N DIMENSIONS is found in Georg Cantor, "Sur divers théorèmes de lat théorie des ensembles de points situés dans un espace continu à n dimensions, Première communication," Acta Mathematica 2 (1883) [James A. Landau].
N-GON is found in 1867-78 in J. Wolstenholme, Math. Probl. (ed. 2): "In the moving circle is described a regular m-gon. .. The same epicycloid may also be generated by the corners of a regular n-gon" (OED2).
N-VARIATE is found in J. W. Mauchly, "Significance test for sphericity of a normal n-variate distribution," Ann. Math. Statist. 1 (1940).
NABLA (as a name for the "del" or Hamiltonian operator). According to Cargill Gilston Knott in Life and Scientific Work of Peter Guthrie Tait (1911),
From the resemblance of this inverted delta to an Assyrian harp Robertson Smith suggested the name Nabla. The name was used in playful intercourse between Tait and Clerk Maxwell, who in a letter of uncertain date finished a brief sketch of a particular problem in orthogonal surfaces by the remark "It is neater and perhaps wiser to compose a nablody on this theme which is well suited for this species of composition." [...]
It was probably this reluctance on the part of Maxwell to use the term Nabla in serious writings which prevented Tait from introducing the word earlier than he did. The one published use of the word by Maxwell is in the title to his homorous Tyndallic Ode, which is dedicated to the "Chief Musician upon Nabla," that is, Tait.
In a letter from Maxwell to Tait on Nov. 7, 1870, Maxwell wrote, "What do you call this? Atled?"
In a letter from Maxwell to Tait on Jan. 23, 1871, Maxwell began with, "Still harping on that Nabla?"
The term nabla was used by both Heaviside and Hamilton (Cajori vol. 2, page 135; Kline, page 780).
Webster's Third New International Dictionary defines nabla as "an ancient stringed instrument probably like a Hebrew harp of 10 or 12 strings -- called also nebel." The dictionary also says that the mathematical operator is "probably so called from the resemblance of its symbol, the inverted Greek delta, to a harp."
NAPIERIAN LOGARITHM. Phil. Trans. (1750) 46, 562 has: "The Logarithms to that Spiral which cuts its Rays at Angles of 45Degrees, are of the Napierian Kind" [Alan Hughes, Associate Editor of the OED].
Naper's Logarithms and Napier's logarithm appear in the 1771 edition of the Encyclopaedia Britannica [James A. Landau].
Napierian logarithm appears in Traité élémentaire de calcul différentiel et de calcul intégral (1797-1800) by Lacroix: "Le nombre e se présente souvent dans les recherches analytiques; on le prend pour base d'un système logarithmique, que j'ai appelé Népérien, du nom de Néper, inventeur des logarithmes."
Napierian logarithm appears in English in 1816 in a translation of Lacroix's Differential and Integral Calculus: "A system of logarithms, which we shall call Naperian, from the name of Naper their inventor" (OED2).
NAPIER'S CONSTANT has been suggested for the number 2.718..., according to e: The Story of a Number by Eli Maor (1994). Maor writes that Napier came close to discovering the constant.
The term Napier's constant was used twice in an episode of the television series The X-Files titled "Paper Clip," which originally aired on Sept. 29, 1995.
NATURAL LOGARITHM (early sense). According ot the DSB, Thomas Fantet de Lagny (1660-1734) "attempted to establish trigonometric tables through the use of transcription into binary arithmetic, which he termed 'natural logarithm' and the properties of which he discovered independently of Leibniz."
NATURAL LOGARITHM (modern sense). Boyer (page 430) implies that Pietro Mengoli (1625-1686) coined the term: "Mercator took over from Mengoli the name 'natural logarithm' for values that are derived by means of this series."
Natural logarithm was used by De Moivre in Phil. Trans. (1746) 43, 70 [Alan Hughes, Associate Editor of the OED].
Euler called these logarithms "natural or hyperbolic" in 1748 in his Introductio, according to Dunham (page 26), who provides a reference to Vol. I, page 97, of the Introductio.
Natural logarithm is found in the first edition of the Encyclopaedia Britannica (1768-1771) [James A. Landau].
NATURAL NUMBER. Chuquet (1484) used the term progression naturelle for the sequence 1, 2, 3, 4, etc.
In 1727, Bernard le Bouyer de Fontenelle (1657-1757) wrote in Elemens de la Geometrie de l'infini: "Pour mieux concevoir l'Infini, je considere la suite naturelle des nombres, dont l'origine est 0 ou 1" [Gunnar Berg].
Natural number appears in 1763 in The method of increments by William Emerson: "To find the product of all natural numbers from 1 to 100" (OED2).
Natural number, defined as the numbers 1, 2, 3, 4, 5, etc., appears in the 1771 Encyclopaedia Britannica in the Logarithm article.
In 1889, Elements of Algebra by G. A. Wentworth has: "The natural series of numbers begins with 0; each succeeding number is obtained by adding one to the preceding number, and the series is infinite."
Apparently, except for the various Random House dictionaries, all modern dictionaries define the term natural number to exclude 0. [The term whole number is defined to include 0.]
Gerald A. Edgar wrote in sci.math that he has found that algebra texts tend to include 0 and analysis texts tend to exclude it.
Bertrand Russell in his 1919 Introduction to Mathematical Philosophy writes:
To the average educated person of the present day, the obvious starting-point of mathematics would be the series of whole numbers,
1, 2, 3, 4, . . . etc.
