An early use of

This result is usually called Maclaurin's series, having been given in his[James A. Landau]Fluxions(1742). It had, however, been previously published by Stirling in hisMeth. Diff.(1717); but neither Stirling nor Maclaurin laid any claim to the theorem as being original, both referring it to Taylor.

**MAGIC SQUARE** is found in the title *Des quassez ou tables magiques*
by Frenicle de Bessy (1605-1675).

The first citation in the OED2 is in 1704 in *Lexicon technicum, or
an universal English dictionary of arts and sciences* by John Harris.

Benjamin Franklin used the term in his autobiography.

The term **MANDELBROT SET** was coined by Adrien Douady, according
to an Internet web page.

**MANIFOLD** was apparently introduced as *Mannigfaltigkeit*
by Bernhard Riemann (1826-1866).

**MANTISSA** is a late Latin term of Etruscan origin, originally
meaning an addition, a makeweight, or something of minor value, and was
written *mantisa.* In the 16th century it came to be written *mantissa*
and to mean appendix (Smith vol. 2, page 514).

Numerous sources, including Smith (vol. 2, page 524), Boyer (page 345),
the *Century Dictionary* (1889-97), and *Webster's New International
Dictionary* (1909), claim that *mantissa* was introduced by Henry
Briggs (1561-1631) in 1624 in *Arithmetica logarithmica.* However,
this information apparently is incorrect. Johannes Tropfke in his "Geschichte
der Elementar-Mathematik, vol. 2, 3rd edition 1933, says "Das Fachwort
Mantisse hatte Briggs noch nicht" (p. 252). [Christoph J. Scriba]

According to Cajori (1919, page 152), the word *mantissa* was first
used by John Wallis in 1693:

Ejusque partes decimales abscissas,The citation above is from "Opera mathematica," vol. 2, Oxoniae, 1693 (appendicemvoco, sive mantissam.

*Mantissa* was also used by Leonhard Euler in 1748:

Constat ergo logarithmus quisque ex numero integro et fractione decimali et ille numerus integer vocari solet characteristica, fractio decimalis autem mantissa. (The logarithm consists of an integral part, called the characteristic, and a decimal fraction, called the mantissa.)The citation above is from Euler's

Gauss suggested using the word for the fractional part of all decimals:
"Si fractio communis in decimalem convertitur, seriem figurarum decimalium
... fractionis *mantissam* vocamus ..." (Smith vol. 2, page 514).

**MAPPING** is found in "On the Metric Geometry of the Plane *N*-Line,"
F. Morley, *Transactions of the American Mathematical Society,* Vol.
1, No. 2. (Apr., 1900).

**MARKOV CHAIN.** The phrase "les châines de Markoff" is found in
1930 by Kaucky [James A. Landau].

**MARKOV PROCESS** occurs in Kosaku Yosida and Shizuo Kakutani, "Markoff
process with an enumerable infinite number of possible states," *Jap.
J. Math.* 16 (1939).

The term also appears in Shizuo Kakutani, "Some results in the operator-theoretical
treatment of the Markoff process," *Proc. Imp. Acad. Jap.* 15 (1939).

The term **MARRIAGE THEOREM** was introduced by Hermann Weyl (1885-1955)
in "Almost periodic invariant vector sets in a metric vector space", *Amer.
J. Math.* 71 (1949), 178-205, according to Konrad Jacobs in *Measure
and Integral,* Academic Press, 1978. The theorem is also called "Hall's
theorem" or "Hall's marriage theorem" since it was first proved by Philip
Hall in 1935 [Carlos César de Araújo].

**MATH** is dated ca. 1878 in MWCD10.

The phrase "Math: books" is found in the writings of Isaac Newton, but apparently the colon indicates this is an abbreviation [James A. Landau, Axel Harvey].

**MATHS.** The first citation in the OED2 is 1911: "The Answers to
Maths. Ques. were given us all this morning." This citation is from the
collected letters of Wilfred Edward Salter Owen, published in 1967. The
next OED2 citation is from *Wireless World* in 1917: "Extremely 'rusty'
in 'maths'." Apparently there is not a period in this use of the word.

