Earliest Known Uses of Some of the Words of Mathematics (N)

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 Roberton 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. 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."

Neperian logarithm appears in English in An Elementary Treatise on Curves, Functions and Forces (1846) by Benjamin Peirce: "The number e is the base of Neper's system of logarithms, and the logarithms taken in this system are called the Neperian logarithms."

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 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."

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 1771 edition of the Encyclopaedia Britannica [James A. Landau].

NATURAL NUMBER. Chuquet (1484) used the term progression naturelle for the sequence 1, 2, 3, 4, etc.

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.

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.]

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."

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, ... .

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 "absurd." 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 Kersey in 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 appears in English in 1891 in G. L. Cathcart's translation of Harnack's Introductory Study of Elementary Differential and Integral Calculus (OED2).

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.

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).

NEWTON'S METHOD. In 1695, John 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. Florian 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.

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).

In 1865 Brande & Cox, Dict. Sci., etc. 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).

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).

The term NOETHER'S THEOREM is found in Sybil D. Jervis, "On professor Elliott's proof of Noether's theorem," Tohoku Math. J. 39 (1934).

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.

NON-CANTORIAN appears in Paul J. Cohen and Reuben Hersh, "Non-Cantorian Set Theory," Scientific American, December 1967 [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-NORMAL appears in R. C. Geary, "The distribution of 'Student's' ratio for non-normal samples," Supp. J. R. Statist. Soc. 3 (1936) [James A. Landau].

NONPARAMETRIC (referring to a statistical inference) first appears in a 1942 paper by Jacob Wolfowitz (1910-1981), according to the University of St. Andrews website.

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.

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 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 CURVE. According to Walker (p. 185), Karl Pearson did not coin this term. She writes, "Galton used it, as did also Lexis, and the writer has not found any reference which seems to be its first use."

The DSB says, "...Pearson's consistent and exclusive use of this term in his epoch-making publications led to its adoption throughout the statistical community."

The OED2 shows a use of "normal probability curve" 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."

"Normal curve" was used by Pearson 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 DEVIATES is found in Herman Wold, "Random normal deviates. 25,000 items compiled from Tract No. XXIV (M. G. Kendall and B. Babington Smith's tables of random sampling numbers), Tracts for Computers XXV (1948).

The term NORMAL DISTRIBUTION was first used by Galton in 1889, according to an Internet web page.

Normal distribution appears in 1897 in Proc. R. Soc. LXII. 176: "A random selection from a normal distribution" (OED2).

The term NORMAL EQUATION was introduced by Gauss, according to Stigler in Statistics Before 1827 [James A. Landau].

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 is found in Reinhold Baer, Groups with descending chain condition for normal subgroups, Duke Math. J. 16, 1-22 (1949).

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 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 n^th 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 HYPOTHESIS is used in 1935 by Ronald Aylmer Fisher in 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."

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.

NUMBER LINE is dated 1960 in MWCD10.

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 1912 in the Bulletin of the American Mathematical Society (OED2).

The words 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. 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.