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Sarah Yeust
April 2011
Honors Geometry Research Paper
The History of Proof Attempts of Euclid’s Fifth Postulate and the Advent of Non-Euclidean
Geometry
When Italian-French Mathematician Joseph-Louis Lagrange stood up in front of a body
of his French peers in 1800, he believed he was about to do something that Euclid, ancient Greek
and Roman scholars, mathematical intellectuals throughout the years, and many of the
mathematicians that surrounded him had failed at in their attempts. His audience was held in
anticipation as they awaited the work that was soon to be presented. Everyone knew that what
Lagrange was about to do would be perhaps the greatest mathematical discovery yet. However,
as Lagrange began to address the crowd, he uttered words that would mark the course of
Geometric history: “ ‘Il faut que j’y songe encore’ ” (“I shall have to think it over again”).
Lagrange had a simple mistake in his proof, and a majority in the audience saw it immediately.
He put the manuscript in his pocket and left the podium (Bardi 1-8).
Lagrange was last in a long line of mathematicians who had tried to attempt to prove
Euclids’s Fifth Postulate directly (Bardi 8). Euclid’s geometric system consists of five postulates
about flat space. These contain assumptions about the way points and straight lines lie in space.
In general, the First Postulate states that “there is exactly one straight line connecting any two
distinct points.” This means that there is at least one and no more than one line through two
points, but nothing can be assumed about straight lines (Rucker 18-19).
From the Second Postulate, Euclid says that “Every straight line can be continued
endlessly.” Space has no boundaries or edges. There is not a point beyond which a given straight
line cannot be continued. The Third Postulate states that “It is possible to draw a circle with any
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given center and radius.” Although this postulate does not seem at first to concern points and
straight lines, part of it leads to the inference that the distance in space is defined in such a way
that a line segment’s length does not change when it is moved from one place to another (Rucker
19).
From the Fourth Postulate, one can find that “All right angles are equal to each other.” To
use Euclid’s definitions, a right angle is “When a straight line set up on a straight line makes the
adjacent angles equal to one another.” This postulate expresses that space is locally flat, since
even a small enough region will not have any curvature (Rucker 19-20).
The first four of Euclid’s Postulates can be accepted in the normally considered space.
However, the Fifth Postulate has always been the focus of much debate and uncertainty. In his
Fifth Postulate, Euclid states that “Given a line m and a point P not on m, there is exactly one
line n that passes through P and is parallel to m.” (See Figure I)
Figure I
It is understood that in this case, lines are described as parallel when they do not intersect.
However, the Fifth Postulate could fail under two different circumstances. First, there might be
no lines through P that are parallel to m or there might be more than one. Both of these make
sense when the right “space” is considered (Rucker 20-21).
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This Fifth Postulate stands out because it does not follow logically from experience.
Many centuries passed in which scholars felt that the Fifth Postulate could not be false in
normally considered space. They did not believe that God would have “botched” His
wonderfully eternal absolute form of space by including converging and diverging collections of
straight lines that would be required to contradict the Fifth Postulate. The German philosopher
Immanuel Kant is largely responsible for the perception that space is flat. Kant had written that
space is a creation of the mind, and non-Euclidean space cannot be imagined. Therefore, space is
Euclidean, meaning that the Fifth Postulate is satisfied. Kant believed that no one could imagine
Non-Euclidean space because no one at his time of writing had done so. Since the alternative to
the Fifth Postulate was supposedly unimaginable, he concluded that space must necessarily
satisfy the Fifth Postulate (Rucker 20-21).
Euclid listed his Postulates in his Elements. This book was seen as the guidebook to truth
in elementary geometry. To read Euclid’s writings meant that a person knew geometry, and
knowledge of geometry was knowledge of reality. Lagrange had failed with his proof for the
same reason that everyone before him had- the Fifth Postulate could not be proven (Bardi 9).
This problem had been considered by many over a period of 2,000 years. Archimedes’ lost
treatise On parallel lines was perhaps the first work devoted to the issue, and this title is one item
in the list of Archimedes’ works contained in the Arabic Book of bibliography of the sciences.
