[FOM] Understanding Euclid

[Michael Barany]

The use of diagrams in Euclid makes for a very interesting case study.
 Thomas Heath claims that there were diagrams already in the first
printed translations (in Latin) or the Elements at the end of the
fifteenth century, and there was most certainly a trade in them in
manuscript editions well before then.  It's historically unclear how
Euclid himself would have used diagrams (the paper Jeremy gives below
mentions Netz's work on the subject).  I've been working of late with
the first English translations of Euclid, circa 1550-1570, and by then
the diagrams are both varied enough to reflect a wide variety of
source material already using diagrams of different sorts and uniform
enough to suggest a large degree of standardization as to what sort of
diagram was appropriate for which arguments.  In all but one of the
translations I'm examining the diagram for the Pythagorean theorem is
largely identical with the iconic one everyone seems to use today.

Netz makes a point which it pays to point out here, which is that in
Greek mathematics they were generally more concerned with specific
problem-solutions, rather than with universal generalities.  The
simplest explanation for why all the problems with new points and
antipodes and so forth doesn't seem to be an issue for early
Euclideans is that those problems just never came up.  This is
especially plausible if, as Jeremy seems to argue below, they were
more or less always working with neatly bounded diagrammatic sketches.
 Such was almost certainly the case in the millenia of pedagogy
responsible for the transmission of the Elements as we know them
today.

Vaughan mentioned something about distinguishing lines and line
segments in another email.  The sixteenth century authors I'm looking
at seem to have no concept of an infinite line, though some think of
lines which can be infinitely extended. (I would distinguish this from
statements that they can be extended arbitrarily far, which others
make.)  For them, lines are segments of smooth curves, as distinct
from right lines (i.e. the straight lines we see in the rest of the
text).  It can be confusing because they stop specifying that lines
need to be right lines very early in the translation.  It's still an
important distinction, e.g. for Recorde in 1551, who criticizes Durer
for stating a theorem about parallel lines when Recorde claimed it
only held for parallel right lines.  Some of these distinctions appear
almost silly to the modern eye, but by that virtue they realize how
intellectually dangerous it is to try too hard to impose our modern
notions, and even our modern language, on older authors.

I think a lot of the discussion on Vaughan's question has been quite
illuminating, but we mustn't forget that often the simplest
explanation is most historically valid: it simply didn't occur to them
that there were these sorts of problems with their arguments.  Then
you get into the fun debates about whether everyone shared certain
unstated assumptions or whether their statements of the problems
predisposed them to think one way or another, etc.

Smiles,

Michael

On Sat, Dec 6, 2008 at 5:42 PM, Jeremy Avigad <avigad at cmu.edu> wrote:
>
> Of course, there is a lot that Euclid does not state in his postulates
> and common notions, but, rather, takes to be "diagrammatically obvious,"
> or implicit in his definitions. Ed Dean, John Mumma, and I have recently
> finished a detailed analysis of Euclidean inference, where we try to
> spell out the details:
>
>   "A formal system for Euclid's Elements"
>   http://arxiv.org/abs/0810.4315
>
> That might help shed some light on where and how non-spherical
> assumptions enter into his reasoning.
>
> Best wishes,
>
> Jeremy
>