Probably only a person with some mathematical knowledge would think of beginning with 0 instead of with 1, but we will presume this degree of knowledge; we will take as our starting-point the series:
0, 1, 2, 3, . . . n, n + 1, . . .
and it is this series that we shall mean when we speak of the "series of natural numbers."
John Conway writes:
The older nomenclature was that "natural number" meant "positive integer." This was unfortunate, because the set of non-negative integers is really much more "natural" in the sense that it has simpler properties. So starting in about the 1960s lots of people (including me) started to use "natural number" in the inclusive sense. After all, for the positive integers we have the much better term "positive integer."
Stephen Cole Kleene (1909-1994), in his Mathematical logic (John Wiley & Sons, 1967) wrote (p. 176):
Some authors use "natural numbers" as a synonym for "positive integers" 1, 2, 3, ..., obliging one to use the more cumbersome name "nonnegative integers" for 0, 1, 2, .... Besides, in this age, we should accept 0 on the same footing with 1, 2, 3, ... .
[Carlos César de Araújo, Sam Kutler]
NEGATIVE. According to Burton (page 245), Brahmagupta introduced negative numbers and a word equivalent to negative.
In his Ars Magna (1545) Cardano referred to negative numbers as numeri ficti. In speaking of the roots of an equation, he wrote, "una semper est rei uera aestimatio, altera ei aequalis, ficta" (Smith vol. 2, page 260).
Stifel (1544) called negative numbers numeri absurdi. He spoke of zero as "quod mediat inter numeros veros et numeros absurdos" (Smith vol. 2, page 260).
The word negative was used by Scheubel (1551) (Smith vol. 2, page 260).
Robert Recorde in Whetstone of Witt (1557) used absurde nomber: "8 - 12 is an Absurde nomber. For it betokeneth lesse then nought by 4" (OED2).
Napier (c. 1600) used the adjective defectivi to designate negative numbers (Smith vol. 2, page 260).
Negative was used in English in 1673 by John Kersey in Elements of Algebra: "A negative Root (which Cartesius calls a false Root) expresseth a Quantity whose Denomination is opposite to an affirmative, as -5 or -20" (OED2).
NEGATIVE BINOMIAL DISTRIBUTION occurs in R. A. Fisher, "The negative binomial distribution," Ann. Eugen., 11 (1941) [James A. Landau].
NEIGHBORHOOD OF A POINT. In an 1877 paper in the Messenger of Mathematics, Arthur Cayley wrote, "The lines in the neighbourhood of a point of maximum, or of minimum, paameter are ovals surrounding the point in question, each oval being itself surrounded by the consecutive oval" [University of Michigan Historical Math Collection].
Neighborhood of a point appears in 1891 in George L. Cathcart's translation of An Introduction to the Study of the Elements of the Differential and Integral Calculus by Axel Harnack:
With the foregoing considerations we have attained the conception of the Region or Neighbourhood of a point. By it we mean an arbitrarily small but still always finite interval at both sides of the value x.
NEPHROID was used for the bicuspid epicycloid by Richard Anthony Proctor (1837-1888) in 1878 in A treatise on the cycloid and all forms of cycloidal curves:
The epicycloid with two cusps (the dotted curve of fig. 39, which, from its shape, we may call the nephroid) presents also many interesting relations.
This citation was provided by Julio González Cabillón. According to E. W. Lockwood, in 1879 Freeth used the same name for another curve, Freeth's Nephroid.
The term NET was coined by J. Kelley. He had considered using the term "way" so the analog of subsequence would be "subway." E. J. McShane also proposed the term "stream" since he thought it was intuitive to think of the relation of the directed set as "being downstream from" (McShane, Partial Orderings and Moore-Smith Limits, 1950, p. 282).
NEW MATH. New mathematics is found in Time magazine of Feb. 3, 1958, in the heading, "The new mathematics" (OED2).
New math is dated 1964 in MWCD10.
New math is found in 1966 in the title Fun with the New Math by Meyer and Hanlon (OED2).
NEWTON'S METHOD. In 1685 John Wallis wrote in Algebra: "Another Method of Approximation, by Mr. Isaac Newton" (OED2).
In 1695, Wallis used the phrase "Mr. Newton's Method of Approximation for the Extracting of Roots" in Phil. Trans. XIX (OED2).
In the second edition of Theory and Solution of Algebraical Equations (1843) by J. R. Young is a section heading, "Newton's method of approximating the incommensurable roots of an equation."
NEWTON-RAPHSON METHOD appears in 1916 in William Oughtred: A Great Seventeenth-Century Teacher of Mathematics by Florian Cajori.
Cajori writes on page 203 of his History of Mathematics (1919):
Perhaps the name "Newton-Raphson method" would be a designation more nearly representing the facts of history.
Cajori subsequently uses the term throughout the book. He may have written the same thing in the 1893 edition, which I have not seen.
Cajori used the term in the title, "Historical note on the Newton-Raphson method of approximation," Amer. Math. Monthly 18.
NEXT TO CONVEX. In a post to the geometry forum, Michael E. Gage wrote that he believed the term was coined by Gromov in "Hyberbolic manifolds, groups and actions," Annals of Mathematics Studies, v. 97, page 183.
NILPOTENT. See idempotent.
