Earliest Known Uses of Some of the Words of Mathematics

LAGRANGE MULTIPLIER. The term "Lagrange's method of undetermined multipliers" appears in J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics (1912) [James A. Landau].

The term "Lagrange multiplier rule" appears in "The Problem of Mayer with Variable End Points," Gilbert Ames Bliss, Transactions of the American Mathematical Society, Vol. 19, No. 3. (Jul., 1918).

Lagrange multiplier is found in "Necessary Conditions in the Problems of Mayer in the Calculus of Variations," Gillie A. Larew, Transactions of the American Mathematical Society, Vol. 20, No. 1. (Jan., 1919): "The [lambda]'s appearing in this sum are the functions of x sometimes called Lagrange multipliers."

LAGRANGE'S THEOREM. Formule de Lagrange appears in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800) by Lacroix.

Lagrange's theorem appears in An Elementary Treatise on Curves, Functions and Forms (1846) by Benjamin Peirce: "The theorem (650) under this form of application, has been often called Laplace's Theorem; but, regarding this change as obvious and insignificant, we do not hesitate to discard the latter name, and give the whole honor of the theorem to its true author, Lagrange."

Lagrange's formula for interpolation appears in 1849 in An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson.

Lagrange's method of approximation occurs in the third edition of An Elementary Treatise on the Theory of Equations (1875) by Isaac Todhunter.

LAGRANGIAN (as a noun) occurs in Th. Muir, "Note on the Lagrangian of a special unit determinant," Transactions Royal Soc. South Africa (1929).

LAPLACE'S COEFFICIENTS. According to Todhunter (1873), "the name Laplace's coefficients appears to have been first used" by William Whewell (1794-1866) [Chris Linton].

Laplace's coefficients appears in the title Mathematical tracts Part I: On Laplace's coefficients, the figure of the earth, the motion of a rigid body about its center of gravity, and precession and nutation (1840) by Matthew O'Brien.

LAPLACE'S EQUATION appears in 1845 in the Encyclopedia Metropolitana.

LAPLACE'S FUNCTIONS appears in 1855 in the title "On the solution of the equation of Laplace's functions" by Charles Graves (Bishop of Limerick). The paper was published in Proc. Dublin [James A. Landau].

Laplace's functions also appears in 1860 in the title On attractions, Laplace's functions and the figure of the Earth by John Henry Pratt (1809-1871). Todhunter (1873) writes, "The distinction between the coefficients and the functions is given for the first time to my knowledge in Pratt's Figure of the Earth" [Chris Linton].

The term LAPLACE'S OPERATOR was used in 1873 by James Clerk Maxwell in a Treatise on Electricity and Magnetism: "...an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator" (OED2).

The term LAPLACE TRANSFORM was used by Boole and Poincaré. According to the website of the University of St. Andrews, Boole and Poincaré might otherwise have used the term Petzval transform but they were influenced by a student of Józeph Miksa Petzval (1807-1891) who, after a falling out with his instructor, claimed incorrectly that Petzval had plagiarised Laplace's work.

LAPLACIAN (as a noun, for the differential operator) was used in 1935 by Pauling and Wilson in Introd. Quantum Mech. (OED2).

LATENT VALUE and VECTOR. See Eigenvalue.

The term LATIN SQUARE was named by Euler (as quarré latin) in 1782 in Verh. uitgegeven door het Zeeuwsch Genootschap d. Wetensch. te Vlissingen.

Latin square appears in English in 1890 in the title of a paper by Arthur Cayley, "On Latin Squares" in Messenger of Mathematics.

The term was introduced into statistics by R. A. Fisher, according to Tankard (p. 112). Fisher used the term in 1925 in Statistical Methods Res. Workers (OED2).

Graeco-Latin square appears in 1934 in R. A. Fisher and F. Yates, "The 6 x 6 Latin Squares," Proceedings of the Cambridge Philosophical Society 30, 492-507.

LATITUDE and LONGITUDE. Henry of Ghent used the word latitudo in connection with the concept of latitude of forms.

Nicole Oresme (1320-1382) used the terms latitude and longitude approximately in the sense of abscissa and ordinate.

LATTICE POINT is found in 1857 in Cayley, Coll. Math. Papers (1890) III. 40: "Imagine now in a plane, a rectangular system of coordinates (x, y) and the whole plane divided by lines parallel to the axes at distances = 1 from each other into squares of the dimension = 1. And let the angles which do not lie on the axes of coordinates be called lattice points (OED2).

The term LATUS RECTUM was used by Gilles Personne de Roberval (1602-1675) in his lectures on Conic Sections. The lectures were printed posthumously under the title Propositum locum geometricum ad aequationem analyticam revocare,... in 1693 [Barnabas Hughes].

LAW OF COSINES is found in 1895 in Plane and spherical trigonometry, surveying and tables by George Albert Wentworth: "Law of Cosines. ... The square of any side of a triangle is equal to the sum of the squares of the other two sides, diminished by twice their product into the cosine of the included angle" [University of Michigan Digital Library].

The term LAW OF INTERTIA OF QUADRATIC FORMS is due to James Joseph Sylvester (DSB).

LAW OF LARGE NUMBERS. La loi de grands nombres appears in 1835 in Siméon-Denis Poisson (1781-1840), "Recherches sur la Probabilité des Jugements, Principalement en Matiére Criminelle," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences, 1, 473-494 (James, 1998).

According to Porter (p. 12), Poisson coined the term in 1835.

LAW OF SINES (Snell's law). The law of sines is found in 1851-54 in Hand-books of natural philosophy and astronomy by Dionysius Lardner [University of Michigan Digital Library].

LAW OF SINES (trigonometry) is found in 1895 in Plane and spherical trigonometry, surveying and tables by George Albert Wentworth: "...the Law of Sines, which may be thus stated: The sides of a triangle are proportional to the sines of the opposite angles [University of Michigan Digital Library].

LAW OF TANGENTS is found in 1895 in Plane and spherical trigonometry, surveying and tables by George Albert Wentworth: "Hence the Law of Tangents: The difference of two sides of a triangle is to their sum as the tangent of half the difference of the opposite angles is to the tangent of half their sum" [University of Michigan Digital Library].

LAW OF THE ITERATED LOGARITHM is found in Philip Hartman and Aurel Wintner, "On the law of the iterated logarithm," Am. J. Math. 63, 169-176 (1941).

The term LEAST ACTION was used by Lagrange (DSB).

LEAST COMMON MULTIPLE. Common denominator appears in English in 1594 in Exercises by Blundevil: "Multiply the Denominators the one into the other, and the Product thereof shall bee a common Denominator to both fractions" (OED2).

Common divisor was used in 1674 by Samuel Jeake in Arithmetick, published in 1696: "Commensurable, called also Symmetral, is when the given Numbers have a Common Divisor" (OED2).

Least common multiple is found in 1823 in J. Mitchell, Dict. Math. & Phys. Sci.: "To find the least common Multiple of several Numbers" (OED2).

Least common denominator is found in 1844 in Introduction to The national arithmetic, on the inductive system by Benjamin Greenleaf: "RULE. - Reduce the fractions, if necessary, to the least common denominator. Then find the greatest common divisor of the numerators, which, written over the least common denominator, will give the greatest common divisor required" [University of Michigan Digital Library].

