Asimov, Isaac Left Hand of the Electron [pdf]

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agree that "saying a" is equal to "saying b" in this case is consist-
ent with all the rest of geometry, so we can get away with it.
Let us now say that two lines, such as EF and CD, which do not
intersect with each other when extended any finite distance, how-
ever great, are "parallel."
Clearly, there is only one line passing through L that can be
parallel to line CD, and that is line EF. Any line through L that
does not coincide with line EF is (however slightly) either of the
type of line GH or of line JK, with an interior angle on one side
or the other that is less than a right angle. This argument is sleight
of hand, and not rigorous, but it allows us to see the point and
say: Given a straight line, and a point outside that line, it is pos-
sible to draw one and only one straight line through that point
parallel to the given line.
This statement is entirely equivalent to Euclid's fifth postulate.
128 THE PROBLEM OF NUMBERS AND LINES
If Euclid's fifth postulate is removed and this statement put in
its place, the entire structure of Euclidean geometry remains
standing without as much as a quiver.
The version of the postulate that refers to parallel lines sounds
clearer and easier to understand than the way Euclid puts it, be-
cause even the beginning student has some notion of what paral-
lel lines look like, whereas he may not have the foggiest idea of
what interior angles are. That is why it is in this "parallel" form
that you usually see the postulate in elementary geometry books.
Actually, though, it isn't really simpler and clearer in this form,
for as soon as you try to explain what you mean by "parallel"
you're going to run into the matter of interior angles. Or, if you
try to avoid that, you'll run into the problem of talking about
lines of infinite length, of intersections at an infinite distance be-
ing equivalent to no intersection, and that's even worse.
But look, just because I didn't prove the fifth postulate doesn't
mean it can't be proven. Perhaps by some line of argument, ex-
ceedingly lengthy, subtle and ingenious, it is possible to prove
the fifth postulate by use of the other four postulates and the five
common notions (or by use of some additional axiom not in-
cluded in the list which, however, is much simpler and more
"obvious" than the fifth postulate is).
Alas, no. For two thousand years mathematicians have now and
then tried to prove the fifth postulate from the other axioms
simply because that cursed fifth postulate was so long and so un-
obvious that it didn't seem possible that it could be an axiom.
Well, they always failed and it seems certain they must fail. The
fifth postulate is just not contained in the other axioms or in any
list of axioms useful in geometry and simpler than itself.
It can be argued, in fact, that the fifth postulate is Euclid's
greatest achievement. By some remarkable leap of insight, he
realized that, given the nine brief and clearly "obvious" axioms,
he could not prove the fifth postulate and that he could not do
without it either, and that, therefore, long and complicated though
the fifth postulate was, he had to include it among his assumptions.
EUCLI D' S F I F T H 129
So for two thousand years the fifth postulate stood there: long,
ungainly, puzzling. It was like a flaw in perfection, a standing
reproach to a line of argument otherwise infinitely stately. It
bothered the very devil out of mathematicians.
And then, in 1733, an Italian priest, Girolamo Saccheri, got
the most brilliant notion concerning the fifth postulate that any-
one had had since the time of Euclid, but wasn't brilliant enough
himself to handle i t -
Let's go into that in the next chapter.
11 THE PLANE TRUTH
There are occasionally problems in immersing myself in these
science essays I write. For instance, I watched a luncheon com-
panion sprinkle salt on his dish after an unsatisfactory forkful,
try another bite, and say with satisfaction, "That's much better."
I stirred uneasily and said, "Actually, what you mean is, 'I like
that much better.' In saying merely, "That's much better,' you are
making the unwarranted assumption that food can be objectively
better or worse in taste and the further assumption that your own
subjective sensation of taste is a sure guide to the objective
situation."
I think I came within a quarter of an inch of getting that dish,
salted to perfection as it was, right in the face; and would have
well deserved it, too. The trouble, you see, was that I had just
written the previous chapter and was brimful on the subject of
assumptions.
So let's get back to that. The subject under consideration is
Euclid's "fifth postulate," which I will repeat here so that you
won't have to refer back to it:
If a straight line falling on two straight lines makes 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 two right angles.
All Euclid's other axioms are extremely simple but he appar-
ently realized that this fifth postulate, complicated as it seemed,
could not be proved from the other axioms, and must therefore
be included as an axiom itself.
For two thousand years after Euclid other geometers kept try-
THE PLANE TRUTH
131
ing to prove Euclid too hasty in having given up, and strove to
find some ingenious method of proving the fifth postulate from
the other axioms, so that it might therefore be removed from the
list if only because it was too long, too complicated, and too not
immediately obvious to seem a good axiom.
One system of approaching the problem was to consider the
following quadrilateral: [ Pobierz całość w formacie PDF ]

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