The term NINE-POINT CIRCLE first appears in 1820 in "Recherches sur la détermination d'une hyperbola équilatére, au moyen de quatres conditions données" by Charles-Julien Brianchon (1783-1864) and Poncelet (DSB).
Nine-points circle appears in English in 1865 in Brande & Cox, Dict. Sci., etc., which has: "The circle which passes through the middle points of the sides of a triangle is referred to by Continental writers as 'the nine-points circle' (OED2).
Nine-point circle appears in W. F. Mc Michael, "Elementary proof of the contact of the nine-point circle with the inscribed and escribed circles," Mess. (1881).
Nine-point circle also appears in English in C. Pendlebury, "Proof of a nine-point circle theorem," Ed. Times (1881).
In Plane Geometry (1904), James Sturgeon Mackay wrote, "The designation often given to the circle, namely, Euler's circle, is quite erroneous" [Ken Pledger].
According to Eves (page 437), the circle is sometimes called Euler's circle because of misplaced credit, and in Germany the circle is called Feuerbach's circle.
NODE. In 1753 Daniel Bernoulli used the Latin word noeud [James A. Landau].
NOETHERIAN RING occurs in "On integrally closed Noetherian rings" and "The intersection theorem on Noetherian rings," both of which are by Yoshida, Michio; Sakuma, Motoyoshi in J. Sci. Hiroshima Univ., Ser. A 17, 311-315 (1954).
NOETHER'S THEOREM. Noether's fundamental theorem is found in English in H. J. Baker, "On Noether's fundamental theorem," Math. Ann. (1893).
Noether's theorem is found in English in F. S. Macaulay, "The theorem of residuation, Noether's theorem, and the Riemann-Roch theorem," Lond. M. S. Proc. (1899).
NOME. The OED2 indicates this word comes to English from the French nôme, the second element in binôme (binomial).
In 1665 Collins used the term in English in correspondence: "The limits of such equations as have but two nomes."
In 1704 Harris's Lexicon technicum has: "Nome, in Algebra, is any Quantity with a Sign prefixed to it, and by which 'tis usually connected with some other Quantity, and then the whole is called a Binomial, a Trinomial, &c."
About 1727 nome appears in English meaning "one of the 36 territorial divisions of ancient Egypt." Most recent dictionaries indicate the word is derived from the Greek nomos meaning "a pasture or district"; however, the OED2 indicates the word is derived a Greek word meaning "to divide."
According to a post by Eric Conrad to the history of mathematics mailing list, "the term nome for the argument q in elliptic function theory comes from a Greek word meaning to partition or divide. Many identities in elliptic function theory have combinatorial interpretations involving partitions. For example, the Fourier expansion of the elliptic function dn leads to an identity which counts partitions of an integer into a sum of four squares."
The term NOMOGRAPHY is due to Philbert Maurice d'Ocagne (1862-1938). It occurs in his Traité de nomographie (1899) (DSB).
NONAGON is dated 1639 in MWCD10.
In 1688, R. Holme, Armoury has: "An Henneagon, or Enneagon or Nonagon, a fort of nyne corners" (OED2).
The 1828 Webster dictionary defines nonagon and enneagon.
NON-CANTORIAN appears in Paul J. Cohen and Reuben Hersh, "Non-Cantorian Set Theory," Scientific American, December 1967.
The term also appears in 1967 in the article "Set Theory" in The Encyclopedia of Philosophy [James A. Landau].
The term NON-EUCLIDEAN GEOMETRY was introduced by Carl Friedrich Gauss (1777-1855) (Sommerville). Gauss had earlier used the terms anti-Euclidean geometry and astral geometry (Kline, page 872).
Non-Euclidean geometry is found in 1868 in the title of the paper "Saggio di interpretazione della geometria non-euclidea" by Eugenio Beltrami.
Non-Euclidean geometry is found in English in 1872 in Mathematische Ann. vol. 5. p. 630: "The theory of Non-Euclidean Geometry as developed in Dr. Klein's paper 'Uber die Nicht-Euclidische Geometrie' may be illustrated by showing how in such a system we actually measure a distance and an angle" [David B. Shirt, Senior Science Editor, OED].
NON-NORMAL appears in 1929 in Biometrika in the heading: "On the distribution of the ratio of mean to standard deviation in small samples from non-normal universes" (OED2).
NONPARAMETRIC (referring to a statistical inference) is found in 1942 in Jacob Wolfowitz (1910-1981), "Additive Partition Functions and a Class of Statistical Hypotheses," Annals of Mathematical Statistics, 13, 247-279 (David, 1995).
NON-TERMINATING DECIMAL. Interminate decimal is found in 1714 in S. Cunn, Treat. Fractions Pref.: "The Reverend Mr. Brown, in his System of Decimal arithmetick, manages such interminate Decimals as have a single Digit continually repeated" (OED2).
Non-terminating decimal is found in 1905 in Ann. Math. VI. 175: "An example of a non-denumerable class is the class of all non-terminating decimal fractions" (OED2).
NORM in algebra was introduced in 1832 as the Latin norma by Carl Friedrich Gauss (1777-1855) in Commentationes Recentiones Soc. R. Scient. Gottingensis VII. Class. math. 98. He used the term for a2 + b2.
In 1856 Hamilton used norm to refer to a2 + b2 for the complex number a + ib.
In 1921 the term norm refers to sqrt (a2 + b2) for the complex number a + ib in Proceedings of the National Academy of Science VII. 84.
In 1949 A. Albert defines the norm of a vector P as the inner product P.P in Solid Analytic Geometry (OED2).
