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

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 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. The term appears in English in W. Burnside Theory of Groups of Finite Order (1897). Burnside (p. 380) reports the corresponding French term as “groupe des opérations échangeables.” [Peter M. Neumann and Julia Tompson]

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." [These citations were provided by staff at The Correspondence of William Henry Fox Talbot at Glasgow University. The University will soon have these letters available on line.]

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 (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.

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 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."

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.”

Searches at Google books give absolute Werth in 1831 and 1844, absolute Grösse in 1836, 1841, and 1853, and absolute länge in 1853.

Absolute value is found in English in 1850 in The elements of analytical geometry; comprehending the doctrine of the conic sections, and the general theory of curves and surfaces of the second order by John Radford Young (1799-1885): "we have AF the positive value of x equal to BA - BF, and for the negative value, BF must exceed BA, that is, F must be on the other side of A, as at F', hence making AF' equal to the absolute value of the negative root of the equation" [University of Michigan Digital Library].

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 H. S. Vandiver, "Theory of Finite Algebras," Transactions of the American Mathematical Society, Jul., 1912.

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, . . .

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 appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "An acute angle is that, which is lesse then a right angle"; "an obtuse angle is that which is greater then a right angle" (OED2).

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 (th)at (th)er ben .7. spices or partes of (th)is craft. The first is called addicion, (th)e 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 (th)e rote.

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.

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's équation adjointé, so I used the word adjunct, suggested by the German adjungirte, and not unlike the French adjointe. It seems better now, however, to employ the word associate, or, when speaking simply of Lagrange's équation adjointé, the word adjoint.

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 ith row and jth column is the cofactor of the element in the jth row and ith 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]

ALGEBRA comes from the title of a work written in Arabic about 825 by al-Khowarizmi, al-jabr w'al-muqâbalah, in which al-jabr means "the reunion of broken parts." When this was translated from Arabic into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.

In 1140 Robert of Chester translated the Arabic title into Latin as Liber algebrae et almucabala.

In the 16th century it is found in English as algiebar and almachabel, and in various other forms but was finally shortened to algebra. The words mean "restoration and opposition."

In Kholâsat al-Hisâb (Essence of Arithmetic), Behâ Eddîn (c. 1600) writes, "The member which is affected by a minus sign will be increased and the same added to the other member, this being algebra; the homogeneous and equal terms will then be canceled, this being al-muqâbala."

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 earliest known use of the word algebra in English in its mathematical sense is by Robert Recorde 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."

The phrase an algebra is found in 1849 Trigonometry and Double Algebra by Augustus de Morgan: "Ordinary langauge has methods of instantaneously assigning meaning to contadictory phrases: and thus it has stronger analogies with an algebra (if there were such a thing) in which there are preorganized rules for explaining new contradictory symbols as they arise, than with one in which a single instance of them demands an immediate revision of the whole dictionary" [University of Michigan Historical Math Collection].

Sylvester wrote, "It is an algebra upon an algebra" [I do not have the citation].

Algebras (in the plural) appears in 1849 Trigonometry and Double Algebra by Augustus de Morgan: "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 [University of Michigan Historic Math Collection].

It was formerly thought 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...."

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. Algebraical geometry appears in 1823 in the title An elementary Treatise on algebraical Geometry by Rev. Dionysius Lardner.

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 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 (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 velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, 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.)

The 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 [Acta Eruditorum, vol. 3, pp. 467-473, October 1684], page 469. Both terms "Algorithmo" and "differentialem" are italicized 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 Godel, 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 r was "apparently introduced" by B. V. Levin in "A one-dimensional sieve," (Russian) 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. W. Folkingham, Art of Survey ii. v. 55 (1610) has: "To Rectifie the Plot: diagone alternate angles" (OED2).

Alternate angles appears in 1660 in Barrow's translation of Euclid: "If a right line falling upon two right lines make the alternate angles equal" (OED2).

In 1828, Elements of Geometry and Trigonometry (1832) by David Brewster (a translation of Legendre) has:

AGO, GOC, we have already named interior angles on the same side; BGO, GOD, have the same name; AGO, GOD, are called alternate interior angles, or simply alternate; so also, are BGO, GOC: and lastly EGB, GOD, or EGA, GOC, are called, respectively, the opposite exterior and interior angles; and EGB, COF, or AGE, DOF, the alternate exterior angles.

ALTERNATING GROUP appears in Camille Jordan, "Sur la limite de transitivité des groupes non alternés," Bull. Soc. math. de France 1 (1873).

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]

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.

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. 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."

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, 'ambitus' implies 'ambition', 'religio' 'religion', or 'homines honesti', a set of honorable men."

Analysis with the new meaning had entered English by 1902 when E. T. Whittaker published A Course of Modern Analysis. He wrote in the preface, "The first half of this book contains an account of those methods and processes of higher mathematical analysis, which seem to be of greatest importance at the present time; .. it is chiefly concerned with the properties of infinite series and complex integrals, and their applications to the analytical expression of functions." (OED)

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.

