FOM: Re: different sizes of infinity. W.Taylor at
Thu Jun 20 03:29:01 EDT 2002

Just a lurker's comment about early discoveries of the equipollence of
sets and their proper subsets.

I seem to recall reading in a popularization that the classical Greeks
had also observed this in passing.  Specifically, they noted that two
parallel line segments, with corresponding endpoints projected from
a single point of projection, (a very common projective geometry sketch!)
had that same property.  By pairing each point with its mate from
the projectivity, they noted that the two unequal segenments must have
"the same number of points" in them.  Like Galileo, they didn't really
know what to do about this, so they ignored it.

Does anyone else recall this appearing in any classic text, or being
relayed by a modern author?

Also; many thanks to Elizabeth Theta Brown for her reference to Albert
of Saxony; judging from her middle name Elizabeth must have had a good
early start in math or Greek (or both).  Thanks also to the poster with
the Grossteste reference.  (I'd heard of Grossteste before, I wonder if
he fulfilled *his* name!)

        Bill Taylor                  W.Taylor at

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