**MATHEMATICAL EXPECTATION** was used by DeMorgan in 1838 in *An
Essay on Probabilities* (1841) 97: "The balance is the average required,
and is known by the name of mathematical expectation" (OED2).

See also *expectation.*

The term **MATHEMATICAL INDUCTION** was introduced by Augustus de
Morgan (1806-1871) in 1838 in the article *Induction (Mathematics)*
which he wrote for the *Penny Cyclopedia.* De Morgan had suggested
the name *successive induction* in the same article and only used
the term *mathematical induction* incidentally. The expression *complete
induction* attained popularity in Germany after Dedekind used it in
a paper of 1887 (Burton, page 440).

See also *complete induction.*

**MATHEMATICAL LOGIC** occurs in 1858 in *On Syllogisms* by
A. DeMorgan (OED2).

According to the University of St. Andrews website, Ernst Schröder (1841-1902) "seems to be the first to use the term mathematical logic."

**MATHEMATICAL RIGOR.** Leonhard Euler used a term in 1755 in *Institutiones
calculi differentialis* which is rendered "mathematical rigor" in an
English translation.

**MATHEMATICIAN** is first found in Higden's Polychronicon, translated
1432-50. (The word is spelled "mathematicions.") (OED2).

**MATHEMATICS.** Pythagoras is said to have coined the words *philosophy*
for "love of wisdom" and *mathematics* for "that which is learned."

*Mathematics* is found in English in 1581 in *Positions, wherein
those primitive circumstances be examined, which are necessarie for the
training up of children* by Richard Mulcaster. (The word is spelled
"mathematikes.") (OED2)

The term **MATRIX** was coined in 1850 by James Joseph Sylvester
(1814-1897):

[...] For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, ofThe citation above is from "Additions to the Articlesmlines andncolumns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a numberp,and selecting at willplines andpcolumns, the squares corresponding ofpth order.

Kline (page 804) says, "The word matrix was first used by Sylvester when in fact he wished to refer to a rectangular array of numbers and could not use the word determinant, though he was at that time concerned only with the determinants that could be formed from the elements of the rectangular arry."

Katz (1993) says: "The English word *matrix* meant 'the place from
which something else originates.' Sylvester himself made no use of the
term at the time. It was his friend Cayley who put the terminology to use
in papers of 1855 and 1858."

In 1851 Sylvester informally uses the term *matrix* as follows:

Form the rectangular matrix consisting ofThe citation above is from "An essay on canonical forms, supplement to a sketch of a memoir on elimination, transformation and canonical forms", London, 1851. Reprinted in Sylvester'snrows and (n+1) columns[matrix] Then all the

n+1 determinants that can be formed by rejecting anyonecolumn at pleasure out of this matrix are identically zero.

[Randy K. Schwartz and Julio González Cabillón]

**MATROID.** In a effort to axiomatize the notion of "independence"
that arises in graph theory and in vector spaces theory, Hassler Whitney
coined the term "matroid" and introduced it in his fundamental paper *On
the abstract properties of linear independence*, Amer. J. Math. 57 (1935)
509-533. The choice of the name arose because he took as an initial model
the finite sets of linearly independent column vectors of a *matrix*
over a field. In his paper Whitney gave several equivalent characterizations
of a matroid, but the general idea is that of a finite set endowed with
a "independence structure" (just as a topological space is a set endowed
with a "closeness structure"). Extensions to infinite sets and additional
contributions were made by Saunders Mac Lane (1936), R. Rado (1942), W.
T. Tutte (1961) and many others. [Carlos César de Araújo]

The term **MAXIMUM LIKELIHOOD** was introduced by Sir Ronald Aylmer
Fisher in his paper "On the Mathematical Foundations of Theoretical Statistics,"
in *Philosophical Transactions of the Royal Society,* April 19, 1922.
In this paper he made clear for the first time the distinction between
the mathematical properties of "likelihoods" and "probabilities" (DSB).

