Subject: [HM] translating segments (was: angle names)
From: Gordon Fisher (gfisher@shentel.net)
Date: Wed Dec 22 1999 - 18:01:13 EST
[ Gordon Fisher ]
> To what extent should the ancients be credited with a knowledge or
> implicit use of linear transformations? In the above, one might argue
> that a rotation is being applied. There is, for example, Euclid's
> notorious proof ... And he wasn't at all adverse to translating
> segments, etc.
[ David Fowler ]
> Euclid sets up the translation of segments right at the beginning,
> Book I, Props 1-3.
[ Christian Marinus Taisbak ]
> Does he, now ? If I understand "translation", it means some kind
> of movement, which is certainly forbidden in the Elements. Euclid
> demonstrates how to =copy= a given segment by his magnificent
> copy-machine in I.2 (the copy even comes out upside down !)
> Do not bother about the apparent movement in I.4. That can be
> "explained away" (as I did in Luminy Oct. '95 and will repeat in
> my comments on the =Data= if I live.)
Actually, I was using the word "translation" here in a general sense,
referring to a kind of idealized movement (in one's mind, say), rather
than in a modern, technical sense, as in a formulation using linear
algebra, or the concept of homeomorphism, or continuity in some form,
etc.
[ David Fowler ]
> I'm reluctant to talk of 'linear transformations'. This is an
> importation of today into the past, and it may not stop there.
As to David's comment on my comments on this question, I'm well aware
of problems with anachronism in studying ancient Greek mathematics.
I didn't intend in any way to suggest that the Greeks had somehow
anticipated later developments in geometry of various European kinds,
or even that they somehow knew what we know already. I was rather
wondering to what extent one might attribute to them an unacknowledged
or even unconscious use of motion in their geometric thinking. I think
it might even be anachronistic to suggest that they didn't do so, perhaps
because of their philosophical concerns about the nature of time, or
something of that sort. Christian, you say you can explain the apparent
use of movement in I.4. I don't doubt this, but can you justify with
documentary evidence that this was the sort of thing Euclid, or possibly
one or more of his later editors, had in mind?
Gordon Fisher gfisher@shentel.net
This archive was generated by hypermail 2b28 : Tue Jan 04 2000 - 09:28:19 EST