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CM360X Notes — Non-Euclidean Geometry.
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Some views.
Most people are unaware that around a century and a half ago a revolution took
place in the field of geometry that was as scientifically profound as the Copernican revolution in astronomy and, in its impact, as philosophically important
as the Darwinian theory of evolution. [Greenberg (1974), ix.]
If a straight line, falling on two other straight lines, makes the two interior
angles on one side less than two right angles, then the two straight lines, produced indefinitely, will meet on the side on which are the two angles less than
two right angles. [Euclid, Elements, Postulate 5.]
This [postulate 5] ought to be struck from the postulates altogether. For
it is a theorem — one that raises many questions, which Ptolemy proposed to
resolve in one of his books — and requires for its demonstration a number of
definitions as well as theorems. [Proclus, (1970), cited in Fauvel and Gray 3.B1.]
It is no more conceivable, again, that the infinite should exist as a whole
of similar parts. For, in the first place, there is no other (straight) movement
beyond those mentioned: we must therefore give it one of them. And if so, we
shall have to admit either infinite weight or infinite lightness. Nor, secondly,
could the body whose movement is circular be infinite, since it is impossible for
the infinite to move in a circle. This, indeed, would be as good as saying that
the heavens are infinite, which we have shown to be impossible. [Aristotle, De
Caelo, I.7.]
As lines (so loves) oblique may well
Themselves in every corner greet:
But ours so truly parallel,
Though infinite, can never meet.
[Andrew Marvell, The Definition of Love (c.1650).
I discovered [about 1890] that in addition to Euclidean geometry there were
various non-Euclidean geometries, and that no one knew which was right. If
mathematics was doubtful, how much more doubtful ethics must be! [Russell,
cited in Richards (1988), p.204-5.]
In fact, one sees not only that no contradiction is reached, but one soon
feels oneself facing an open deduction. While a problem given a proof by contradiction should head fairly quickly for a conclusion where the contradiction
is clear, the deductive work of Lobachevsky’s dialectic settles itself more and
more solidly in the mind of the reader. Psychologically speaking, there is no
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more reason to expect a contradiction from Lobachevsky than from Euclid. This
equivalence will no doubt later be technically proved thanks to the work of Klein
and Poincaré; but it is already present at the psychological level. [Bachelard
(1934) p.30.]
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Some questions.
1. What is geometry about — what is its subject-matter?
2. How do we know that its results are true?
3. How does the theorem that the angles of a triangle add up to two right angles
follow from the parallel postulate?
4. How would you prove Girard’s theorem on ‘spherical excess’; that, on a sphere
of radius R, in a triangle ABC whose sides are arcs of great circles (‘straight
lines’):
angle A + angle B +angle C −π is equal to [area (ABC)]/R2 ?
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The story — standard version.
1. From the beginning of Euclid’s geometry, and posssibly even earlier, dissatisfaction with his apparently perfect system was centred on the so-called ‘parallel
postulate’. This says (in one version) that if the angles α, β in Fig.1 add up
to less than two right angles, then the lines l, l0 meet. Another version, which
is perhaps easier to understand, is ‘Playfair’s axiom’: there is a unique straight
line through A which is parallel to l0 , (does not meet it); and this line makes
the angles add up to two right angles as stated. It was felt that this was not
intuitively obvious, and should be provable using the other axioms, or from ‘first
principles’.
[It was, on the other hand, reasoned that if the angles added up to two right
angles exactly, then AB, DE would not meet (they would be parallel). A quick
way of ‘seeing’ this is as follows. If the angles on one side are two right angles,
so are the angles on the other. If the lines meet on one side, then by symmetry
they must meet on the other side too. But this implies that there is not a
unique straight line joining two points (the two points where they meet), which
is unreasonable.]
2. For roughly two millennia there were attempts to prove the postulate.
Recorded efforts were made by Proclus (5th cent.), Thābit ibn Qurra (9th cent.),
ibn al-Haytham (10th cent.) Khayyam (11th cent.), Nas.ir al-Dīn al-T
. ūs.ī (13th
cent.); and, in ’modern’ times, by a number of writers some well-known, others
obscure. It’s worth noticing that the ‘parallels problem’ was never regarded
as a key question in mathematics. Obviously it was more important to those
(like the Arabs) who valued the Greek classics, but even so, it was often seen
as rather a blind alley, pursued by eccentrics.