Lowest common denominator appears in 1854 in Arithmetic, oral and written, practically applied by means of suggestive questions by Thomas H. Palmer: "Suggestive Questions. - Are all the underlined factors to be found in the denominators of the fractions marked a and b? Should they be omitted, then, in finding the lowest common denominator? What is the product of the factors that are not underlined? (80·3·5.) Has this product every factor contained in all the given denominators? Will it form their common denominator, then? Does it contain no more factors than they do? Will it form, then, their lowest common denominator?" [University of Michigan Digital Library].

Least common dividend appears in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

Lowest common multiple appears in 1873 in Test examples in algebra, especially adapted for use in connection with Olney's School, or University algebra by Edward Olney [University of Michigan Digital Library].

The term LEBESGUE INTEGRAL was coined by William Henry Young (1863-1942), according to Hardy in his obituary of Young, which is quoted in Kramer, p. 643.

In 1912, James Pierpont writes in Lectures on the Theory of Functions of Real Variables, volume II:

The author has chosen a definition which occurred to him many years ago, and which to him seems far more natural. In volume I it is shown that if the metric field be divided into a finite number of metric sets delta[1] ...What then is more natural than to ask what will happen if the cells delta[1], delta [2],... are infinite instead of finite in number? Form this apparently trivial question results a theory of L-integrals which contains the Lebesgue integrals as a special case, and which, furthermore, has the great advantage that not only is the relation of the new integrals to the ordinary or Riemannian integrals perfectly obvious, but also the form of reasoning employed in Riemann's theory may be taken over to develop the properties of the new integrals.

The citation above, which was provided by James A. Landau, is from the preface, page iv.

Lebesgue integral appears in the title of N. J. Lennes, "Note on Lebesgue and Pierpont integral," Amer. Math. Soc. Bull. (1913). Landau believes it is likely that what Pierpont calls a Lebesgue integral is what Lennes calls a Pierpont integral.

Other forms of this term appear in these titles:

Ch. J. de la Vallée Poussin, "Intégrales de Lebesgue. Fonctions d'ensemble. Classes de Baire" (Paris, 1916).

Ch. J. de la Vallée Poussin, "Sur l'intégrale de Lebesgue," Transactions Amer. Math. Soc. 16 (1916).

F. Riesz, "Sur l'intégrale de Lebesgue," Acta. Math. 42 (1919-1920).

A. Denjoy, "Une extension de l'intégrale de M. Lebesgue," Comptes Rendus Acad. Sci. Paris 154 (1912).

Burton H. Camp, "Lebesgue Integrals Containing a Parameter, with Applications," Transactions of the American Mathematical Society, 15 (Jan., 1914).

The term may occur in W. H. Young, "On a new method in the theory of integration," Proc. London Math. Soc. 9 (1910). [James A. Landau]

LEG for a side of a right triangle other than the hypotenuse is found in English in 1659 in Joseph Moxon, Globes (OED2).

Leg is used in the sense of one of the congruent sides of an isosceles triangle in 1702 Ralphson's Math. Dict.: "Isosceles Triangle is a Triangle that has two equal Legs" (OED2).

LEMMA appears in English in the 1570 translation by Sir Henry Billingsley of Euclid's Elements (OED2). [The plural of lemma can be written lemmas or lemmata.]

LEMNISCATE. Jacob Bernoulli named this curve the lemniscus in Acta Eruditorum in 1694. He wrote, "...formam refert jacentis notae octonarii [infinity symbol], seu complicitae in nodum fasciae, sive lemnisci" (Smith vol. 2, page 329).

LEMOINE POINT. See symmedian point.

LEPTOKURTIC (and platykurtic and mesokurtic) were introduced by Karl Pearson, who wrote in Biometrika (1905) IV. 173: "Given two frequency distributions which have the same variability as measured by the standard deviation, they may be relatively more or less flat-topped than the normal curve. If more flat-topped I term them platykurtic, if less flat-topped leptokurtic, and if equally flat-topped mesokurtic" (OED2).

L'HOSPITAL'S RULE. In Differential and Integral Calculus (1902) by Virgil Snyder and John Irwin Hutchinson, the procedure is termed "evaluation by differentiation." The same term is used in Elementary Textbook of the Calculus (1912) by the same authors.

de l'Hosptial's theorem on indeterminate forms is found in approximately 1904 in the E. R. Hedrick translation of volume I of A Course in Mathematical Analysis by Edouard Goursat. The translation carries the date 1904, although a footnote references a work dated 1905 [James A. Landau].

In Differential and Integral Calculus (1908) by Daniel A. Murray, the procedure is shown but is not named.

James A. Landau has found in J. W. Mellor, Higher Mathematics for Students of Chemistry and Physics, 4th ed. (1912), the sentence, "This is the so-called rule of l'Hopital."

The rule is named for Guillaume-Francois-Antoine de l'Hospital (1661-1704), although the rule was discovered by Johann Bernoulli. The rule and its proof appear in a 1694 letter from him to l'Hospital.

The family later changed the spelling of the name to l'Hôpital.

LIE GROUP appears in L. Autonne, "Sur une application des groupes de M. Lie.," C. R. CXII. 570-573 (1891).

Lie group appears in English in H. B. Newson, "A new theory of collineations und their Lie groups," American J. 24, 109-172.

Lie group also appears in English in S. D. Zeldin, "On the quadratic ternary partial differential equation admitting Lie-groups of orders four and five.," American M. S. Bull. (1923).

LIKELIHOOD. The term was first used in its modern sense in R. A. Fisher's "On the 'Probable Error' of a Coefficient of Correlation Deduced from a Small Sample," Metron, 1, (1921), 3-32.

Formerly, likelihood was a synonym for probability, as it still is in everyday English. (See the entry on maximum likelihood and the passage quoted there for Fisher's attempt to distinguish the two. In 1921 Fisher referred to the value that maximizes the likelihood as "the optimum.")

Likelihood first appeared in a Bayesian context in H. Jeffreys's Theory of Probability (1939) [John Aldrich, based on David (2001)].

LIKELIHOOD PRINCIPLE. This expression burst into print in 1962, appearing in "Likelihood Inference and Time Series" by G. A. Barnard, G. M. Jenkins, C. B. Winsten (Journal of the Royal Statistical Society A, 125, 321-372), "On the Foundations of Statistical Inference" by A. Birnbaum (Journal of the American Statistical Association, 57, 269-306), and L. J. Savage et al, (1962) The Foundations of Statistical Inference. It must have been current for some time because the Savage volume records a conference in 1959; the term appears in Savage's contribution so the expression may have been his coining.

The principle (without a name) can be traced back to R. A. Fisher's writings of the 1920s though its clearest earlier manifestation is in Barnard's 1949 "Statistical Inference" (Journal of the Royal Statistical Society. Series B, 11, 115-149). On these earlier outings the principle attracted little attention.

The LIKELIHOOD RATIO figured in the test theory of J. Neyman and E. S. Pearson from the beginning, "On the Use of Certain Test Criteria for Purposes of Statistical Inference, Part I" Biometrika, (1928), 20A, 175-240. They usually referred to it as the likelihood although the phrase "likelihood ratio" appears incidentally in their "Problem of k Samples," Bulletin Académie Polonaise des Sciences et Lettres, A, (1931) 460-481. This phrase was more often used by others writing about Neyman and Pearson's work, e.g. Brandner "A Test of the Significance of the Difference of the Correlation Coefficients in Normal Bivariate Samples," Biometrika, 25, (1933), 102-109.