Archimedes most likely used a definition of parallel lines much different than Euclid did, and
Archimedes probably based his on distance. This definition might have been similar to the one
used by the mathematician Posidonius, which stated: “Parallel lines are lines in the same plane
that come neither near nor apart, so that all the perpendiculars from the points of one of them to
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the other are equal.” From such a definition, the parallel postulate is a simple consequence
(Rosenfeld 41).
Ptolemy was responsible for another “proof” of the parallel line postulate. Ptolemy
proved that if two parallel lines are cut by a transversal, then the interior angles on the same side
add to two right angles. This is equivalent to the Fifth Postulate and is a proof by contradiction.
For example, first suppose that the interior angles on one side of the transversal are less than two
right angles (see Figure II). The corresponding angles on the other side of the transversal are
supplementary, and thus, they must add up to more than two right angles.
Figure II
The lines on one side of the transversal are “no more parallel” than the same lines on its
other side. It is a false conclusion that if the interior angles on one side of the transversal add up
to less than two right angles, then so do the interior angles on the other side. The contradiction
proves Ptolemy’s assertion. From it, he argues by contradiction and obtains the parallel postulate.
Proclus then gave his own “proof” of the postulate (Rosenfeld 41-42).
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Figure III
A
E
C
H
B
D
F
He considers interior angles BEF and EFD on the same side of transversal EF through
AB and CD. These two angles add up to less than two right angles (see Figure III).
Through E, draw a line EH parallel to CD. Proclus argues that because the distance
between the points on the sides of the angle BEH can be made arbitrarily large by moving
sufficiently far away from E, the distance will most likely exceed the distance between
the parallels CD and EH. Consequently, the side EB of angle EBH is bound to intersect
CD. Proclus is guilty of petition principii for his assumption that the distance between
nonintersecting coplanar lines (called parallel lines) is bounded is equivalent to the
postulate he wanted to prove (see Posidonius above). This is that the distance between
nonintersecting coplanar straight lines can be arbitrarily large and is a principle in
Lobačevskian geometry (Rosenfeld 40-42).
The first preserved “proof” attempt by a scholar of the medieval East was by ‘Abbās ibn
Sa`īd al-Jawharī from Baghdad in the first half of the ninth century. His work was included in
Improvement of the book the “Elements” and was preserved in the treatise about parallel lines by
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Na īr al-Dīn al-Tūsī. Al-Jawharī’s principle included that if one is given any two unequal lines
and takes away from the longer one half of the shorter line, and then a quarter of the shorter line
is taken away from the longer line many times, eventually, the original longer line will be shorter
than the original shorter line. Al-Jawharī added a proposition that if one draws from any point
three straight lines in different directions, then the three resulting angles equal four right angles.
In total, Al-Jawharī added six propositions to Elements. His thirty-second proposition read: “If
we draw in an arbitrary angle a line passing through its vertex and choose on that line an
arbitrary point, then” a line can be drawn in both directions through this point that is the base of
the given triangle (Figure IV). With this proposition in mind, Al-Jawharī constructed his thirtythird proposition, which was the “proof” of the parallel postulate (Rosenfeld 46-49).
Figure IV
If two lines are extended from a line (see Figure V) in the direction of angles less than
two right angles, then they intersect on that side. AB and CD are drawn from the line BD
in the direction of angle ABD and angle CDB, which are less than two right angles.
Claim that if one extends them in their direction, then they intersect. Extend BD in its
direction to the points E and H and lay off BF, equal to BD (by proposition 3). Angles
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ABD and CDB are by assumption less than two right angles. Therefore, angle ABE is
greater than angle CDB. On BA at B, construct angle ABK equal to angle CDB. Then
ABD and ABK are together less than two right angles, KBL. Pass the line KL through
point F, and KL is a base of angle KBL (by proposition 32). From proposition 16, angle
KFB, an exterior angle of triangle FBL, is greater than the interior angle FBL. Construct
on BF at F the angle BFO equal to FBL. Angle KBA is equal to angle CDB. Therefore,
angles BFO and OBF are equal to angles ABD and CDB, respectively. BF was laid off
equal to BD. By superimposing BD on BF, equal to it, angle CDB is superimposed on
OBF and equal to it, and ABD is superimposed on BFO and equal to it. Hence, BA and
DC are superimposed on lines FO and BO which intersect at point O.
