Views of
Euclid's Parallel Postulate

in Ancient Greece and in Medieval
Islam

Michelle Eder

History of Mathematics

Rutgers, Spring 2000

Throughout the course of history there have been many remarkable advances, both intellectual and physical, which have changed our conceptual framework. One area in which this is apparent is Mathematics. In some cases mathematicians have spent years of their lives trying to solve a single problem. Such are Euclid, Proclus, John Wallis, N.I. Lobachevsky and Abu' Ali Ibn al-Haytham, who will be considered here in connection with the history of Euclid's parallel postulate.

Little or nothing is reliably known about Euclid's life. It is believed
that he lived in Alexandria, Greece around 300 B.C. (Varadarajan,
page 3). Some say that he was the most successful textbook writer the
world has ever known, whose manuscripts dominat *E* the teaching of the
subject (Smith, page 103). In the writing of his *Elements*,
Euclid "successfully incorporated all the essential parts of the
accumulated mathematical knowledge of his time" (Sarton, page
104). And although he was no t the first of Greek mathematicians to
consolidate the materials of geometry into a text, he did so so
"perfectly" that it came to replace the works of his
predecessors (Morrow, page *xxii*). Every step to the proofs of his
theorems was justified by referring back to a previous definition,
axiom, theorem or proof of a theorem. However, though Euclid's
*Elements* became the "tool-box" for Greek mathematics,
his Parallel Postulate, postulate V, raises a great deal of
controversy within the mathematical field. Euclid's formulation of
the parallel postulate was as follows:

(Heath, page 202)

This states:
*That, if a straight line falling on two straight
lines make the interior angles on the same side less than two right
angles, the two straight lines, if produced indefinitely, meet on that
side on which are the angles less than the tw o right angles.
*
(Heath, page 202). This postulate, one of the most controversial
topics in the history of mathematics, is one that geometers have tried
to eliminate for more than two thousand years.

Among the
first to explore other options to the parallel postulate were the
Greeks. The Greek geometers of the 7th to 3rd centuries B.C. helped
to enrich the science with new facts and took important steps toward
the formulation of a
"rigorou s logical sequence
"
(Pogorelov, page 186). They saw the parallel postulate as a theorem
involving many difficulties whose proof required a number of
definitions and theorems. In comparison to Euclid's other postulates
the parallel postulate was com plicated and unclear. In addition,
some found it difficult to accept on an intuitive basis. Even Euclid
himself must have been displeased with it, for he made an effort to
prove some of his other propositions without the use of the parallel
postulate an *D* only began using it when it became absolutely necessary
(Varadarajan, page 5). From his view,
"there was no way out but
to accept it as a postulate and go ahead.
" (Sarton, page 39).

In the course of many failed attempts to modify this postulate, mathematicians have tried desperately to find an easier way to deal with the parallel postulate. a person of average intelligence might say that the proposition is evident and needs no proof (Sarton, page 39). However, from a more sophisticated mathematical viewpoint one would realize the need of a proof and attempt to give it (Sarton, page 39). In the attempt to clarify the status of this postulate some mathematicians tried to eliminate it altogether by replacing it with a simpler, more convincing axiom, while others tried to deduce it from other axioms. In these attempts, all these people proved that the fifth postulate is not necessary if one accepts another postulate "rendering the same service "; however, the use of them would seem "artificial " (Sarton, page 40). It is because of this that Euclid, seeing the necessity of the postulate, selected what he apparently found to be the simplest form of it as his fifth postulate (Sarton, page 40).

Among those who attempted a proof of the parallel postulate was
Proclus, who lived 410 to 485 A.D. (Heath, page 29), receiving his
training in Alexandria, Greece and afterwards Athens, where he became
a "prolific writer " (Smith, page 139). His works, a
valuable source of information on the history of Greek geometry,
included a commentary on Book I of Euclid's *Elements*. This
commentary may not have been written with the intention of correcting
or improving upon Euclid, but there is one instance in which he
attempts to alter a "difficulty " he finds in Euclid's
*Elements* (Heath, page 31). This difficulty is what we commonly
refer to as the "parallel postulate ".

