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 ab?lien in 1870 in Trait? des Substitutions et des Equations Alg?braiques. 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 ab?lien" and "?quation ab?lienne." 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 f?hig 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]
[Dave Rusin pointed out in sci.math that "Abel is bestowed the unique honor of having his name used without capitalization. (Other authors, particularly those writing in French, extend this treatment to other mathematicians' names.)" Indeed, Merriam-Webster's Collegiate Dictionary, 10th ed., defines only three words named for mathematicians which are not capitalized: abelian, algorithm, and biotite, although "abelian" is labeled "often capitalized." The shortness of the name "Abel" may be a factor. Cartesian is often seen uncapitalized, but it is capitalized in MWCD10.]
ABELIAN INTEGRAL appears in English in 1847 in the Report of the British Association for the Advancement of Science of 1846 (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 was coined by Gottfried Wilhelm Leibniz (1646-1716), who used the Latin term 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."
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.ABSOLUTE CONVERGENCE was defined by Cauchy (DSB).
ABSOLUTE VALUE (in German as "absoluten Betrag") was coined by Karl Weierstrass (1815-1897). In his memoir, "Zur Theorie der eindeutigen analytischen Functionen" [published in Abhandlungen der Koeniglich Akademie der Wissenschaften, pp. 11-60, Berlin, 1876; reprinted in Zweiter Band (volume II) of his "Mathematische Werke" (1895)], there is a footnote which reads:
Ich bezeichne den absoluten Betrag einer complex Groesse x mit |x|. [I denote the absolute value of complex number x by |x|.]Note that Weierstrass first applied the term and symbolism to complex numbers. [Julio Gonz?lez Cabill?n]
Earlier, Leibniz used the Latin term moles with the meaning of "absolute value."
The term ABSTRACT GROUP was used by Cayley, and apparently coined by him.
ABUNDANT NUMBER. Theon of Smyrna (about A. D. 130) distinguished between perfect, abundant, and deficient numbers.
A translation of Chapter XIV of Book I of Nicomachus has:
Now the superabundant number is one which has, over and above the factors which belong to it and fall to its share, others in addition, just as if an animal should be created with too many parts or limbs, with 10 tongues as the poet says, and 10 mouths, or with 9 lips, or 3 rows of teeth, or 100 hands, or too many fingers on one hand. Similarly, if when all the factors in a number are examined and added together in one sum, it proves upon investigation that the number's own factors exceed the number itself, this is called a superabundant number, for it oversteps the symmetry which exists between the perfect and its own parts. Such are 12, 24, and certain others, . . .This citation was provided by Sam Kutler.
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).
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.
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.In the above, (th) is a thorn. The citation is from "The crafte of nombrynge" (ca. 1300), one of the earliest manuscripts [Egerton ms. 2622] in the English language that refers to mathematics. The transcription was carried out by Robert Steele (1860-1944), and it was first privately printed in 1894 by the Early English Text Society (London). It was later included in "The Earliest arithmetics in English," a sourcebook edited by Robert Steele, and published for the same Society, by H. Milford, Oxford University Press, xviii, 84 pages, London, 1922. [Julio Gonz?lez Cabill?n]
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].
ADJOINT EQUATION. Lagrange used the term ?quation adjointe.
ADJOINT OF A MATRIX appears in Introduction to Higher Algebra by Maxime B?cher (1867-1918) (OED2).
The term also appears in "On the Group Defined for any Given Field by the Multiplication Table of Any Given Finite Group," Leonard Eugene Dickson, Transactions of the American Mathematical Society, Vol. 3, No. 3. (Jul., 1902).
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 defined "curvas affines" in section 442. The words continued to be used a little throughout the 19th century, for example in M?bius "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 about 825 by al-Khowarizmi, al-jebr w'al-muq?balah, in which al-jebr means "the reunion of broken parts." When this was translated into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.
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 Khol?sat al-His?b (Essence of Arithmetic), Beh? Edd?n (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-muq?bala."
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."