NORMAL (perpendicular) appears as a geometry term in the phrase "Normal Line" about 1696 in The English Euclide by Edmund Scarburgh, although the OED2 shows a slightly earlier non-mathematical use of the term to mean "right" or "rectangular."
NORMAL (statistics). Normal was used by F. Galton in 1889 in Natural Inheritance. David (1995) writes that Stigler informs him that this is the first use of "normal" unambiguously as a term for the distribution.
Normal probability curve was used by Karl Pearson (1857-1936) in 1893 in Nature 26 Oct. 615/2: "As verification note that for the normal probability curve 3µ22 = µ4 and µ3 = 0" (OED2).
Pearson used normal curve in 1894 in "Contributions to the Mathematical Theory of Evolution":
When a series of measurements gives rise to a normal curve, we may probably assume something approaching a stable condition; there is production and destruction impartially around the mean.
The above quotation is from Porter.
Pearson used normal curve in 1894 in Phil. Trans. R. Soc. A. CLXXXV. 72: "A frequency-curve, which for practical purposes, can be represented by the error curve, will for the remainder of this paper be termed a normal curve."
Normal distribution appears in 1897 in Proc. R. Soc. LXII. 176: "A random selection from a normal distribution" (OED2).
According to Hald, p. 356:
The new error distribution was first of all called the law of error, but many other names came to be used, such as the law of facility of errors, the law of frequency of errors, the Gaussian law of errors, the exponential law, and the typical law of errors. In his paper "Typical laws of heredity" Galton (1877) studied biological variation, and he therefore replaced the term "error" with "deviation," and referring to Quetelet, he called the distribution "the mathematical law of deviation." Chapter 5 in Galton's Natural Inheritance (1889a) is entitled "Normal Variability," and he writes consistently about "The Normal Curve of Distributions," an expression that caught on.
According to Walker (p. 185), Karl Pearson did not coin the term normal curve. She writes, "Galton used it, as did also Lexis, and the writer has not found any reference which seems to be its first use."
Nevertheless, "...Pearson's consistent and exclusive use of this term in his epoch-making publications led to its adoption throughout the statistical community" (DSB).
However, Porter (p. 312) calls normal curve a "Pearsonian neologism."
NORMAL CORRRELATION appears in W. F. Sheppard, "On the application of the theory of error to cases of normal distributions and normal correlations," Phil. Trans. A, 192, page 1091, and Proc. Roy. Soc. 62, page 170 (1898) [James A. Landau].
NORMAL DEVIATE is found in 1925 in R. A. Fisher, Statistical Methods: "Table I. shows that the normal deviate falls outside the range +/-1.598193 in 10 per cent of cases" (OED2).
The term NORMAL EQUATION was introduced by Gauss, according to Stigler in Statistics Before 1827 [James A. Landau].
NORMAL LAW was coined by Karl Pearson in 1894, according to Porter (p. 13).
NORMAL POPULATION appears in E. S. Pearson, "A further note on the distribution of range in samples taken from a normal population," Biometrika 18, page 173 (1926) [James A. Landau]. Also see extreme value.
NORMAL SAMPLES is found in R. A. Fisher, "The moments of the distribution for normal samples of measures of departure from normality," Proc. Roy. Soc. A, 130 (1930).
NORMAL SUBGROUP. According to Kramer (p. 388), Galois used the adjective "invariant" referring to a normal subgroup.
According to The Genesis of the Abstract Group Concept (1984) by Hans Wussing, "The German Normalteiler (normal subgroup) goes back to Weber [H., Lehrbuch der Algebra, vol. 1, Braunschweig, 1895. p.511] and is possibly linked to Dedekind's term Teiler (divisor), which was employed in ideal theory" [Dirk Schlimm].
Normal subgroup is found in English in 1908 in An Introduction to the Theory of Groups of Finite Order by Harold Hilton: "Similarly, if every element of G transforms a subgroup H into itself, H is called a normal, self-conjugate, or invariant subgroup of G (or 'a subgroup normal in G')."
G. A. Miller writes in Historical Introduction to Mathematical Literature (1916), "In the newer subjects the tendency is especially strong to use different terms for the same concept. For instance, in the theory of groups the following seven terms have been used by various writers to denote a single concept: invariant subgroup, self-conjugate subgroup, normal divisor, monotypic subgroup, proper divisor, distinguished subgroup, autojug."
NORMAL UNIVERSE is found in A. T. MacKay, "The distribution of the difference between the extreme observation and the sample mean in samples of n from a normal universe," Biometrika 27 (1935) [James A. Landau].
NORMAL VARIATE is found in A. A. Anis and E. H. Lloyd, "Range of partial sums of normal variates," Biometrika 40 (1953) [James A. Landau].
NORMALITY appears in R. A. Fisher, "The moments of the distribution for normal samples of measures of departure from normality," Proc. Roy. Soc. A, 130 (1930).
Normality also appears in E. S. Pearson, "A further development of tests for normality," Biometrika 22 (1930) [James A. Landau].
NORMALIZER appears in German as Normalisator in Theorie der Gruppen von endlicher Ordnung, 2nd ed., by A. Speiser [Huw Davies].
NTH. Mth is found in 1763 in The Method of Increments by William Emerson: "The infinitinomial 1 + By + Cy2 &c. is to be raised to the mth power" (OED2).