3. The last major serious ‘proof’ within the context of classical geometry was
due to an Italian priest, Gerolamo Saccheri, published in 1733. This refined a
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framework for the problem (division into three cases) which is perhaps originally
due to al-T
. ūs.ī. We start by constructing a quadrilateral ABCD, (Fig. 2) with
the angles at B and C both right angles, and the sides AB and CD equal.
It’s then easy to show that the angles at A and D are equal. If we have the
parallels postulate, we can deduce that they are right angles (try to see how);
but without it, we don’t know this. Saccheri distinguished cases according to
whether these two angles are right angles. acute or obtuse; and describes these
as the ‘hypothesis of the right (acute, obtuse) angle’ — HRA, HAA, HOA for
short. HRA corresponds to Euclid’s geometry with postulate 5 included; it’s
what we normally take to be true. His idea was to get a contradiction by
carefully spelling out the consequences of the HOA and HAA, so that the HRA
would be left as the only true geometry. The proof which he thought he had was
in fact wrong, but the idea of the three hypotheses was to be very useful; and in
developing his ‘proof’ he deduced a great many consequences which must follow
if we assume the HAA; this, as we’ll see, is the difficult case, which amounts to
denying postulate 5.
Saccheri proved that in fact these are three mutually exclusive choices: if,
say, the HAA is true for one quadrilateral then it’s true for all. There are various
other ways of looking at this distinction. For example, with the HOA there are
no parallels (we shall consider how this can happen later), while with the HAA
there are an infinite number of lines through a point P which don’t meet a given
line l. Again, with the HOA (the HAA), all triangles have angle-sum greater
than (less than) two right angles.
4. After Saccheri, attempts at proof continued, but gradually new elements
involving explicit measurement (such as trigonometry) were brought in — and
at the same time we see an increasing tendency to doubt the possibility of
effectively proving the postulate. Gauss1 in particular became convinced (some
time around 1800) that a consistent geometry in which the postulate was untrue
could be constructed; but he confined his thoughts to private correspondence.
5. In the 1820s two independent researchers, N.I.Lobachevsky and Janos
Bolyai, both of whom had been trying to prove the postulate, chose a different
aim: to construct a consistent geometry based on the ‘acute angle’ hypothesis. Note the similarity, though, to Saccheri’s programme. In each case the
idea was to assume such a geometry possible, but Saccheri hoped to deduce a
contradiction, while Lobachevsky and Bolyai did not. Both published their results in obscure places (in Russia and Hungary) in the 1820’s2 ; and both works
were more or less forgotten. However, each of them proved some important and
unexpected properties of the alternative ‘non-Euclidean’ geometry, and went a
long way towards making it an interesting study in its own right. This is the
‘Copernican revolution’ referred to in our opening quote.
6. Although Lobachevsky and Bolyai had constructed their non-euclidean
1 Who
should have much more than a passing reference; he was the dominant mathematician
in almost all fields in the years from 1800 to 1840.
2 To be precise: Lobachevsky’s first memoir, ‘Theory of Parallels’ in Russian, was in the
Kazan Messenger in 1829; Bolyai’s ‘Science Absolute of Space’, in Latin, was published in
1831 as an appendix to his father’s Tentamen.
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geometries, they had not proved them consistent. This is not a merely pedantic
point; it would theoretically still be possible to prove that non-euclidean geometry was inconsistent, and so deduce postulate 5 after all. A wider variety
of geometries (more or fewer dimensions, varying rules of measurement) were
outlined by Riemann in his groundbreaking paper of 1854, and publicized in
the years which followed by Helmholtz; and in particular, the meaning of ‘noneuclidean’ was clarified. Proof of consistency was achieved in stages through the
later nineteenth century by Beltrami, Klein and Poincaré among others, by the
characteristically modern method of defining ‘models’ for the new geometry.
