he developed that is independent of the coordinate system. The subject is now usually called TENSOR ANALYSIS.

**ABSOLUTE VALUE.**In his

*Disquisitiones arithmeticae*(1801), Gauss used

*magnitudo absoluta*(article 361, page 454).

An 1807 French translation of Euler's

*Elements of Algebra*uses

*valeur absolue.*An English translation of the French is: "We have supposed that the numbers

*p*and

*q*must both be positive; of course if one makes them both negative, it does not make any change in the absolute value of the proposed formula; it does not make any change of sign in the case where the exponent is odd; and it is absolutely the same in the case where the exponent is even; thus it is not important which signs on gives to the numbers

*p*and

*q,*as long as we suppose both are of the same sign."

*Absolute value*is found in English in 1816 in a translation of

*An elementary treatise on the differential and integral calculus*by Lacroix. [OED]

Searches at Google print give

*absolute Werth*in 1831 and 1844,

*absolute Grösse*in 1836, 1841, and 1853, and

*absolute länge*in 1853.

*Absolute Betrag*is found in 1851 in

*Die Reihenentwickelungen der Differenzial- und Integralrechnung*by Dr. Oskar Schlömilch. This is a use in the modern sense.

Cayley used the term

*absolute magnitude*in 1856,

*Coll. Works.*Vol II page 359: "...it follows that

*x*will be in absolute magnitude greater or less than 1, i.e., the points in question will be imaginary or real, according as

*a*> 1 or

*a*< 1."

The

*Century Dictionary*(1889-1897) has

*absolute magnitude,*rather than

*absolute value.*

James A. Landau, Wilfried Neumaier, and Julio González Cabillón contributed to this entry. See also MODULUS and the entry on the Earliest Uses of Function Symbols.

**ABSTRACT ALGEBRA**is found in 1860 in

*The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science*: "It is the essential principle of abstract algebra to furnish rules and symbols of operation which are proper for calculating independently of the knowledge of relative magnitudes." [OED]

**ABUNDANT NUMBER.**Theon of Smyrna (about A. D. 130) distinguished between perfect, abundant, and deficient numbers.

A translation of Chapter XIV of Book I of Nicomachus has:

Now the superabundant number is one which has, over and above the factors which belong to it and fall to its share, others in addition, just as if an animal should be created with too many parts or limbs, with 10 tongues as the poet says, and 10 mouths, or with 9 lips, or 3 rows of teeth, or 100 hands, or too many fingers on one hand. Similarly, if when all the factors in a number are examined and added together in one sum, it proves upon investigation that the number's own factors exceed the number itself, this is called a superabundant number, for it oversteps the symmetry which exists between the perfect and its own parts. Such are 12, 24, and certain others, . . .This citation was provided by Sam Kutler.

*Abundaunt number*is found in English in 1557 in

*Whetstone of Witte*by Robert Recorde:

Imperfecte nombers be suche, whose partes added together, doe make either more or lesse, then the whole nomber it self... And if the partes make more then the whole nomber, then is that nomber called superfluouse, or abundaunt (OED2).

**ACUTE ANGLE.**See the entry

**RIGHT, OBTUSE and ACUTE ANGLES**.

**ACUTE TRIANGLE.**

*Oxygon*appears in its Latin form in 1570 in Sir Henry Billingsley's translation of Euclid: "An oxigonium or an acuteangled triangle, is a triangle which hath all his three angles acute" (OED2).

*Acutangle triangle*appears in 1685 in R. Williams,

*Euclid*: "Oxygone, or Acutangle triangle is that whose angles are all acute" (OED2).

*Oxygon*is found in 1838 in Sir William Rowan Hamilton,

*Logic,*: "Oxygon, i.e. triangle which has its three angles acute" (OED2).

*Acutangular*appears in

*Chambers Cyclopaedia*in 1727-41: "If all the angles be acute..the triangle is said to be acutangular, or oxygonous" (OED2).

**ADDEND.**Johann Scheubel (1494-1570) in an arithmetic published in 1545 wrote

*numeri addendi*(numbers to be added).

Orontius Fineus (1530) used

*addendi*alone (1555 ed., fol. 3), according to Smith (vol. 2, page 89).

*Addend*was used in English by Samuel Jeake (1623-1690), in

*Logisicelogia, or arithmetic surveighed and reviewed,*written in 1674 but first published in 1696: "Place the Addends in rank and file one directly under another" (OED2).

**ADDITION.**Fibonacci used the Latin

*additio,*although he also used

*compositio*and

*collectio*for this operation (Smith vol. 2, page 89).

The word is found in English in about 1300 in the following passage:

Here tells þat þer ben .7. spices or partes of (th)is craft. The first is called addicion, þe secunde is called subtraccion. The thyrd is called duplacion. The 4. is called dimydicion. The 5. is called multiplicacion. The 6. is called diuision. The 7. is called extraccion of þe rote.The citation is from "The crafte of nombrynge" (ca. 1300), one of the earliest manuscripts [Egerton ms. 2622] in the English language that refers to mathematics. The transcription was carried out by Robert Steele (1860-1944), and it was first privately printed in 1894 by the

*Early English Text Society*(London). It was later included in "The Earliest arithmetics in English," a sourcebook edited by Robert Steele, and published for the same Society, by H. Milford, Oxford University Press, xviii, 84 pages, London, 1922. [Julio González Cabillón]

**ADDITIVE IDENTITY**is found in 1953 in

*First Course in Abstract Algebra*by Richard E. Johnson [James A. Landau].

**ADDITIVE INVERSE**is found in the 1953 edition of

*A Survey of Modern Algebra*by Garrett Birkhoff and Saunders MacLane. It may be in the first edition of 1941 [John Harper]. It is also found in 1953 in

*First Course in Abstract Algebra*by Richard E. Johnson [James A. Landau].

**ADDITIVITY**(in ANOVA) is found in 1947 in C. Eisenhart, "The Assumptions Underlying the Analysis of Variance,"

*Biometrics,*3, 1-21 (David, 1995).

See VARIANCE.

**ADEQUALITY.**The Latin word

*adaequalitas*was used by Pierre de Fermat.

André Weil writes in

*Number Theory, An approach through history from Hammurapi to Legendre*:

[Fermat] introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.According Mikhail Katz, Diophantus coined the term

*parisotes*to refer to an approximate equality. The term was rendered as

*adaequalitas*in Claude Gaspard Bachet de Méziriac's Latin translation of Diophantus, and adéquation and adégaler in Paul Tannery's French translation of Fermat's Latin treatises on maxima and minima and related problems.

[Information from the Wikipedia article Adequality.]

**ADJOINT EQUATION.**Lagrange used the term

*équation adjointe.*

In 1889, T. Craig wrote in

*Treat. Linear Differential Equations*:

When Chapter I was written..I had not seen Forsyth's memoir, and had not been able to find an adopted English term for Lagrange'sThis citation was taken from the OED2.équation adjointé,so I used the wordadjunct,suggested by the Germanadjungirte,and not unlike the Frenchadjointe.It seems better now, however, to employ the wordassociate,or, when speaking simply of Lagrange'séquation adjointé,the wordadjoint.

**ADJOINT LINEAR FORM**appears in 1856 in Arthur Cayley, "A Third Memoir Upon Quantics,"

*Philosophical Transactions of the Royal Society of London*Vol. 146. (1856), pp. 627-647 reprinted in

*Papers II*p. 310 [University of Michigan Historical Math Collection].

**ADJOINT MATRIX.**The

*OED*quotes from M. Bôcher's

*Introduction to Higher Algebra*of 1907: "By the adjoint

*A*of a matrix

*a*is understood another matrix of the same order in which the element in the

*i*th row and

*j*th column is the cofactor of the element in the

*j*th row and

*i*th column of

*a.*" (p. 77)

**ADMISSIBLE ESTIMATE**and

*admissible system of (acceptance) regions*appear in Wald's 1939 "Contributions to the Theory of Statistical Estimation and Testing Hypotheses,"

*Annals of Mathematical Statistics,*

**10**, 299-326 [John Aldrich, based on David (2001)].

See DECISION THEORY.

**AFFINE.**

*Affinis*and

*affinitas*were first used by Leonhard Euler in

*Introductio in analysin infinitorum*(1748) Chapter XVIII: "De similitudine et affinitate linearum curvarum." He also wrote (II. xviii. 239): "Quia Curvae hoc modo ortae inter se quandam Affinitatem tenent, has Curvas affines vocabimus."