The statement Proclus proves instead of the parallel postulate is,
"Given
a
+
b
<
2d , prove that the straight lines g'
and g''
meet at a certain point *C*."

In his proof of this,
Proclus draws a straight line, g'''
through a given
point a parallel to
g'.
Then taking a point *B* on g
''
he drops a
perpendicular to g
'''
from it. From this he reasons that since the distance
from
g
'''
increases without limit as the distance between a and *B* grows and the
distance between
g'
and
g'''
is constant then there must be a point *C* on
g''
belonging to
g'. And it
is this point where
g'
and g''
meet, thus completing his proof. However, as with most
of the other alternatives to the parallel postulate, this one had
faults. It is observed by Pogorelov that the parallel straight lines
this proof relies on are not explicitly contained in the other
postulates or axioms and therefore cannot be deduced from them.
(Pogorelov, page 188).

Another person who attempted a proof to the parallel postulate was John Wallis. Wallis studied at Emmanuel College in Cambridge where he earned both a B.A. and a M.A in Theology in 1637 and 1640, respectively (Smith, page 407). Although his degre es were in theology, his "taste " was in the line of physics and mathematics (Smith, page 407). In 1649 he was elected to the Savilian professorship of geometry at Oxford (Smith, page 407). In his interest in Mathematics, Wallis was one of the first to recognize the "significance of the generalization of exponents to include negative and fractional, as well as positive and integral, numbers. " (Smith, page 408).

In addition to Wallis' recognition of the significance of exponents, he also attempted a proof to the parallel postulate. However, instead of proving the theorem directly with neutral geometry, he proposed a new axiom. This postulate expressed the idea was that one could either magnify or shrink a triangle as much as one likes without distortion. Using this, Wallis proves the parallel postulate as follows.

He begins with two straight lines making, with a
third infinite straight line, two interior angles, less than two right
angles.
He then
"slides
" one angle down the line AF until it reaches a designated position
ab, cutting the first line at
p
Then using his first postulate, he claims that the two triangles
*aC
b* and *ACP* are similar,
thus showing that *AB* and *CD* meet at a point *P*,
and proving the theorem.
However, this too had a fault. In fact, the original postulate that
he based the proof on was logically equivalent to Euclid's fifth
postulate. (Heath, page 210). Therefore, he had assumed what he was
trying to prove, which makes his proof invalid.

In addition to Proclus' and Wallis' proofs, in 1826 another mathematician's replacement of the parallel postulate lead to the discovery of Non-Euclidean geometry. This mathematician was N.I. Lobachevsky. Lobachevsky was a Russian mathematician wh o lived 1792 to 1856. For his proof to the parallel postulate, Lobachevsky proved that "Atleast two straight lines not intersecting a given one pass through an outside point. " In proving this he hoped to find a contradiction in the "Eucli dean corollary system ". However, in the development of his theory, Lobachevsky, instead, saw that the system was "non-contradictory ". From this he drew the conclusion that there existed a geometry, different from Euclidean, with the fifth postulate not holding. This geometry became known as "Non-Euclidean " geometry (Pogorelov, page 190).

Another group to comment on Euclid's parallel postulate was the
Medieval Islams. From the ninth to the fifteenth centuries, extensive
mathematical activity revived only in the large cosmopolitan cities in
Islam. Arabic thinkers cultivated mathema tics in at least two ways.
The first was by "the preservation and transmission of older
knowledge " (Calinger, page 166). And the second was by original
contributions to arithmetic, algebra and geometry. In Islam, society
became more firmly es tablished and they began to focus their energies
more toward educational developments in Mathematics. To them
mathematics was closely linked to astronomy, astrology, cosmology,
geography, natural philosophy, and optics (Calinger, page 169). From
this, Islamic society shifted its interest to Greek thought. The
first of the Greek texts to be translated, *Euclid's Elements,*
brought the issues involved in the parallel postulate to the attention
of Islamic mathematicians, and they, too, as the Greeks before them,
began exploring the possibility of proving this postulate.