**MEAN** (mean terms in a proportion) is found in English in 1571
in *A geometrical practise named Pantometria* by Thomas Digges (1546?-1595):
"When foure magnitudes are...in continual proportion, the first and the
fourth are the extremes, and the second and thirde the meanes" (OED2).

**MEAN ERROR.** The 1845 *Encyclopedia Metropolitana* has "mean
risk of error" (OED2).

In 1878, Petrie, in *Jrnl. Anthrop. Inst.* wrote, "the mean error
being 7 inches on 130 feet" (OED2).

In 1894 in *Phil. Trans. Roy. Soc,* Karl Pearson has "error of
mean square" as an alternate term for "standard-deviation" (OED2).

In *Higher Mathematics for Students of Chemistry and Physics* (1912),
J. W. Mellor writes:

In Germany, the favourite method is to employ theIn a footnote, Mellor writes, "Some writers call our "average error" the "mean error," and our "mean error" the "error of mean square" [James A. Landau].mean error,which is defined asthe error whose square is the mean of the squares of all the errors,or the "error which, if it alone were assumed in all the observations indifferently, would give the same sum of the squares of the errors as that which actually exists." ...The mean error must not be confused with the "mean of the errors," or, as it is sometimes called, the

average error,another standard of comparison defined as the mean of all the errors regardless of sign.

**MEANS.** According to Smith (vol. 2, page 483), "The terms 'means,'
'antecedent,' and 'consequent' are due to the Latin translators of Euclid."

The term **MEAN SQUARE DEVIATION** (apparently meaning *variance*)
appears in a paper published by Sir Ronald Aylmer Fisher in 1920 [James
A. Landau].

The term **MEAN VALUE THEOREM** is found in "A Simple Proof of the
Fundamental Cauchy-Goursat Theorem," Eliakim Hastings Moore, *Transactions
of the American Mathematical Society,* Vol. 1, No. 4. (Oct., 1900).

The term **MEASURABLE FUNCTION** was used by Arnaud Denjoy (1884-1974)
(Kramer, p. 648).

An early use of the term is N. Lusin, "Sur les propriétés des fonctions
mesurables," *Comptes Rendua Acad. Sci. Paris,* 154 (1912).

**MEASURE (of a set)** is found in "On Non-Measurable Sets of Points,
with an Example," Edward B. Van Vleck, *Transactions of the American
Mathematical Society,* Vol. 9, No. 2 (Apr., 1908): "Lebesgue's theory
of integration is based on the notion of the *measure* of a set of
points, a notion introduced by BOREL and subsequently refined by LEBESGUE
himself."

Émile Borel (1871-1956), who created the theory of the measure of sets of points, wrote: "La définition de la mesure des ensembles linéaires bien définis m'est entièrement due" (The definition of the measure of well defined linear sets, is entirely due to me) [Udai Venedem].

**MEASURE THEORY** is found in H. Steinhaus, "Les probabilités dénombrables
et leur rapport à la théorie de la mesure," *Fund. Math.* 4 (1923)
[James A. Landau].

**MECHANICAL QUADRATURE** is found in F. G. Mehler, "Bemerkungen
zur Theorie der mechanischen Quadraturen," *J. Reine angew. Math*
63 (1864) [James A. Landau].

**MEDIAN** (in statistics) was used by Francis Galton in *Report
of the British Association for the Advancement of Science* in 1881:
"The Median, in height, weight, or any other attribute, is the value which
is exceeded by one-half of an infinitely large group, and which the other
half fall short of" (OED2).

**MEDIAN** (of a triangle) appears in the *Encyclopaedia Britannica*
of 1883 (OED2).

**MERSENNE NUMBER** is found in É. Lucas, *Récréations Mathématiques,*
tome II, Note II, "Sur les nombres de Fermat et de Mersenne" (1883).

*Mersenne's number* is found in English in R. E. Powers, "On Mersenne's
Numbers," *P. London Math. Soc.* (1914).

*Mersenne number* is found in English in D. H. Lehmer, "Note on
Mersenne numbers," *Bull. Am. Math. Soc.* 38 (1932) and A. E. Western,
"On Lucas's and Pepin's Tests for the Primeness of Mersenne Numbers," *J.
London Math. Soc.* 7 (1932).