Algebras (in the plural), and even the expression "an algebra" (note the indefinite article) both appeared "probably for the very first time anywhere in English language" in Benjamim Peirce's Linear associative algebra (first published in 1870-71). This was conjectured by Salomon Bochner (1899-1982) ("a discerning antiquarian", in his own words) in his "Mathematical Reflections" (Amer. Math. Monthly, 1974, p. 830) [Carlos C?sar de Ara?jo].
ALGEBRAIC CURVE. See transcendental curve.
ALGEBRAIC NUMBER appears in the 1902 translation by E. J. Townsend of David Hilbert's Gottingen lectures on geometry.
James A. Landau, who provided this citation, writes that "it is not clear what Hilbert meant by 'algebraic number,' but there is the possibility that Hilbert (or his translator) used the term to mean, not the root of a polynomial with integer coefficients, but a real number that could be constructed with compass and straightedge. For example, sqrt(2 + sqrt(2) ) would qualify, but cube root of 2 would not."
ALGEBRAIC FUNCTION appears in the 1872 second edition of A Treatise on the Calculus of Finite Differences by George Boole [James A. Landau].
The term ALGEBRAIC LOGIC was introduced by Paul Halmos in a sequence of four papers, beginning with "Algebraic logic, I. Monadic Boolean Algebras", Compositio Mathematica, vol. 12 (1955), pp. 217-249. These papers and six others were collected in his book "Algebraic Logic", Chelsea, NY, 1962. In "Mathematical foundations for mathematics" (Amer. Math. Monthly, 1971, 463-487), Leon Henkin wrote: "Although the term 'algebraic logic' (coined by Halmos) is less than ten years old, the ideas of the subject are easily traced back along a continuous path to Boole's work in 1850." [Carlos C?sar de Ara?jo]
ALGORITHM (a systematic technique for solving a problem) was used by Gottfried Wilhelm Leibniz (1646-1716):
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.
Apparently the earliest English translation was carried out by Joseph Raphson in The Theory of Fluxions, Shewing in a compendious manner The first Rise of, and various Improvements made in that Incomparable Method, London, 1715: "Now from this being known as the Algorithm, as I may say of this Calculus, which I call differential, ..." (p.23). The word was then taken up by Euler, for instance in his article 'De usu novi algorithmi in problemate Pelliano solvendo', and its use was then firmly established. [Julio Gonz?lez Cabill?n, David Fowler]
The word algorithm is derived from the much older word algorism, and "influenced by the Greek word arithmos (number)," according to the OED2. Algorism (meaning "the Hindu-Arabic system of numeration or calculations using it") 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).
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."
The phrase ALIQUOT PART occurs in 1570 in Sir Henry Billingsley's translation of Euclid's Elements (OED2).
The term ALLOTRIOUS FACTOR was coined by James Joseph Sylvester.
The term ALPHAMETIC was coined in 1955 by J. A. H. Hunter (Schwartzman).
ALTERNATING GROUP appears in Camille Jordan, "Sur la limite de transitivit? des groupes non altern?s," Bull. Soc. math. de France 1 (1873).
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 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 term ANALYSIS was introduced by Theon of Alexandria, according to Kline (page 279).
Fran?ois Vi?te used the term analytic art for algebra in 1591 in In artem analyticem isagoge. Schwartzman (page 23) says that around 1590 "the French mathematician Vi?te 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 Knuth, according to an Internet website.
