...
>1. To my knowledge, the infinity symbol was first introduced by John Wallis
>(1616-1703) in 1655 in his _De sectionibus conicis_ (On Conic Sections) as
>follows:
>...
>David E. Smith (1860-1944) remarked that "the symbol for infinity (oo) is
>first found in print in the _Arithmetica Infinitorum_ published in 1655..."
>["History of Mathematics", vol. II, New York: Dover Publications,
>Inc., 1958. See p. 413]
I have long been trying to find out why I have seen (or rather, heard) this
assertion made, and am very pleased to have, at last, this reference. Does
anyone know if it originated with Smith?
>Now, whereas the infinity symbol certainly appears in the _De sectionibus
>conicis_ (1655) and in the _Arithmetica infinitorum_ (1655/56?), some people
>believe that the latter was published earlier. I do not actually know
>which opus first saw the light; however, it must be noted that on page 3
>of the _Arithmetica_, when Wallis discusses a 'corollarium' on triangles
>and parallelograms, he explicitly quotes his treatise ['libri nostri']
>_De sectionibus conicis_ as if this tract were (or should be) already
>known to the readers:
If I remember correctly, I did check the official history of Oxford
University Press (I don't have the reference here, in France, but can find
it when I return), which said that they did come out on the dates on their
title pages:
_De sectionibus conicis_ in 1655
_Arithmetica infinitorum_ in 1656
There is a photograph of one of the tables in the _Arithmetica infinitorum_,
which contains an infinity symbol, in my article
The binomial coefficient function, American Mathematical Monthly, 103
(January 1996), 1-17,
though this is not the first place in the treatise where the symbol
appears. This reproduction comes from the fairly rare first edition; this
is a little octavo volume, and the printer obviously had to do a lot of
improvisation to set the table. The second edition, in the Opera, is a much
better printed large folio. (Again, if I remember correctly.)
I assert in my article that Wallis has stumbled on a corner of something
that few people seem to be aware of today, and that he is the first to
publish it until ... (?, perhaps even this article) that, if we generalise
(n choose m) to (x choose y) by generalising n! to x!, then, for all y,
(x choose y) -> infinity in a rather complicated way as x -> 0. (The
initial figures in this article gave been mangled by the printer and I can
send proper ones to anyone who wants.)
>2. As far as I know, Wallis never explained why *his* symbol for infinity
>was shaped as such. Therefore, my answer to your implicit question is that
>we simply do not *really* know why Wallis made his choice.
I'd put that more strongly and say, in the absence of any explanation by
Wallis, and if there is no clearly undeniable explanation, we simply do not
know why Wallis made his choice. I.e. omit the *really*.
...
>Julio GC
As someone who has scratched around trying to find out this kind of
information, I'd say that this note is something worth publishing
somewhere. I'm sure it would also be of interest and use to others.
David Fowler