3.1 - The Concept of Parallelism
Non-Euclidean geometry – History
Article by: J J O'Connor and E F Robertson
In about 300 BC Euclid wrote The Elements, a book which was to become one of the
most famous books ever written. Euclid stated five postulates on which he based all his
theorems:
1.
2.
3.
4.
5.
To draw a straight line from any point to any other.
To produce a finite straight line continuously in a straight line.
To describe a circle with any centre and distance.
That all right angles are equal to each other.
That, if a straight line falling on two straight lines make the interior angles on the
same side less than two right angles, if produced indefinitely, meet on that side on
which are the angles less than the two right angles.
It is clear that the fifth postulate is different from the other four. It did not satisfy Euclid
and he tried to avoid its use as long as possible - in fact the first 28 propositions of The
Elements are proved without using it. Another comment worth making at this point is that
Euclid, and many that were to follow him, assumed that straight lines were infinite.
Proclus (410-485) wrote a commentary on The Elements where he comments on
attempted proofs to deduce the fifth postulate from the other four, in particular he notes
that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his
own. However he did give the following postulate which is equivalent to the fifth
postulate.
Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly
one line through the given point parallel to the line.
Although known from the time of Proclus, this became known as Playfair's Axiom after
John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed
replacing Euclid's fifth postulate by this axiom.
Many attempts were made to prove the fifth postulate from the other four, many of them
being accepted as proofs for long periods of time until the mistake was found. Invariably
the mistake was assuming some 'obvious' property which turned out to be equivalent to
the fifth postulate. One such 'proof' was given by Wallis in 1663 when he thought he had
deduced the fifth postulate, but he had actually shown it to be equivalent to:To each triangle, there exists a similar triangle of arbitrary magnitude.
One of the attempted proofs turned out to be more important than most others. It was
produced in 1697 by Girolamo Saccheri. The importance of Saccheri's work was that he
assumed the fifth postulate false and attempted to derive a contradiction.
Here is the Saccheri quadrilateral
In this figure Saccheri proved that the summit
angles at D and C were equal.The proof uses
properties of congruent triangles which Euclid
proved in Propositions 4 and 8 which are proved
before the fifth postulate is used. Saccheri has
shown:
a) The summit angles are > 90 (hypothesis of the
obtuse angle).
b) The summit angles are < 90 (hypothesis of the
acute angle).
c) The summit angles are = 90 (hypothesis of the
right angle).
Euclid's fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle
implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the
hypothesis of the acute angle and derived many theorems of non-Euclidean geometry
without realising what he was doing. However he eventually 'proved' that the hypothesis
of the acute angle led to a contradiction by assuming that there is a 'point at infinity'
which lies on a plane.
In 1766 Lambert followed a similar line to Saccheri. However he did not fall into the trap
that Saccheri fell into and investigated the hypothesis of the acute angle without
obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a
triangle increased as the area of the triangle decreased.
Legendre spent 40 years of his life working on the parallel postulate and the work
appears in appendices to various editions of his highly successful geometry book
Eléments de Géométrie. Legendre proved that Euclid's fifth postulate is equivalent to:The sum of the angles of a triangle is equal to two right angles.
Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a
triangle cannot be greater than two right angles. This, again like Saccheri, rested on the
fact that straight lines were infinite. In trying to show that the angle sum cannot be less
than 180 Legendre assumed that through any point in the interior of an angle it is always
possible to draw a line which meets both sides of the angle. This turns out to be another
equivalent form of the fifth postulate, but Legendre never realised his error himself.
Elementary geometry was by this time engulfed in the problems of the parallel postulate.
d'Alembert, in 1767, called it the scandal of elementary geometry.
The first person to really come to understand the problem of the parallels was Gauss. He
began work on the fifth postulate in 1792 while only 15 years old, at first attempting to
prove the parallels postulate from the other four. By 1813 he had made little progress and
wrote:
In the theory of parallels we are even now not further than Euclid. This is a shameful part
of mathematics...
However by 1817 Gauss had become convinced that the fifth postulate was independent
of the other four postulates. He began to work out the consequences of a geometry in
which more than one line can be drawn through a given point parallel to a given line.
Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At
this time thinking was dominated by Kant who had stated that Euclidean geometry is the
inevitable necessity of thought and Gauss disliked controversy.
Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai
who made several false proofs of the parallel postulate. Farkas Bolyai taught his son,
János Bolyai, mathematics but, despite advising his son not to waste one hour's time on
that problem of the problem of the fifth postulate, János Bolyai did work on the problem.
In 1823 Bolyai wrote to his father saying I have discovered things so wonderful that I
was astounded ... out of nothing I have created a strange new world. However it took
Bolyai a further two years before it was all written down and he published his strange
new world as a 24 page appendix to his father's book, although just to confuse future
generations the appendix was published before the book itself.
Gauss, after reading the 24 pages, described János Bolyai in these words while writing to
a friend: I regard this young geometer Bolyai as a genius of the first order . However in
some sense Bolyai only assumed that the new geometry was possible. He then followed
the consequences in a not too dissimilar fashion from those who had chosen to assume
the fifth postulate was false and seek a contradiction. However the real breakthrough was
the belief that the new geometry was possible. Gauss, however impressed he sounded in
the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had
discovered all this earlier but had not published. Although this must undoubtedly be true,
it detracts in no way from Bolyai's incredible breakthrough.
Nor is Bolyai's work diminished because Lobachevsky published a work on nonEuclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work,
mainly because it was only published in Russian in the Kazan Messenger a local
university publication. Lobachevsky's attempt to reach a wider audience had failed when
his paper was rejected by Ostrogradski.
In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his
momentous work. He published Geometrical investigations on the theory of parallels in
1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work. The
publication of an account in French in Crelle's Journal in 1837 brought his work on nonEuclidean geometry to a wide audience but the mathematical community was not ready
to accept ideas so revolutionary.
In Lobachevsky's 1840 booklet he explains clearly how his non-Euclidean geometry
works.
All straight lines which in a plane 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 non-cutting.
The boundary lines of the one and the other class of those lines will be called parallel to
the given line.
Here is the
Lobachevsky's
diagram
Hence Lobachevsky has replaced the fifth postulate of Euclid by:Lobachevsky's Parallel Postulate. There exist two lines parallel to a given line through a
given point not on the line.
Lobachevsky went on to develop many trigonometric identities for triangles which held
in this geometry, showing that as the triangle became small the identities tended to the
usual trigonometric identities.
Riemann, who wrote his doctoral dissertation under Gauss's supervision, gave an
inaugural lecture on 10 June 1854 in which he reformulated the whole concept of
geometry which he saw as a space with enough extra structure to be able to measure
things like length. This lecture was not published until 1868, two years after Riemann's
death but was to have a profound influence on the development of a wealth of different
geometries. Riemann briefly discussed a 'spherical' geometry in which every line through
a point P not on a line AB meets the line AB. In this geometry no parallels are possible.
It is important to realise that neither Bolyai's nor Lobachevsky's description of their new
geometry had been proved to be consistent. In fact it was no different from Euclidean
geometry in this respect although the many centuries of work with Euclidean geometry
was sufficient to convince mathematicians that no contradiction would ever appear within
it.
The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same
footing as Euclidean geometry was Eugenio Beltrami (1835-1900). In 1868 he wrote a
paper Essay on the interpretation of non-Euclidean geometry which produced a model for
2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry. The
model was obtained on the surface of revolution of a tractrix about its asymptote. This is
sometimes called a pseudo-sphere.
In fact Beltrami's model was incomplete but it certainly gave a final decision on the fifth
postulate of Euclid since the model provided a setting in which Euclid's first four
postulates held but the fifth did not hold. It reduced the problem of consistency of the
axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean
geometry.
Beltrami's work on a model of Bolyai - Lobachevsky's non-Euclidean geometry was
completed by Klein in 1871. Klein went further than this and gave models of other nonEuclidean geometries such as Riemann's spherical geometry. Klein's work was based on a
notion of distance defined by Cayley in 1859 when he proposed a generalised definition
for distance.
Klein showed that there are three basically different types of geometry. In the Bolyai Lobachevsky type of geometry, straight lines have two infinitely distant points. In the
Riemann type of spherical geometry, lines have no (or more precisely two imaginary)
infinitely distant points. Euclidean geometry is a limiting case between the two where for
each line there are two coincident infinitely distant points.
An Axiomatic system consists of:

