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this quadrilateral, and side AD is equal in length to side BC.
Given these facts, it is possible to prove that side DC is equal to
side AB and that angles ADC and DCB are also right angles (so
that the quadrilateral is actually a rectangle) if Euclid's fifth
postulate is used.
If Euclid's fifth postulate is not used, then by using only the
other axioms, all one can do is to prove that angles ADC and
DCB are equal, but not that they are actually right angles.
The problem then arises whether it is possible to show that
from the fact that angles ADC and DCB are equal, it is possible
to show that they are also right angles. If one could do that, it
would then follow from the fact that quadrilateral ABCD is a
rectangle, that the fifth postulate is true. This would have been
proven from the other axioms only and it would no longer be
necessary to include Euclid's fifth among them.
Such an attempt was first made by the medieval Arabs, who
carried on the traditions of Greek geometry while Western Eu-
rope was sunk in darkness. The first to draw this quadrilateral and
132 THE PROBLEM OF NUMBERS AND LINES
labor over its right angles was none other than Omar Khayyam
(1050-1123).*
Omar pointed out that if angles ADC and DCB were equal,
then there were three possibilities: 1) they were each a right
angle, 2) they were each less than a right angle, that is "acute,"
or 3) they were each more than a right angle, or "obtuse."
He then went through a line of argument to show that the
acute and obtuse cases were absurd, based on the assumption
that two converging lines must intersect.
To be sure, it is perfectly commonsensical to suppose that two
converging lines must intersect, but, unfortunately, commonsense
or not, that assumption is mathematically equivalent to Euclid's
fifth postulate. Omar Khayyam ended, therefore, by "proving" the
fifth postulate by assuming it to be true as one of the conditions
of the proof. This is called either "arguing in a circle" or "begging
the question," but whatever it is called, it is not allowed in
mathematics.
Another Arabian mathematician, Nasir Eddin al-Tus (1201-
74), made a similar attempt on the quadrilateral, using a differ-
ent and more complicated assumption to outlaw the acute and
obtuse cases. Alas, his assumption was also mathematically equiva-
lent to Euclid's fifth.
Which brings us down to the Italian, Girolamo Saccheri (1667-
1733), whom I referred to at the end of the previous chapter and
who was both a professor of mathematics at the University of Pisa,
and a Jesuit priest.
He knew of Nasir Eddin's work and he, too, tackled the quadri-
lateral. Saccheri, however, introduced something altogether new,
something that in two thousand years no one had thought of do-
ing in connection with Euclid's fifth.
Until then, people had omitted Euclid's fifth to see what would
happen, or else had made assumptions that turned out to be
* He wrote clever quatrains which Edward FitzGerald even more cleverly
translated into English in 1859, making Omar forever famous as a hedo-
nistic and agnostic poet, but the fact is that he ought to be remembered as
a great mathematician and astronomer.
THE PLANE TRUTH I33
equivalent to Euclid's fifth. What Saccheri did was to begin by
assuming Euclid's fifth to be false, and to substitute for it some
other postulate that was contradictory to it. He planned then to
try to build up a geometry based on Euclid's other axioms plus
the "alternate fifth" until he came to a contradiction (proving
that a particular theorem was both true and false, for instance).
When the contradiction was reached, the "alternate fifth"
would have to be thrown out. If every possible "alternate fifth" is
eliminated in this fashion, then Euclid's fifth must be true. This
method of proving a theorem by showing all other possibilities to
be absurd is a perfectly acceptable mathematical technique* and
Saccheri was on the right road.
Working on this system, Saccheri therefore started by assuming
that the angles ADC and DCB were both greater than a right
angle. With this assumption, plus all the axioms of Euclid other
than the fifth, he began working his way through what we might
call "obtuse geometry." Quickly, he came across a contradiction.
This meant that obtuse geometry could not be true and that
angles ADC and DCB could not each be greater than a right
angle.
This accomplishment was so important that the quadrilateral
which Omar Khayyam had first used in connection with Euclid's
fifth is now called the "Saccheri quadrilateral."
Greatly cheered by this, Saccheri then tackled "acute geome-
try," beginning with the assumption that angles ADC and DCB
were each smaller than a right angle. He must have begun the
task lightheartedly, sure that, as in the case of obtuse geometry,
he would quickly find a contradiction in acute geometry. If that
were so, Euclid's fifth would stand proven and his "right-angle
geometry" would no longer require that uncomfortably long state-
ment as an axiom.
As Saccheri went on from proposition to proposition in his
acute geometry, his feeling of pleasure gave way to increasing
* This is equivalent to Sherlock Holmes's famous dictum that when the
impossible has been eliminated, whatever remains, however improbable, must
be true.
T H E
134 PROBLEM OF NUMBERS AND LINES
anxiety, for he did not come across any contradiction. More and
more he found himself faced with the possibility that one could
build up a thoroughly self-consistent geometry which was based
on at least one axiom that directly contradicted a Euclidean
axiom. The result would be a "non-Euclidean" geometry which
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