The words continued to be used a little throughout the 19th century, for example in Möbius "Der barycentrische Calcul" (1827) Chapter 3: "Von der Affinitaet." There is a brief historical summary of this stage by E. Papperitz in the

*Encyclopaedie der mathematische Wissenschaften*III A B 6 (1909), p.517. [Ken Pledger]

*Affine*is found in English in 1895 in

*Annals of Mathematics*: "Let us find the differential equations of the first order which are invariant under the G1 of the affine transformations in the plane." [OED]

**ALGEBRA**comes from a phrase in the title of a work written in Arabic around 825 by al-Khowarizmi, al-jabr w'al-muqâbalah. Wikipedia. Katz explains the use of the terms:

*al-jabr*refers to the operation of transposing a subtracted quantity on one side of an equation to the other side where it becomes an added quantity;

*al-muqâbalah*refers to the reduction of a positive term by subtracting equal amounts from both sides of the equation.

The word

*al-jabr*means "the reunion of broken parts" and when the Moors took the word

*al-jabr*into Spain, an

*algebrista*being a restorer, one who resets broken bones. Thus in

*Don Quixote*(II, chap. 15), mention is made of "

*un algebrista*who attended to the luckless Samson." At one time it was not unusual to see over the entrance to a barber shop the words "Algebrista y Sangrador" (bonesetter and bloodletter) (Smith vol. 2, pages 389-90). The word in this sense also had some currency in English. Thus the

*OED*has a quotation from a book on surgery from 1541: "The helpes of Algebra & of dislocations."

Returning to mathematics, when the work of al-Khwarizmi was translated into Latin—around 1140 by Robert of Chester—the words in the title were not translated, instead the Arabic words were borrowed. Thus the title became

*Liber algebrae et almucabala*. Viète disliked the word

*algebra*because it had no meaning in any European language and promoted instead the term

*analytic art*: see the entry ANALYSIS. The original Arabic/Latin phrase is echoed in English in John Dee's Preface to Billingsley's translation (1570) of Euclid's

*Elements*: "The Science of workyng Algiebar and Almachabel, that is, the Science of findyng an vnknowen number, by Addyng of a Number, and Diuision and æquation." In the text Billingsley drops the second half of the original phrase and uses the first half in a more general way: "That more secret and subtill part of Arithmetike, commonly called Algebra." Robert Recorde had done the same a few years before in

*The Pathwaie to Knowledge*in 1551: "Also the rule of false position, with dyvers examples not onely vulgar, but some appertayning to the rule of Algeber." These quotations are from the

*OED*.

It was once believed that the word

*algebra*was derived from the name Gabir ben Allah of Sevilla, commonly called Geber, a celebrated astronomer apparently skilled in algebra. Joseph Raphson's

*Mathematical Dictionary*(1702) has: "

*Algebra,*from

*Al*in Arabick which signifies Excellent, and

*Geber*the Name of the supposed Inventor of it...." This etymology based on EPONYMY is not totally absurd: compare the word ALGORITHM.

The

*algebra*of al-Khwarizmi was the name for an operation used in solving an equation but by 1600 the word was applied to the study in which that operation was used. The word continued to be applied to a study, but one whose scope increased enormously especially in the 19

^{th}and 20

^{th}centuries. See the Historical Survey in the

*Encyclopaedia of Mathematics*and the entries GROUP, FIELD, MATRIX, RING, VECTOR SPACE and other algebra topics.

In the 19

^{th}and 20

^{th}centuries the word

**algebra**acquired further meanings. It is sometimes used to mean a formal calculus as in the title of Schröder's book of 1890

*Vorlesungen über die Algebra der Logik*. An early expression of this point of view is found in Augustus de Morgan's

*Trigonometry and Double Algebra*(1849, p. 101): "It is most important that the student should bear in mind that,

*with one exception,*no word nor sign of arithmetic or algebra has one atom of meaning throughout this chapter, the object of which is

*symbols, and their laws of combination,*giving a

*symbolic algebra*(page 92) which may hereafter become the grammar of a hundred distinct

*significant algebras*." The term is also used in some very specific constructions, e.g.

**Boolean algebra**and

**algebra of sets**(in measure theory). A

*JSTOR*search found

*Boolean algebra*in the title of a 1913 article by Sheffer while the expression

*algebra of sets*(an abbreviation for

*Boolean algebra of sets*) was popularised by Halmos's

*Measure Theory*(1950). See the

*Encyclopaedia of Mathematics*entries Boolean algebra and Algebra of sets.

**ALGEBRAIC CURVE.**See TRANSCENDENTAL CURVE.

**ALGEBRAIC FUNCTION**is found in English in 1794 in

*An Essay on the Principles of Human Knowledge*by E. Waring, M. D.: "8. Infinite series: e. g. let a quantity denoting the ordinate, be an algebraic function of the abscissa

*x*, by the common methods of division and extraction of roots, reduce it into an infinite series ascending or descending according to the dimensions of

*x*, and then find the integral of each of the resdulting terms." [Google print search]

**ALGEBRAIC GEOMETRY**is found in 1821 in

*Cambridge Problems.*[OED]

**ALGEBRAIC LOGIC.**The following is from "From Paraconsistent Logic to Universal Logic" by Jean-Yves Béziau:

The road leading from the algebra of logic to algebraic logic is an interesting object of study for the historian of modern logic which has yet to be fully examined. Curry stands in the middle of the road, he was the first to use the expression "algebraic logic" in (Curry 1952) and not Halmos as erroneously stated in (Blok/Pigozzi 1991, p. 365), but what he meant by it was still close to algebra of logic. Halmos introduced this expression rather to denote the algebraic treatement of first-order logic, but nowadays the expression "algebraic logic" is used to include both the zero and the first-order levels.

**ALGEBRAIC NUMBER.**According to an Internet web page, this term was used by Abel.

*Algebraic quantity*appears in 1673 in

*Elements of Algebra*(1725) by John Kersey: "Two or more Algebraic quantities" (OED2).

*Algebraic*(in the sense of an algebraic number) is found in 1840 in

*Mathematical Dissertations*(1841) by J. R. Young [James A. Landau].

*Algebraic number*appears in 1870 in the title

*Kritische Untersuchungen über die Theorie der algebraischen Zahlen*by H. Schwarz.

**ALGEBRAIC TOPOLOGY**appears in 1942 in the title

*Algebraic Topology*by Solomon Lefschetz. A

*JSTOR*search found a slightly earlier appearance in Norman E. Steenrod "Universal Homology Groups,"

*American Journal of Mathematics*,

**58**, (1936), 661-701.

**ALGORITHM**is derived via Latin from the Arabic

*al-Khowarazmi*, the native of Khwarazm (Khiva), surname of the Arab mathematician and astronomer Abu Ja'far Mohammed Ben Musa Al-Khwarizmi (c. 780 - c.850).

The

*OED*has separate entries for

*algorithm*and the earlier

*algorism.*It says that

*algorithm*is derived from the earlier word

*algorism*and "influenced by the Greek word

*arithmos*(number)." However, other dictionaries show

*algorism*as a Middle English spelling of

*algorithm.*

The term originally meant "arithmetic" or "the Hindu-Arabic numerals."

In the twelfth century, the arithmetic of al-Khowarazmi was translated into Latin by Robert of Chester and it bore the name

*Algoritmi de numero indorum*meaning

*al-Khowarazmi*

*on the Hindu numbers*.

In Middle English

*augrim, algorisme,*and other spellings are found. Chaucer used

*augrym.*Recorde used

*augrim*and

*algorisme.*

In 1503 in

*Margarita philosophica*Gregor Reisch used the headings

*Algorithmus de minutijs vulgaribus, Algorithmus de minutijs physicalibus,*and

*Algorithmus cum denarijs piectilibus: seu calcularis.*Under these headings are rules for handling fractions, astronomical fractions, and rules for computing on the reckoning board.

*Algorithmus*appears in 1534 in

*Algorithmus demonstratus*by Johann Schoner.

*Algorithmus*(in the sense of a systematic technique for solving a problem) was used by Gottfried Wilhelm Leibniz in 1684:

Ex cognito hoc velutThe citation above is from "Nova Methodvs pro maximis et minimis, itemque tangentibus, quae nec fractas, nec irrationales quantitates moratur, & singulare pro illis calculi genus, per G.G.L." (A new method for maxima and minima, as well as tangents, which is not obstructed by fractional or irrational quantities), Leibniz' first published account of the calculus [Algorithmo,ut ita dicam, calculi hujus, quem vocodifferentialem,omnes aliae aequationes differentiales inveniri possunt per calculum communem, maximae que & minimae, item que tangentes haberi, ita ut opus non sit tolli fractas aut irrationales, aut alia vincula, quod tamen faciendum fuit secundum Methodos hactenus editas. (From this rule, known as an algorithm, so to speak, of this calculus, which I call differential, all other differential equations may be found by means of a general calculus, and maxima and minima, as well as tangents [may be] obtained, so that there may be no need of removing fractions, nor irrationals, nor other aggregates, which nevertheless formerly had to be done in accordance with the methods published up to the present.)