Undefined terms – point, line, plane, on

Defimed terms

Axioms

System of logic

Theorems
Axiomatic systems are said to be Consistent if there do not exists any two axioms or
an axiom and a theorem, or any two theorems that contradict each other.
A set of axioms is said to be independent if they are not be derivable from one another.
Over the centuries, in trying to prove that Euclid’s fifth postulate was dependent on the
first four, mathematicians proved the following statements which we now know to be
logically equivalent to fifth postulate:
Exercise: Prove that Playfair’s Postulate is equivalent to Euclid’s fifth postulate.
Definition: Saccheri quadrilateral ABCD – Two congruent sides AC and BD are drawn
perpendicular to the base AB. CD is called the summit, and angles ACD and BDC are
called the summit angles.
Other Geometries
The first four Euclidean Axioms with the:
Hyperbolic Axiom: Through a point not on a given line, more than one parallel may be
drawn to the given line;
- characterizes Hyperbolic Geometry.
Euclidean Axiom: Through a point not on a given line, exactly one parallel may be
drawn to the given line;
- characterizes Euclidean geometry.
Elliptic Axiom: Through a point not on a given line, no parallels may be drawn to the
given line;
- characterizes Elliptic (spherical) geometry
By embedding an infinite hyperbolic space in a finite Euclidean plane, Henri Poincare
formulated a model for hyperbolic geometry
Exercises: Prove the following using Euclid’s first 4 axioms.
1. The segment joining the midpoints of the summit and base of a Saccheri
quadrilateral is perpendicular to both.
2. The summit angles of a Saccheri quadrilateral are congruent.
3. In a Saccheri quadrilateral, the segment joining the midpoints of the lateral
sides is perpendicular to the segment joining the midpoints of the base and
the and the summit.
4. Prove that each of the following statements are equivalent to Playfair’s
postulate:
a) The sum of the angles of a triangle is equal to two right angles
b) There exists a pair of lines everywhere equidistant from one another
c) If three angles of a quadrilateral are right angles, then the fourth
angle is also a right angle
d) Straight lines parallel to the same straight line are parallel to one
another.