| I know that the infinity symbol is attributed to John Wallis, but can't
| find anything on why it's shaped as it is. I have heard that it might have
| been derived from the lowercase omega, but cannot document that.
|
| Thanks for any help you can provide.
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:
P A R S P R I M A [page 4]
PROP. I
De Figuris planis juxta Indivisibilium
methodum considerandis
Suppono in limine (juxta^ Bonaventurae Cavallerii _Geometriam
Indivisibilium_) Planum quodlibet quasi ex infinitis lineis
parallelis conflari: Vel potiu\s (quod ego mallem) ex infinitis
Prallelogrammis [sic] aeque\ altis; quorum quidem singulorum
altitudo sit totius altitudinis 1/oo [symbol], sive alicuota pars
infinite parva; (esto enim oo [symbol] nota numeri infiniti;)
adeo/q; omnium simul altitude aequalis altitudini figurae.
John Wallis also used the infinity symbol in various passages of his
_Arithmetica infinitorum_ (Arithmetic of Infinites). For instance, he
wrote (p. 70):
Cum enim primus terminus in serie Primanorum sit 0, primus
terminus in serie reciproca erit oo [symbol] vel infinitus:
(sicut, in divisione, si diviso sit 0, quotiens erit infinitus)
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]
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:
[...] ut ostendimus ad pr. 1. & 2. libri nostri de Conicis Sectionibus;
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.
3. Conjectures.
As Ken Pledger suggested on-list (and John Bibby, off-list), Cajori's
"History of Mathematical Notations" seems to be a first step to start
a search for your query:
"The conjecture [*] has been made that Wallis, who was a classical
scholar, adopted this sign from late Roman symbol oo for 1,000."
[*] W. Wattenbach, _Anleitung zur lateinischen Pala"ographie_ 2. Aufl.,
Leipzig: S. Hirzel, 1872. Appendix: p. 41.
[Cajori, Florian (1859-1930): "A History of Mathematical Notations",
vol. II (two volumes bound as one), New York: Dover Publications, Inc.,
1993. See p. 44]
By the way, the conjecture of Wilhelm Wattenbach (1819-1897) seems to be
tacitly accepted by Karl Menninger when he remarked:
"For 1000 the Romans could also use the curious form oo [symbol],
which ever since the English mathematician Wallis proposed it in
1655 has been accepted as the mathematical symbol for infinity
(see Fig. 73)."
[Incidentally, Fig. 73 is a photograph of an inscription (Scavi di Ostia,
Rome) dating from the year 36 AD where the Roman numeral for 100 million
consists of the symbol oo placed within a frame, which itself originated
from an extension of the numeral for 100,000. Cf. Menninger, Karl: "Number
Words and Number Symbols: A Cultural History of Numbers", New York: Dover
Publications, Inc., 1969; see p. 245. Please mind that this edition is NOT
an unabridged translation of the German edition of "Zahlwort und Ziffer:
eine Kulturgeschichte der Zahlen", which was originally published in two
volumes by Vandenhoek & Ruprecht Publishing Company, Go"tingen, Germany,
1957-58. For instance, the bibliography has been left out in the English
edition!]
A plausible explanation of how the late Roman symbol oo develops into the
'modern' Roman M (= 1000) can be found, for instance, on page 27 of Sanchez
Perez's "La aritmetica en Roma, en India y en Arabia" [Madrid: Consejo
Superior de Investigaciones Cientificas, Instituto _Miguel Asin_, Escuelas
de Estudios Arabes de Madrid y Granada, 1949].
Other writers have also speculated that it is somehow plausible that Wallis
got the idea for oo from the lowercase omega -- being this the last letter
of the Greek alphabet it could well be imagined as a "last" number. Hmmm...
4. The notations oo [symbol] and lowercase omega can be found in Cantor's
writings. It might be worth recalling that Georg Cantor (1845-1918) used
the symbol oo in "Ueber unendliche, lineare Punktmannigfaltigkeiten [2]",
_Mathematische Annalen_, vol. 17, pp. 355-358, 1880; see p. 357.
About two years later, Cantor introduced his lowercase omega instead of
the oo. As a matter of fact, this seemingly innocent shift in notation
meant a remarkable conceptual change! Cf. "Ueber unendliche, lineare
Punktmannigfaltigkeiten [5]", _Mathematische Annalen_, vol. 21, pp. 545-586,
1883; see p. 577.
Julio GC