*Acta Eruditorum,*vol. 3, pp. 467-473, October 1684], page 469. Both terms "Algorithmo" and "differentialem" are italicised in the original. The English translation is from Evelyn Walker's translation of extracts from Leibniz' memoir found on p. 623 of Smith's

*Source Book in Mathematics*(1929), vol. 2.

*Algorithm*is found in English in 1699 in

*Phil. Trans.*XXI. 263: "The Algorithm or Numeral Figures now in use" (OED2).

*Algorithm*is found in English in 1715 in

*The Theory of Fluxions*by Joseph Raphson: "Now from this being known as the Algorithm, as I may say of this Calculus, which I call differential, ..." (p.23).

*Algorithm*was used by Euler, for instance in his article "De usu novi algorithmi in problemate Pelliano solvendo," and its use was then firmly established.

According to the Theseus Logic, Inc., website, "The term algorithm was not, apparently, a commonly used mathematical term in America or Europe before Markov, a Russian, introduced it. None of the other investigators, Herbrand and Gödel, Post, Turing or Church used the term. The term however caught on very quickly in the computing community."

[Julio González Cabillón, Heinz Lueneburg, David Fowler, and John Aldrich contributed to this entry.]

**ALIAS**as a term in the theory of experimental design appears in D. J. Finney's "The Fractional Replication of Factorial Arrangements,"

*Annals of Eugenics*,

**12**, (1945), 291-301. (David (2001))

**ALIQUOT PART.**

*Aliqi*is an adjective in Latin meaning "some" or "any."

St. Augustine of Hippo (354-430) wrote in

*De Civitate Dei*: "Itemque in denario quaternarius est aliqua pars eius..."

*Aliquot part*occurs in English in 1570 in Sir Henry Billingsley's translation of Euclid's

*Elements*: "This..is called .. a measuring part .. and of the barbarous it is called .. an aliquote part" (OED2).

The term

**ALLOTRIOUS FACTOR**was coined by James Joseph Sylvester.

The term

**ALMOST PRIME OF ORDER**was "apparently introduced" by B. V. Levin in "A one-dimensional sieve," (Russian)

*r**Acta Arith.*10 (1964/65), 387-397. MR 31, 4774, according to H. Halberstam and H.-E. Richert in

*Sieve Methods*(1974) [Paul Pollack].

The term

**ALPHAMETIC**was coined in 1955 by J. A. H. Hunter (Schwartzman). In the Dec. 23, 1955, Toronto

*Globe & Mail*Hunter wrote, "These alphametics seem set to take the place of crosswords as a new craze... Don't forget that each letter stands for a particular figure" (OED2).

**ALTERNATE ANGLE.**Henry Billingsley's 1570 translation of Euclid's

*Elements*has: "If a right line falling vpon two right lines, do make the alternate angles equall the one to the other: those two right lines are parallels the one to the other." [Alan Hughes]

In 1828,

*Elements of Geometry and Trigonometry*(1832) by David Brewster (a translation of Legendre) has:

AGO, GOC, we have already namedIn May 1940, a review ininterior angles on the same side; BGO, GOD, have the same name; AGO, GOD, are calledalternate interiorangles, or simplyalternate; so also, are BGO, GOC: and lastly EGB, GOD, or EGA, GOC, are called, respectively, the oppositeexterior and interiorangles; and EGB, COF, or AGE, DOF, thealternate exteriorangles.

*The Mathematics Teacher*has: "While the writer is correct in saying that the abbreviated description is in general use for 'the exterior angles on the same side of the transversal,' he is wrong about the 'interior angles on the same side of the transversal.' At least two textbooks use the satisfactory description 'consecutive interior angles' as a substitute for the latter clumsy expression. The reference to the exterior angles on the same side of the transversal is so infrequent as not to warrant a need for an abbreviated description."

**ALTERNATING GROUP**appears in Camille Jordan, "Sur la limite de transitivité des groupes non alternés,"

*Bull. Soc. math. de France*1 (1873).

*Alternating group*is found in English in 1886 in

*American Journal of Mathematics*8/275: "The Borchardt moduli...are irrational to a considerable degree, and their group is 32-1-edric isomorphic with the alternating group of the

*x*'s." [OED]

**ALTERNATING SERIES**is found in 1817 in

*Philosophy of Arithmetic*by Sir John Leslie: "The original fraction...is converted by substitution into the alternating series, [etc.]." [OED]

**ALTERNATIVE HYPOTHESIS.**See HYPOTHESIS and HYPOTHESIS TESTING.

**AMBIGUOUS CASE.**In 1761

*Navigation New Modelled*by Henry Wilson has "...where the Case is so ambiguous by Sol. 8..." [Google print search]

*Ambiguous case*is found in 1827 in

*Mathematical and astronomical tables*by William Galbraith: "If in this triangle the side

*B*be greater than

*C*, there may be two triangles formed, constituting what is called the ambiguous case, that is, it admits of two solutions, either of which answers the conditions required, unless from some known circumstances one of htem must be adopted in preference to the other." [Google print search]

**AMENABLE**semigroups. In his "Amenable semigroups,"

*Illinois Journal of Mathematics*,

**1**, 509-544 M. M. Day gives the definition, "A semigroup

*Σ*is called amenable if there is a mean

*μ*on

*m*(

*Σ*) which is both left and right invariant." A good pun needs no explanation and so Day does not point out that the term fuses "amenable" in its usual meaning with the barbarism "meanable." Day's term was an improvement on "messbar" (measurable) used in J. von Neumann (1929) Zur allgemeinen Theorie des Maßes,

*Fundamenta Mathematicae*,

**13**, 73-116. Day reflected on the success of the paper and its title in This week's citation classic June 27 1983.

In a 2010 email, Peter Flor writes:

The English word amenable was intended by M. Day, who introduced the term, to remind the audience of the word "mean" since the concept of amenability is defined by the existence of a certain mean. Now, "mean" (in the mathematical sense) is "Mittel" in German, and "moyenne" in French. Accordingly, amenable (semi-)groups were called "mittelbar" in German and "moyennable" in French, since about 1950. J.-P. Pier, author of the bookSee alsoAmenable Locally Compact Groups,called them "groupes moyennables" in his earlier publications in French. The German word "mittelbar" might be called a barbarism, just like your characterization of the non-existing word "meanable" which may have induced Day to coin the term "amenable". However, "mittelbar" is certainly more suitable than von Neumann's "messbar" which usually means measurable.

*Wikipedia*amenable group.

**AMICABLE NUMBERS.**The philosopher Iamblichus of Chalcis (c. 250-330) wrote that the Pythagoreans called certain numbers amicable numbers. An Internet web page claims Pythagoras coined the term.

Ibn Khaldun (1332-1406) used a term which is translated "amicable (or sympathetic) numbers."

In his 1796 mathematical dictionary, Hutton says he believes the term

*amicable number*was coined by Frans van Schooten (1615-1660).

The term appears in the title

*De numeris amicabilibus*by Euler (1747) in Opera mathematica. Series prima. Volumen II p. 59.

Sometimes the terms

*amiable*or

*agreeable*are used.

**AMPLITUDE**(of an elliptic integral, meaning the angle subtended by the elliptical arc) appears in French (same spelling as in English) in 1786 in Legendre's paper

*Second Mémoire sur les Intégrations par arcs d'ellipses*, which was published in the

*Histoire de l'Académie royale des sciences, Année MDCCLXXXVI avec les mémoires de Mathématique & de Physique.*[James A. Landau]

The term

**ANALLAGMATIC**was used by James Joseph Sylvester (1814-1897) (Schwartzman).

The OED2 has an 1869 citation, Clifford,

*Brit. Assoc. Rep.*8: "On the Umbilici of Anallagmatic Surfaces."

**ANALYSIS**is a term with a long history in Mathematics and its meaning has changed several times. The word comes via Medieval Latin from the Greek, derived from the verb to unloose, undo. T. Heath writes in

*A History of Greek Mathematics, volume 1*(1921, p. 291), "analysis [was] according to the ancient view nothing more than a series of successive reductions of a theorem or a problem till it is finally reduced to a theorem or problem already known." Kline (page 279) states that the term was introduced by Theon of Alexandria.

François Viète used the term

*analytic art*for algebra in 1591 in

*In artem analyticem isagoge.*Schwartzman (page 23) says that "the French mathematician Viète opted for the term

*analysis*rather than

*algebra,*claiming that

*algebra*doesn't mean anything in any European language. He didn't succeed in driving out the word

*algebra,*but he did popularize

*analysis*to the point where it has stayed with us." See the entry ALGEBRA.

The earliest quotations in the OED illustrate this sense of

*analysis*as

*algebra*: "Simple Analysis is that employed in solving problems reducible to simple equations" (

*Chambers's Cyclopedia Supplement*of 1753); "

*Analysis*or the

*Analytic Method*.. is that which is commonly used in Algebra" (Charles Hutton

*A Course of Mathematics*of 1827).

