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Science Fiction World , v. 1, issue 4, August 1946

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SCIENCE FICTION WORLD (1)
EXCURSIONS IN SCIENCE #1
The case of the unadmitted axion.
Narrated by T.D. Clarence, and presented as the first of a series by S F W.
[illo: line drawings of an acute angle; triangle with interior angles marked; two crossing lines; a square; and a circle]
About 2200 years ago a brilliant fellow by the name of Euclid assembled the by then well advanced science of geometry into a remarkable textbook which has stood as a model during all the time since. Euclid prefaced his text with ten assumptions which he considered to be self-evident truths. The first nine of these assumptions were so easily and clearly stated that even Lil Abner could have understood them. As for the tenth --- well, here's what Euclid said :
"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 two angles less then two right angles."
That statement stood out like a supernova and practically challenged dissent. It's equivalent meaning can be paraphrased in a variety of ways such as
"Through a given point can be drawn only one parallel to a given line." -- Playfairs axiom -- or
"There exists a pair of straight lines everywhere equally distant from each other."
Of course a great many fellows set out to prove this 'fifth postulate' as it was called. The name was derived through a division of the original ten basic premises into two groups of five each, the first group being called 'common assumptions' and the second group being called 'postulates' of which the one in question was of course the 'fifth'. Although no one actually succeeded in a proof, a great many thought that they did. The end result of their chimera chasing was the discovery of the Non-Euclidean geometries.
In hyperbolic geometry we toss out the 'fifth' and proceed with this sense twister:
"Through a given point not on a given line more than one line can be drawn not intersecting the given line."
You might take that to mean that, in hyper-geo, there were dozens of lines through a given point, which were parallel to a given line.
There are only two! All the rest are labelled 'non-intersecting lines' and act entirely in a different manner than do paralleles.
In Elliptic Geometry we take this premise --
"Two straight lines always intersect one another," and add something else to make it a little more complex :
"Every straight line is boundless."
The latter statement means only that all lines return upon themselves. Straight lines that is.

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