The standing of "likelihood ratio" was confirmed by S. S. Wilks's "The Large-Sample Distribution of the Likelihood Ratio for Testing Composite Hypotheses," Annals of Mathematical Statistics, 9, (1938), 60-620 [John Aldrich, based on David (2001)].

The term LIMAÇON was coined in 1650 by Gilles Persone de Roberval (1602-1675) (Encyclopaedia Britannica, article: "Geometry"). It is sometimes called Pascal's limaçon, for Étienne Pascal (1588?-1651), the first person to study it. Boyer (page 395) writes that "on the suggestion of Roberval" the curve is named for Pascal.

LIMIT. Gregory of St. Vincent (1584-1667) used terminus to mean the limit of a progression, according to Carl B. Boyer in The History of the Calculus and its Conceptual Development.

Limit was used by Isaac Newton: "Quibus Terminis, sive Limitibus respondent semicirculi Limites, sive Termini." This citation is from a. 1727, Opuscula I (OED2).

Gregory used terminatio for limit of a series (DSB).

In 1922 in Introduction to the Calculus, William F. Osgood writes: "Some writers find it convenient to use the expression 'a variable approaches a limit' to include the case that the variable becomes infinite. We shall not adopt this mode of expression, but shall understand the words 'approaches a limit' in their strict sense."

LIMIT POINT. Cantor used Häufungspunkt (accumulation point) in an 1872 paper "Über die Ausdehnung eines Satzes der Theorie der trigonometrischen Reihen," which appeared in Mathematische Annalen, Band 5, pp. 122-132 [Roger Cooke].

Limit point is found in English in "A Simple Proof of the Fundamental Cauchy-Goursat Theorem," Transactions of the American Mathematical Society, Vol. 1, No. 4. (Oct., 1900), pp. 499-506.

Point of accumulation appears in English in in E. W. Chittenden, "On the classification of points of accumulation in the theory of abstract sets," Bulletin A. M. S. 32 (1926).

LINE FUNCTION was the term used for functional by Vito Volterra (1860-1940), according to the DSB.

LINE GRAPH is dated ca. 1924 in MWCD10.

The term LINE INTEGRAL was used in in 1873 by James Clerk Maxwell in a Treatise on Electricity and Magnetism in the phrase "Line-Integral of Electric Force, or Electromotive Force along an Art of a Curve" (OED2).

The term LINE OF EQUAL POWER was coined by Steiner.

LINEAR ALGEBRA. The DSB seems to imply that the term algebra linearia is used by Rafael Bombelli (1526-1572) in Book IV of his Algebra to refer to the application of geometrical methods to algebra.

Linear associative algebra appears in 1870 as the title of a paper, "Linear Associative Algebra" by Benjamin Peirce. The paper was read before the National Academy of Sciences in Washington [James A. Landau].

Linear algebra occurs in 1875 in the title, "On the uses and transformations of linear algebra" by Benjamin Peirce, published in American Acad. Proc. 2 [James A. Landau].

LINEAR COMBINATION occurs in "On the Extension of Delaunay's Method in the Lunar Theory to the General Problem of Planetary Motion," G. W. Hill, Transactions of the American Mathematical Society, Vol. 1, No. 2. (Apr., 1900).

LINEAR DEPENDENCE appears in the 1907 edition of Introduction to Higher Algebra by Maxime Bôcher [James A. Landau].

LINEAR DIFFERENTIAL EQUATION appears in J. L. Lagrange, "Recherches sur les suites récurrentes don't les termed varient de plusieurs manières différentes, ou sur l'intégration des équations linéaires aux différences finies et partielles; et sur l'usage de ces équations dans la théorie des hasards," Nouv. Mém. Acad. R. Sci. Berlin 6 (1777) [James A. Landau].

LINEAR EQUATION appears in English the 1816 translation of Lacroix's Differential and Integral Calculus (OED2).

LINEAR FUNCTION. An 1857 English language translation of Gauss's Theoria motus has "...it is possible to assign a linear function alpha P + beta Q + gamma R + delta S + etc" and "And when it can be assumed that these are so small that their squares and products may be neglected, the corresponding changes, produced in the computed geocentric places of a heavenly body, can be obtained by means of the differential formulas given in the Second Section of the First Book. The computed places, therefore, which we obtain from the corrected elements, will be expressed by linear functions of the corrections of the elements, and their comparison with the observed places according to the principles before explained, will lead to the determination of the most probable values."

Linear function is found in English in volume I of An Elementary Treatise on Curves, Functions and Forces by Benjamin Peirce. The title page of this work has 1852; the copyright date on the reverse of the title page is 1841 [James A. Landau].

Linear function is also found in 1850 in The elements of analytical geometry by John Radford Young [University of Michigan Digital Library].

LINEAR INDEPENDENCE is found in 1901 in Linear Groups, with an exposition of the Galois field theory by Leonard Eugene Dickson [James A. Landau].

LINEAR OPERATOR. Linear operation appears in 1837 in Robert Murphy, "First Memoir on the Theory of Analytic Operations," Philosophical Transactions of the Royal Society of London, 127, 179-210. Murphy used "linear operation" in the sense of the modern term "linear operator" [Robert Emmett Bradley].

LINEAR PRODUCT. This term was used by Hermann Grassman in his Ausdehnungslehre (1844).

LINEAR PROGRAMMING. See programming.

LINEAR TRANSFORMATION appears in 1845 in Arthur Cayley, "On the Theory of Linear Transformations," Cambridge Math. J. 4, 193-209 [Romulo Lins].

LINEARLY DEPENDENT was used in 1893 in "A Doubly Infinite System of Simple Groups" by Eliakim Hastings Moore. The paper was read in 1893 and published in 1896 [James A. Landau].

LINEARLY INDEPENDENT is found in 1847 in "On the Theory of Involution in Geometry" by Arthur Cayley in the Cambridge and Dublin Mathematical Journal [University of Michigan Historical Math Collection].

The term LITUUS (Latin for the curved wand used by the Roman pagan priests known as augurs) was chosen by Roger Cotes (1682-1716) for the locus of a point moving such that the area of a circular sector remains constant, and it appears in his Harmonia Mensurarum, published posthumously in Cambridge, 1722 [Julio González Cabillón].

The term LOCAL PROBABILITY is due to Morgan W. Crofton (1826-1915) (Cajori 1919, page 379).

The term appears in the title of his 1868 paper, "On the Theory of Local Probability, applied to Straight Lines drawn at random in a plane; the methods used being also extended to the proof of certain new Theorems in the Integral Calculus," Trans. of London.

LOCUS is a Latin translation of the Greek word topos. Both words mean "place."

According to Pappus, Aristaeus (c. 370 to c. 300 BC) wrote a work called On Solid Loci (Topwn sterewn).

Pappus also mentions Euclid in connection with locus problems.

Apollonius mentioned the "locus for three and four lines" ("...ton epi treis kai tessaras grammas topon...") in the extant letter opening Book I of the Conica. Apollonius said in the first book that the third book contains propositions (III.54-56) relevant to the 3 and 4 line locus problem (and, since these propositions are new, Apollonius claimed Euclid could not have solved the problem completely--a claim that caused Pappus to call Apollonius a braggard (alazonikos). In Book III itself there is no mention of the locus problem [Michael N. Fried].