Figure V
L
This proof had a crude error as associated with the first proposition, but it was the first to
prove the possibility of drawing a line through an arbitrary interior point of an angle that
intersects both sides of it in connection to the parallel line postulate (Rosenfeld 46-49).
Persian astronomer and mathematician Nasiraddin (1201-1274) compiled an Arabic
version of Euclid. He also wrote a treatise on the Euclidean postulate. From historical analysis
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and comparison, it seems as though he was the first to direct focus on the theorem of the sum of
angles in a triangle in relationship to the Fifth Postulate. His attempt to prove the Fifth Postulate
provided the seeds for lines of thought and approach in following years (Wolfe 28-29).
Figure VI
A
E
G
I
B
D
C
F
H
J
Consider two straight lines, AB and CD (see Figure VI), where successive
perpendiculars such as EF, GH, and IJ from points E, G, and I to CD, always make
unequal angles with AB. These angles are always acute on the side toward B and always
obtuse on the side toward A. Lines AB and CD continually diverge in the direction of A
and C, and they converge in the direction of B and D. Therefore, the perpendiculars
continually grow larger in the direction of A and C, and consequently, they become
shorter in length in the direction of B and D. Conversely, if the perpendiculars become
continually longer in one direction and shorter in the other, then the lines AB and CD
diverge in the first direction and converge in the second. Also, the perpendiculars will
make unequal angles with AB, with obtuse angles lying on the side toward divergence,
and acute angles on the side of convergence (Wolfe 28-29).
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After this, Nasiraddin introduced a figure that would become famous. At the extremities of AB,
he constructed equal perpendiculars AD and BC on the same side, then joining C and D (see
Figure VII).
Figure VII
D
C
A
B
He attempted to prove that angles CDA and DCB are right angles by using the
assumption concerning the length of the perpendiculars. Therefore, if angle DCB were acute, DA
would be shorter than CB, contrary to fact. Hence, angle DCB is not acute. However, it is not
obtuse. He assumed that when angle DCB is acute, angle CDA must be obtuse. This argument
led to the conclusion that all four angles of the quadrilateral are right angles. If DB is drawn, the
triangles ABD and CDB are congruent. The angle sum of each is equal to two right angles.
If everything had been complete and satisfactory to this point, then the Fifth Postulate
would have followed. Nasiraddin presented an elaborate proof of this. Unfortunately, his work
contained flaws because his initial statements concerning the measurement of the angles (such as
CDA and DCB) had no fundamental proof. The assumptions made at the beginning lack just as
much proof as the Fifth Postulate. However, it was from these assumptions that he followed with
his argument leading to “proof” of the Fifth Postulate. Thus, in Figure VII, the fact that angle
CDA is obtuse does not follow from the assumption that angle DCB is acute (Wolfe 28-29).
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John Wallis (1616-1703) offered his own proof of the Fifth Postulate in 1663 after
becoming interested in Nasiraddin’s work. His proof is another example of one that makes use of
an assumption that is equivalent to the Fifth Postulate. This assumption is found in his statement
about the measures of interior angles, DHG and BGH, which cannot be proven.
Figure VIII
E
A
G
I
J
L
K
B
H
D
C
F
Wallis suggested that given a triangle, it is possible to construct another triangle similar
to it of any size. He argues that given line AB and CD (see Figure VIII), EF is a transversal that
intersects AB and CD at points G and H respectively. Also, the sum of angles BGH and DHG is
less than two right angles. He wanted to show that AB and CD will meet if sufficiently produced.
Clearly, angle EGB is greater than GHD. If segment HG is moved along EF, with HG attached to
it, until H coincides with the initial position of G, HD takes the position GI entirely above GB.