*Mersenne prime* is found in English in D. H. Lehmer (Editor &
Reviewer), "A New Mersenne Prime," Note 138, *MTAC* 6 (1952).

**MESSENGER PROBLEM.** In 1930, Karl Menger (1902-1985) mentioned
the *messenger problem,* referring to the problem of finding the shortest
Hamiltonian path, according to an Internet web page.

**METABELIAN GROUP** appears in William Benjamin Fite, "On Metabelian
Groups," *Transactions of the American Mathematical Society* 3 (July,
1902): "We define a *Metabelian Group* as *a group whose group of
cogredient isomorphisms is abelian.*"

The term **METAMATHEMATICS** goes back to the 1870s where it was
used as a pejorative (intending to put it in the same light as metaphysics)
in discussions of non-Euclidean geometries.

In the 1890 *Funk & Wagnalls* Dictionary the word is defined
as "The philosophy or metaphysics of mathematics."

It was first used in its modern sense by David Hilbert (1862-1943) in a 1922 lecture. [Michael Detlefsen, Carlos César de Araújo]

**METHOD OF EXHAUSTION.** The Flemish Jesuit mathematician Gregorius
a Sancto Vincentio (or Gregory St. Vincent) (1584-1667) was "probably the
first to use the word *exhaurire* in a geometrical sense" (Cajori
1919). The term *method of exhaustion* arose from this word.

**METHOD OF LEAST SQUARES.** *Sur la méthode des moindres carrés*
by Adrien Marie Legendre (1752-1833) was published in August 1820. According
to a post in sci.math by Paul Gardner, Legendre coined this term in 1805.

The term **METRIC SPACE** is due to Felix Hausdorff (1869-1942).
The term *metrischer raum* is found in *Grundzüge der Mengenlehre*
(1914).

**METRIC SYSTEM** is found in English in the "Metric Weights and
Measures Act, 1864."

The metric system is explained in Noah Webster's 1806 dictionary under the heading "New French Weights and Measures."

*Gram* is found in English in Aug. 1797 in *Nicholson's Journal*
where it is spelled "gramme." *Kilogram* and *liter* are found
in English in Aug. 1797 in *Journal of Natural Philosophy.* *Kilometer,
milliliter, millimeter,* and *milligram* are found in English in
Noah Webster's 1806 *A Compendious Dictionary of the English Language,*
although kilometer is spelled "chiliometer." *Metric ton* is dated
ca. 1890 in MWCD10.

**MILLER-RABIN TEST** is found in H. W. Lenstra, Jr. "Primality testing,"
Number theory and computers, Studyweek, Math. Cent. Amsterdam 1980, and
in Louis Monier, "Evaluation and comparison of two efficient probabilistic
primality testing algorithms," *Theor. Comput. Sci.,* 12 (1980).

Related terms are found in H. W. Lenstra, Jr., "Miller's primality test,"
*Inf.
Process. Lett.* 8 (1979) and Tore Herlestam, "A note on Rabin's probabilistic
primality test," *BIT, Nord. Tidskr. Informationsbehandling* 20 (1980).

**MILLIARD.** Gulielmus Budaeus (1467-1540) used the term in his
*De
Asse et Partibus eius Libri V.* In the Paris edition of 1532, the following
appears: "hoc est denas myriadu myriadas, quod vno verbo nostrates abaci
studiosi Milliartu appellat, quasi millionu millione" (Smith vol. 2, page
85).

**MILLION.** According to Smith (vol. 2, page 81), Maximus Planudes
(c. 1340) "seems to have been among the first of the mathematicians to
use the word."

Ghaligai wrote that Maestro Paulo da Pisa "La settima dice numero di milione" (read the seventh order as millions). Smith (vol. 2, page 81) writes that this Paulo may have been Paolo Dagomari (b. 1281; d., 1365 or 1374).

*Million* was used in English in 1362 in *Piers Plowman* by
William Langland (c. 1334-c.1400): "Coueyte not his goodes / For millions
of moneye."