The phrase ANALYSIS OF VARIANCE appears in a paper by Sir Ronald Aylmer Fisher published in 1924, used as if Fisher expected the reader to know that an analysis of variance was. In a 1920 paper, Fisher used the phrase "analysis of total variance" as if he had to explain what such a procedure is. In The History of Statistics: The Measurement of Uncertainty before 1900, Stephen M. Stigler writes, "Yule derived what we now, following Fisher, call the analysis of variance breakdown." [James A. Landau]
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 Gonz?lez Cabill?n]
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 Trait? 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 Th?orie des Fonctions Analytiques. According to Georges Valiron's essay in "Great currents of mathematical thought," Edited by Fran?ois 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 Fran?ois Lacroix (1765-1843) wrote in Trait? du calcul diff?rentiel et du calcul int?gral: "There exists a manner of viewing geometry that could be called g?om?trie 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. Dieudonn? protested:
It is absolutely intolerable to use analytical geometry for linear algebra with coordinates, still called analytical geometry in the elementary books. Analytical geometry in this sense never existed. There are only people who do linear algebra badly, by taking coordinates and this they call analytical geometry. Out with them! Everyone knows that analytical geometry is the theory of analytical spaces, one of the deepest and most difficult theories of all mathematics.[Julio Gonz?lez Cabill?n, Carlos C?sar de Ara?jo]
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 "Aper?u historique .... des M?thodes en G?om?trie," 1837, p.35. He uses essentially the same definition as M?bius, but without the signs, so a harmonic range gives the value +1 rather than -1. Then he says "Ce rapport ?tant dit harmonique dans le cas particulier o? il est ?gal a l'unit?, nous l'appellerons, dans le cas g?n?ral, rapport ou fonction anharmonique" [Ken Pledger].
See also cross-ratio.
ANNULUS is found in 1834 in the Penny Cyclopedia (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 appears in Higher Mathematics for Students of Chemistry and Physics (1912) by J. W. Mellor: "The integral, f' (x) dx, is sometimes called an anti-differential" [James A. Landau].
The term ANTILOGARITHM was introduced by John Napier in his masterpiece:
And they are also the Logarithmes of the complements of the arches and sines towards the right hand, which we call Antilogarithmes.Cf. page 11, 'first Booke' (Chap. 3), in John Napier's "A description of the Admirable Table of Logarithmes ...", London: Printed for Simon Waterson, 1618. The term antilogarithm may also appear in the first English translation (1616), which has not been consulted. [Julio Gonz?lez Cabill?n]
APOTHEM is dated ca. 1856 in MWCD10.
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.
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.In English the term is found in M. G. Gaba, "On a generalization of the arbelos," Am. Math. Mon. 47 (1940).
...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'), ...
[Ken Pledger, Antreas P. Hatzipolakis, Julio Gonz?lez Cabill?n]
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).
ARGAND DIAGRAM appears in 1908 in Math. Theory Electr. & Magnet. by Sir James Hopwood Jeans (1877-1946) (OED2).
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 ARITHMETIC PROGRESSION was used by Michael Stifel in 1543: "Divisio in Arethmeticis progressionibus respondet extractionibus radicum in progressionibus Geometricis" [James A. Landau].
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 the 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, : 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.The citation above is from Sir William Rowan Hamilton, by Thomas Hankins, chapter 23 ("The Fate of Quaternions").
(These citations were provided by David Wilkins.)
ASTROID. According to E. H. Lockwood (1961):
The astroid seems to have acquired its present name only in 1838, in a book published in Vienna; it went, even after that time, under various other names, such as cubocycloid, paracycle, four-cusp-curve, and so on. The equation x2/3 + y2/3 == a2/3 can, however, be found in Leibniz's correspondence as early as 1715.This quote was taken from Xah Lee's Visual Dictionary of Special Plane Curves website.
ASYMPTOTE was used by Apollonius, with a broader meaning than its current definition, referring to any lines which do not meet, in whatever direction they are produced (Smith).
The first citation of the word in the OED is in 1656 in Hobbes' Elements of Philosophy by Thomas Hobbes.
AUGMENTED MATRIX is dated ca. 1949 in MWCD10.
The term AUTOMORPHIC FORM was introduced by Jules Henri Poincar? (1854-1912). Kramer (p. 382) implies he also used the term automorphic function.
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."
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."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 "Unersuchungen ?ber 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 "Chesi?res, 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 M?glichkeit 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 setsIn the same paper Zermelo adds:
IV. Axiom. A set S that can be decomposed into a set of disjoint parts A, B, C..., each containing at least one element, possesses at least one subset 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.All of the above material is from Heijenoort (1967). The translations of Zermelo in this book are by Stefan Bauer-Mengelberg.
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 ? which takes one and only one member from each member of kappa." This we shall define as the "multiplicative axiom."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).