Nth is found in William George Horner, "A new method of solving numerical equations of all orders, by continuous approximation," Philosophical Transactions of the Royal Society of London (1819): "We perceive also, that some advance has been made toward arithmetical facility; for all the figurate coefficients here employed are lower by one order than those which naturally occur in transforming an equation of the nth degree."
The above excerpt was taken from A Source Book in Mathematics by David Eugene Smith.
Nth is found in 1820 in Functional Equations by Babbage: "To find periodic functions of the nth order...."
Nth is found in 1831 in Elements of the Integral Calculus (1839) by J. R. Young:
In all the foregoing examples, the first differential coefficient is given to determine the primitive function from which it has been derived; when, however, it is not the first, but the nth differential coefficient which is given, then by a first integration, we shall arrive at the preceding or n - 1th differential coefficient; by a second integration we get the n - 2th coefficient; and, by thus continuing the integration, we at length arrive at the original function.
NULL CLASS is found in "Sets of Independent Postulates for the Algebra of Logic," Edward V. Huntington, Transactions of the American Mathematical Society, Vol. 5, No. 3. (Jul., 1904).
NULL HYPOTHESIS is used in 1935 by Ronald Aylmer Fisher in The Design of Experiments. He writes, "We may speak of this hypothesis as the 'null hypothesis,' and it should be noted that the null hypothesis is never proved or established, but is possibly disproved, in the course of experimentation."
The "null hypothesis" is often identified with the "hypothesis tested" of J. Neyman and E. S. Pearson's 1933 paper, "On the Problems of the Most Efficient Tests of Statistical Hypotheses" Phil. Trans. Roy. Soc. A (1933), 289-337, and represented by their symbol H0. Neyman did not like the "null hypothesis," arguing (First Course in Probability and Statistics, 1950, p. 259) that "the original term 'hypothesis tested' seems more descriptive." It is not clear, however, that "hypothesis tested" was ever floated as a technical term [John Aldrich].
NULLITY appears in 1884 in James Joseph Sylvester, Amer. Jrnl. Math. VI. 274:
The absolute zero for matrices of any order is the matrix all of whose elements are zero. It possesses so far as regards multiplication .. the distinguishing property of the zero, viz. that when entering into composition with any other matrix .. the product .. is itself over again... This is the highest degree of nullity which any matrix can possess, and (regarded as an integer) will be called [omega], the order of the matrix... In general.., if all the minors of order [omega] - i + 1 vanish, but the minors of order [omega] - i do not all vanish, the nullity will be said to be i.
NULL SET. Null-set appears in 1906 in Theory of Sets and Points by W. H. and G. C. Young (OED2).
NUMBER LINE. An earlier term was scale of numbers.
In 1912, First Year Algebra by Webster Wells and Walter W. Hart has: "Thus, the positive and negative numbers together form a complete scale extending in both directions from zero."
Lawrence M. Henderson, "An Alternative Technique for Teaching Subtraction of Signed Numbers," The Mathematics Teacher, Nov. 1945 has: "In teaching subtraction of signed numbers, I first draw a number scale."
Number line is found in January 1956 in "An exploratory approach to solving equations" by Max Beberman and Bruce E. Meserve in The Mathematics Teacher: "In an earlier paper we described a procedure by which students could 'solve' equations and inequalities using a number line. The set of points on a number line is in one-to-one correspondence with the set of real numbers."
NUMBER THEORY. According to Diogenes Laertius, Xenocrates of Chalcedon (396 BC - 314 BC) wrote a book titled The theory of numbers.
A letter written by Blaise Pascal to Fermat dated July 29, 1654, includes the sentence, "The Chevalier de Mèré said to me that he found a falsehood in the theory of numbers for the following reason."
The term appears in 1798 in the title Essai sur la théorie des nombres by Adrien-Marie Legendre (1752-1833).
In English, theory of numbers appears in 1811 in the title An elementary investigation of the theory of numbers by Peter Barlow.
Number theory appears in 1853 in Manual of Greek literature from the earliest authentic periods to the close of the Byzantine era by Charles Anthon: "The ethics of the Pythagoreans consisted more in ascetic practice and in maxims for the restraint of the passions, especially of anger, and the cultivation of the power of endurance, than in scientific theory. What of the latter they had was, as might be expected, intimately connected with their number-theory" [University of Michigan Digital Library].
Number theory appears in 1864 in A history of philosophy in epitome by Dr. Albert Schwegler, translated from the original German by Julius H. Seelye: "Not only the old Pythagoreans, who have spoken of him, delighted in the mysterious and esoteric, but even his new-Platonistic biographers, Porphyry and Jamblichus, have treated his life as a historico-philosophical romance. We have the same uncertainty in reference to his doctrines, i. e. in reference to his share in the number-theory. Aristotle, e. g. does not ascribe this to Pythagoras himself, but only to the Pythagoreans generally, i. e. to their school" [University of Michigan Digital Library].
Number theory appears in 1912 in the Bulletin of the American Mathematical Society (OED2).
The words NUMERATOR and DENOMINATOR are found in 1202 in Liber abbaci by Leonardo of Pisa:
Cum super quemlibet numerum quedam uirgula protracta fuerit, et super ipsam quilibet alius numerus descriptus fuerit, superior numerus partem uel partes inferioris numeri affirmat; nam inferior denominatus, et superior denominans appellatur. Vt si super binarium protracta fuerit uirgula, et super ipsam unitas descripta sit ipsa unitas unam partem de duabus partibus unius integri affirmat, hoc est medietatem sic 1/2 [When above any number a line is drawn, and above that is written any other number, the superior number stands for the part or parts of the inferior number; the inferior is called the denominator, the superior the numerator. Thus, if above the two a line is drawn, and above that unity is written, this unity stands for one part of two parts of an integer, i. e., for a half, thus 1/2].