Therefore, during its motion, at some time, HD must cut GB at L. Now, if one constructs a
triangle similar to triangle GJL on base GH - assumed to be possible- HD must cut GB (Wolfe
29-30).
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The discovery of Non-Euclidean Geometry, long attributed to Wolfgang Bolyai and
Nikolai Ivanovich Lobachewsky, had all but been made by Italian Jesuit priest Gerolamo
Saccheri. His book, Euclides ab omni naevo vindicatus (Euclid Freed of Every Flaw), was
published in Milan in 1733, but was forgotten until it resurfaced in 1889. Saccheri had read
Euclid’s Elements and was impressed by his use of reduction ad absurdum, the strategy of first
assuming the hypothesis to be proven is false. Naturally, Saccheri tried to apply this to the
problem of the Fifth Postulate. As far as history shows, this was the first time anyone had tried to
use this method with the Fifth Postulate. With a background in work such as definitions and
postulates, Saccheri was prepared to tackle this challenge. Saccheri also used the figure of the
isosceles quadrilateral with the two base angles as right angles (see Figure VII). In quadrilateral
ABCD, assuming that AD and BC were equal and that angles A and B were right angles,
Saccheri proved that angles C and D were equal and that the line joining the midpoints of AB
and DC was perpendicular to the two lines. He did this without using the Fifth Postulate. Angles
C and D are known to be right angles under the Euclidean hypothesis. Assuming that they are
acute or obtuse would imply the falsity of the Postulate. Saccheri’s plan required these
assumptions. Saccheri focused on three main hypotheses: hypothesis of the right angle,
hypothesis of the obtuse angle, and hypothesis of the acute angle. He stated and proved
propositions including the following:
1. If one of the hypotheses is true for a single quadrilateral, of the type under
consideration, it is true for every such quadrilateral.
2. On the hypothesis of the right angle, the obtuse angle or the acute angle, the sum of
the angles of a triangle is always equal to, greater than or less than two right angles.
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3. If there exists a single triangle for which the sum of the angles is equal to, greater
than, or less that two right angles, then follows the truth of the hypothesis of the right
angle, the obtuse angle, or the acute angle.
4. Two straight lines lying in the same plane either have (even on the hypothesis of the
acute angle) a common perpendicular or, if produced in the same direction, either
meet one another once at a finite distance or else continually approach one another.
Saccheri adapted Euclid’s assumption that a straight line is infinite, and using this, he disposed
the hypothesis of the obtuse angle. From this, he was able to prove the Fifth Postulate, which
implies that the sum of the angles of a triangle is equal to two right angles, contradicting the
hypothesis. If he had not assumed the infinitude of the line, the contradiction could never have
been reached (Wolfe 30-33).
The hypothesis of the acute angle was harder. There was no contradiction to come. In his
process, Saccheri found many of the propositions that would become fundamental in NonEuclidean geometry, but he concluded that there exist two straight lines which, when produced to
infinity, merge into one straight line with a common perpendicular at infinity. He then tried a
second proof, but with no greater success. It is important to note that if Saccheri had suspected
that he reached no conclusion because there was none to be reached, the advent of NonEuclidean Geometry would have come about a century before it did (Wolfe 30-33).
Inspired by a dissertation from Georgius Simon Klügel that appeared in 1763, Johann
Heinrich Lambert (1728-1777) was another example of someone who came very close to the
discovery of Non-Euclidean Geometry. Klügel appears to have been the first to express some
doubt as to whether proving the Fifth Postulate was possible. Saccheri’s Euclides Vindicatus and
Lambert’s Theorie der Parallelinien bear a striking resemblance, but Lambert’s work was
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published posthumously. Lambert chose a quadrilateral with three right angles (one-half the
isosceles quadrilateral used by Saccheri) as his fundamental figure. He proposed three
hypotheses in which the remaining angle was right, obtuse, or acute. In two of his hypotheses, he
was able to advance much further than Saccheri had advanced. Lambert proved that the area of a
triangle is proportional to the difference between the sum of its angles and two right angles. This
was to excess in the case of the hypothesis for the obtuse angle and to deficit in the case for the
hypothesis for the acute angle. Lambert also noticed that the geometry associated with the
second hypothesis (obtuse) was related to spherical geometry in which the area of a triangle is
proportional to its spherical excess. Making a bold move, he leaned toward the conclusion that
the geometry based upon his third hypothesis (acute) could be verified on a sphere with an
imaginary radius. However, he realized that the arguments against the geometry based upon the
third hypothesis were substantiated by results of tradition and sentiment. He claimed that such
arguments should be banished from geometry and all of science (Wolfe 33-34).