[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 Robveral 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]

The term ANALYTIC GEOMETRY was apparently first used (as geometria analytica) in Geometria analytica sive specimina artis analyticae, published by Samuel Horsley (1733-1806) in volume 1 of Isaaci Newtoni opera quae exstant omnia. Commentariis illustrabat Samuel Horsley (1779).

In 1797 Sylvestre François Lacroix (1765-1843) wrote in 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.

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 OF DEPRESSION and ELEVATION appear in 1790 in Roy, "An Account of the Trigonometrical Operation, Whereby the Distance between the Meridians of the Royal Observatories of Greenwich and Paris Has Been Determined," Phil. Trans. LXXX, p. 154: "By means of this piece of mechanism in the eye-end of the telescope ... small angles of elevation or depression may be determined with great accuracy" (OED2).

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, is the same as that of y, η, y, .” [James A. Landau]

See also CROSS-RATIO.

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).

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.

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 (as a word, with no space) 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].

Arcsine appears in Cajori (1928-29): "In 1729 he used 'A. S.' to represent 'arcsine.'"

Arctangent appears in Hedrick (above). It does not appear in running text; however, the index has the entry "Arctangent, series for."

Arctangent appears in Cajori (1928-29): "Euler in 1736 introduced 'A t' for 'arctangent' ..."

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.

The term appears in English appears in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "The plane is chosen as the field on which these points are to lie; and the figure formed by representing points in this way is referred to as Argand's diagram. ... In the Argand diagram, to each point z are attached all the corresponding values of w."

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 amplitudein 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. It passed into Latin as arithmetica. According to Smith (vol. 2, page 8):

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 extra r, 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 droppping the initial and using the form "Rismetirca." The extra r is 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.

The term arithmetical mean is found in E. Halley "A Most Compendious and Facile Method for Constructing the Logarithms, Exemplified and Demonstrated from the Nature of Numbers, without any Regard to the Hyperbola, with a Speedy Method for Finding the Number from the Logarithm Given," Philosophical Transactions of the Royal Society of London, Vol. 19. (1695 - 1697), pp. 58-67. (JSTOR)

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].

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

Arithmetic series is found in 1851 in a collection of math problems, The principles of the solution of the Senate-house 'riders,' exemplified by the solution of those proposed in the earlier parts of the examinations of the years 1848-1851 by Francis James Jameson (1828-1869): "(A) Find the sum of an arithmetic series" [University of Michigan Digital Library].

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 the associative character of the operation....

The citation above is from "On a new Species of Imaginary Quantities connected with a theory of Quaternions" Royal Irish Academy, 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 the Transactions of 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, Transactions 21, [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.

(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.

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.

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. The term seems to have been introduced in this paper [Rod Gow].

Augmented matrix is found in 1907 in Introduction to Higher Algebra by Maxime Bôcher: "We will call a the matrix of the system of equations, b the augmented matrix."

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.

The term AUTOMORPHIC FORM was introduced by Jules Henri Poincaré (1854-1912). Kramer (p. 392) implies he also used the term automorphic function.

AUTOMORPHIC FUNCTION was introduced by Felix Klein. He used the term automorphe funktion in 1890 in Nachrichten v. d. königlichen Ges. d. Wissenschaften, Göttingen 1890.

AUTOMORPHIC TRANSFORMATION was used by Arthur Cayley in 1883 in the title "On the automorphic transformation of the binary cubic function," Lond. M. S. Proc.

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.

(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 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 English aver 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."

The term 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, 22, (1883), 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.

AXIOM OF CHOICE. In 1904 in " Beweis, dass jede Menge wohlgeordnet werden kann" ["Proof that every set can be well-ordered"], Mathematische Annalen, 59, 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:

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'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.

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."

In the paper " Untersuchungen über die Grundlagen der Mengenlehre I" ["Investigations in the foundations of set theory I"], Mathematische Annalen, 65, 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 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 S₁ having one and only one element in common with each element of T. [The original German read "Axiom der Auswahl".]

In another paper also published in 1908, " Neuer Beweis für die Möglichkeit einer Wohlordnung" ["A new proof of the possibility of a well-ordering"], Mathematische Annalen, 65, 107-28, Zermelo writes

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

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 S₁ having exactly one element in common with each of the parts A,B,C,... considered.

In the same paper Zermelo adds:

Even Peano's 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.

All of the above material is from Heijenoort (1967). The translations of Zermelo in this book are by Stefan Bauer-Mengelberg.

In 1908 Bertrand Russell wrote in his "Mathematical Logic as Based on the Theory of Types," American Journal of Mathematics, 30, 222-262

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)

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.

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, 163, (1916), 688-691. Gregory Moore 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 the axiom was referred to as “Zermelo’s axiom” or “Zermelo’s axiom of choice” but eventually 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).