# [FOM] : Who coined the terms "natural number"?

Irving ianellis at iupui.edu
Sat Aug 21 13:40:49 EDT 2010

```Arnold Neumaier wrote:

I have two historical questions:

1. Who coined the term ''natural number''?

2. Who was the first to call 0 a natural number(rather than a cardinal
number)?

Nicolas Chuquet (1445-1488) used the term "progression naturelle" in
Triparty en la science des nombres (Lyon, 1484) for the sequence 1, 2,
3, 4, etc.; see p. 350, Moritz Cantor, Vorlesungen über Geschichte der
Mathematik: Von 1200-1668 (Leipzig: B. G. Teubner, 2te Aufl., 1900).

Bernard le Bouyer de Fontenelle (1657-1757) wrote in Elemens de la
Geometrie de l'infini (Paris, Imprimerie royale, 1727), p. 29: "Pour
mieux concevoir l'Infini, je considere la suite naturelle des nombres,
dont l'origine est 0 ou 1." The expression "la suite naturelle des
nombres" is introduced on p. 13.

"Natural number" appears in The Method of Increments (London: Nours,
1763) by William Emerson (1701-1782), p. 113: "To find the product of
all natural numbers from 1 to 100."

"Natural number", defined as the numbers 1, 2, 3, 4, 5, etc., appears
in the 1771 Encyclopaedia Britannica in the "Logarithm" article.

In Elements of Algebra (Boston: Ginn & Co., 1881; 1896) by George
Albert Wentworth (1835-1906), p. 2 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."

Bertrand Russell in his Introduction to Mathematical Philosophy
(London: Unwin/N.Y. Macmillan 1919, 2nd ed., 1920; Dever reprint, 1993)
wrote (pp. 2-3):

"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.""

The term cardinal number (Kardinalzahl), at least as we understand it
and as I am guessing is meant in the context of the question, is much
later, starting withn Georg Cantor's "Ueber eine Eigenschaft des
Inbegriffes aller reellen algebraischen Zahlen" of 1874; see esp.
Cantor's "Mitteilungen zur Lehre vom Transfiniten", Zeitschrift für
Philosophie und philosophische Kritik 91 (1887), 81-125, 92 (1888),
240–265.

However, Richard Percival in 1591 in Bibliotheca Hispanica wrote: "The
numerals are either Cardinall, that is, principall, vpon which the rest
depend, ...."

Richard Percival in Richard Percival & Thomas D'Oylie, Bibliotheca
Hispanica: containing a grammar, with a dictionarie in Spanish,
English, and Latine, gathered out of diuers good authors, very
profitable for the studious of the Spanish toong (London, By Iohn
Iackson, for Richard Watkins, 1591) wrote: "The numerals are either
Cardinall, that is, principall, vpon which the rest depend, ...."

Irving H. Anellis
Visiting Research Associate
Peirce Edition, Institute for American Thought
902 W. New York St.
Indiana University-Purdue University at Indianapolis
Indianapolis, IN 46202-5159
USA
URL: http://www.irvinganellis.info

```