In

*The Purloined Letter*(1845), Edgar Allan Poe wrote:

The mathematicians, I grant you, have done their best to promulgate the popular error to which you allude, and which is none the less an error for its promulgation as truth. With an art worthy a better cause, for example, they have insinuated the term 'analysis' into application to algebra. The French are the originators of this particular deception; but if a term is of any importance -- if words derive any value from applicability -- then 'analysis' conveys 'algebra' about as much as, in Latin, 'Poe was out of date for the French were again redefiningambitus' implies 'ambition', 'religio' 'religion', or 'homines honesti', a set ofhonorablemen.

*analysis*. Perhaps Lagrange had set the scene for analysis to invade calculus with his project to found calculus on algebra; see his lectures

*Théorie des fonctions analytiques contenant les principes du calcul différentiel dégagés de toute considération d'infiniment petits et d'évanouissans, de limites ou de fluxions et réduits à l'analyse algébrique des quantités finies*(1797). Beginning with Cauchy's

*Cours d'analyse de l'école Royale Polytechnique*of 1821 (Oeuvres II.3),

*Cours d'analyse*was the title French authors chose for their advanced works on calculus.

*Analysis*with its modern meaning in English is found in 1802 in a review of J. E. Montucla, "A History of Mathematics," in

*The Critical Review; or, Annals of Literature; Extended & Improved*: "Several new branches of analysis having arisen out of the foregoing, the author treats of them to such extent as the nature of his work will permit; such as the calculus of finite differences; that of circular, logarithmic, and imaginary quantities; the methods of limits, of analytic functions, of variations, of partial differentials, of infinite series, of eliminations, of interpolations, of continued fractions, &c." [Google print search, James A. Landau]

In the later 20

^{th}century the scope of analysis increased as newcomers such as FUNCTIONAL ANALYSIS were accommodated. Numerous sub-fields are now distinguished--REAL ANALYSIS, COMPLEX ANALYSIS, classical analysis, modern analysis etc.

For a list of entries associated with analysis see here.

[John Aldrich contributed to this entry.]

The term

**ANALYSIS OF ALGORITHMS**was coined by Donald E. Knuth. The term appears in the preface to his

*The Art of Computer Programming,*Volume 1 [Tim Nikkel].

**ANALYSIS OF VARIANCE.**See VARIANCE.

**ANALYSIS SITUS.**See TOPOLOGY.

**ANALYTIC ARITHMETIC.**At the end of the sixteenth century, a few authors, following Viete's interest in analysis, attached the word "analytic" to some terms related to math. For instance, Nicholas Reimers Ursus (1551-1600) used the expression

*arithmetica analytica*in

*Nicol. Raimari arithmetica analytica vulgo Cosa, oder Algebra,*published posthumously in 1601. [Julio González Cabillón]

The term

**ANALYTIC EQUATION**was used by Roberval in

*De geometrica planarum & cubicarum aequationum,*printed posthumously in 1693: "Dicitur locus aliquis geometricus ad aequationem analyticam revocari, cum ex una aliqua, vel ex pluribus ex illlius proprietatibus specificis, quaedam deducitur aequatio analytica, in qua una vel duae vel tres ad summum sint magnitudines incognitae." [Barnabas Hughes]

In modern analysis the term

**ANALYTIC FUNCTION**is used in two ways: (of a complex function) having a complex derivative at every point of its domain, and in consequence possessing derivatives of all orders and agreeing with its Taylor series locally; (of a real function) possessing derivatives of all orders and agreeing with its Taylor series locally. (Borowski & Borwein)

According to Youschkevitch (pages 37-84), the term was first used by Marquis de Condorcet (1743-1794) in his unpublished

*Traité du Calcul Integral*and then more conspicuously by Joseph Louis Lagrange (1736-1813) in his

*Théorie des Fonctions Analytiques*(1797). For these authors an analytic function simply signified a function of the kind treated in analysis. [Giovanni Ferraro].

The connection between the usage of Lagrange and modern usage is explained by Judith V. Grabiner in her

*The Origins of Cauchy's Rigorous Calculus*: "For Lagrange, all the applications of calculus ... rested on those properties of functions which could be learned by studying their Taylor series developments ... Once the question of convergence has been dealt with, in fact a great deal of information can be obtained about functions by studying their Taylor-series expansions. Weierstrass later exploited this idea in his theory of functions of a complex variable, retaining Lagrange's term "analytic function" to designate, for Weierstrass, a function of a complex variable with a convergent Taylor series." (pp. 52 & 194) See the

*Encyclopedia article*Analytic function. [John Aldrich]

**ANALYTIC GEOMETRY.**The following appears as a footnote in

*Descartes's Mathematical Thought*by Chikara Sasaki:

According to Boyer, the terminology "analytic geometry" was used probably for the first time by Michel Rolle in his "De l'evanouissement des quantitez inconnues dans la geometrie analytique,"In 1797 Sylvestre François Lacroix (1765-1843) wrote inMémoires de l'Académie royale des sciences(Paris), Année 1709 (1709. 9. Aoust), pp. 419-450. See Carl B. Boyer,History of Analytic Geometry(New York, 1956), p. 155: "Incidentally, in Rolle's discussion of this problem, the term 'analytic geometry' appeared in print, perhaps for the first time, in a sense analogous that of today." On p. 141 of the same book Boyer also refers to Newton's treatise titled "Artis Analyticae Specimina vel Geometria Analytica," which was published inOperae quae exstant omnia,Vol. I, pp. 389-518, under the editorship of Samuel Horsley in London in 1779. The title is ascribed not to the author Newton but to a copyist William Jones. The tract which Horsley edited asArtis analyticae specimina vel Geometria analyticais basically identical to "De Methodus serierum et fluxionum," written in Winter 1670-71, in D. T. Whiteside, ed,The Mathematical Papers of Isaac Newton,Vol. III (1670-1673), (Cambridge, 1969). Jones copied Newton's manuscript about 1710. See Whiteside's Introduction,Op. cit.,pp. 11-13. See also Whiteside's "General Introduction," inThe Mathematical Papers of Isaac Newton,Vol. I (1664-1666), (Cambridge, 1967), pp. xxiii and xxviii, notes (26) and (34). I am skeptical about Newton's use of "analytic geometry" in other works as well.

*Traité du calcul différentiel et du calcul intégral*: "There exists a manner of viewing geometry that could be called

*géométrie analytiques,*and which would consist in deducing the properties of extension from the least possible number of principles, and by truly analytic methods." This quote was taken from the DSB. The Compact DSB states that Lacroix "was first to propose the term analytic geometry."

In "The work of Nicholas Bourbaki" (

*Amer. Math. Monthly,*1970, p. 140), J. A. Dieudonné protested:

It is absolutely intolerable to use analytical geometry for linear algebra with coordinates, still called analytical geometry in the elementary books. Analytical geometry in this sense never existed. There are only people who do linear algebra badly, by taking coordinates and this they call analytical geometry. Out with them! Everyone knows that analytical geometry is the theory of analytical spaces, one of the deepest and most difficult theories of all mathematics.[Julio González Cabillón, Carlos César de Araújo]

**ANCILLARY**in the theory of statistical estimation. The term "ancillary statistic" first appears in R. A. Fisher's 1925 "Theory of Statistical Estimation.,"

*Proc. Cambr. Philos. Soc.*22. p. 724, although interest in ancillary statistics only gathered momentum in the mid-1930s when Fisher returned to the topic in

*Two New Properties of Mathematical Likelihood*and other authors started contributing to it [John Aldrich, David (1995)].

See ESTIMATION.

**ANGLE**is the subject of Definition 8 of Book 1 of Euclid's

*Elements.*The modern English word derives ultimately from the Latin

*angulus*which could mean

*corner*as well as

*angle*.

The

*OED's*first citation in English for

*angle*in the geometric sense is from John Trevisa's translation (1398) of Bartholomaeus Anglicus

*De Proprietatibus Rerum*: "Of dyuers touchyng of lynes comeþ diuers anglis. For som angle hatte

*rectus angulus*, and somme

*obliquus*and

*reflexus.*" Trevisa does not give English expressions for the three kinds of angle; for these, see the entry

**RIGHT, OBTUSE and ACUTE ANGLES**.

**ANGLE OF DEPRESSION.**A Google print search finds

*angle of depression*in 1723 in

*A System of Mathematics...*by James Hodgson: "From the Parallax of Altitude or Angle of Depression, by the means of which the Planets appear at a greater distance from the Zenith than otherwise they would do...." [James A. Landau]

**ANGLE OF ELEVATION.**A Google print search by James A. Landau finds

*angle of elevation*in a 1686 paper by E. Halley, "A Discourse Concerning Gravity, and its Properties, wherein the Descent of Heavy Bodies, and the motion of Projects is briefly, but fully handled: Together with the Solution of a Problem of great Use in GUNNERY" in the

*Philosophical Transactions*:

Prop.IX. Problem: AProjectionbeing made as you please, having the Distance and Altitude, or Descent of anObject,through which theProjectpasses, together with theAngleofElevationof thelineofDirection; to find theParameterandVelocity,that is (inFig.2.) having theAngleFGB, GM, and MX.

