# [FOM] Who coined the term ''natural number''?

Vaughan Pratt pratt at cs.stanford.edu
Sun Aug 22 03:28:19 EDT 2010

```On 8/20/2010 3:02 AM, Arnold Neumaier wrote:
> I have two historical questions:
>
> 1. Who coined the term ''natural number''?

To answer this literally one would have to start with Napier, who used
the term in 1614 to distinguish numbers from their logarithms, which he
viewed as artificial numbers.  But of course that's not what you meant.

It's not clear to me whether Euler had Napier's meaning in mind in his
1743 monograph "De summis serierum reciprocarum ex potestatibus
numerorum naturalium ortarum," "On the sums of reciprocal series arising
from the powers of the natural numbers," which expressed exp(z) as a sum
of powers of z where z could be complex (and so obtained the trig
functions), or whether he was referring to the exponents of the powers,
which were nonnegative integers, or something else again (since he also
had infinite and infinitesimal numbers in his treatment, namely when he
sets exp(z) = (1 + z/n)^n "when n emerges an infinite number.").

In any event Jacques Bernoulli predates him in his posthumously
published work "Ars Conjectandi," 1715, which among other things
introduces the Bernoulli numbers.  Bernoulli writes (in Latin) "Let the
series of natural numbers 1, 2, 3, 4, 5, etc. up to n be given, ..."
(Translated by Prof. Jekuthiel Ginsburg of Yeshiva College, NYC --- I
don't have the Latin to confirm that the original for "of natural
numbers" was "numerorum naturalium" but it is at least plausible given
Euler's example above.)

Some 16 pages later in the same volume, still in Latin but now
translated by Mary M. Taylor, U. Pittsburgh, Bernoulli writes "From
this, with no great difficulty, we infer that the single terms of all
the series form a group of ones; the binaries a series of positive
integers (or natural numbers); ..."

In both of the Bernoulli examples it is clear that he excluded 0 as a
natural number, at least on those two occasions.

All four of these items can be found in David Eugene Smith's "A Source
Book in Mathematics," on pages 151, 96, 87, and 273 respectively in the
Dover edition.

None of this establishes that Bernoulli was first.  However Robert
Recorde in the Grounde of Artes in 1540 refers to whole numbers as
"unities."  Furthermore Napier would appear to have seen no
terminological conflict in 1614 in referring to "non-logarithms" as
being "natural numbers."  So this would seem to narrow the introduction
of the term with more or less its present meaning down to the window
1614-1705, the latter being the year of Bernoulli's death.

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

Apparently not Bernoulli.  But this might be hard to answer were there a
period when the question was not considered important enough to insist
on ruling out zero as a natural number.  Perhaps even Bernoulli himself
might not have insisted on it.

Vaughan Pratt
```