One mathematician from this time who contributed to clarification of
Euclid's
parallel postulate was Abu' Ali Ibn al-Haytham,
an Arabic physicist, mathematician, and astronomer.
He begins his proof by first addressing Euclid's definition
of parallel lines:

which is phrased so that all the "lines" involved will be

To
continue his proof, al-Haytham needs to show that line *CD* is equal to
*EF*, and that both are greater than *AB*.
Using what we now refer to as
the *Side-Angle-Side Theorem*,
he says that since *CA* is
equal to *AE*, angle *CAB* is equal to angle *EAB*
(right angles), and the side
*AB* is common, therefore the triangles *CAB* and
*EAB* must also be equal. Thus
line *CB* is equal to *EB*
and the two remaining angles must also be
equal. al-Haytham continues that angle *CBA* and angle *EBA*
are equal,
and since angles *ABD* and *ABF* are equal, therefore
the angles *CBD* and *EBF* must
also be equal. Next, by what we now refer to as the Side-Angle-Side
Theorem, he claims that since the angles *CDB* and *EFB*
are equal and sides *DB* and *BF* are equal,
therefore the triangles *CDB* and *EFB* must be equal.
Therefore, *CD* and *EF* are also equal. Then, since *CD*
is greater than *AB* (by assumption) *EF* must also be greater than *AB*. Next in
al-Haytham's proof, he says to imagine *EF* moving along *FB*
so that the angle *EFB*
remains a right angle throughout the motion, with *EF* remaining
perpendicular to *FB*.
Then when point *F* coincides with point *B*, line
*EF* will be "superposed" onto *AB*.
But he claims that since
the magnitude of *EF* is greater than that of *AB*,
point *E* will lie
outside *AB* (on the same side with *A*).
Thus at this point *EF* is equal to *HB*.
Next al-Haytham slides line BH along BD. If
in this process point *B* coincides with point *D* then *BH*
will be "superposed" on *DC* (because angles *HBF* and *CDB*
are equal). Then
since *BH* =
*EF* = *CD*,
al-Haytham claims that *H*
coincides with *C*. Thus al-Haytham has showed that if *EF*
is put in motion along *FD*, then the points *E* and *F*
will coincide with *C* and *D*,
respectively. Next he notes that if any straight line moves in this
way then it's ends will describe a straight line. Thus the point E
describes the straight line *EHC*. al-Haytham concludes that since H
does not lie on *AB* it can not coincide with point a and therefore
there must exist a surface bound by two straight lines which he finds
to be &q uot;absurd ", therefore proving that *CD* is neither
greater nor less than A. Thus al-Haytham has showed that *CD*
and all other perpendicular lines dropped from *AC* to *BD*
are equal to *AB*.
(Rosenfield, pages 59-62).

In conclusion, throughout the past 2300 years of mathematical history many mathematicians from all around the world have unsuccessfully been trying to prove Euclid's parallel postulate. Although these attempted proofs did not lead to the desired result, they did play a part in the development of geometry, enriching it with new theorems that were not based on the fifth postulate, as well as leading to the construction of a new geometry, Non-Euclidean geometry, not based on the parallel postulate.

Works Cited

- Heath, Sir Thomas L..
*Euclid: The Thirteen Books of Elements.*Dover Publications, Inc.: New York, 1956. - Morrow, Glenn R..
*Proclus: A Commentary on the First Book* - Pogorelov, A.
*Geometry*. Mir Publishers: Moscow, 1987. - Rosenfield, B. A..
*A History of Non-Euclidean Geometry:**Evolution of the Concept of a Geometric Space.*Springer-Verlag: New York, 1988. - Sarton, George.
*Hellenistic Science and Culture in the Last Three Centuries B. C.*Dover Publications, Inc.: New York, 1959. - Smith, D. E.
*History of Mathematics.*Dover Publications, Inc.: New York, 1951. - Varadarajan, V.S..
*Algebra in Ancient and Modern Times*, Hindustan Book Agency, 1998.