**ANHARMONIC RATIO**and

**ANHARMONIC FUNCTION.**Michel Chasles coined the terms

*rapport anharmonique*and

*fonction anharmonique*(Smith vol. 2, page 334).

He used the terms in his "Aperçu historique .... des Méthodes en Géométrie," 1837, p.35. He uses essentially the same definition as Möbius, but without the signs, so a harmonic range gives the value +1 rather than -1. Then he says "Ce rapport étant dit

*harmonique*dans le cas particulier où il est égal a l'unité, nous l'appellerons, dans le cas général,

*rapport*ou

*fonction anharmonique*" [Ken Pledger].

*Anharmonic relation*is found in 1843 in "Demonstration of Pascal's Theorem" by Arthur Cayley in the

*Cambridge Mathematical Journal,*vol. IV: "The demonstration of Chasles' form of Pascal's Theorem (viz. that the anharmonic relation of the planes 61, 62, 63, 64 is the same with that of 51, 52, 53, 54), is very much simpler; but as it would require some preparatory information with reference to the analytical definition of the similarity of anharmonic relation, I must defer it to another opportunity" [University of Michigan Digital Library].

*Anharmonic ratio*is found in English in in Arthur Cayley, "On the Triple Tangent Planes of Surfaces of the Third Order,"

*The Cambridge and Dublin Mathematical Journal,*vol. IV (1849), pp. 118-132: "Which remains unaltered for cyclical permutations of

*l, m, n,*ie. the anharmonic ratio of

*x, ξ, x, x*is the same as that of

*y, η, y, y*." [James A. Landau]

See also CROSS-RATIO.

**ANNULUS.**A Google print search finds the phrase

*annulus nascens*(growing ring) in 1739 in an article, "Hydraulicks" in the

*Philosophical Transactions,*although here it is perhaps not a mathematical term. [James A. Landau]

A Google print search shows that Andrew Motte's 1803 translation of Newton's

*Principia*has "...so as for form a fluid annulus, or ring, of a round figure, and concentrical to the body T; and the several parts of this annulus, performing their motions by the same law as the body P, will draw nearer to the body T, and move swifter in the conjunction and opposition of themselves and the body S, than in the quadratures." [James A. Landau]

*Annulus*is found in 1834 in the

*Penny Cyclopedia,*where it is described as "The name of a ring, or solid formed by the revolution of a circle about a straight line exterior to its circumference as an axis, and in the plane of the said circle" (OED2).

**ANSATZ**. In everyday German this word has several meanings related to the idea of

*beginning*. Two—at least—went into German and then into English mathematics. One is found in the expression

*Hibert's Ansatz*, an adaptation of

*der Hilbertische Ansatz*which is found in Hilbert and Bernays

*Grundlagen der Mathematik*(1939) II. p. 21 though the technique is older, going back to Hilbert's work of the early 1920s. Here

*Ansatz*means

*formulation*or

*approach*.

In mathematical physics

*ansatz*is often used in another sense, as a "mathematical assumption, especially about the form of an unknown function, which is made in order to facilitate solution of an equation or other problem." (

*OED*). A paper by Hans Bethe is often cited: "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette,"

*Zeitschrift für Physik A*,

**71**, (1931), 205-226. Bethe uses the term

*Ansatz*(see p. 208) without any explanation which suggests that the usage was already established. In English E. A. Uehling and G. E. Uhlenbeck wrote the word with an initial capital letter, as in German, and in quotes to signal the word's foreign origin: see "Transport Phenomena in Einstein-Bose and Fermi-Dirac Gases. I,"

*Physical Review*,

**43**, (1933), p. 553. Today the word is written like an ordinary English word. [John Aldrich]

**ANTIDERIVATIVE.**

*Anti-derivative*appears in 1903 in "Integration as a Summation" by George R. Dean in

*The American Mathematical Monthly.*[

*JSTOR*search] The word may be older.

**ANTI-DIFFERENTIAL**appears in 1904 in

*Elements of the Differential and Integral Calculus*by William Anthony Granville with Percey F. Smith. A footnote to the word

*integral*reads: "Called

*anti-differential*by some writers" [James A. Landau].

The term

**ANTILOGARITHM**was introduced by John Napier. It appears in

*A Description of the Admirable Table of Logarithmes*(1616), an English translation by Edward Wright of Napier's work:

And they are also the Logarithmes of the complements of the arches and sines towards the right hand, which we call Antilogarithmes.

**APARTMENT, BUILDING,**and

**CHAMBER**in group theory. These are translations of the everyday French words,

*appartement, immeuble,*and

*chambre*. J. Tits writes in "Buildings and their Applications,"

*Proceedings of the International Congress of Mathematicians*: Vancouver, 1974 vol. 1, p. 209:

The buildings considered in this talk are particular simplicial complexes naturally associated to algebraic simple groups. The "real estate" terminology due to N. Bourbaki [In the English translation, N. Bourbakiéléments de mathématique. Groupes et algèbres de Lie ch. 4-6.1968], originated in the fact that the maximal simplexes of these complexes are called "chambers" (in French, "chambre", that is, "room") because of their close connection with the "Weyl chamber" in the theory of root systems.

*Lie groups and Lie algebras chapters 4-6*(available on Google books), the concepts are defined on p. 35. [John Aldrich]

The term

**APOLAR**is due to Th. Reye, according to Maxime Bôcher in

*Modern Higher Algebra*(1885).

**APOTHEM**is found in 1822 in

*Elements of Geoemtry and Trigonometry*translated from the French of M. A. Legendre, edited by David Brewster. [Google print search]

The word does not appear in Euclid or any other ancient Greek text and appears to be a scholarly neologism [Antreas P. Hatzipolakis].

*Apothem*is commonly mispronounced; the primary stress is on the first syllable.

**APPLIED MATHEMATICS.**See the entry PURE and APPLIED MATHEMATICS.

The term

**ARBELOS**is apparently due to Archimedes. Thomas L. Heath, in his "The Works of Archimedes" [Cambridge: At the University Press, 1897], remarks that:

... we have a collection of Lemmas (Liber Assumptorum) which has reached us through the Arabic. [...] The Lemmas cannot, however, have been written by Archimedes in their present form, because his name is quoted in them more than once.

... though it is quite likely that some of the propositions were of Archimedean origin, e.g. those concerning the geometrical figures called respectively [arbelos, in Greek] (literally 'shoemaker's knife') and [salinon, in Greek] (probably a 'salt-cellar'), ...

*Arbelos*is found in English in May 1859 in "On Contact, Centres of Similitude, and Radical Axes" by Matthew Collins in the

*Mathematical Monthly*: "This case is Pappus' Theorem XVI., and was referred to by him as the

*Arbelos.*" [Google print search]

[Ken Pledger, Antreas P. Hatzipolakis, Julio González Cabillón]

**ARBORESCENCE**is found in Y.-j. Chu and L. Tseng-hong,

*On the shortest arborescence of a directed graph,*Sci. Sin. 14, 1396-1400 (1965).

**ARBORICITY**occurs in G. Chartrand, H. V. Kronk, and C. E. Wall,

*The point-arboricity of a graph,*Isr. J. Math. 6, 169-175 (1968).

**ARCSINE.**For early uses of the symbol arc sin (or variations thereof), see the trigonometry page of the companion Math Symbols site.

This entry will show uses of

*arcsine*as a word, spelled with the final

*e.*

A Google print search by James A. Landau finds that "Discussion of Inverse Functions" by Cooper D. Schmitt in

*American Mathematical Monthly,*April 1899, has:

We read it, "xequals the arc whose sine isa,or more briefly, "arc-sinea,"i. e.xequals arc-sinea.The expression is sometimes read "xequals the inverse sine ofa," or "the anti-sine ofa."

*Arcsine*is found in approximately 1904 in the E. R. Hedrick translation of volume I of

*A Course in Mathematical Analysis*by Edouard Goursat. The translation carries the date 1904, although a footnote references a work dated 1905. The word appears in running text in the phrase "we obtain the following development for the arcsine" [James A. Landau].

*Arctangent*appears in Hedrick (above).

**ARGAND DIAGRAM.**Jean-Robert Argand presented his diagram in a small book published in 1806. The book is reprinted with additional material from 1813-4 in

*Essai sur une manière de représenter les quantités imaginaires dans les constructions géométriques*.