According to Smith (vol. 2, page 81), the word first appeared in a printed
work in the Treviso arithmetic of 1478. According to Johnson (page 157),
it first appeared in print in 1494 in *Summa de Arithmetica,* by Luca
Paciola (1445-1514).

*Million* appears in the King James Bible: "And they blessed Rebekah,
and said unto her, Thou art our sister, be thou the mother of thousands
of millions, and let thy seed possess the gate of those which hate them"
(Gen. 24: 60). The word also appears in Shakespeare.

To avoid confusion, mathematicians tended to use "thousand thousand" into the sixteenth century.

The term **MINOR** was apparently coined by James Joseph Sylvester,
who wrote in *Philos. Mag.* Nov. 1850:

Now conceive any one line and any one column to be struck out, we get ... a square, one term less in breadth and depth than the original square; and by varying in every possible manner the selection of the line and column excluded, we obtain, supposing the original square to consist ofSylvester also usednlines andncolumns,n^{2}such minor squares, each of which will represent what I term a First Minor Determinant relative to the principal or complete determinant. Now suppose two lines and two columns struck out from the original square ... These constitute what I term a system of Second Minor Determinants; and ... we can form a system ofrth minor determinants by the exclusion ofrlines andrcolumns.

**MINUEND** is an abbreviation of the Latin *numerus minuendus*
(number to be diminished), which was used by Johannes Hispalensis (c. 1140)
(Smith vol. 2, page 96).

In English, *minuend* was used in 1706 by William Jones in *Synopsis
palmariorum matheseos, or a new introduction to the mathematics* (OED2).

**MINUS.** See *plus.*

The term **MINUS SIGN** is dated 1668 in MWCD10.

**MÖBIUS STRIP** appears in 1904 in E. R. Hedrick, translation of
*Goursat's
Course in Mathematical Analysis* (as "Möbius' strip) (OED2).

**MODE** was coined by Karl Pearson (1857-1936). In 1895 he wrote
in the *Philosophical Transactions of the Royal Society,* "I have
found it convenient to use the term *mode* for the abscissa corresponding
to the ordinate of maximum frequency. Thus the "mean," the "mode," and
the "median" have all distinct characters."

The term **MODULAR ARITHMETIC** was coined by Gauss, according to
an Internet website. The term is dated 1959, in English, in MWCD10.

**MODULAR CURVE** appears in 1883 in the title *The Modular Curves
of an Uneven Order* by H. J. S. Smith (OED2).

The term **MODULAR EQUATION** was introduced by Jacobi [*Encyclopaedia
Britannica* (1902), article "Infinitesimal Calculus"]. The OED2 shows
a use of the term in 1845 by DeMorgan in *Encyclopaedia Metropolitana.*

**MODULAR FORM** occurs in the heading "Definite Modular Forms" in
"Definite Forms in a Finite Field," Leonard Eugene Dickson, *Transactions
of the American Mathematical Society,* Vol. 10, No. 1. (Jan., 1909).

**MODULAR FUNCTION.** Christoph Gudermann (1798-1852) called elliptical
functions "Modularfunctionen" (DSB).

**MODULO** is dated 1897 in MWCD10.

Carlos César de Araújo notes that besides its use as a technical term,
*modulo*
is being widely used by mathematicians in a related charming sense as a
slang expression. He provides these examples:

- "The following proof is self-contained modulo the standard material on operators and inner-product spaces."
- "He called them continuous functionals. It was clear that modulo unimportant differences these two classes of functionals were equivalent."
- "Turing's work showed that, modulo a universal Turing machine, hardware and software are interchangeable."

One could almost imagine a journal of experimental number theory. For example, there are papers published by number theorists which are, mathematicians say, "modulo the Riemann hypothesis." That is to say, they're taking the Riemann hypothesis as an axiom, but instead of calling it a new axiom they're calling it a hypothesis.That is, if you have a mathematical sentence S the proof of which required RH (Riemann hypothesis) as a hypothesis, then "S is proved modulo RH" is taken to mean "RH -> S" is proved.

