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) (DSB).

Abelian function appears in the title "Zur Theorie der Abelschen Functionen" by Karl Weierstrass (1815-1897) in Crelle's Journal, 47 (1854).

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

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. [Peter M. Neumann and Julia Tompson]

ABELIAN INTEGRAL appears in English in 1847 in the Report of the British Association for the Advancement of Science (1846): "What are the corresponding functions to which the hyper-elliptic or Abelian integrals are inverse, and how by means of them can Abel’s theorem be stated?" (OED2).

Weierstrass wrote a memoir, Abelian Integrals, which appeared in the catalogue of studies for 1848-49 of the Royal Catholic Gymnasium at Braunsberg (Kramer, p. 548).

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 VALUE. Leibniz used the Latin term moles with the meaning of absolute value of a real number.

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 value was coined in German as absoluten Betrag by Karl Weierstrass (1815-1897), who wrote:

Ich bezeichne den absoluten Betrag einer complex Groesse x mit |x|. [I denote the absolute value of complex number x by |x|.]

Absoluten Betrage appears in 1876 in the title "Ein allgemeiner Ausdruck für die ihrem absoluten Betrage nach kleinste Wurzel der Gleichung n Grades" by J. König.

In 1889, Elements of Algebra by G. A. Wentworth has: "Every algebraic number, as +4 or -4, consists of a sign + or - and the absolute value of the number; in this case 4. The sign shows whether the number belongs to the positive or negative series of numbers; the absolute value shows what place the number has in the positive or negative series."

The Century Dictionary (1889-1897) has absolute magnitude, rather than absolute value.

ABSTRACT ALGEBRA is found in H. S. Vandiver, "Theory of Finite Algebras," Transactions of the American Mathematical Society, Jul., 1912.

The term ABSTRACT GROUP was apparently coined by Cayley. In an 1854 paper he defined an abstract group.

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).

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 [University of Michigan Historical Math Collection].

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)].

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.570. [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].

ALGEBRAIC CURVE. See transcendental curve.

ALGEBRAIC FUNCTION appears in 1819 in G. Peacock, View Fluxional Calculus: "Where the fluent or integral is expressed by an algebraic function" (OED2).

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.

ALGORISM, meaning "arithmetic" or "the Hindu-Arabic numerals," is found in several languages, including middle English, in which various spellings are found. Chaucer used augrym; Recorde used augrim and algorisme. The word is derived 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).

ALGORITHM is derived from the much older word algorism, and "influenced by the Greek word arithmos (number)," according to the OED2. However, some scholars believe it was not so influenced.

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.)

Algorithm is found in English in 1699, although meaning "algorism", 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, and David Fowler contributed to this entry.]

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 appears in 1933 in J. Neyman and E. S. Pearson, "On the Problem of the Most Efficient Tests of Statistical Hypotheses," Philosophical Transactions of the Royal Society of London, Ser. A, 231, 289-337 (David, 1995).

AMBIGUOUS CASE. The first edition of the Encyclopaedia Britannica (1768-1771) has: "The 10th, 11th, and 12th cases are ambiguous; since it cannot be determined by the data, whether A, B, C, and A C, be greater or less than 90 degrees each."

Ambiguous case is found in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

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).

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."

The term ANALYSIS was introduced by Theon of Alexandria, according to Kline (page 279).

François Viète used the term analytic art for algebra in 1591 in In artem analyticem isagoge. Schwartzman (page 23) says that around 1590 "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 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 appears in 1918 in Sir Ronald Aylmer Fisher, "The Causes of Human Variability," Eugenics Review, 10, 213-220 (David, 1995).

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]

The term ANALYTIC FUNCTION was first used by Marquis de Condorcet (1743-1794) in his unpublished Traité du calcul integral (Youschkevitch, pages 37-84) [Giovanni Ferraro].

The term analytic function was also used in 1797 by Joseph Louis Lagrange (1736-1813) in Théorie des Fonctions Analytiques. According to Georges Valiron's essay in "Great currents of mathematical thought," Edited by François Le Lionnais (1901-1984), New York, Dover Publications, 1971, this is the earliest use, but it is later than that of Condorcet.

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. 700-725, although interest in ancillary statistics only gathered momentum in the mid-1930s when Fisher returned to the topic and other authors started contributing to it [John Aldrich, David (1995)].

ANGLE OF DEPRESSION and ELEVATION appear in 1790 in Ray, Phil. Trans. LXXX: "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 ratio is found in English in 1863 in Salmon Conic Sect. 57: "This ratio is called the Anharmonic ratio of the pencil" [OED2].

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 appears in "General Mean Value and Remainder Theorems with Applications to Mechanical Differentiation and Quadrature," George David Birkhoff, Transactions of the American Mathematical Society, Vol. 7, No. 1. (Jan., 1906).

ANTI-DIFFERENTIAL and ANTI-DIFFERENTIATION appear in Differential and Integral Calculus (1908) by Daniel A. Murray.

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

The radius OT of the inscribed circle is nothing else than the perpendicular let fall from the centre on one of the sides: it is sometimes named the apothem of the polygon.

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

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'), ...

In English arbelos is found in M. G. Gaba, "On a generalization of the arbelos," Am. Math. Mon. 47 (1940).

[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 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."

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.

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)

... 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 ...

The expression Arithmetisirung der Mathematik was introduced by Felix Klein in 1895 during his address before the Royal Society of Sciences in Goettingen [Julio González Cabillón, Michael Detlefsen].

ASSOCIATION (in statistics) is found in 1900 in G. U. Yule, "On the Association of Attributes in Statistics," Philosophical Transactions of the Royal Society of London, Ser. A, 194, 257-319 (David, 1998).

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....

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."

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].

The term AUTOMORPHIC FORM was introduced by Jules Henri Poincaré (1854-1912). Kramer (p. 382) 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. Yule had no special term (David, 2001).

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 was used by the Stoic philosophers and Aristotle himself (Smith vol. 2, page 280).

Smith writes, "Euclid seems to have used the term 'common notion' to designate an axiom, although the may have used the term 'axiom' also."

According to Paul Carus in The Foundations of Mathematics (1908), Euclid did not use the term axiom.

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

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.

In the paper "Unersuchungen über die Grundlagen der Mengenlehre I" ["Investigations in the foundations of set theory I"], Mathematische Annalen 65, 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" ["The possibility of a well-ordering"], Mathematische Annalen 65, 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.

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.

Axiom of choice appears in Waclaw Sierpinski, "Les exemples effectifs et l'axiome du choix," Fundamenta Mathematicae 2, pp 112-118 (1921).

See also Jules Richard, "Sur un paradoxe de la théorie des ensembles et sur l'axiome Zermelo," Enseignement mathématique 9, pp 94-98 (1907).

Whitehead and Russell in Principia Mathematica section *88 write:

"If kappa is a class of mutually exclusive classes, no one of which is null, there is at least one class µ which takes one and only one member from each member of kappa." This we shall define as the "multiplicative axiom."

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).