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Last revision: Aug. 3, 1999

An early use of MACLAURIN'S SERIES in English is in 1902 in the Encyclopaedia Britannica in the article "Infinitesimal Calculus":
This result is usually called Maclaurin's series, having been given in his Fluxions (1742). It had, however, been previously published by Stirling in his Meth. Diff. (1717); but neither Stirling nor Maclaurin laid any claim to the theorem as being original, both referring it to Taylor.
[James A. Landau]

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, appendicem voco, sive mantissam.
The citation above is from "Opera mathematica," vol. 2, Oxoniae, 1693 (De Algebra tractatus), page 41. This is in the Latin edition, and not in the original edition of 1685, in which Wallis uses the English word "appendage." According to Julio González Cabillón, this is the first use of the term to mean "the decimal part of any number."

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 Introductio in analysin infinitorum, vol. 1, page 83 (Lausannae 1748). According to Julio González Cabillón, this is the first use of the term to mean "the decimal part of a logarithm." According to Smith (vol. 2, page 514), the word was not commonly used until its adoption by Euler.

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, of m lines and n columns. 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 number p, and selecting at will p lines and p columns, the squares corresponding of pth order.
The citation above is from "Additions to the Articles On a new class of theorems, and On Pascal's theorem," Philosophical Magazine, pp. 363-370, 1850. Reprinted in Sylvester's Collected Mathematical Papers, vol. 1, pp. 145-151, Cambridge (At the University Press), 1904, page 150.

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 of n rows and (n+1) columns

Then all the n+1 determinants that can be formed by rejecting any one column at pleasure out of this matrix are identically zero.

The 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's Collected Mathematical Papers, vol. 1, Cambridge (At the University Press), 1904, page 209.

[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 the mean error, which is defined as the 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.

In 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].

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 of n lines and n columns, n2 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 of rth minor determinants by the exclusion of r lines and r columns.
Sylvester also used minor as a noun in the same article: "The whole of a system of rth minors being zero" and "We shall have only to deal with a system of first minors" (OED).

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:

In all these examples, "modulo" can be replaced by "taking for granted" or else "except for". He has not found any explicit discussion of this usage in print, modulo a short passage in Gregory Chaitin's homepage, where he says:
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).

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