Subject: Re: [HM] Kant and non-Euclidean geometry
From: Colin McLarty (cxm7@po.cwru.edu)
Date: Sun Mar 12 2000 - 16:22:58 EST
The case of Lambert on non-euclidean geometry is very subtle, it
seems to me. Lambert's posthumous paper on it (I do not have the cites at
home with me, but it is the one published in 1786 referred to before)
examines whether Euclid's parallel postulate can be derived from Euclid's
other postulates. It is unfinished and especially the later parts are clearly
casting about to find the decisive considerations. He nicely shows that if
you take all of Euclid's postulates except the parallel postulate, you can
prove that just one of three things must happen:
The hypothesis of the obtuse angle. In short, this says the sum of the
angles in any triangle is more than 180 degrees.
The hypothesis of the right angle. The Euclidean case. The sum of angles in
a triangle is always 180 degrees.
The hypothesis of the acute angle. In short, the angles of a triangle sum
to less than 180.
Lambert proves the impossibility of "the obtuse angle" (i.e. geometry
with no parallel lines, where the angles of a triangle sum to more that 180
degrees). He does that by in effect showing it contradicts the postulate
that a line can always be extended--which he understands to mean that a
line segment can always be extended WITHOUT RETURNING TO ITSELF.
Of course he was extremely familiar with geometry on a sphere, and had
vast experience with spherical trigonometry. He knew very well that a
triangle of great circles on a sphere has angle sum over 180, and that two
segments of great circles will enclose a region on the sphere. And that an
arc of great circle does return to itself if extended far enough.
So Lambert knew well that such a "space" was possible, and even
important in astronomy; although it violates another postulate of Euclid's
besides the parallel postulate, and so is ruled out in this investigation
of the parallel postulate. Did Lambert think physical space could have such a
geometry? And did he influence Kant on this? Before going to that, let me
describe the other non-Euclidean hypothesis.
Lambert was left with the "acute angle" (where the angles of a triangle
sum to less than 180--our current hyperbolic geometry). He made a number of
efforts to prove this impossible. I think I agree with Jeremy Gray in
saying that the dominant tone at the end of the paper is that he feels it
is impossible. But he makes an intriguing remark, that such a hypothesis
looks like "geometry on a sphere of imaginary radius".
Given Lambert's other work, it is extremely likely that he was serious
about this and had a good idea what it meant: at least that it meant doing
an analogue to spherical trigonometry, but with imaginary rather than real
radius. But I think he supposed that even if this hypothesis of the acute
angle WAS realized on imaginary spheres, geometry on those "spheres" would
contradict other Euclidean postulates. He just could not find how.
So, did Lambert think his non-Euclidean hypotheses could apply to
physical space? Along with much of his math, and his COSMOLOGICAL LETTERS,
I have read his philosophic books the NEW ORGANON and the FOUNDATIONS OF
ARCHITECTONIC, and much else in his Oeuvres, and I found no serious clues.
Nor do I find any significant trace of the questions in a place that Kant
could have seen before 1786.
I will say that Kant was well equipped to understand the issue, at
least in the treatment of the hypothesis of the obtuse angle--and that is
the one that would have influenced his choice of examples in the CRITIQUE
OF PURE REASON as Bill Tait cited. In my heart I suspect that Kant did hear
about Lambert's work on this at third hand. But surely Lambert and Kant
never met personally (there is no evidence that they did, and it is hardly
likely that Lambert dropped by Koenigsberg without leaving a trace of it).
And as I said I do not see it in their correspondence nor in publications of
Lambert during his lifetime. And surely there were few people at that time
capable of understanding the issue well enough to convey it from Lambert to
Kant.
Of course I'd be thrilled if recent Kant scholarship has turned up
documentary sources unknown in the early 1980s, when I worked on this. What
someone could do, and perhaps someone has, is to see if Kant's views on
Euclidean geometry changed around 1786--since Kant might every well have
seen Lambert's paper when it was printed.
Colin McLarty
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