Adrien Marie Legendre (1753-1833) contributed extensive writings to the Fifth Postulate
problem. Most of his results had already been made by his predecessors. However, his
straightforward writings seem to make concepts simpler at a time when people were on the verge
of some discoveries (Wolfe 34-35). For example, his work Théorie des nombres, was considered
a “light” read for Georg Riemann, who would go on to publish his own influential materials
about number theory (Mlodinow 136-137). Also, Semen Emel’yanovič Gur’ev (1746-1813),
who would go on to publish influential geometry works, found an error in Legendre’s Elements
of Geometry that led him to propose a proof of the parallel postulate beginning with two straight
lines cut by another that is perpendicular to one of the original lines (Rosenfield 106-107). This
made some of Legendre’s proofs of permanent value since they were of such an elegant style.
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Legendre attacked the parallel postulate much like Saccheri did, and his results were very
similar. However, he chose to place emphasis on the angle-sum of the triangle. He proposed
three hypotheses in which the sum was equal to, greater than, and less than two right angles. He
had hoped to reject the last two.
Unconsciously, Legendre assumed the straight line was infinite, but by doing this, he was
able to eliminate geometry based upon the second condition. He did this by proving that the sum
of the three angles in a triangle cannot be greater than two right angles (Wolfe 34-35).
Figure IX
Assume that the sum of the angles of a triangle ABC (see Figure IX) is 180°+α and that
the angle CAB is not greater than either of the others. Join A to D, the midpoint of BC.
Produce AD to E so that DE is equal to AD. Draw CE. Triangles BDA and CDE are
congruent. It follows that the sum of the angles of triangle AEC is equal to the sum of the
angles of triangle ABC, namely to 180°+α, and that either angle CAE or CEA is equal to
or less than one-half angle CAB. By applying the same process to triangle AEC, one
obtains a third triangle whose angle-sum is 180°+α and one of its angles is equal to or
less than
1
of angle CAB. After this construction is made n times, a triangle is reached
22
which has the sum of its angles equal to 180°+α and one of its angles equal to or less than
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1
of angle CAB. By the Postulate of Archimedes, there is a finite multiple of α,
2n
however small α may be, which exceeds angle CAB. In other words, angle CAB is equal
to k• α. If n is chosen so that k is less than 2n, then
1
is less than angle CAB which is
2n
less than α, and the sum of the two of the angles of the triangle last obtained is greater
than two right angles. However, this is impossible (Wolfe 35).
Although he tried, Legendre could not dispose of the third hypothesis. His proofs are interesting
to examine.
Figure X
Begin by assuming that the sum of the three angles of triangle ABC (see Figure X) is
180°-α and that angle BAC is not greater than either of the others. Construct a triangle
BCD on side BC such that BCD is congruent to triangle ABC. Angles DBC and DCB are
equal to angles BCA and CBA, respectively. Draw through D any line which cuts AB
and AC produced in E and F, respectively. In triangle BCD, the sum of the angles is also
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180°-α. Since the sum of the angles of a triangle cannot be greater than two right angles
(as proved above), the sum of the angles of triangle BDE and CDF can therefore, not be
greater than 180°. It follows that the sum of all of the angles of all four triangles cannot
be greater than 720°-2α. Therefore, the sum of the three angles of triangle AEF cannot be
greater than 180°-2α.
After repeating this construction until n such triangles are formed, the last one will have
its angle sum not greater than 180°-2α. Since a finite multiple of α can be found which is
greater than two right angles, n can be chosen so large that a triangle will be reached
whose sum of angles is negative.