Locus appears in the title of a 1636 paper by Fermat, "Ad Locos Planos et Solidos Isagoge" ("Introduction to Plane and Solid Loci").

In English, locus is found in 1727-41 in Chambers Cyclopedia: "A locus is a line, any point of which may equally solve an indeterminate problem. ... All loci of the second degree are conic sections" (OED2).

Locus geometricus is an entry in the 1771 Encyclopaedia Britannica.

LOGARITHM. Before he coined the term logarithmus Napier called these numbers numeri artificiales, and the arguments of his logarithmic function were numeri naturales [Heinz Lueneburg].

Logarithmus was coined (in Latin) by John Napier (1550-1617) and appears in 1614 in his Mirifici Logarithmorum Canonis descriptio.

According to the OED2, "Napier does not explain his view of the literal meaning of logarithmus. It is commonly taken to mean 'ratio-number', and as thus interpreted it is not inappropriate, though its fitness is not obvious without explanation. Perhaps, however, Napier may have used logos merely in the sense of 'reckoning', 'calculation.'"

According to Briggs in Arithmetica logarithmica (1624), Napier used the term because logarithms exhibit numbers which preserve always the same ratio to one another.

According to Hacker (1970):

It undoubtedly was Napier's observation that logarithms of proportionals are "equidifferent" that led him to coin the name "logarithm," which occurs throughout the Descriptio but only in the title of the Constructio, which clearly was drafted first although published later. The many-meaning Greek word logos is therefore used in the sense of ratio. But there is an amusing play on words to which we might call attention since it does not seem to have been noticed. It is interesting that the Greeks also employed logos to distinguish reckoning, or that is to say mere calculation, from arithmos, which was generally reserved by them to indicate the use of number in the higher context of what today we call the theory of numbers. Napier's "logarithms" have indeed served both purposes.

Logarithm appears in English in a letter of March 10, 1615, from Henry Briggs to James Ussher: "Napper, Lord of Markinston, hath set my Head and Hands a Work, with his new and admirable Logarithms. I hope to see him this summer, if it please God, for I never saw a book which pleased me better or made me more wonder."

Logarithm appears in English in 1616 in E. Wright's English translation of the Descriptio: "This new course of Logarithmes doth cleane take away all the difficultye that heretofore hath beene in mathematicall calculations. [...] The Logarithmes of proportionall numbers are equally differing."

In the Constructio, which was drafted before the Descriptio, the term "artificial number" is used, rather than "logarithm." Napier adopted the term logarithmus before his discovery was announced.

Jobst Bürgi called the logarithm Die Rothe Zahl since the logarithms were printed in red and the antilogarithms in black in his Progress Tabulen, published in 1620 but conceived some years earlier (Smith vol. 2, page 523).

[Older English-language dictionaries pronounce logarithm with an unvoiced th, as in thick and arithmetic.]

LOGARITHMIC CURVE. Huygens proposed the terms hemihyperbola and linea logarithmica sive Neperiana.

Christiaan Huygens used logarithmica when he wrote in Latin and logarithmique when he wrote in French.

Johann Bernoulli used a phrase which is translated "logarithmic curve" in 1691/92 in Opera omnia (Struik, page 328).

Logarithmic curve is an entry in the 1771 edition of the Encyclopaedia Britannica [James A. Landau].

LOGARITHMIC FUNCTION. Lacroix used fonctions logarithmiques in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800).

Logarithmic function appears in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "Thus, a^x, a log x, sin x, &c., are transcendental functions: the first is an exponential function, the second a logarithmic function, and the third a circular function" [James A. Landau]

The term LOGARITHMIC POTENTIAL was coined by Carl Gottfried Neumann (1832-1925) (DSB).

The term LOGARITHMIC SPIRAL was introduced by Pierre Varignon (1654-1722) in a paper he presented to the Paris Academy in 1704 and published in 1722 (Cajori 1919, page 156).

Another term for this curve is equiangular spiral.

Jakob Bernoulli called the curve spira mirabilis (marvelous spiral).

LOGIC. According to the University of St. Andrews website, the term logic was introduced by Xenocrates of Chalcedon (396 BC - 314 BC). Aristotle's name for logic was analytics.

The term LOGISTIC CURVE is attributed to Edward Wright (ca. 1558-1615) (Thompson 1992, page 145). Wright was apparently referring to the logarithmic curve and was not using the term in the modern sense.

Pierre Francois Verhulst (1804-1849) introduced the term logistique as applied to the sigmoid curve [Julio González Cabillón]. Bonnie Shulman believes that "logistic," as coined by Verhulst, refers to the "log-like" qualities of the curve.

LOGNORMAL. Logarithmic-normal was used in 1919 by S. Nydell in "The Mean Errors of the Characteristics in Logarithmic-Normal Distributions," Skandinavisk Aktuarietidskrift, 2, 134-144 (David, 1995).

Lognormal was used by J. H. Gaddun in Nature on Oct. 20, 1945: "It is proposed to call the distribution of x 'lognormal' when the distribution of log x is normal" (OED2).

LONG DIVISION is found in 1827 in A Course of Mathematics by Charles Hutton: "Divide by the whole divisor at once, after the manner of Long division" (OED2).

LOSS and LOSS FUNCTION in statistical decision theory. In the paper establishing the subject ("Contributions to the Theory of Statistical Estimation and Testing Hypotheses," Annals of Mathematical Statistics, 10, 299-326) Wald referred to "loss" but used "weight function" for the (modern) loss function. He continued to use weight function, for instance in his book Statistical Decision Functions (1950), while others adopted loss function. Arrow, Blackwell & Girshick’s "Bayes and Minimax Solutions of Sequential Decision Problems" (Econometrica, 17, (1949) 213-244) wrote L rather than W for the function and called it the loss function. A paper by Hodges & Lehmann ("Some Problems in Minimax Point Estimation," Annals of Mathematical Statistics, 21, (1950), 182-197) used loss function more freely but retained Wald’s W. [John Aldrich, based on David (2001) and JSTOR]

The term LOWER SEMICONTINUITY was used by René-Louis Baire (1874-1932), according to Kramer (p. 575), who implies he coined the term.

The phrase LOWEST TERMS appears in about 1675 in Cocker's Arithmetic, written by Edward Cocker (1631-1676): "Reduce a fraction to its lowest terms at the first Work" (OED2). (There is some dispute about whether Cocker in fact was the author of the work.)

LOXODROME. Pedro Nunez (Pedro Nonius) (1492-1577) announced his discovery and analysis of the curve in De arte navigandi. He called the curve the rumbus (Catholic Encyclopedia).

The term loxodrome is due to Willebrord Snell van Roijen (1581-1626) and was coined in 1624 (Smith and DSB, article: "Nunez Salaciense).

LUCAS-LEHMER TEST occurs in the title, "The Lucas-Lehmer test for Mersenne numbers," by S. Kravitz in the Fibonacci Quarterly 8, 1-3 (1970).

The term Lucas's test was used in 1932 by A. E. Western in "On Lucas's and Pepin's tests for the primeness of Mersenne's numbers," J. London Math. Soc. 7 (1932), and in 1935 by D. H. Lehmer in "On Lucas's test for the primality of Mersenne's numbers," J. London Math. Soc. 10 (1935).

The term LUCAS PSEUDOPRIME occurs in the title "Lucas Pseudoprimes" by Robert Baillie and Samuel S. Wagstaff Jr. in Math. Comput. 35, 1391-1417 (1980): "If n is composite, but (1) still holds, then we call n a Lucas pseudoprime with parameters P and Q ..." [Paul Pollack].

