"A Finite Quantity (as AB) may be supposed (by such continual Bijections)
divisible into a number of parts Infinitely many (or, more than
any Finite number assignable.) For there is no stint beyond which
such division may not be supposed to be continued; (for still the last,
how small soever, will have two halves.) And, all those Parts were in
the Undivided whole; (else, where should they be had?)"
> The words "may be supposed" could be compared with the aristotelian
> "become" (infinite).There is a motion involved in the process of drawing
> the line. Among others Galilei had the same opinion. I suppose Cavalieri's
> line was more static and was composed of an infinite number of
> (zero-dimensional, indivisible) points.
>
> It seems to me that Wallis' infinity includes a nice mixture between the
> potential infinity of Aristoteles and the indivisibles of Cavalieri. Do
> you agree?
I think Wallis's discussion will clarify things:
"...usually in Euclide, and all after him, by 'Infinite' is meant
but, 'More than any assignable Finite', though not Absolutely Infinite,
or the greatest possible. Nor do they mean, when Infinites are propoosed,
that they should actually Be, or be possible to be performed; but only,
that they be supposed. (It being usual with them, upon supposition of
things Impossible, to infer useful Truths.) And Euclid (in his second
Postulate) requiring, the producing a steight line Infinitely, either way;
did not mean, that it should be actually performed (for it is not possible
for any man to produce a streight line Infinitely;) but that it be supposed.
... Again, when (by Euclide's tenth Proposition) the same AB may be Bisected
in M, and each of the halves in m, and so onwards, Infinitely: it is not his
meaning (when such continual section is proposed) that it should be actually
done, (for, who can do it?), but that it be supposed. And upon such
(supposed) section infinitely continued, the parts must be (supposed)
infinitely many; for no Finite number of parts would suffice for Infinite
sections. ... And this I say, to shew that the supposition of Infinites
(with these attendants) is not so new, or so Peculiar to Cavallerius or Dr.
Wallis, but that Euclide admits it, and all Mathematicians with him; as at
least supposable, whether Possible or not."
> Another question: Is there any English translation of Wallis' Arithmetica
> Infinitorum?
>
> Best greetings
>
> Staffan Rodhe
> Department of Mathematics
> University of Uppsala
If you send me a mailing address, I could put a copy of the Wallis
paper into the mail to you.
-Mark McKinzie-
mckinzie@math.wisc.edu