This, of course, is illogical. The problem with this proof is the false assumption that
through any point within an angle less than two thirds of a right angle, there can always be drawn
a straight line which meets both sides of the angle. As shown above, this assumption is
equivalent to the Fifth Postulate (Wolfe 36-37).
Another Legendre proof addresses the theorem that states that if the sum of the angles of
a triangle is equal to two right angles, the same is true for all triangles obtained from it by
drawing lines through vertices to points on the opposite sides.
Figure XI
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In triangle ABC (see Figure XI), if the sum of the angles is equal to two right angles, then
the same must be true for triangle ABD, one of the two triangles into which triangle ABC
is subdivided by the line joining vertex B to point D on the opposite side. For the sum of
the angles of the triangle cannot be greater that two right angles (as proven above with
the assumption of the infinitude of the straight line). If the sum of the angles were less
than two right angles, that for triangle BDC would have to be greater than two right
angles (Wolfe 37).
Legendre also worked on a proof of the theorem stating that if there exists a triangle with
the sum of its angles equal to two right angles, an isosceles right triangle can be constructed with
the sum of its angles equal to two right angles and the legs greater in length than any given line
segment. He did this by drawing an altitude (see Figure XII), and if neither of the triangles was
isosceles, measuring off on the longer leg of one of them a segment equal to the shorter. Joining
together the hypotenuses of two isosceles right triangles, a quadrilateral will be formed with all
right angles and all equal sides. If another such quadrilateral is formed with sides of twice the
length of the original, and this process is repeated, one eventually obtains a quadrilateral with its
sides greater than any given line segment (Wolfe 38).
Figure XII
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Finally, he addressed the theorem stating that if there exists a single triangle with the sum of its
angles equal to two right angles, then the sum of the angles of every triangle will be equal to two
right angles.
Figure XIII
Start with a given triangle with the sum of its three angles equal to two right angles, and
it is proved that any other triangle ABC has its angle sum equal to two right angles. It
may be assumed that ABC is a right triangle (see Figure XIII), since any triangle can be
divided into two right triangles. By the previous theorem, there can be constructed an
isosceles right triangle DEF, whose sum of angles is equal to two right angles and its
equal legs are greater than the legs of triangle ABC. Construct CA and CB to A' and B',
respectively, so that CA' = CB' = ED = EF, and join A' to B and to B'. Because triangles
A'CB' and DEF are congruent, the former has the sum of its angles equal to two right
angles and the same is also true for triangle A'BC and finally for ABC. Legendre
obtained the following as a consequence of these results: “If there exists a single triangle
with the sum of its angles less than two right angles, then the sum of the angles of every
triangle will be less than two right angles (Wolfe 38-39).”
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When considering the proofs offered thus far, some have depended upon a conscious or
unconscious use of an equivalent to the Fifth Postulate, and others have made use of the
reduction ad absurdam (proof by contradiction) method. In each case, the results have been
unconvincing. However, there are two important proofs left that should be mentioned and use
unique approaches. The first is from Bernhard Friedrich Thibaut (1775-1832) and has been
periodically endorsed.
Figure XIV
c
b
a
b
c
a
b
c
c
In triangle ABC (see Figure XIV), side AB is allowed to rotate about A clockwise. It
rotates until it coincides with CA produced to L. Let CL rotate clockwise about C until it
coincides with BC to produce M. When BM has rotated clockwise about B until it
coincides with AB produced to N, it appears as though AB has undergone a complete
rotation through four right angles. However, three angles of the rotation are the exterior
angles of the triangle. Their sum is equal to four right angles, so the sum of the interior
angles must be equal to two right angles.
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This proof is a good example of one that depends upon direction. Both rotation and
translation are involved, since the rotations take place about different points on the rotating line.