LUDOLPHIAN NUMBER. The number 3.14159... was often called the Ludolphische Zahl in Germany, for Ludolph van Ceulen (1540-1610).

In English, Ludolphian number is found in 1886 in G. S. Carr, Synopsis Pure & Applied Math (OED2).

In English, Ludolph's number is found in 1894 in History of Mathematics by Florian Cajori (OED2).

LUNE. Lunula appears in A Geometricall Practise named Pantometria by Thomas Digges (1571): "Ye last figure called a Lunula" (OED2).

Lune appears in English in 1704 in Lexicon technicum, or an universal English dictionary of arts and sciences by John Harris (OED2).

MACLAURIN'S SERIES. Maclaurin's theorem appears in 1820 in Collection of Examples of the Applications of the Differential and Integral Calculus by G. Peacock [Mark Dunn].

In 1849, An Introduction to the Differential and Integral Calculus, 2nd ed., by James Thomson has: "A particular case of this formula is commonly called Maclaurin's theorem, because it was first made generally known by that writer. It had been given previously, however, by Stirling, another Scotch mathematician; and therefore, if a particular case of Taylor's general theorem should be named after any other mathematician, this ought to be called Stirling's theorem." Thomson subsequently uses the term Stirling's theorem throughout the book.

McLaurin's formula is found in English in 1855 in Elements of the differential and integral calculus by Albert Ensign Church [University of Michigan Digital Library].

Les séries de Taylor et de Maclaurin is found in 1870 in J. Bourget, "Note sur les séries de Taylor et de Maclaurin," Nouv. Ann.

Maclaurin's series is found in English in 1831 in the second edition of Elements of the Differential Calculus (1836) by John Radford Young: "All that is meant is, that the function in particular states may fail to be developable according to Taylor's series, and under particular forms it may fail to be developable according to Maclaurin's series; so that, in fact, these theorems fail to give the true development only when that development is impossible" [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:

This latter station was the more agreeable to me, as I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so unentertaining that I was induc'd to amuse myself with making magic squares or circles, or any thing to avoid weariness; and I conceiv'd my becoming a member would enlarge my power of doing good.

Franklin also used the term in a letter in which he wrote, "I make no question, but you will readily allow the square of 16 to be the most magically magical of any magic square ever made by any magician" (Cajori 1919, page 170).

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

MANIFOLD was introduced as Mannigfaltigkeit by Bernhard Riemann (1826-1866) in Grundlagen füreine Allgemeine Theorie der Functionen, published (posthumously) in 1867 [Mark Dunn].

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).

MANY-VALUED is found in 1893 in J. Harkness and F. Morley, Treatise on the Theory of Functions 36 (OED Online).

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].

The term is found in English in 1938 in American Mathematical Monthly, vol. 45, p. 410 [Mark Dunn, JSTOR].

MARKOV PROCESS occurs in 1938 in Trans. American Math. Soc. vol. 44, p. 102 [Mark Dunn, JSTOR].

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].

MARTINGALE. The original sense is given in the OED: "a system in gambling which consists in doubling the stake when losing in the hope of eventually recouping oneself." The oldest quotation is from 1815 but the nicest is from 1854: Thackeray in The Newcomes I. 266 "You have not played as yet? Do not do so; above all avoid a martingale if you do."

J. Venn in his Logic of Chance (1888) wrote that the possibility that "by mere persistency [the martingale player] may accumulate any sum of money he pleases, in apparent defiance of all that is meant by luck" has been "a source of perplexity to persons of considerable acutenesss."

There was an early discussion by C. Babbage ("An Examination of Some Questions Connected with Games of Chance" Trans. Royal Soc. Edinburgh, 9 (1821) 153-177).

The martingale of modern probability theory is a mathematical model of a fair game and so is different from the martingale as a gambling system. The connection is a theorem that the martingale system will not change a fair game into an unfair game--an old martingale is a new martingale. J. Ville's Étude Critique de la Notion de Collectif (1939) begins by discussing old martingale in the context of von Mises's requirement that with a random sequence a successful gambling system is impossible and goes on to define a (new) martingale as "un jeu équitable."

J. L. Doob's Stochastic Processes (1954) made the martingale an important chapter of probability theory. In 1940 Doob wrote about "chance variables with the property E" ("Regularity Properties of Certain Families of Chance Variables," Transactions of the American Mathematical Society, 47, 455-486.) [This entry was contributed by John Aldrich.]

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

The first citation for maths 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.

Maths is found in Wireless World in 1917: "Extremely 'rusty' in 'maths'" (OED2). It is unclear whether a period is used to indicate an abbreviation or the end of the sentence or both.

The earliest use of math in the OED2 in which it is clear that no period is intended is in 1924 in P. Marks, Plastic Age: "I'm talking about the copying of math problems and the using of trots." However, there are a number of earlier uses in which the word ends a sentence, so that it is unclear whether the writer would have used a period to indicate an abbreviation.

The earliest use of maths in the OED2 in which a period is clearly absent is in the Times of Sept. 8, 1959: "Royal Australian Air Force. Education Officers required with Majors in Maths or Physics."

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 likelihood.

MEAN occurs in English in the sense of a geometric mean in a Middle English manuscript of circa 1450 known as The Art of Numbering: "Lede the rote of o quadrat into the rote of the oþer quadrat, and þan wolle þe meene shew" [Mark Dunn].

In 1571, A geometrical practise named Pantometria by Thomas Digges (1546?-1595) has: "When foure magnitudes are...in continual proportion, the first and the fourth are the extremes, and the second and thirde the meanes" (OED2).

Mean is found in 1755 in Thomas Simpson, "An ATTEMPT to shew the Advantage, arising by Taking the Mean of a Number of Observations, in practical Astronomy," Philosophical Transactions of the Royal Society of London.

MEAN CURVATURE appears in 1840 in J. R. Young, Mathematical Dissertations (1841). (The preface is dated Nov. 25, 1840.) According to James A. Landau, who provided this citation, Young specialized in introducing recent French developments in geometry (particularly those of Monge) to English-speaking readers, so that it is possible that this is the first appearance of "mean curvature" in English.

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

Mean error is found in 1853 in A dictionary of arts, manufactures, and mines; containing a clear exposition of their principles and practice by Andrew Ure [University of Michigan Digital Library].

Mean error is found in English in an 1857 translation of Gauss's Theoria motus: Consequently, if we desire the greatest accuracy, it will be necessary to compute the geocentric place from the elements for the same time, and afterwards to free it from the mean error A, in order that the most accurate position may be obtained. But it will in general be abundantly sufficient if the mean error is referred to the observation nearest to the mean time" [University of Michigan Digital Library].

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."

MEAN SQUARE is found in 1845 Encycl. Metrop. (OED2).

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).

In Differential and Integral Calculus by Virgil Snyder and J. L. Hutchinson (1902) the theorem is called "the theorem of mean value."

In Advanced Calculus by Edwin Bidwell Wilson (1912), the theorem is called the "theorem of the mean."

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. The following articles feature some uses of the term measure.

Giuseppe Vitali, Sul problema della misura dei gruppi di punti di una retta Bologna: Tip. Gamberini e Parmeggiani (1905).

"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."

Nikolai Luzin, "Sur les propriétês des fonctions mesurables," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 154 (1912).