A Google print search finds

*Argand diagram*in 1889 in

*Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges*by George Chrystal: "If we draw the Argand diagram for

*w*= 1 +

*x*+

*yi*, we see that when

*r*is given

*w*describes a circle of radius

*r,*whose centre is the point (1, 0)." [James A. Landau]

**ARGUMENT.**This word derived from the Latin entered English in the 14th century with two basic meanings, one in logic and one in astronomy. The quotations given in the

*OED*are from the work of Geoffrey Chaucer who wrote stories about students of these subjects as well as an unfinished book on the astrolabe, an instrument used by astronomers; this appears to have been the earliest technical manual in English.

To illustrate the meaning "a statement or fact advanced for the purpose of influencing the mind" the

*OED*quotes from Chaucer's

*Franklin's Tale*line 178 "Clerkes wol seyn as hem leste By Argumentz that al is for the beste." Related meanings include dispute and process of reasoning.

To illustrate the meaning "the angle, arc, or other mathematical quantity, from which another required quantity may be deduced, or on which its calculation depends" the

*OED*quotes from the

*Franklin's Tale*line 549 "Hise othere geeris, As been his centris and hise Argumentz" and from the

*Treatise on the Astrolabe.*xliv. 54 "To knowe the mene mote and the argumentis of any planete."

The related meanings here include the argument (or phase or amplitude) of a complex number and the argument (or INDEPENDENT VARIABLE) of a function.

For the first see Briot & Bouquet's

*Théorie des fonctions elliptiques*(1859, p. 1). Today

*argument*is the most popular English term but a century ago there were several alternatives: "We find [the angle] designated as argument, declination, arcus, anomalie, amplitude" wrote the translator of Burkhardt's German treatise

*Theory of Functions of a Complex Variable*(1913, p. 15). He translated Burkhardt's

*arcus*as

*amplitude*. G. H. Hardy also chose

*amplitude*in his

*Course of Pure Mathematics*(1908, p. 82), apparently to keep

*argument*for an independent variable.

For the second the

*OED*gives a quotation from the 1879 edition of the

*Encyclopedia Britannica*, "In each case

*u*is the independent variable or argument of the function." A

*JSTOR*search found Gompertz writing in 1825, "These tables represent the logarithm of the present values of annuities for every value of a certain argument." ("On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies"

*Abstracts of Papers printed in the Philosophical Transactions of the Royal Society*vol. 2, p. 253.)

This entry was contributed by John Aldrich. See also ABSOLUTE VALUE, ARGAND DIAGRAM and MODULUS.

**ARITHMETIC**is a Greek word transliterated into English as

*arithmetike.*

*Arithmetike*derives from

*arithmos*, the word for number and it passed into Latin as

*arithmetica.*Smith (vol. 2, page 7) writes that the Greeks used

*arithmetike*for the theory of numbers and

*logistike*(logistic) for the art of calculating. In modern English the more aristocratic name is applied to both disciplines and most often to the more practical one. See the entry NUMBER THEORY.

Smith (vol. 2, page 8) describes some of the confusions around the word's spelling:

The word "arithmetic," like most other words, has undergone many vicissitudes. In the Middle Ages, through a mistaken idea of its etymology, it took an extrar,as if it had to do with "metric." So we find Plato of Tivoli, in his translation (1116) of Abraham Savasorda, speaking of "Boetius in arismetricis." The title of the work of Johannes Hispalensis, a few years later (c. 1140), is given as "Arismetrica," and fifty years later than this we find Fibonacci dropping the initial and using the form "Rismetirca." The extraris generally found in the Italian literature until the time of printing. From Italy it passed over to Germany, where it is not uncommonly found in the books of the 16th century, and to France, where it is found less frequently. The ordinary variations in spelling have less significance, merely illustrating, as in the case with many other mathematical terms, the vagaries of pronunciation in the uncritical periods of the world's literatures.

**ARITHMETIC MEAN.**The arithmetic mean was one of the three means studied by the Pythagoreans in Antiquity. See MEAN.

A Google print search by James A. Landau finds

*arithmetical mean*in 1687 in

*The Gaugers Magazine, Wherein the Foundation of his Art is briefly Expalin'd and Illustrated with such Figures as may render the Whole intelligible to a mean Capacity*by William Hunt, Gauger:

Take an Arithmetical mean between GH and CD, Diameters at the Top and Bottom of the Crown, viz. VW, and multiply the Area answerable thereto by IF the Altitude of that Segment, the Product is the Content of GHDC from which subduct the Content of the Crown CFD, the remainder is the Liquor lying about the Crown when just covered, which added to the upper part, the Sum is the Content of the whole Copper.The arithmetic mean has played an important role in the combination of observations since the time of Galileo. In the

*Theoria Motus*(

*Theory of the Motion of Heavenly Bodies moving around the Sun in Conic Sections*) (1809, p. 244) Gauss took it as axiomatic that the arithmetic mean affords the most probable value ("valorem maxime probabilem") of the unknown. This has been called the

**postulate of the arithmetic mean**(E. T. Whittaker & G. Robinson

*Calculus of Observations*(1924, p. 215)) Gauss found that the postulate implied that the errors follow the normal distribution. See ERROR, NORMAL, and GAUSSIAN.

See also AVERAGE and MEAN.

The term

**ARITHMETIC PROGRESSION**was used by Michael Stifel in 1543: "Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis" [James A. Landau].

*Arithmetic series*is found in 1723 in

*A System of the Mathematics*James Hodgson [Google print search, James A. Landau].

**ARITHMETIZATION.**

*Arithmeticize*is found in 1859 in

*Pestalozzi and Pestalozzianism : life, educational principles, and methods, of John Henry Pestalozzi, with biographical sketches of several of his assistants and disciples / reprinted from the American journal of education*edited by Henry Barnard: "[number] alone leads to infallible results; and, if geometry makes the same claim, it can be only be means of the application of arithmetic, and in conjunction with it; that is, it is infallible, as long as it arithmeticizes" [University of Michigan Digital Library].

*Arithmetisirung*is found in 1879 in a review of McColl's "Calculus of Equivalent Statements,"

*Proc. LMS,*vol X, pp 16-28:

In diesem dritten Artikel (die frueheren s.F.d.M. X. 34, JFM10. 0034.02) setzt Herr McColl seine zuerst unter dem Namen "Symbolical Language" veroeffentlichte und zunaechst zur Verwendung auf mathematische Probleme bestimmte Arithmetisirung der Logik fort und bringt sie zu einem gewissen Abschluss. (JFM11.0049.01)

*Arithmetisierung*appears in an 1887 essay by Leopold Kronecker (1823-1891) "Ueber den Zahlbegriff" in

*Crelle's Journal*101 (in

*Werke*p. 253):

... ich glaube auch, dass es dereinst gelingen wird, den gesammten Inhalt aller dieser mathematischen Disciplinen zu "arithmetisiren", d.h. einzig und allein auf den im engsten Sinne genommenen Zahlbegriff zu gruenden, also die Modificationen und Erweiterungen dieses Begriffs wieder abzustreifen ...

*Arithmetize*is found in English in 1892 in H. B. Fine in

*Bull. N.Y. Math. Soc.*I. 175: "It is not merely that the purely arithmetical problems growing out of algebra were attractive to him [sc. Kronecker] -- he 'arithmetized' algebra itself" [OED2].

The phrase

*Arithmetisirung der Mathematik*is the title of Felix Klein's 1895 address before the Royal Society of Sciences in Goettingen

*Gesammelte mathematische Abhandlungen 2*p. 232. [Julio González Cabillón, Michael Detlefsen].

G. U. Yule introduced the term

**ASSOCIATION**of attributes to "distinguish it from the 'correlation' of continuous variables" (p. 257n) in 1900 ("On the Association of Attributes in Statistics,"

*Philosophical Transactions of the Royal Society of London,*Ser. A, 194, 257-319.) (David, 1998.)

See CORRELATION.

**ASSOCIATIVE**"seems to be due to W. R. Hamilton" (Cajori 1919, page 273). Hamilton used the term as follows:

However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called theThe citation above is from "On a new Species of Imaginary Quantities connected with a theory of Quaternions" Royal Irish Academy,associativecharacter of the operation....

*Proceedings,*Nov. 13, 1843, vol. 2, 424-434.

In his biography of Hamilton, Thomas Hankins discusses this paper:

This paper is marked as having been communicated on Nov. 13, 1843, but it closed with a note that it had been abstracted from a larger paper to appear in theThe citation above is fromTransactionsof the academy (Math. Papers, 3:116). That longer paper, also dated Nov. 13, 1843 was not published until 1848, and when it did appear it concluded with a note mentioning works written as late as June 1847 ("Researches respecting Quaternions: First Series," Royal Irish Academy,Transactions21, [1848]: 199-296, in Math. Papers, 3:159-216, "note A," pp. 217-26). In a note Hamilton stated that his presentation on Nov. 13, 1843 had been "in great part oral" (Math. Papers, 3:225); therefore it is possible that his statement of and naming of the associative law was added later when his original communication was to be printed in 1844. It is thus possible that he first recognized the importance of the associative law in 1844, when he began work on Graves's octaves.