Segment AB is eventually translated along AB through a distance equal to the perimeter of the
triangle. Only in Euclidean Geometry can the translations be ignored, and the assumption
amounts to taking the Fifth Postulate for granted. A similar argument can be used for a spherical
triangle. Assuming that the translations would take place about a specific point would only lead
one to the assumption that if two lines intersect two given lines and make equal corresponding
angles with them, then they make equal corresponding angles with each other. However, this is
the proposition to be proved, since the two lines would be parallel. Furthermore, even if one
attempts to rotate about a point, such as A, this proof still does not succeed. For example, if PQ
is drawn through A, making angle PAL equal to angle MCA, it cannot be concluded that angle
PAB will be equal to angle CBN, since this would essentially be equivalent to the proposition to
be proved (Wolfe 40-41).
The proof of Swiss mathematician Louis Bertrand (1731-1812) also captured attention.
He used the following argument to attempt to prove the Fifth Postulate directly.
Figure XV
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Given lines AP1 and A1B1 (see Figure XV) cut by transversal AA1 in such a way that the
sum of angles P1AA1 and AA1B1 is less than two right angles, AP1 and A1B1 meet if
sufficiently produced. Draw AB so that angle BAA1 equals angle B1A1A2, where A2 is a
point on AA1 produced through A1. AP1 will lie within angle BAA1, since angle P1AA1 is
less than angle B1A1A2. Construct AP2, AP3, …APn so that angles P1AP2, P2AP3, …Pn1APn
are all equal to angle BAP1. Since an integral multiple of angle BAP1 can be found
which exceeds angle BAA1, n can be chosen so large that APn will fall below AA1 and
angle BAPn greater than angle BAA1. Since the infinite sectors BAP1, P1AP2,…Pn-1APn
can be superposed, they have equal areas. Each has an area equal to that of the infinite
sector BAPn divided by n.
On AA1 produced through A1, measure A1A2, A2A3,…An-1An all equal to AA1, and
construct A2B2, A3B3, …AnBn so that they make with AAn the same angle which A1B1
makes with that line. Then the infinite strips BAA1B1, B1A1A2B2, …. Bn-1An-1AnBn can
be superposed and thus have equal areas, each equal to the area of the infinite strip
BAAnBn divided by n. Since the infinite sector BAPn includes the infinite strip BAAnBn,
it follows that the area of the sector BAP1 is greater than that of the strip BAA1B1, and
therefore, AP1 must intersect A1B1 if produced sufficiently (Wolfe 42-43).
The fallacy in this proof lies in the treatment of infinite magnitudes as finite. The idea of
congruence as used above was not even formally defined. Any comparison of infinite
magnitudes must be made to depend upon the process of finding the limit of a fraction. Both the
numerator and the denominator will become infinite (Wolfe 43).
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When János Bolyai (1802-1860) announced his work and findings in non-Euclidean
geometry, his father warned him that if he had found a solution, it should be published quickly
since it might be easily copied by someone else and also since it might be the appropriate time
for this information to be welcomed into the math world. János Bolyai published his discoveries
as a twenty-six page appendix to a book by his father, who sent it to German mathematician
friend Carl Friendrich Gauss (1777-1855). However, János was disappointed when Gauss
replied, saying that he himself had found similar results but found few people willing to receive
them. János imagined that his father had informed Gauss of the results, and that Gauss was trying
to claim them as his own. János became depressed and never again published his work
(Greenberg 140-142).
There is evidence that Gauss had been working on non-Euclidean geometry concepts
since age fifteen (1792). In response to a question, he had once indicated that he had investigated
what would happen if the sum of the angles of a triangle was less than 180 degrees. He stated
that it led to a curious geometry, in which he could solve every problem with the exception of
the determination of a constant, which could not be designated a priori (not from experience).
The greater the constant, the nearer one gets to Euclidean geometry. However, if this nonEuclidean geometry were true, it could be determined a posteriori (from experience) (Greenberg
143-144).
Gauss was afraid to make his discoveries public, for fear of the “metaphysicians,” who
were followers of Immanuel Kant. Kant believed that Euclidean space is inherent in the human
mind. Gauss also was a perfectionist and never liked to make his work public until he had
checked it thoroughly (Greenberg 144-145). In general, Gauss’ discoveries came by rejecting the
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Fifth Postulate, assuming that through a point, more than one parallel can be drawn to a given
point (Wolfe 48).