Waclaw Sierpinski, "Sur quelques problèmes qui impliquent des fonctions non-mesurables," Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences 164 (1917).

Henri Lebesgue, "Remarques sur les théories de la mesure et de l'intégration," Annales Scientifiques de l'Ecole Normale Supérieure (3) 35, pp 191-250 (1918) [James A. Landau].

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].

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). Valeur médiane was used by Antoine A. Cournot in 1843 in Exposition de la Théorie des Chances et des Probabilités (David, 1998).

Median was used in English 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) is found in 1876 in Lessons in elementary mechanics. Introductory to the study of physical science by Sir Philip Magnus, with emendations and introduction by Prof. DeVolson Wood: "In the same way it may be shown that the centre of gravity of the triangle is in the median CE (fig. 109). Hence the centre of gravity of the triangle is at G, where the two medians intersect" [University of Michigan Digital Library].

MEDIATE is found in Dorothy Wrinch, "On Mediate Cardinals," American Journal of Mathematics 45 (1923) [James A. Landau].

MENTAL ARITHMETIC is found in 1766 in H. Brooke, Fool of Quality, vol. I., p. 260: "I cast up, in a pleasing kind of mental arithmetic, how much my weekly twenty guineas would amount to at the year's end" [Mark Dunn].

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 the title "Mersenne's numbers" by W. W. Rouse Ball in Messenger of Mathematics in 1891.

Mersenne number is found in English in the 1911 Encyclopaedia Britannica: "Similar difficulties are encountered when we examine Mersenne's numbers, which are those of the form 2^p - 1, with p a prime; the known cases for which a Mersenne number is prime correspond to p = 2, 3, 5, 7, 13, 17, 19, 31, 61" (OED2).

Mersenne prime is found in English in 1943 in American Math. Monthly, vol. 50, p. 29 [Mark Dunn, JSTOR].

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."

The word was first used in its modern sense by David Hilbert (1862-1943) in a 1922 lecture and it appears, as metamathematik, in 1923 in Math. Ann. LXXXVIII. 153) [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.

Vincentio used the term in 1647, according to A Concise History of Mathematics by Dirk J. Struik, third edition.

Method of exhaustions appears in English in 1685 in Treat. Algebra by John Wallis: "It will be necessary to premise somewhat concerning (what is wont to be called) the Method of Exhaustions" (OED2).

The term METHOD OF LEAST SQUARES was coined by Adrien Marie Legendre (1752-1833), appearing in Sur la Méthode des moindres quarrés [On the method of least squares], the title of an appendix to Nouvelles méthodes pour la détermination des orbites des comètes (1805). The appendix is dated March 6, 1805 [James A. Landau].

"Minimum" and "small" were the early English translations of moindres (David, 1995).

Method of least squares occurs in English in 1825 in the title "On the Method of Least Squares" by J. Ivory in Philosophical Magazine, 65, 3-10.

The term METRIC SPACE is due to Felix Hausdorff (1869-1942).

Metrischer raum is found in Grundzüge der Mengenlehre (1914).

Metric space is found in English in V. W. Niemytski, "On the third axiom of metric space," Bulletin A. M. S. (1926).

METRIC SYSTEM. Noah Webster's 1806 dictionary has the heading "New French Weights and Measures."

In 1821 John Quincy Adams used the terms French system and French metrology.

Webster's dictionary of 1828 refers to French measure.

Decimal system appears in January 1844 in The Southern quarterly review: "These units, multiplied or divided by ten, ad infinitum, formed the beautiful decimal system of the French, which surpasses all others."

In May 1854, Debow's review, Agricultural, commercial, industrial progress and resources uses the terms the decimal system of measures, French metrical system, metrical-decimal system, and decimal-metrical system of France.

The term French decimal system is used in 1857 in Mathematical Dictionary and Cyclopedia of Mathematical Science.

Metric system appears in the title Davies' Metric System, the date of which is unknown but it is advertised in a book published in 1851.

Metric system appears in the (London) Times of Sept. 10, 1862 (OED Online).

Metric system is found in English on July 11, 1863, in The Living Age: "The French people themselves did not seem to admire the metric system at all in the commencement, and it took a long time before it found favor, particularly with the lower classes."

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 found in 1871 in Chemistry, general, medical, and pharmaceutical, including the chemistry of the U.S. pharmacopoeia by John Attfield: "The Metric Ton of 1000 Kilo-grammes = 19 cwt. 2 qrs. 20 lbs. 10 ozs" [University of Michigan Digital Library Project].

Micron (one millionth of a meter) was coined by Johann Benedict Listing (1808-1882), according to Breitenberger (1999). The OED2 shows a use of the word in French in 1880 in Procès-Verbaux des Séances du Comité Internat. des Poids et Mesures 1879.

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, BILLION, etc. The following is taken from Smith (vol. 2, pages 80-86):

One of the most striking features of ancient arithmetic is the rarity of large numbers. There are exceptions, as in some of the Hindu traditions of Buddha's skill with numbers, in the records on some of the Babylonian tablets, and in the Sand Reckoner of Archimedes with its number system extending to 10⁶³, but these are all cases in which the élite of the mathematical world were concerned; the people, and indeed the substantial mathematicians in most cases, had little need for or interest in numbers of any considerable size.

The word "million," for example, is not found before the 13th century, and seems to have come into use in England even later. William Langland (c. 1334-c. 1400), in Piers Plowman, says,

Coueyte not his goodes
For millions of moneye,

but Maximus Planudes (c. 1340) seems to have been among the first of the mathematicians to use the word. By the 15th century it was known to the Italian arithmeticians, for Ghaligai (1521; 1552 ed., fol. 3) relates that "Maestro Paulo da Pisa" read the seventh order as millions. It first appeared in a printed work in the Treviso arithmetic of 1478. Thereafter it found place in the works of most of the important popular Italian writers, such as Borghi (1484), Pellos (1492), and Pacioli (1494), but outside of Italy and France it was for a long time used only sparingly. Thus, Gemma Frisius (1540) used "thousand thousand" in his Latin editions, which were published in the North, while in the Italian translation (1567) the word millioni appears. Similarly, Clavius carried his German ideas along with him when he went to Rome, and when (1583) he wished to speak of a thousand thousand he almost apologized for using "million," referring to it as an Italian form which needed some explanation.

In Spain the word cuento was early used for 10⁶, the word million being reserved for 10¹². When the latter word was adopted by mathematicians, it was slow in coming into general use.

France early took the word "million" from Italy, as when Chuquet (14848) used it, being followed by De la Roche (1520), after which it became fairly common.

The conservative Latin writers of the 16th century were very slow in adopting the word. Even Tonstall (1522), who followed such eminent Italian writers as Pacioli, did not commonly use it. He seems to have been influenced by the fact that the Romans had no use for large numbers; or by the fact that, for common purposes, it sufficed to say "thousand thousand" as had been done for many generations. He simply mentions the word as a piece of foreign slang to be avoided. Other Latin writers were content to say "thousand thousand."

The German writers were equally slow in abandoning "thousand thousand" for "million," most of the writers of the 16th century preferring the older form. The Dutch were even more conservative, continuing the old form later than the writers in the neighboring countries. Indeed, for the ordinary needs of business in the 16th century, the word "million" was a luxury rather than a necessity.