**MODULUS** (the factor to change a logarithm to a different base)
was used by Roger Cotes (1682-1716) in 1722 in *Harmonia Mensurarum:*
Pro diversa magnitudine quantitatis assumptae *M,* quae adeo vocetur
systematis *Modulus.* Cotes also coined the term *ratio modularis*
(modular ratio) in this work.

**MODULUS** (a coefficient that expresses the degree to which a body
possesses a particular property) appears in the 1738 edition of *The
Doctrine of Chances: or, a Method of Calculating the Probability of Events
in Play* by Abraham De Moivre (1667-1754) [James A. Landau].

**MODULUS** (in number theory) was introduced by Gauss in 1801 in
*Disquisitiones
arithmeticae* (OED2).

**MODULUS** (of a complex number) was introduced by Augustin-Louis
Cauchy
(1789-1857) in 1821.

**MODULUS** (a constant multiplier) was used by MacCullagh in 1843
in the *Proceedings of the Royal Irish Academy* (OED2).

**MODULUS** (of a function) was used by De Morgan in 1845 in "Calculus
of Functions" in *Encyclopaedia Metropolitana* (OED2).

**MODULUS** (the length of the vector *a* + *bi*) is due
to Jean Robert Argand (1768-1822) (Cajori 1919, page 265).

The term **MODULUS OF TRANSFORMATION** was used in 1882 by George
M. Minchin in *Uniplanar Kinematics of Solids and Fluids:* "It will
be convenient to speak of this quantity *K* as a modulus of transformation"
(OED2).

The term **MONOGENIC** (for a function having a single derivative
at a point) was introduced by Augustin-Louis Cauchy (1789-1857).

**MONOMIAL** is dated ca. 1706 in MWCD10.

**MONOTONIC** is found in 1901 in *Ann. Math.* II: "It follows
that *f*(*s*) is a monotonic function that actually decreases
in parts of the interval..." (OED2).

It is also found in W. F. Osgood, "On the Existence of a Minimum of
the Integral...," *Transactions of the American Mathematical Society,*
2 (Apr., 1901). The term is probably considerably older.

**MONTE CARLO.** The method as well as the name for it were apparently
first suggested by John von Neumann and S. M. Ulam, according to information
in *Mathematical Tables and Other Aids to Computation* (1949). An
Internet web page says the name and the systematic development of the method
date to about 1944.

The term occurs in 1949 in the title "The Monte Carlo Method" by Nicholas
Metropolis in the *Journal of the American Statistical Association*
44 (1949).

**MOORE SPACE.** This name was introduced by F. Burton Jones in *Concerning
normal and completely normal spaces* (Bull. Amer. Math. Soc. 43 (1937)
671-677, p.675) for a topological space satisfying "Axiom 0 and parts 1,
2, and 3 of Axiom 1 of R. L. Moore’s *Foundations of Point
Set Theory*" (Amer. Math. Soc. Coll. Publ. 13, NY, 1932). It was in
that paper (p. 676) that Jones stated for the first time the famous *normal
Moore space conjecture*: "Is every normal Moore space *M* metric
[metrizable]?" Despite considerable effort spent in seeking a solution,
the question was "settled" only in 1970, when Tall and Silver (by using
a Cohen model) showed its undecidability from traditional set theory. [Carlos
César de Araújo]

The term **MORAL EXPECTATION** was used by Daniel Bernoulli.

**MULTIPLY** was used in English as a verb ("multiply by two") about
1391 by Chaucer in *A Treatise on the Astrolabe* (OED2).

**MULTIPLICATION** was used by Chaucer in a non-mathematical sense
about 1384 and in a mathematical sense in 1390 by John Gower in *Confessio
amantis* III 89 (OED2).

**MULTIPLICATIVE IDENTITY** and **MULTIPLICATIVE INVERSE** are
found in 1953 in *First Course in Abstract Algebra* by Richard E.
Johnson [James A. Landau].

**MULTIVARIATE** is found in Solomon Kullback, "On samples from a
multivariate normal population," *Ann. Math. Statist.* 6 (1935).

The term also appears in A. C. Aitken, "Note on selection from a multivariate
normal population," *Proc. Edinb. Math. Soc.,* II. Ser. 4 (1935).