Numerator and denominator are found in Algorismus proportionum by Nicole Oresme (ca. 1323-1382). The work is in Latin but the words are spelled as they are in English, and are defined as "the number above the line" and "the number below the line" (Cajori vol. 1, page 91).
NUMERICAL ANALYSIS appears in 1853 in A dictionary of science, literature & art in a discussion of the rule of false position: "The rules given for the solution of questions in double position, are founded on certain principles of algebra, which may be applied with much greater facility to the immediate solution of the questions themselves. Such questions, therefore, cannot be considered as properly belong to arithmetic, but to what may be denominated numerical analysis" [University of Michigan Digital Library].
Numerical analysis appears in S. Réalis, "Questions d'analyse numérique," N. C. M. V. (1879).
Numerical mathematical analysis occurs in 1930 in the title Numerical mathematical analysis by J. B. Scarborough (OED2).
Numerical analysis occurs in Randolph Church, "Numerical analysis of certain free distributive structures," Duke Math. J. 6 (1940).
NUMERICAL DIFFERENTIATION is found in W. G. Bickley, "Formulae for Numerical Differentiation," Math. Gaz. 25 (1941) [James A. Landau].
NUMERICAL INTEGRATION occurs in the title A Course in Integration and Numerical Integration (1915) by David Gibb.
OBLIQUE ANGLE. In 1594 Thomas Blundevil referred to "oblique angled Triangles" in Exercises.
Oblique angle appears in English in 1695 in Geometry Epitomized by William Alingham: "An Oblique Angle, is either Acute or Obtuse" (OED2).
OBTUSE ANGLE appears in English in 1570 in Sir Henry Billingsley's translation of Euclid's Elements: "An acute angle is that, which is lesse then a right angle"; "an obtuse angle is that which is greater then a right angle" (OED2).
OBTUSE TRIANGLE. Amblygonium and obtuse angled triangle appear in 1570 Billingsley's translation of Euclid: "An ambligonium or an obtuse angled triangle" (OED2).
Amblygone is found in English in 1623; amblygon is found in 1706.
Obtusangulous is found in English in 1680; obtusangular is found in 1706.
In 1828 Webster's dictionary has: "If one of the angles is obtuse, the triangle is called obtusangular or amblygonous."
OCTAGON is dated 1639 in MWCD10.
ODD FUNCTION is found in 1849 in Trigonometry and Double Algebra by Augustus De Morgan: "But the sine is what is called an odd function; that is, it changes sign when [theta] is changed into -[theta]; or sin (-[theta] = -sin [theta]" [University of Michigan Historical Math Collection].
ODD NUMBER and EVEN NUMBER. The Pythagoreans knew of the distinction between odd and even numbers. The Pythagoreans used the term gnomon for the odd number.
A fragment of Philolaus (c. 425 B. C.) says that "numbers are of two special kinds, odd and even, with a third, even-odd, arising from a mixture of the two."
Euclid, Book 7, definition 6 is "An even number is that which is divisible into two parts."
Odd and even are found in English in various Middle Age documents including Art of Nombrynge (around 1430) "Compt the nombre of the figures, and wete yf it be ode or even" (OED2).
In English, gonomon is found in 1660 in Stanley, Hist. Philos. (1701): "Odd Numbers they called Gnomons, because being added to Squares, they keep the same Figures; so Gnomons do in Geometry" (OED2).
ODD PERMUTATION and EVEN PERMUTATION are found in "On the Relation between the Three-Parameter Groups of a Cubic Space Curve and a Quadric Surface," A. B. Coble, Transactions of the American Mathematical Society, Vol. 7, No. 1. (Jan., 1906).
Odd permutation is found in the fourth edition of Determinants (1906) by Laenas Gifford Weld, and may appear in earlier editions.
The term OMEGA RULE was first used by A. Grzegorczyk, A. Mostowski, A. and C. Ryll-Nardzewski in "The Classical and the w-Complete Arithmetic", JSL (1958), 188-206, according to Edgard G. K. López-Escobar e Ítala M. Loffredo D'Ottaviano, "A regra-w : passado, presente e futuro", Centro de lógica, epistemologia e história da ciéncia, Campinas-São Paulo, 1987.
An earlier term for this rule of inference, "Carnap's rule" was first used by J. Barkley Rosser in "Gödel-Theorems for Non-Constructive Logics (JSL 2 (1937), 129-37), where he alludes to Carnap's "Ein Gültigkeitskriterium für die Sätze der Klassischen Mathematik" (Monatsheffe für Mathematik und Physik 42 (1935), 163-90). However, as it was pointed in López-Escobar,
Infelizmente, isto está longe da realidade, porque Carnap não fala de uma regra de dedução que tenha qualquer semelhança com a regra-w . [Unhappily, this is far from the reality, because Carnap doesn't speak about a deduction rule that has any resemblance with the w-rule.]