England adopted the Italian word more readily than the other countries, probably owing to the influence of Recorde (c. 1542). It is interesting to see that Poland was also among the first to recognize its value, the word appearing in the arithmetic of Klos in 1538.

Until the World War of 1914-1918 taught the world to think in billions there was not much need for number names beyond millions. Numbers could be expressed in figures, and an astronomer could write a number like 9.15 · 10⁷, or 2.5 · 10²⁰, without caring anything about the name. Because of this fact there was no uniformity in the use of the word "billion." It meant a thousand million (10⁹) in the United States and a million million (10¹²) in England, while France commonly used milliard for 10⁹, with billion as an alternative term.

Historically the billion first appears as 10¹², as the English use the term. It is found in this sense in Chuquet's number scheme (1484), and this scheme was used by De la Roche (1520), who simply copied parts of Chuquet's unpublished manuscript, but it was not common in France at this time, and it was not until the latter part of the 17th century that it found place in Germany. Although Italy had been the first country to make use of the word "million," it was slow in adopting the word "billion." Even in the 1592 edition of Tartaglia's arithmetic the word does not appear. Cataldi (1602) was the first Italian writer of any prominence to use the term, but he suggested it as a curiosity rather than a word of practical value. About the same time the term appeared in Holland, but it was not often recognized by writers there or elsewhere until the 18th century, and even then it was not used outside the schools. Even as good an arithmetician as Guido Grandi (1671-1742) preferred to speak of a million million rather than use the shorter term.

The French use of milliard, for 10⁹, with billion as an alternative, is relatively late. The word appears at least as early as the beginning of the 16th century as the equivalent both of 10⁹ and of 10¹², the latter being the billion of England today. By the 17th century, however, it was used in Holland to mean 10⁹, and no doubt it was about this time that the usage began to change in France.

As to the American usage, taking a billion to mean a thousand million and running the subsequent names by thousands, it should be said that this is due in part to French influence after the Revolutionary War, although our earliest native American arithmetic, the Greenwood book of 1729, gave the billion as 10⁹, the trillion as 10¹², and so on. Names for large numbers were the fashion in early days, Pike's well-known arithmetic (1788), for example, proceeding to duodecillions before taking up addition.

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). (This is translated "many millions" in the Living Bible.)

Million was also used by Shakespeare a number of times.

The number 200,000,000 appears in the Living Bible in Rev. 9:16. It is translated as "two hundred thousand thousand" in the King James version (1611), "twice ten thousand times ten thousand" in Darby (1890) and RSV (1946), "two myriads of myriads" in Young's Literal Translation (1898), and "two hundred million" in the New International Version (1973).

Billion first occurs, with the meaning 10¹², in French in 1484 in Le Triparty en la Science des Nombres by Nicolas Chuquet (1445?-1500?). He used the words byllion, tryllion, quadrillion, quyllion, sixlion, septyllion, ottyllion, and nonyllion. A translation has: "The first dot indicates million, the second dot billion, the third dot trillion, the fourth dot quadrillion...and so on as far as one may wish to go."

The OED2 has:

The name [billion] appears not to have been adopted in Eng. before the end of the 17th c. .... Subsequently the application of the word was changed by French arithmeticians, figures being divided in numeration into groups of threes, instead of sixes, so that F. billion, trillion, denoted not the second and third powers of a million, but a thousand millions and a thousand thousand millions. In the 19th century, the U.S. adopted the French convention, but Britain retained the original and etymological use (to which France reverted in 1948). Since 1951 the U.S. value, a thousand millions, has been increasingly used in Britain, especially in technical writing and, more recently, in journalism; but the older sense "a million millions" is still common.]

Decillion occurs in English in 1847.

Centillionth, with an imprecise meaning, appears in English in 1852 in Tait's Magazine: "There existed not a centillionth of the blessing."

Centillion is found in English in 1863 in The Normal: or, Methods of Teaching the Common Branches, Orthoepy, Orthography, Grammar, Geography, Arithmetic and Elocution by Alfred Holbrook, which has the following:

Names of the periods. - 1st, Units. 2d, Thousands. 3d, Millions. 4th, Billions. 5th, Trillions. 6th, Quadrillions. 7th, Quintillions. 8th, Sextillions. 9th, Septillions. 10th, Octillions. 11th, Nonillions. 12th, Decillions. 13th, Undecillions. 14th, Duodecillions. 15th, Tridecillions. 16th, Quadrodecillions. 17th, Quindecillions. 18th, Sexdecillions. 19th, Septodecillions. 20th, Octodecillions. 21st, Nonodecillions. 22d, Vigintillions. 23d, Unvingintillions. 24th, Duo-vingintillions, etc. 32d, Trigintillions. 42d, Quadrogintillions. 52d, Quingintillions. 62d, Sexagintillions. 72d, Septuagintillions. 82d, Octogintillions. 92d, Ninogintillions. 102d, Centillions. 103d, Uncentillions. 104th, Duocentillions, etc. 202d, Duocentillions, etc. 1002d, Millillions, etc.

The term MINIMAL BASIS is due to Felix Klein, according to Harkness and Morley in A Treatise on the Theory of Functions.

MINIMAX (earlier meaning). In the sense of a saddle point of a surface or similar concept in higher dimensions, Poincaré wrote in 1899 in Méthodes Nouvelles de la Mécanique Céleste III. 246: "J'appelle minimax, à l'exemple des Anglais, un point pour lequel..."

Alan M. Hughes, Associate Editor of the OED, reports that, despite Poincare's comment, no earlier English usage has been traced.

Mark Dunn writes that the earliest English use appears to be in 1917 in Trans. American Math. Soc., vol. 18, p. 240. Most later examples of this meaning in English refer to this 1917 article as though it is the first use.

MINIMAX (later meaning). In 1928 J. von Neumann wrote in Mathematische Annalen C. 307 the heading "Beweis des Satzes Max Min = Min Max" (OED2).

Min-max is found in English in 1944 in J. Von Neumann & Morgenstern, Theory of Games: "A slightly more general form of this Min-Max problem arises in another question of mathematical economics" (OED2).

Minimax solution to a statistical decision problem appears in 1947 in Wald’s "Foundations of a General Theory of Sequential Decision Functions," Econometrica, 15, 279-313 but the concept had appeared in his 1939 paper under the guise of the "best estimate."

Minimax estimate appears in Hodges & Lehmann’s "Some Problems in Minimax Point Estimation", Annals of Mathematical Statistics, 21, (1950), 182-197 [John Aldrich, based on David (2001)].

Maximin is dated 1951 in MWCD10.

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.

MINUS SIGN. Negative sign appears in 1668 in T. Brancker, Introd. Algebra: "The Sign for Subtraction is - i.e. Minus, or the Negative Sign.

Minus sign occurs in 1851 in Scientific American, 6 Sept., p. 407 [Alan M. Hughes, Associate Editor of the OED].

Minus sign appears in 1856 in Outlines of Physical Geography by George William Fitch: "In the above Table, a minus sign (-) placed before a degree of lat. indicates that the lat. is south, and placed before a degree of long. denotes that is is east long."

Minus sign appears in 1889 in Elements of Algebra by G. A. Wentworth: "When an expression within a parenthesis is preceded by a minus sign, the parenthesis may be removed if the sign of every term within the parenthesis be changed."

MIXED NUMBER appears in English in 1542 in The Ground of Artes by Robert Recorde: "mixt numbers (that is whole numbers with fractions)" (OED2).

