There is an interesting article by Antoni Malet "Barrow, Wallis, and the
Remaking of the Seventeenth Century Indivisibles." (Centaurus 1997, ISSN
0008-8994).
We can read about Thomas Hobbes' critics of Wallis' infinitesimals.
Hobbes asked Wallis
"whether a Finite Quantity can be divided into an Infinite Number of lesser
Quantities, or a Finite Quantity consist of an Infinite Number of Parts."
Wallis answer in the Philosophical Transactions in 1671 was
"A finite Quantity -- may be supposed -- divisible into a number of parts
Infinitely many (or, more than any Finite number assignable) ... And, all
those Parts were in the undivided whole; (else, where should they be had?)"
I suppose the three dots (in the middle of the answer) means that some of
Wallis' answer is excluded by Mr Malet. Can anyone tell me the missing
words?
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?
Another question: Is there any English translation of Wallis' Arithmetica
Infinitorum?
Best greetings
Staffan Rodhe
Department of Mathematics
University of Uppsala