Russian mathematician Nikolai Ivanovich Lobachevsky (1792-1856) would also become
a major player in the non-Euclidean geometry evolution. He was the first to publish an account
of non-Euclidean geometry (1829), but his work attracted little attention because it appeared in
Russia, where those who read it were critical. When he published a treatise in German in 1840, it
caught Gauss’ attention. He challenged Kantian doctrine of space as a subjective intuition, but he
was not widely appreciated, and was even fired from the University of Kazan, despite twenty
years of service as a teacher (Greenberg 147).
The world began to take non-Euclidean concepts seriously in 1855, at the time of Gauss’
death. Other mathematicians began to take up interest in the area and work to expand it. Finally,
in 1868, Beltrami proved that there was no proof of the Fifth Postulate possible. He did this by
proving that non-Euclidean and Euclidean geometry are just as consistent (Greenberg 147),
meaning that it is proved that there is no possible way to derive a contradiction (Greenberg 178).
This resulted in the Metamathematical Theorem 1, which states that “if Euclidean geometry is
consistent, then so is hyperbolic geometry (Greenberg 178).” This lead to Corollary 2, which
states that “If Euclidean geometry is consistent, then no proof or disproof of the parallel postulate
from the rest of Hilbert’s postulates will ever be found, i.e., the parallel postulate is independent
of the other postulates (Greenberg 179).” To prove this, assume the contrary- that a proof of the
parallel postulate exists. Then hyperbolic geometry would be inconsistent, since the hyperbolic
axiom contradicts a proved result. However, hyperbolic geometry should be consistent relative to
Euclidean. This contradiction proves that no proof of the parallel postulate exists (Greenbery
179).
Yeust 24
Technically speaking, non-Euclidean geometry is any geometry different from Euclid’s
geometry. Today, many of these geometries are known. The geometry discovered by Gauss, J.
Bolyai, and Lobachevsky was hyperbolic geometry, which by definition, is “the geometry you
get by assuming all the axioms for neutral geometry” and replacing the Parallel/Fifth Postulate (a
form of which is referred to as Hilbert’s parallel postulate) by its negation. This is called the
hyperbolic axiom (Greenberg 148). The Hyperbolic Axiom states that “In hyperbolic geometry
there exist a line l and a point P not on l such that at least two distinct lines parallel to l pass
through P (Greenberg 148).”
Figure XVI
This axiom (refer to Figure XVI) immediately identifies the flaw in Legendre’s attempted proof
of the parallel postulate. The entire line l lies in the interior of angle APB without meeting either
side. Legendre had assumed this to be impossible. However, the first important consequence of
the hyperbolic axiom is a lemma which states that there exists a triangle whose angle sum is less
than 180 degrees. From this, the Universal Hyperbolic Theorem follows, which states that “In
hyperbolic geometry, for every line l and every point P not on l, there pass through P at least two
Yeust 25
distinct parallels to l (Greenberg 149).” To prove this, drop perpendicular PQ to l and erect line
m through P perpendicular to PQ (see Figure XVII).
Figure XVII
Let R be another point on l, erect perpendicular t to l through R, and drop perpendicular
PS to t. Now PS is parallel to l since they are both perpendicular to t. Claim that m and PS
are distinct lines. Assume that S lies on m. Then PQRS is a rectangle. It was previously
shown in hyperbolic work that if one rectangle exists, then all triangles have an angle
sum of 180 degrees. This however, contradicts the Lemma (Greenberg 149).
This proof and information provide just a brief glimpse to the introduction of nonEuclidean geometry concepts and the advent of this area of thought. In conclusion, after
centuries of challenges and intrigue concerning the search for a proof of Euclid’s infamous Fifth
Postulate, the answer was actually the opposite of what everyone had attempted. It was
determined that the Fifth Postulate could not be proven, and from this point on, mathematicians
had the green light to begin to develop systems of geometry around the resulting conclusions of
Yeust 26
the Fifth Postulate if it was not able to be proven. These are the foundations of what has come to
be known today as non-Euclidean geometry.
Yeust 27
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