[FOM] Paolo Mancosu's monograph on 17th century mathematics

[wwtx@earthlink.net]

On Monday, March 24, 2003, at 04:17  PM, Martin Davis wrote:

> I found most striking the general reaction to a result by Torricelli 
> (the scientist who took over Galileo's chair in Florence). Torricelli 
> computed the finite volume enclosed by a particular infinite surface 
> of revolution. (In modern terms: the solid is formed by rotating about 
> the X-axis the rectangular hyperbola y = 1/x with x >= 1.)  This 
> result (today a homework problem in a standard calculus course) 
> created a sensation. I recommend to all Mancosu's discussion of the 
> way in which Torricelli's result played havoc with the received wisdom 
> of the day about the actual infinite.

Without in the least taking away from Martin's point in the  paragraph 
following this one, I want to note that this kind of example (i.e. 
unbounded objects of finite magnitude) actually had its roots in the 
14th century. In his _Liber calculationum_, Richard Swineshead [I can 
picture him now] gives the following example. Imagine a rod [0,1]. Its 
density of heat is constantly 1 degree on  [0, 1/2], 2 degrees on [1/2, 
3/4], etc. He argues that the total quantity of heat  in the rod is 2. 
So there is no absolute bound on the density of heat of the segments, 
but the total heat is still finite. He does not make the geometric 
representation, which again would be an unbounded figure with finite 
area;  but it is fairly clear that it was in the background [Namely, 
his proof was clearly an infinite cut-and-past construction). (Why 
didn't he give the geometric case? Maybe because it would have fallen 
afoul Aristotelian/Church doctrine that space is bounded.) He also 
noted that if one added any fixed positive epsilon to the increase of 
heat  of each segment over its predecessor, then the total heat is 
infinite.

There is a brief discussion of this in John E. Murdoch's ``Infinity and 
continuity'' in _the Cambridge History of Later Medieval Philosophy_ 
(ed. N. Kretzmann et al).

Bill Tait