[FOM] Understanding Euclid

[William Tait]

On Dec 5, 2008, at 3:13 AM, Vaughan Pratt wrote:

> Bill,
>
> Derivable from what?  Proposition 32?  In my message I fingered
> Proposition 32 as a counterexample to spherical geometry along with
> Proposition 47.  We're in agreement that Proposition 32 does not  
> hold in
> spherical geometry.
>
> The difficulty I'm having is that I don't see how any of Euclid's five
> postulates could fail in spherical geometry.  My point was that
> propositions such as 32 and 47, which clearly don't hold in spherical
> geometry, can't follow logically from postulates all of which hold in
> spherical geometry.
>
> Which one of Euclid's five postulates fails in spherical geometry?

The first postulate, as it is actually used (and likely as it was  
intended by Euclid: Who knows?), fails. Namely, the straight line  
connecting two distinct points will not be unique in spherical  
geometry when the points are antipodal.  Postulate 3, that there is a  
circle with a given point as center and a given straight line segment  
as radius, is true only if one admits a point to be a circle. (Let the  
radius be half a great circle.)

What goes wrong with the proof of Proposition 32 is in the  
construction of a parallel to a line from a point not on it. This  
depends on the construction of a perpendicular to a straight line from  
a point on it: Proposition 11. This depends on Proposition 1 which  
assumes APPARENTLY WITHOUT PROOF that, if A and B are distinct points,  
the circle with center A and radius AB intersects the circle with  
center B and the same radius. This of course fails in spherical  
geometry with A and B antipodal  points.

If Euclid intended that the circle be a line (i.e. curve), one can  
construe the definitions to imply something like Dedekind continuity  
for lines (understanding the statement that the 'extremities' of lines  
are points to mean that however you cut the line, it determines a  
point). On this reading, Postulate 3 is false in spherical geometry  
and Euclid's proof of Proposition 1 is OK.

So Postulates 1 and 3 COULD be wrong in spherical geometry, depending  
on what Euclid meant. But taken literally in the version that we have,  
I agree that they hold and that the proof of Proposition 1 and  
therefore of Proposition 32 that is defective,.

Bill