> 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 complete text goes like this (ABC is a triangle with base BC):
"1. There may be supposed a row of quantities infinitely many, and
continually increasing, (as the supposed parallels in the triangle ABC,
reckoning downwards from A to BC,) whereof the last (BC) is given.
2. A finite quantity (as AB) may be supposed (by such continual bisections)
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?).
3. Of supposed infinites, one may be supposed greater than another. As a,
supposed, infinite number of men, may be supposed to have a greater number
of eyes.
4. A surface, or solid, may be supposed so constituted, as to be infinitely
long, but finitely great, (the breadth continually decreasing in greater in
greater proportion than the length increaseth,) and so as to have no center
of gravity. Such is Torricellio's Solidum hyperbolicum acutum; and others
innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But to
determine this, requires more of geometry, and logick (whatever it to do of
the latin tongue) than Mr. Hobs is master of.
5. There may be supposed a number infinite; that is, greater than any
assignable finite: As the supposed number of parts, arising from a supposed
section nifinitely continued.
6. There is therefore no reason, why the doctrine of Euclid, Cavallerius, or
Dr. Wallis, should be rejected as of no use."
> Another question: Is there any English translation of Wallis' Arithmetica
> Infinitorum?
As ar as I know there is no such translation, but you will find much of the
contents repeated in Wallis's "A treatise of Algebra" (1685).
Yours sincerely
Siegmund Probst
-----------------------------------
Siegmund Probst
Niedersaechsische Landesbibliothek
Leibniz-Archiv
Waterloostr. 8
D-30167 Hannover
Siegmund.Probst@zb.nlb-hannover.de
probstsieg@aol.com