Another term for this rule, "Novikov's rule," was used by Schoenfield, Mostowski and Kreisel, who mention Novikov's "On the Consistency of Certain Logical Calculi" (Math. Sbornik 12 (1943), 231-61). But López-Escobar points out again (p. 3) that this name is inappropriate because Novik's paper deals "only with infinite formulas with recursive conjunctions and disjunctions", not with the w-rule.
ONE-TO-ONE CORRESPONDENCE is found in H. G. Zeuthen, "Sur les points fondamentaux de deux surfaces dont les points se correspondent un à un," C. R. LXX. 742. (1870).
One-to-one correspondence is found in English in 1873 in Proc. Lond. Math. Soc. IV. 252: "The equations .. being supposed to establish a 'one-to-one' correspondence between the two integral spaces" (OED2).
ONTO was used as a preposition in 1940 by C. C. MacDuffee in Introd. Abstract Algebra: "If a homomorphism of A onto B exists, we write A ~ B."
Onto was used as an adjective in 1942 by S. Lefschetz in Algebraic Topology: "If a transformation is 'onto,' ..." (OED2).
OPEN SET. The phrase "open set of points" appears in W. H. Young, "On the density of linear sets of points," Proc. London Math. Soc. 34 (1902) (OED2).
The concept of open set emerged with Lebesgue's 1905 paper on analytically representable functions. He wrote (p. 157): "les ensembles overts, c'est-a-dire ceux qui sont les complementaires des ensembles fermes."
Open set is found in 1906 in W. H. & G. C. Young, Theory of Sets of Points: "The set of all the points inside a triangle is called a triangular domain, or the interior of the triangle, and is a simple case of a region .... The points of a domain always form an open set" (OED2). Here the term is used to mean "not closed" rather than "complement of closed" as Legesgue, and mathematicians now, use the term.
Weierstrass used connected open sets in n-space and called such a set a "Gebiet."
Open set is found in 1902 in Proc. Lond. Math. Soc. XXXIV. 289 in the heading, "Open sets of points" (OED2).
Hausdorff in 1914 took over Weierstrass's term "Gebiet," but changed it to mean an open set, and explicitly referred to Weierstrass's earlier usage of the term to mean open connected set.
Open set is found in 1939 in M. H. A. Newman, Elem. Topology of Plane Sets of Points: "The sum of any set of open sets is an open set" (OED2).
[Much of this entry was taken from a mailing list message by Gregory H. Moore.]
OPERAND appears in Sir William Rowan Hamilton, Lectures on Quaternions (London: Whittaker & Co, 1853): "...in order to justify the subsequent abstraction of the operand step-set, or the abridgement (compare [25]) of this formula of successive operation to the following..." [James A. Landau].
OPERATION. Christopher Clavius (1537-1612) used the term operationes in his Algebra Christophori Clavii Bambergensis of 1608 (Smith vol. 2).
In English operation appears in 1713 in Introd. Math. by J. Ward: "If the whole Æquation..be now taken, and we proceed to a Second Operation, the Value of a may be increas'd with twelve Places of Figures more, and those may be obtain’d by plain Division only (OED2).
ORDER (degree) is found in English in 1706 in Ditton, Fluxions 22: "An Infinitesimal of another Order or Degree" and ibid. 123: "These sorts of [Exponential] Quantities are of several Orders or Degrees" (OED2).
Gradus appears in the title De Constructione Aequationum Differentialium Primi Gradus (1707) by Gabriele Manfredi (1681-1761).
Degree is found in Dictionarium Britannicum (1730-6): "Parodic Degree (in Algebra) is the index or exponent of any power; so in numbers, 1. is the parodick degree, or exponent of the root or side; 2. of the square, 3. of the cube, etc" (OED2).
Euler used gradus in the title "Nova methodus innumerabiles aequationes differentiales secundi gradus reducendi ad aequationes differentiales primi gradus," which was published in Commentarii academiae scientarum imperialis Petropolitanae, Tomus III ad annum 1728, Petropoli, 1732, pp. 124-137. The title is translated "A New Method of reducing innumerable differential equations of the second degree to differential equations of the first degree," although the modern terminology is "order" of a differential equation.
Order is found in English in 1743 in William Emerson, The Doctrine of Fluxions: "In any Fluxionary Equation, a Quantity of the first Order is that which has only one first Fluxion in it; a Quantity of the second Order has either one second Fluxion or two First Fluxions: Quantities of the third Order, are third Fluxions, product of three first Fluxions, product of a first and second Fluxion, etc." (OED2).
ORDER OF MAGNITUDE appears in 1853 in A dictionary of science, literature & art: "It is therefore called the modulus of elasticity, or coefficient of elasticity; and is of the same nature and order of magnitude as the cohesive resistance which bodies oppose to rupture on being crushed, and may therefore be expressed as so many pounds acting on a square inch of surface" [University of Michigan Digital Library].
Order of magnitude appears in 1851-54 in Hand-books of natural philosophy and astronomy by Dionysius Lardner: "As spaces or times which are extremely great or extremely small are more difficult to conceive than those which are of the order of magnitude that most commonly falls under the observation of the senses" [University of Michigan Digital Library].
Order of magnitude appears in March 1857 in I. T. Danson, "Connection between American Slavery and the British Cotton Manufacture," Debow's review, Agricultural, commercial, industrial progress and resources: "The two customers next on the list, when arranged in order of magnitude, are the United States and the United Kingdom."