7. As a result of these developments, a new outlook came into being, according to which there was no unique geometry The way was open to the modern
understanding of a ‘geometry’ as the study of an axiom-system which asserts
certain properties of objects called (e.g.) ‘points’, ‘lines’, etc, without reference to what these names may mean. The unique geometry of Euclid has been
replaced by a multiplicity of geometries, which are equally valid as objects of
mathematical study. Because they were the first to suggest an alternative to
Euclid, Lobachevsky and Bolyai can be seen as the founding fathers of this
revolution. ...
[For a ‘picture’ of non-Euclidean geometry see Fig.3 — Poincaré’s model.
The circles are to be thought of as ‘straight lines’; the triangles (and the fish)
are all the same size; and the boundary circle is ‘at infinity’.]
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Jeremy Gray’s criticisms of the above story.
However, the standard account is open to several criticisms:
(1) It is well over 100 years from Saccheri’s Euclides Vindicatus to Beltrami’s
Saggio, but the standard account does not explain why the development took
so long.
(2) It does not explain, or adequately discuss, the choice of methods used
at various times but subordinates it to a compilation of results. In Bonola’s
account, for instance, analytic methods appear unheralded in the discussion of
Gauss, Schweikart, and Taurinus. This results in a failur to understand what it
was that the mathematicians were trying to do.
(3) The exact nature of the accomplishment of J. Bolyai and Lobachevskii
is not fully discussed. For, once it is admitted that it is not fully conclusive it
must be asked why it has been found so compelling. This is the problem which
makes it very hard to say who invented, or discovered, non-Euclidean geometry.
(4) Finally, it does not explain why it is that spherical geometry, well-known
throughout the entire period, did not at once settle the question, yet this geometry is now given almost immediately in modern textbooks as an example of a
non-Euclidean geometry...
[Gray (1979), p.155-6.]
These are interesting questions; and yet, in some sense, the ‘standard account’ has survived. Why?
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Extracts.
1. Lambert.
[L]et A, B, C, D, E be right angles, and, assuming the third hypothesis [acute
angle], let G, F, H, J be acute, and indeed H < G, and J < H and likewise
F < G and J < F [Fig.4.] Now I say that the angle G is the measure of the
quadrilateral ADGB, if indeed AB + AD, and likewise the angle J shall be the
measure of the quadrilateral ACJE, if AC = AE....
Therefore the angle G is the absolute measure of the quadrilateral ADGB.
Since the angle has a measure intelligible in itself [i.e. as a fraction of 360◦ ],
if one took e.g. AB = AD as a Paris foot and then the angle G was 80◦ this
is only to say that if one should make the quadrilateral ADGB so big that the
angle G was 80◦ : then one would have the absolute measure of a Paris foot on
AB = AD.
[I]t is not only the case that in every triangle the angle sum is less than 180◦ ,
as we have already seen, but also that the difference from 180◦ increases directly
with the area of the triangle.
[Lambert, in Fauvel and Gray p.517-8.]
2. Lobachevsky.
All straight lines in a plane which go out from a point can, with reference
to a given straight line in the same plane, be divided into two classes — into
cutting and not-cutting.
The boundary lines of the one and the other class of those lines will be called
parallel to the given line.
From the point A let fall upon the line BC the perpendicular AD, to which
again draw the perpendicular AE (fig.5)...
In passing over from the cutting lines, as AF , to the not-cutting lines, as
AG, we must come upon a line AH, parallel to DC, a boundary line, upon one
side of which all lines AG are such as do not meet the line DC, while upon the
other side every straight line AF cuts the line DC.
The angle HAD between the parallel HA and the perpendicular AD is called
the parallel angle (angle of parallelism), which we will here designate by Π(p)
for AD = p.
3. Riemann.
It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case
of a triply extended magnitude. But hence flows as a necessary consequence
that the propositions of geometry cannot be derived from general notions of
magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus
arises the problem, to determine the simplest matters of fact from which the
measure-relations of space can be determined; a problem which from the nature of the case is not fully determinate, since there may be several systems of
matters of fact which suffice to determine the measure-relations of space — the
most important system for our present purpose being that which Euclid has laid
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down as a foundation.
