[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

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