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

MOD and MODULO. The OED2 shows a use of mod. in 1854 in Cambr. & Dublin Math. Jrnl. IX. 85 and a use of mod in 1860 in Rep. Brit. Assoc. Adv. Sci. 1859.

Modulo appears in English in 1887 in American Journal of Math. vol. 10, p. 62 [Mark Dunn, JSTOR].

MODULO (non-technical sense). Modulo is being widely used by mathematicians in a related sense of "(a) taking into account (a particular consideration, aspect, etc.) (b) with respect to an equivalence defined by (some feature)." [This is the definition which will be given by the OED, according to Mark Dunn.]

In the spring of 1953, in a letter to Paul Halmos, Warren Ambrose of Princeton wrote: "[Nash] proceeded to announce that he had solved it, modulo details, and told Mackey he would like to talk about it at the Harvard colloquium." In this citation, modulo means "except for" or "without." This letter, which was critical of John Nash's attempt (later successful) to prove the Riemann Imbedding Theorem, is quoted in A Beautiful Mind by Sylvia Nasar [James A. Landau]

Carlos César de Araújo 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."

MODE was coined by Karl Pearson (1857-1936). He used the term in 1895 in "Skew Variation in Homogeneous Material," Philosophical Transactions of the Royal Society of London, Ser. A, 186, 343-414: "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 1878 in J. J. S. Smith, "On the modular curves," Rep. Brit. Ass.

The term MODULAR EQUATION was introduced by Jacobi [Encyclopaedia Britannica (1902), article "Infinitesimal Calculus"; Smith (1906)].

The term équations modulaires appears on January 12, 1828, in a letter written by Jacobi to Legendre [Emili Bifet].

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).

Joseph Alfred Serret (1819-1885) defined modular functions in 1866 in "Mémoire sur la théorie des congruences suivant un module premier et suivant une fonction modulaire irréductible," Mémoires de l'Acad.: "La fonction irréductible qui intervient ici, joue le rôle de module, et je lui donne en conséquence le nom de fonction modulaire" [Udai Venedem].

Richard Dedekind (1831-1916) used the term elliptic modular function in "Schreiben an Herrn Borchardt ueber die Theorie der elliptischen Modulfunktionen," J. reine angew. Math. 83 (1877), 265-292. According to Klein, this was the origin of the general name modular functions for functions with this or similar invariance [William C. Waterhouse].

MODULUS (in logarithms) 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:

Si numerus a numerorum b, c differentiam metitur, b et c secundum a congrui dicuntur, sin minus, incongrui; ipsum a modulum appelamus. Uterque numerorum b, c priori in casu alterius residuum, in posteriori vero nonresiduum vocatur. [If a number a measure the difference between two numbers b and c, b and c are said to be congruent with respect to a, if not, incongruent; a is called the modulus, and each of the numbers b and c the residue of the other in the first case, the non-residue in the latter case.]

Modulus (in number theory) is found in English in 1811 in An Elementary Investigation of the Theory of Numbers by Peter Barlow [James A. Landau].

Modulus (the length of the vector a + bi) is due to Jean Robert Argand (1768-1822) (Cajori 1919, page 265). The term was first used by him in 1814, according to William F. White in A Scrap-Book of Elementary Mathematics (1908).

Modulus for sqrt (a² + b²) was used by Augustin-Louis Cauchy (1789-1857) in 1821.

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).

MOMENT was used in the obsolete sense of "an infinitesimal increment or decrement of a varying quantity" by Isaac Newton in 1704 in De Quadratura Curvarum: "Momenta id est incrementa momentanea synchrona" (OED2).

Moment appears in English in the obsolete sense of "momentum" in 1706 in Synopsis Palmariorum Matheseos by William Jones: "Moment..is compounded of Velocity..and..Weight" (OED2).

Moment of a force appears in 1830 in A Treatise on Mechanics by Henry Kater and Dionysius Lardner (OED2).

Moment was used in a statistics sense by Karl Pearson in October 1893 in Nature: "Now the centre of gravity of the observation curve is found at once, also its area and its first four moments by easy calculation" (OED2).

The phrase method of moments was used in a statistics sense in the first of Karl Pearson's "Contributions to the Mathematical Theory of Evolution" (Phil. Trans. R. Soc. 1894). The method was used to estimate the parameters of a mixture of normal distributions. For several years Pearson used the method on different problems but the name only gained general currency with the publication of his 1902 Biometrika paper "On the systematic fitting of curves to observations and measurements" (David 1995). In "On the Mathematical Foundations of Theoretical Statistics" (Phil. Trans. R. Soc. 1922), Fisher criticized the method for being inefficient compared to his own maximum likelihood method (Hald pp. 650 and 719). [This paragraph was contributed by John Aldrich.]

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

MONOMIAL appears in English in a 1706 dictionary.

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 Stanislaw M. Ulam. In an unpublished manuscript, "The Origin of the Monte Carlo Method," dated Apr. 12, 1983, Ulam wrote that the method came to him while playing solitaire during an illness in 1946, and that what seems to be the first written account of the method was given by von Neumann in a letter to Robert Richtmyer of Los Alamos in early 1947.

According to W. L. Winston, the term was coined by Ulam and von Neumann in the feasibility project of atomic bomb by simulations of nuclear fission; they gave the code name Monte Carlo for these simulations.

According to several Internet web pages, the term was coined in 1947 by Nicholas Metropolis, inspired by Ulam's interest in poker during the Manhattan Project of World War II.

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

Monte Carlo method also appears in 1949 in Math. Tables & Other Aids to Computation III: "This method of solution of problems in mathematical physics by sampling techniques based on random walk models constitutes what is known as the 'Monte Carlo' method. The method as well as the name for it were apparently first suggested by John von Neumann and S. M. Ulam" (OED2).

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.

The phrase MORALLY CERTAIN was introduced by Jacob (James/Jacques) Bernoulli for a case in which the probability is .99 or perhaps .999 (Walker, 1929, p. 10).

MOVING AVERAGE. This technique for smoothing data points was used for decades before this, or any general term, came into use. In 1909 G. U. Yule (Journal of the Royal Statistical Society, 72, 721-730) described the "instantaneous averages" R. H. Hooker calculated in 1905 as "moving-averages." Yule did not adopt the term in his textbook, but it entered circulation through W. I. King's Elements of Statistical Method (1912).

"Moving average" referring to a type of stochastic process is an abbreviation of H. Wold's "process of moving average" (A Study in the Analysis of Stationary Time Series (1938)). Wold described how special cases of the process had been studied in the 1920s by Yule and Slutsky [John Aldrich].

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).

MULTIPLICATION TABLE. Table of multiplication appears in 1594 in Exercises (1636) by Blundevil: "Before I teach you the true order of multiplying, I thinke it good to set you downe a Table of Multiplication" (OED2).

Multiplication table appears in 1674 in Arithmetic by Samuel Jeake: "To learn by heart the Table commonly called Multiplication Table" (OED2).

The first edition of the Encyclopaedia Britannica (1768-1771) has: "This elementary step may be learned from the following table, commonly called Pythagoras's table of multiplication: which is consulted thus; seek one of the digits or numbers on the head, and the other on the left side, and in the angle of meeting you have their product."

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 J. Wishart, "The generalized product moment distribution in samples from a normal multivariate population," Biometrika 20A, 32 (1928) [James A. Landau].

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