[FOM] FOM Understanding Euclid
George McNulty
mcnulty at math.sc.edu
Mon Dec 8 12:59:57 EST 2008
Vaughan,
I was just looking over one of my copies of Euclid---the one published
by Green Lion Press, Heath's translation, without all the footnotes and
other scholarly stuff---and it seems to me that one could imagine that what
Euclid has in mind with the first three postulates was to describe what
could be done with straight edge and compass. Perhaps Euclid saw his second
postulate as a way of stating:
Given two distinct points p and q first draw the circle with center
at q with p on the circle. Next, take the straight edge and draw the
line segment from p through q to the opposite point r on the circle.
In this way, the line segment joining p and q is extended continuously
in a line to the segment joining p and r. More generally, in a
geometric construction, one must be able to place the straight edge
through p and q and run along it in either direction to discover
points of intersection with other elements of the construction.
Now the debatable thing is whether the point r above is on the segment
whose extremities are p and q. Consider the spherical situation where p and
q are antipodal. What would the construction above produce? The circle with
center at q and p a point on the circle would consist just of p and the
point r above would be the point p or would it? I imagine that Euclid, as
well as most of his readers, understood that this kind of extension of line
segments (or production, as I guess the translation reads) meant proper
extension: new points.
If Euclid had essentially a straightedge-and-compass viewpoint (but the
Fourth Postulate is different) but wanted to express his geometry without
explicit mention of these instruments then this presents an expository
annoyance. One of the troubles, at least from the perspective of several
millenia, is that Euclid appears to take the concepts of ``to produce a line
segment in a straight line'', ``to draw a line segment'' and ``to describe a
circle'' as a fundamental geometric notions. Nowadays we might tend to
simply replace these with blatant existence statements. This might be doing
Euclid's more active algorithmic voice a disservice. Later people introduced
betweenness as a fundamental geometric concept to help get around such
trouble (and get back to the more dignified, academic passive voice?).
It still seems to me an interesting enterprise to imagine that the front
part of the Elements is not Euclid at all and then to take the body of
propositions and try to provide a base, in the voice of Euclid, for the
whole of the Elements. This would seem to be as much a cultural, almost
archeological undertaking, as a mathematical one. It may even be that
the current front end of the Elements was the result of some such effort
perhaps a thousand years ago. Maybe Euclid's own work simply began with
Proposition I of Book I.
George McNulty
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