Order of magnitude appears in 1858 in The Plurality of Worlds by William Whewell: "Applying these considerations to the stars of inferior orders of magnitude, he finally arrives at the following conclusion, which he admits to be of an unexpected character:--that the number of insulated stars is indeed greater than the number of compound systems; but only three times, perhaps only twice as great."
ORDER OF OPERATIONS. In Feb. 1842 in "Remarks on the distinction between algebraical and functional equations" in the Cambridge Mathematical Journal, Robert Leslie Ellis wrote: "As the distinction between functional and common equations depends on the order of operations, it follows that, when part of the solution of an equation does not vary with the nature of the operation subjected to the resolving process, this part is applicable as much to functional equations as to any other" [University of Michigan Historical Math Collection].
Order of operations is found as a chapter heading in 1913 in Second Course in Algebra by Webster Wells and Walter W. Hart.
ORDERED PAIR occurs in "A System of Axioms for Geometry," Oswald Veblen, Transactions of the American Mathematical Society, 5 (Jul., 1904): "Each ordered pair of elements determines a unique element that precedes it, a unique element that follows it and a unique middle element."
ORDINAL. The earliest citation for this term in the OED2 is in 1599 in Percyvall's Dictionarie in Spanish and English enlarged by J. Minsheu, in which the phrase ordinall numerals is found.
ORDINATE. Cajori (1906, page 185) writes: "The Latin term for 'ordinate,' used by Descartes comes from the expression lineae ordinatae, employed by Roman surveyors for parallel lines.
Cajori (1919, page 175) writes: "In the strictly technical sense of analytics as one of the coördinates of a point, the word "ordinate" was used by Leibniz in 1694, but in a less restricted sense such expressions as "ordinatim applicatae" occur much earlier in F. Commandinus and others."
Leibniz used the phrase "per differentias ordinatarum" in a letter to Newton on June 21, 1677 (Scott, page 155).
Leibniz used the term ordinata in 1692 in Acta Eruditorum 11 (Struik, page 272).
ORIGIN. Boyer (page 404) seems to attribute the term origin to Philippe de Lahire (1640-1718).
The term presumably appears in Sections Coniques by Marquis de l'Hospital, since the OED2 shows a use of the term in English in a 1723 translation of this work.
ORTHOCENTER. An earlier term was Archimedean point.
According to John Satterly, "Relations between the portions of the altitudes of a plane triangle," The Mathematical Gazette 46 (Feb. 1962), pp. 50-51, Mathematical Note 2997:
Note: As a matter of historical interest our readers may be reminded that the term "Orthocentre" was invented by two mathematicians, Besant and Ferrers, in 1865, while out for a walk along the Trumpington Road, a road leading out of Cambridge toward London. In those days it was a tree-lined quiet road with a sidewalk, a favourite place for a conversational walk.
This information was provided by Emili Bifet, who reports that Satterly does not provide further reference.
Orthocenter is found in 1869 in Conic sections, treated geometrically by William Henry Besant (1828-1917): "If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre" (OED2).
The 1895 ninth edition of this work (and presumably the first edition also) has: "If a rectangular hyperbola circumscribe a triangle, it passes through the orthocentre. Note: The orthocentre is the point of intersection of the perpendiculars from the angular points on the opposite sides."
ORTHOGONAL is found in English in 1571 in A geometrical practise named Pantometria by Thomas Digges (1546?-1595): "Of straight lined angles there are three kindes, the Orthogonall, the Obtuse and the Acute Angle." (In Billingsley's 1570 translation of Euclid, an orthogon (spelled in Latin orthogonium or orthogonion) is a right triangle.) (OED2).
The term ORTHOGONAL MATRIX was used in 1854 by Charles Hermite (1822-1901) in the Cambridge and Dublin Mathematical Journal, although it was not until 1878 that the formal definition of an orthogonal matrix was published by Frobenius (Kline, page 809).
ORTHOGONAL TRANSFORMATION was used in 1859 by George Salmon in Lessons Introductory to the Modern Higher Algebra: "What we may call the orthogonal transformation is to transform simultaneously a given quadratic function..." (OED2).
OSCILLATING SERIES. In the third edition of A Treatise on Algebra (1892), Charles Smith writes, "Such a series is sometimes called an indeterminate or a neutral series." A footnote reads, "These series are however called divergent series by Cauchy, Bertrand, Laurent and others."
Oscillating series appears in G. H. Hardy, "On certain oscillating series," Quart. J. 38.
Oscillating series appears in 1893 in A Treatise on the Theory of Functions by James Harkness and Frank Morley: "Strictly speaking, an oscillating series is distinct from a divergent series, but it is usual to speak of non-convergent series as divergent."
Oscillating series appears in 1898 in Introduction to the theory of analytic functions by Harkness and Morley:
It is then permissible to introduce parentheses in an infinite series. It is not always permissible to remove them; for example in the series (1 - 1) + (1 - 1) + (1 - 1) + ....., each term is 0 and the limit is 0, but the series 1 - 1 + 1 - 1 + ... oscillates; if we agree to take always an even number of terms the limit is 0; if an odd number, the limit is 1.
A footnote says, "Most English text-books regard oscillating series as not divergent."
The term OSCULATING was used by Leibniz in 1686, Acta Eruditorum, 1686, 289-92 = Math. Schriften, 7, 326-29 (Kline, page 556).
The term OSCULATING PLANE was introduced by John Bernoulli (Kline, page 559).
The term OSCULUM is due to Huygens.
OUTER PRODUCT. See inner product.
