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

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SCIENCE FICTIONWORLD (3)
There are two types of Elliptic geometry:
a... [illo: drawing of the upper half of an ellipse, divided into 2 halves by a vertical line]
b...[illo: drawing of an almost circular ellipse, divided into quarters by horizontal and vertical lines]
The representations, a and b , above, illustrate the two two types of Elliptic geo and simultaneously the proposition which states that all perpendiculars to a given line meet in the same point. The meridians of longitude on the earth which are perpendicular to the Equator again present a handy picture of this state of affairs. The points through which all such perpendiculars pass are called the poles of the line. In type 'b' we operate under the assumption that any given straight line has _two_ poles, a premise incidentally, which gives us that geometry commonly called 'Riemannian geometry'. In type 'b' lines intersect and enclose an area; the sum of two angles of a triangle can be less than the third; a triangle with two angles equal may have the sides opposite them unequal, and the sum of all the angles of a triangle is greater than two right angles.
Type 'b' is designated as 'Double Elliptic' to distinguish it from 'a' called 'Single Elliptic'.
In type 'a' we find that a straight line does not divide the plane into two regions; you can go from one side of the line to the other without crossing the line! Try that on your Mobius strip which is an easily constructed paper figure possessing Single Elliptic properties. Single Elliptic geometry conceives of a surface as having only one side in contrast to two sided conceptions of the other geometries.
Calling the Euclidean sphere again for reference we can distinguish between the workings of Euclidean and Double Elliptic Geometries. In Euclidean we consider the sphere as a three dimensional object in order to work surface problems; in Double Elliptic we are concerned only with the surface and work two dimensionally.
All three main classifications of geometry are logically consistent and serve equally well in the handling of engineering problems but Euclidean has an enormous advantage over the others because of it's comparative simplicity.
If the human race ever gets around to engineering projects that are something like galactic in scope we might then find that one or the other of the geometries is better suited to describe the universe than the others. At present, there are objections to our drawing the conclusion that anyone of them is better than the others one of which is that our methods of measurement and even our instruments themselves must work from one or the other geometry as a basis. An out here would be to get them all together a la Kuttner in a system of variable truths. Fairy geometry!
It makes you wonder all the more what we'll find out when we finally get some of those Van Vogt and E.E. Smith types of space ships. Meanwhile I'm sticking solidly to Euclidean.

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