*Sir William Rowan Hamilton,*by Thomas Hankins, chapter 23 ("The Fate of Quaternions").

(These citations were provided by David Wilkins.)

**ASTROID.**According to E. H. Lockwood (1961):

The astroid seems to have acquired its present name only in 1838, in a book published in Vienna; it went, even after that time, under various other names, such as cubocycloid, paracycle, four-cusp-curve, and so on. The equation $x^{2/3} + y^{2/3} = a^{2/3}$ can, however, be found in Leibniz's correspondence as early as 1715.This quote was taken from Xah Lee's Visual Dictionary of Special Plane Curves website.

**ASYMPTOTE**was used by Apollonius, with a broader meaning than its current definition, referring to any lines which do not meet, in whatever direction they are produced (Smith).

The first citation of the word in the OED is in 1656 in

*Hobbes' Elements of Philosophy*by Thomas Hobbes: "Asymptotes..come still nearer and nearer, but never touch."

**ASYMPTOTIC**in the sense of "pertaining to large-sample behaviour" only became current in Statistics in the 1940s, although asymptotic arguments had been used for centuries. A

*JSTOR*search produced the following:

**asymptotic distribution**in S. S. Wilks "Shortest Average Confidence Intervals from Large Samples"

*Annals of Mathematical Statistics*,

**9**, (1938), pp. 166-175,

**asymptotically normal**in A. M. Mood "The Distribution Theory of Runs"

*Annals of Mathematical Statistics*,

**11**, (1940), pp. 367-392

**asymptotic efficiency**in M. S. Bartlett "On the Theoretical Specification and Sampling Properties of Autocorrelated Time-Series"

*Supplement to the Journal of the Royal Statistical Society*,

**8**, (1946), pp. 27-41.

These were new names for established concepts. However

**asymptotic relative efficiency**was a new concept, introduced and named by E. J. G. Pitman in lectures given in 1948. The term appears in G. E. Noether's "Asymptotic Properties of the Wald-Wolfowitz Test of Randomness"

*Annals of Mathematical Statistics*,

**21**, (1950), pp. 231-246. [John Aldrich, using David (2001)]

See NORMAL, EFFICIENCY, HYPOTHESIS & HYPOTHESIS TESTING, SMALL SAMPLE.

**AUGMENTED MATRIX**is found in an 1861 paper by Henry Smith, "On systems of linear indeterminate equations and congruences,"

*Philosophical Transactions, vol cli*, 293-326. "If to this matrix we add an additional vertical column, composed of the absolute terms of the equations, the resulting matrix we shall term (for brevity) the

*augmented*matrix of the system." [Rod Gow]

**AUTOCORRELATION FUNCTION**in N. Wiener's

*Extrapolation, Interpolation and Smoothing of Stationary Time Series*(1949) refers to a function which was prominent but unnamed in his "Generalized Harmonic Analysis" of 1930 (

*Acta Mathematica,*

**55,**117-258) and in writings back to 1926. The term "auto-correlation" appears in 1933 in a report of a paper by H. T. Davis (

*Econometrica,*

**1,**434) which discusses Wiener's work. Wiener's function was not strictly a correlation for it was not mean-corrected or normalised; it was also defined in continuous time. The

**AUTOCORRELATION COEFFICIENT**of H. Wold's (

*A Study in the Analysis of Stationary Time Series*(1938)) was mean-corrected and normalised and defined for processes in discrete time. The earlier "serial correlation coefficient" has also survived [John Aldrich].

See STOCHASTIC PROCESS and SERIAL CORRELATION.

**AUTOLOGICAL and HETEROLOGICAL.**The

*OED*'s earliest citation is F. P. Ramsey writing in the

*Proceedings of the London Mathematical Society*,

**25**, (1926), 358: "Let us call adjectives whose meanings are predicates of them, like 'short', autological; others heterological." The terms derived from the German

*autologisch*and

*heterologisch*introduced by K. Grelling and L. Nelson "Bemerkungen zu den Paradoxien von Russell und Burali-Forti"

*Abhandlungen der Fries'schen Schule II*, Göttingen 1908, S. 301-334. See

*Wikipedia*on The Grelling-Nelson paradox.

**AUTOMORPHIC FORM, AUTOMORPHIC FUNCTION.**These terms were used by Felix Klein in 1890 in "zur Theorie der Laméschen Functionen" in the 1890 issue of

*Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universität zu Göttingen.*

*Automorphe Functionen*appears once, on page 94:

Bekanntlich sind $\lambda$ und $√f(\lambda)$ in dem soeben gefundenen $\eta$ eindeutig; sie stellen solche eindeutige Functionen von $\eta$ vor, welche sich bei unendlich vielen linearen Substitutionen von η reproduciren (ich möchte vorschlagen, solche Functionen überhaupt automorphe Functionen von $\eta$ zu nennen).Babelfish translates this as "I would like to suggest, such Functionen at all automorphe Functionen of η to call."

*Automorphe Formen*appears once, on page 95:

Bei diesen bleiben dann, $\lambda_{1}, \lambda_{2}$ ungeändert; ich schlage dementsprechend vor, dieselben als automorphe Formen von $F_{1}, F_{2}$ zu bezeichnen.Babelfish translates this as "With these remain then $\lambda_{1}, \lambda_{2}$ ungeandert I suggest accordingly calling the same as automorphe forms of $F$."

Klein also used both

*automorphe functionen*and

*automorphe formen*in

*Vorlesungen über die Theorie der elliptischen Modulfunctionen*(1890), with a footnote reading:

Hr. Poincaré hat über die gedachten Functionen ausser einer grossen Reihe kleinerer mit dem Jahre 1881 beginnender Noten in der Comptes Rendus zunächst 1881 einen Aufsatz in den Mathematischen Annalen (Bd. 19: Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires) .. Hr. Klein bezeichnet die hier in Betracht kommenden Functionen neuerdings als linear-automorphe Functionen oder wohl auch kurzweg als automorphe Functionen.

The 1892 issue of

*Nachrichten*pages 283 – 291 has a paper by Klein "Die Eindeutigen automorphen Formen vom Geschlechte Null" (pp. 283-91) in which he uses

*automorphen Formen, automoörphe Functionen,*and two new expressions,

*automorphe Inegrale*and

*automorphe Reihe.*Furthermore he refers to Poincaré several times and refers to

*Poincarésche Reihe.*

*Automorphic function*appears in English in 1892 in the

*Bulletin of the New York Mathematical Society*: "The theory of automorphic functions, recently developed by Poincare and Klein, affords instances of analytic functions with natural boundaries ..."

[This entry was contributed by James A. Landau.]

**AUTOMORPHIC TRANSFORMATION.**Automorphic transformations appear to have been first investigated by Hermite, who called them (in French)

*transformation en elle-meme,*which translates into English as "transformation into itself."

In "Remarques sur un memoire de M. Cayley relative aux Determinants Gauches" in the

*Cambridge and Dublin Mathematical Journal*(1854), Hermite wrote "Je me propose de donner içi des formules analogues à celles de M. Cayley, pour la transformation en elle-même d'une forme quadratique quelconque."

[I propose to give here formulas analogous to those of Mr. Cayley for the transformation into itself of any quadratic form whatsoever.]

Cayley used

*transformation en elle-même*in a French-language paper he wrote in 1855, but used

*automorphic transformation*when writing in English in 1858. It is not clear whether Cayley introduced the term

*automorphic transformation*into English or whether he was using an existing term.

[This entry was contributed by James A. Landau.]

**AUTOMORPHISM**was used by H. J. Stephen Smith in "Report on the Theory of Numbers—Part III," which appeared in 1862 in the published

*Report of the Thirty-First Meeting of the British Association for the Advancement of Science.*

**AUTOREGRESSION**and

**AUTOREGRESSIVE PROCESS**derive from H. Wold's "process of linear autoregression" (

*A Study in the Analysis of Stationary Time Series*(1938)) which referred to processes of the type analysed by G. U. Yule in "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers,"

*Phil. Trans. Royal Society,*

**226A,**(1927), 267-298. Long before Yule, in 1877, Galton used the first-order process to model inheritance. (David, 2001)

The multivariate generalisation was considered by H. B. Mann & A. Wald "On the Statistical Treatment of Linear Stochastic Difference Equations,"

*Econometrica*,

**11**, No. 3/4, (1943), 173-220. A

*JSTOR*search found the term

**vector autoregressive process**in D. F. Hendry "Maximum Likelihood Estimation of Systems of Simultaneous Regression Equations with Errors Generated by a Vector Autoregressive Process,"

*International Economic Review*,

**12**, No. 2. (1971), 257-272.

See REGRESSION, YULE-WALKER EQUATIONS and STOCHASTIC PROCESS.

