Where did John Wallis get the idea for $\infty$?

Question

I read in an offhand comment in Amir Alexander's 2014 book Infinitesimal (p.280), that John Wallis introduced the symbol $\infty$ for infinity.

Was there any logic, reason, or precedent for this symbol, or did he (as far as we know) just make it up?

Answer 1

from "Earliest Uses of Various Mathematical Symbols"

Infinity. The $\infty$ symbol was introduced by John Wallis (1616-1703) in 1655 in his De sectionibus conicis (On Conic Sections) as follows:

Suppono in limine (juxta Bonaventurae Cavallerii Geometriam Indivisibilium) Planum quodlibet quasi ex infinitis lineis parallelis conflari: Vel potius (quod ego mallem) ex infinitis Prallelogrammis [sic] aeque altis; quorum quidem singulorum altitudo sit totius altitudinis $\frac 1 \infty$, sive aliquota pars infinite parva; (esto enim $\infty$ nota numeri infiniti;) adeoq; omnium simul altitude aequalis altitudini figurae.

Wallis also used the infinity symbol in various passages of his Arithmetica infinitorum (Arithmetic of Infinites) (1655 or 1656). For instance, he wrote (p. 70):

Cum enim primus terminus in serie Primanorum sit $0$, primus terminus in serie reciproca erit $\infty$ vel infinitus: (sicut, in divisione, si diviso sit $0$, quotiens erit infinitus.)

In Zero to Lazy Eight, Alexander Humez, Nicholas Humez, and Joseph Maguire write: "Wallis was a classical scholar and it is possible that he derived $\infty$ [the infinity symbol] from the old Roman sign for 1,000, CD, also written M--though it is also possible that he got the idea from the lowercase omega, omega being the last letter of the Greek alphabet and thus a metaphor of long standing for the upper limit, the end."

Cajori (vol. 2, p 44) says the conjecture has been made that Wallis adopted this symbol from the late Roman symbol $\infty$ for 1,000. He attributes the conjecture to Wilhelm Wattenbach (1819-1897), Anleitung zur lateinischen Paläographie 2. Aufl., Leipzig: S. Hirzel, 1872. Appendix: p. 41.

This conjecture is lent credence by the labels inscribed on a Roman hand abacus stored at the Bibliothèque Nationale in Paris. A plaster cast of this abacus is shown in a photo on page 305 of the English translation of Karl Menninger’s Number Words and Number Symbols; at the time, the cast was held in the Cabinet des Médailles in Paris. The photo reveals that the column devoted to 1000 on this abacus is inscribed with a symbol quite close in shape to the lemniscate symbol, and which Menninger shows would easily have evolved into the symbol M, the eventual Roman symbol for 1000 [Randy K. Schwartz].

(Note, see the actual link for things that don't come out after copy and paste.)

Answer 2

A vague conjecture is that the symbol was taken from a Roman inscription containing the symbol for 100 millions

.

But Wallis never visited Italy. So he can hardly have seen it.

On the other hand, Wallis was a cleric, and 8 is an important element of Christian symbolism, among others standing for eternity. It may have happened that the cleric Wallis just took this 8 and turned it in order to distinguish it from the ordinary 8. (W. Mückenheim: Script to the lecture Die Geschichte des Unendlichen, 7th ed., Maro-Verlag, Augsburg (2011) p. 15)