# [FOM] FOM Understanding Euclid

Colin McLarty colin.mclarty at case.edu
Sun Dec 7 13:25:37 EST 2008

``` ---- Original Message -----
>From  	Vaughan Pratt <pratt at cs.stanford.edu>
Date  	Sat, 06 Dec 2008 02:35:53 -0800
To  	Foundations of Mathematics <fom at cs.nyu.edu>
Subject  	Re: [FOM] FOM Understanding Euclid

When I said Euclid's postulate that a line segment can always be
extended was historically understood to mean "extended to new points,"
and that is how Euclid proves Proposition 16 (in effect ruling out the
"no parallels" case, or spherical geometry) Vaughan wrote

> By "he proves," do you mean Euclid himself or those looking for some
> justification for his reasoning?

Yes, Euclid gives this proof.

> Where does Euclid exploit the no-new-points premise in his proof?

He says to double a certain line segment, and asserts that the
constructed endpoint is not in the original segment.

> How plausible is it, really, that this premise is what Euclid had in
> mind at the time?

The summary that I just gave is virtually verbatim Euclid -- okay, it is
translated into English -- and thus to my mind it is a plausible
interpretation.

> There are just too many missing facts and logical connections to make
> this explanation plausible.
>
> 1.  Unless I'm missing something Euclid says nothing about new points
> anywhere.

When he constructs a point which is not any of the points he started
with, then that is a "new" point.

> 2.  He proves Proposition 16 by appealing to the diagram,

Certainly Euclid is inexplicit on order relations on a line, and this
proof shares the inexplicitness.  If your concern is to show that Euclid
was not an infallible logician then we are all set.  Surely everyone
agrees on that.

But if your concern is to know what Euclid and later geometers were
doing with the parallel postulate, then you need to understand an
historical fact: geometers up to the 19th century were essentially
unanimous in understanding the postulate on extending a line segment
to mean that the points you get that way are not among the points you
started with -- in my terminology they are "new" points.

To my comment that Johann Heinrich Lambert understood Euclid's postulate
on extending lines this way, Vaughan says:

> What was the alternative?  To say that Euclid's argument was unsound?
> Let those who have never argued unsoundly cast the first stone.

Well, Lambert was doing geometry, not history.  His work on the parallel
postulate shows a surprisingly fine understanding of logical axiomatic
rigor, but so far as I recall he was not interested in critiquing
Euclid's rigor.

For Lambert, the alternative to understanding the postulate this way was
to stop doing plane geometry.  Lambert was very good at spherical
geometry, including spherical trigonometry.  He even arrived at the idea
that what we now call hyperbolic geometry was "geometry on a sphere of
imaginary radius (what we would now call negative curvature)."  But he
considered spherical geometry not plane geometry -- and he apparently
thought the same of geometry with imaginary radius but the work is just
unfinished notes and he died young without ever expanding on the idea of

He quite explicitly rules out the "hypothesis of the obtuse angle" (i.e.
what we would call positive curvature) from his plane geometry by
showing it contradicts the extensibility of line segments.

There is more than an historical question here.  The relation between
Euclidean and hyperbolic geometry really is about as simple as one
parallel or many.  But the relation to spherical geometry is much more
complicated, precisely because you have to drop the whole theory of
"betweenness" for points on a line and replace it with a four-place
relation "points x and y separate points u and v."

best, Colin

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