Nabla was the name suggested to [Peter Guthrie] Tait by Robertson Smith because of the similarity of the symbol to an Assyrian harp. See [1; 143]. Maxwell used the word only once in his published writings, and that was in a poem, "To the Chief Musician upon Nabla, A Tyndallic Ode." The "Chief Musician upon Nabla" was Tait. The poem was published in Nature and is given in [1; 171--174].This citation is from note 27 to chapter four of "A history of vector analysis" by Michael J. Crowe (originally published by University of Notre Dame Press in 1967, and republished by Dover in 1985). The note is on page 146 of the Dover edition. Reference [1] above is Cargil Gilston Knott, "Life and Scientific Work of Peter Guthrie Tait", Cambridge, England, 1911. (This information is from Michele Benzi.)

The term nabla was used by both Heaviside and Hamilton (Cajori vol. 2, page 135; Kline, page 780).

According to Schwartzman (p. 142) Hamilton introduced this term.

*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."

The term **NAPIERIAN LOGARITHM** presumably occurs in *Traité de
Calcul differéntiel et intégral* by Sylvestre Francois Lacroix, since
the OED2 shows its use in English in 1816 in Peacock and Herschel's translation
of of this work: "a system of logarithms, which we shall call Napierian."

*Naper's Logarithms* and *Napier's logarithm* appear in the
1771 edition of the *Encyclopaedia Britannica* [James A. Landau].

**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.** 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."

In the sense of a logarithm to be base *e,* Boyer (page 423) implies
that Pietro Mengoli (1625-1686) coined the term.

*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,John Conway writes: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."

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 "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).

Napier (c. 1600) used the adjective *defectivi* to designate negative
numbers (Smith vol. 2, page 260).

**NEIGHBORHOOD OF A POINT** appears 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 theThis 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.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-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.Cajori subsequently uses the term throughout the book. He may have written the same thing in the 1893 edition, which I have not seen.

**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).

**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.

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).

**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.

**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 a^{2} + b^{2}
for the complex number *a* + *ib.*

In 1921 the term *norm* refers to sqrt (a^{2} + b^{2})
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 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 3µ_{2}^{2} = µ_{4}
and µ_{3} = 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."

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

**NORMAL EQUATION** occurs in "On the Reducibility of Linear Groups,"
Leonard Eugene Dickson, *Transactions of the American Mathematical Society,*
Vol. 4, No. 4. (Oct., 1903).

**NORMALIZER** appears in German as *Normalisator* in *Theorie
der Gruppen von endlicher Ordnung,* 2nd ed., by A. Speiser [Huw Davies].

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

**NTH** is dated 1852 in MWCD10.

**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 *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.** Sylvester defined the nullity of a square matrix in
1884.

**NULL SET** appears (as "null-set") in 1906 in *Theory of Sets
and Points* by W. H. and G. C. Young (OED2).

**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** 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 D. Jackson, "The Method of Numerical
Integration in Exterior Ballistics," *War Dept. Document 984,* U.S.
Govt. Printing Office (1921) and in C. Störmer, "Méthode d'intégration
numérique des équations différentielles ordinaires," *C. R. Congr. Intern.
Math. Strasbourg 1920, Toulouse, Privat,* (1921) [James A. Landau].