**AVERAGE**, according to the OED, appeared in English around 1500 originally in connection with maritime trade in the Mediterranean. The dictionary gives a list of successive meanings of the term, as follows:

(1) Originally an impost on goods. (1502)

(2) Any charge in excess of the freight charges, payable by the owner of the goods. (1491)

(3) Loss to the owners from damage at sea. (1611)

(4) Equitable distribution of such expense, when shared by a group. (1598)

(5) Distribution of the aggregate inequalities of a series of things. (1735)

(6) The arithmetic mean so obtained, the medium, amount, the "common run." (1755)

The year is that of the earliest quotation the dictionary gives for that particular usage.
(2) Any charge in excess of the freight charges, payable by the owner of the goods. (1491)

(3) Loss to the owners from damage at sea. (1611)

(4) Equitable distribution of such expense, when shared by a group. (1598)

(5) Distribution of the aggregate inequalities of a series of things. (1735)

(6) The arithmetic mean so obtained, the medium, amount, the "common run." (1755)

(Based on the entry "Average" in Walker (1929, p. 176))

Average is often used as an alternative for both mean (in general) and arithmetic mean. Like mean, it is often used in the sense of "expected value."

See ARITHMETIC MEAN, MEAN, CENTRAL TENDENCY and also Symbols in Statistics on the Symbols in Probability and Statistics page.

**AVOIRDUPOIS.**From Smith vol. 2, page 639:

The word "avoirdupois" is more properly spelled "averdepois," and it so appears in some of the early books. It comes from the Middle EnglishThe termaver de poiz,meaning "goods of weight." In the 16th century it was commonly called "Haberdepoise," as in most of the editions of Recorde's (c. 1542)Ground of Artes.Thus in the Mellis eddition of 1594 we have: "At London & so all England thorugh are vsed two kids of waights and measures, as the Troy waight & the Haberdepoise."

**AXIOM**(in Greek, ) was used by the Stoic philosophers and by Aristotle, although, according to W. & M. Kneale

*The Development of Logic*(1962) p. 145, the Stoic usage does not correspond to the modern "axiom" but to "judgement" or "proposition."

In Euclid's

*Elements*what would now be called "axioms" are designated either as "common notions" or as "postulates." See the list. Kline (p. 59) explains that "the common notions are truths applicable to all sciences, whereas the postulates apply only to geometry." The term "axiom" is not used but in the notes to his edition of the

*Elements*T. L Heath (1926, vol. 1, pp. 120-1) observed that Aristotle used a similar construction to "common notions" as an alternative expression for "axioms." Proclus, an important commentator on the

*Elements*from the 5

^{th}century AD, used "axioms."

The

*OED*records the English word

*axiom*in 1485 used in the sense of "a proposition that commends itself to general acceptance." The word came into English from the Greek via Latin and French.

See POSTULATE.

**AXIOM OF ARCHIMEDES.**According to a message to a math history mailing list by Yaakov Kupitz, an extended account of the axiom of Archimedes is given by F. Enriques: "Prinzipien der Geometrie" in

*Encyklopadie der Mathematischen Wissenschaften. Bd. 3, T.1, H.1*, pp. 34ff. From this we learn that the erroneous name "axiom of Archimedes" was given by Stolz "Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes,"

*Mathematische Annalen,*, 504-519. The suggestion that the axiom be attributed, instead, to Eudoxos was made by Zeuthen at the Heidelberg congress of mathematics (circa 1900), p. 541 of the congress volume. Today Zeuthen's suggestion is universally accepted.

**22**, (1883)**AXIOM OF CHOICE.**In 1904 in "Beweis, dass jede Menge wohlgeordnet werden kann" ["Proof that every set can be well-ordered"],

*Mathematische Annalen,*, 514-6. Ernst Zermelo used the Axiom of Choice to prove that every set can be well-ordered. Zermelo did not use the name "Axiom of Choice". Rather he stated:

**59**The present proof rests upon the assumption that coverings gamma actually do exist, hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one of its elements, or, expressed formally, that the product of an infinite totality of sets, each containing at least one element, itself differs from zero. This logical principle cannot, to be sure, be reduced to a still simpler one, but it is applied without hesitation everywhere in mathematical deduction."gamma" is defined earlier in the paper by

Imagine that with every subset M' there is associated an arbitrary element m'Zermelo adds "I owe to Mr. Erhard Schmidt the idea that, by invoking this principle, we can take an arbitrary covering gamma as a basis for the well-ordering."_{1}that occurs in M' itself...This yields a "covering" gamma of the [set of all subsets M'] by certain elements of the set M.

In the paper "Untersuchungen über die Grundlagen der Mengenlehre I" ["Investigations in the foundations of set theory I"],

*Mathematische Annalen,*, 261-281 Zermelo presents a set of axioms for set theory. (This paper is generally dated 1908 in bibliographies, presumably because that was the year it appeared in

**65***Mathematische Annalen,*but the paper is dated "Chesières, 30 July 1907.") In this paper there appears:

AXIOM VI. (Axiom of choice). If T is a set whose elements all are sets that are different from 0 and mutually disjoint, its union "union of T" includes at least one subset SIn another paper also published in 1908, " Neuer Beweis für die Möglichkeit einer Wohlordnung" ["A new proof of the possibility of a well-ordering"],_{1}having one and only one element in common with each element of T. [The original German read "Axiom der Auswahl".]

*Mathematische Annalen,*, 107-28, Zermelo writes

**65**Now in order to apply our theorem to arbitrary sets, we require only the additional assumption that a simultaneous choice of distinguished elements is in principle always possible for an arbitrary set of sets ...In the same paper Zermelo adds:

IV. Axiom. A set S that can be decomposed into a set of disjoint parts A, B, C..., each containing at least one element, possesses at least one subset S_{1}having exactly one element in common with each of the parts A,B,C,... considered.

Even Peano'sAll of the above material is from Heijenoort (1967). The translations of Zermelo in this book are by Stefan Bauer-Mengelberg.Formulaire,which is an attempt to reduce all of mathematics to "syllogisms" (in the Aristotelian-Scholastics sense), rests upon quite a number of unprovable principles; one of these is equivalent to the principle of choice for a single set and can then be extended syllogistically to an arbitrary finite number of sets.

In 1908 Bertrand Russell wrote in his "Mathematical Logic as Based on the Theory of Types,"

*American Journal of Mathematics*,

**30**, 222-262

According to Grattan-Guinness (2000, p. 341) Russell discovered the multiplicative axiom in 1904 "slightly before the conception and writing of Zermelo's paper, so reversing the priority over finding Russell's paradox." See RUSSELL'S PARADOX.

Many elementary theorems concerning cardinals require the use of the multiplicative axiom [i.e. the axiom that, given a set of mutually exclusive classes, none of which are null, there is at least one class consisting of a member from each class in the set]. It is to be observed that this axiom is equivalent to Zermelo's, and therefore to the assumption that every class can be well ordered. These equivalent assumptions are, apparently, all incapable of proof, though the multiplicative axiom, at least, appears highly self-evident. (pp. 258-9)

The axiom of choice was first used in set theory and the theory of transfinite numbers but it was soon applied to other branches of mathematics. The HAUSDORFF PARADOX was an early example (1914). Many of the applications were due to Polish mathematicians with Wacław Sierpiński in the lead: see e.g. his "Sur le rôle de l'axiome de M. Zermelo dans l'analyse moderne,"

*Comptes Rendus,*, 688-691. Gregory Moore

**163**, (1916)*Zermelo's Axiom of Choice: Its Origins, Development and Influence*(1982, chapters 3 & 4) discusses these applications. A related development was the development of more immediately applicable versions of the axiom. See ZORN'S LEMMA.

For a long time a reference to Zermelo was mandatory as in "the Zermelo Axiom of Choice" in E. W. Chittenden and A. D. Pitcher "On the Theory of Developments of an Abstract Class in Relation to the Calcul Fonctionnel,"

*Transactions of the American Mathematical Society*,

**20**, (3), (1919), p. 215. But eventually the simple

*axiom of choice*became acceptable; see e.g. Sierpiński's "Les exemples effectifs et l'axiome du choix,"

*Fundamenta Mathematicae*,

**2**, (1921), 112-118.

This article was contributed by James A. Landau.

**AXIS**occurs in English in the phrase "the Axis or Altitude of the Cone" in 1571 in

*A Geometricall Practise named Pantometria*by Thomas Digges (1546?-1595) (OED2).