[Riemann (1873),p.14.]
4. Helmholtz.
[Reproduced in (1979), pp.249-250. Helmholtz, to simplify his argument,
considers two-dimensional ‘beings’ constructing geometry from observation of
the world in which they live.]
But intelligent beings...might also live on the surface of a sphere. Their
shortest or straightest line between two points would then be an arc of the great
circle passing through them. Every great circle passing through two points is
divided by them into two parts. If the parts are unequal, the shorter is certainly
the shortest line on the sphere between the two points, but the other, or larger,
arc of the same great circle is also a geodesic, or straightest, line; that is, every
smallest part of it is the shortest line between its ends. Thus the notion of the
geodesic, or straightest, line is not quite identical with that of the shortest line...
Of parallel lines the sphere-dwellers would know nothing. They would declare that any two straightest lines, if sufficiently extended, must finally intersect
not only in one but in two points. The sum of the angles of a triangle would
be always greater than two right angles, increasing as the surface of the triangle grew greater. They could thus have no conception of geometric similarity
between greater and smaller figures of the same kind, for with them a greater
triangle must have greater angles than a smaller one. Their space would be
unlimited, but would be found to be finite or at least represented as such.
It is clear, then, that such beings must set up a very different system of
geometric axioms from that of the inhabitants of a plane or from ours, with our
space of three dimensions, though the logical processes of all were the same; nor
are more examples necessary to show that geometric axioms must vary according
to the kind of space inhabited. But let us proceed still further.
Let us think of reasoning beings existing on the surface of an egg-shaped
body. Shortest lines could be drawn between three points of such a surface and
a triangle constructed. But if the attempt were made to construct congruent
triangles at different points of the surface, it would be found that two triangles
with three pairs of equal sides would not have equal angles. The sum of the
angles of a triangle drawn at the sharper pole of the body would depart further
from two right angles than if the body were drawn at the blunter pole or at the
equator. Hence it appears that not even such a simple figure as a triangle could
be moved on such a surface without change of form.
5. Peano.
We are given thus a category of objects called points. These objects are
not defined. We consider a relation between three given points. This relation,
noted c ∈ ab, is likewise undefined. The reader may understand by the sign 1
any category of objects whatsoever, and by c ∈ ab any relation between three
objects of that category[...] The axioms will be satisfied or not, depending on
the meaning assigned to the undefined signs 1 and c ∈ ab. If a particular group
of axioms is satisfied, all propositions deduced from them will be as well.
[Peano (1889), quoted in Torelli (1978), p.219.]
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Bibliography.
Gaston Bachelard, Le Nouvel esprit scientifique, Paris: PUF, 1934.
Roberto Bonola, Non-Euclidean Geometry: with a Supplement containing ‘The
Theory of Parallels’ by Nicholas Lobachevski and ‘The Science Absolute of Space’
by John Bolyai, New York: Dover, 1955.
Jeremy Gray, Ideas of Space: Euclidean, Non-euclidean and Relativistic, Oxford:
OUP, 1979.
Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries: Development
and History, San Francisco: W.H.Freeman and Company, 1974.
Hermann von Helmholtz, ‘The origins and meaning of Geometric Axioms’, in
Russell Kahl (ed), Selected Writings of Hermann von Helmholtz, Middletown:
Wesleyan University Press, 1979, pp. 246-265.
Proclus, A Commentary on the First Book of Euclid’s Elements, tr. G.R.Morrow,
Princeton: Princeton University Press, 1970.
Joan Richards, Mathematical Visions: The Pursuit of Geometry in Victorian
England New York: Academic Press, 1988.
Bernhard Riemann, ‘On the Hypotheses which lie at the Bases of Geometry’,
tr. William Kingdon Clifford, in Nature vol. VIII (1873), pp.14-17, 36, 37.
Roberto Torrelli, Philosophy of Geometry from Riemann to Poincaré, Dordrecht: Reidel, 1978.
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