**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 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 GroesseNote that Weierstrass first applied the term and symbolism to complex numbers. [Julio Gonz?lez Cabill?n]xmit |x|. [I denote the absolute value of complex numberxby |x|.]

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

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

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 callCf. 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 termAntilogarithmes.

**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 (In English the term is found in M. G. Gaba, "On a generalization of the arbelos,"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'), ...

[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 extraThe termr,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 extraris generally found in the Italian literature until the time of printing. From Italy it passed over to Germany, where it is not uncommonly found in the books of the 16th century, and to France, where it is found less frequently. The ordinary variations in spelling have less significance, merely illustrating, as in the case with many other mathematical terms, the vagaries of pronunciation in the uncritical periods of the world's literatures.

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

However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called theThe citation above is from "On a New Species of Imaginary Quantities Connected with the Theory of Quaternions," Royal Irish Academy,associativecharacter 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 theThe citation above is fromTransactionsof the academy (Math. Papers, 3:116). That longer paper, also dated Nov. 13, 1843 was not published until 1848, and when it did appear it concluded with a note mentioning works written as late as June 1847 ("Researches respecting Quaternions: First Series," Royal Irish Academy,Transactions21, [1848]: 199-296, in Math. Papers, 3:159-216, "note A," pp. 217-26). In a note Hamilton stated that his presentation on Nov. 13, 1843 had been "in great part oral" (Math. Papers, 3:225); therefore it is possible that his statement of and naming of the associative law was added later when his original communication was to be printed in 1844. It is thus possible that he first recognized the importance of the associative law in 1844, when he began work on Graves's octaves.

(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 equationThis quote was taken from Xah Lee's Visual Dictionary of Special Plane Curves website.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.

**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 EnglishThe termaver de poiz,meaning "goods of weight." In the 16th century it was commonly called "Haberdepoise," as in most of the editions of Recorde's (c. 1542)Ground of Artes.Thus in the Mellis eddition of 1594 we have: "At London & so all England thorugh are vsed two kids of waights and measures, as the Troy waight & the Haberdepoise."

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

In the paper "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 SIn another paper also published in 1908, "Neuer Beweis für die M?glichkeit einer Wohlordnung" ["The possibility of a well-ordering"],_{1}having one and only one element in common with each element of T. [The original German read "Axiom der Auswahl".]

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 S

_{1}having exactly one element in common with each of the parts A,B,C,... considered.

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

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).