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# Euclid was wrong!

(Alex and Charlie are sitting down at a library when Tom comes in, with 13 large books)

Alex: You don’t normally take so many books! And such big ones too!

Charlie: You do realise that if you are late in handing these back in the fees will be enormous.

Tom: I was looking for a geometry book when I found this. It’s by some chap called Euclid. Let’s see. If two lines both cross one line, then the two lines meet on the side of the line where the angles add up to less than 180O. What does that even mean? And it’s supposed to be self-evident!

Alex: Let me see. It makes a sort of sense to me. But can it be wrong? I think it probably is. After all, can something so complex really be self-evident?

Charlie: Look at this footnote: This is the same as saying that the angles of a triangle add up to 180O. This makes a lot more sense. By the way, where are you going for the holidays?

Tom: The Maldives.

Charlie: I’m going to the North Pole. But wait... Consider the triangle Pole-Ecuador-Maldives. The angle at the Pole is 150O. The angles at Ecuador and the Maldives are both 90O. These add up to 330O, which is obviously more than 180O.

Tom: So Euclid was wrong. Let’s see this footnote. There are two geometries that do not satisfy the fifth postulate. These are the elliptic and the hyperbolic geometries. So the Earth is elliptic geometry! It seems that some people tried to prove him right. One of these was Girolamo Saccheri. His last work, Euclid Freed of Every Flaw in 1733 attempted to prove that the parallel postulate was true. He considered a symmetric quadrilateral with two right angles. He found three cases: that both other angles were right, obtuse, or acute. Right angles prove the parallel postulate. Obtuse angles lead to lines with finite length, contradicting the second postulate. This leads to elliptic geometry. However, Saccheri wanted to prove it assuming the first four postulates were true, and so he rejected this case. He had less luck with the acute-angled case, which leads to hyperbolic geometry, as this is consistent with the four other postulates. He did manage to prove many interesting results about it though. He finally claimed that it was “repugnant to the nature of a straight line” since many of these were in direct contradiction with Euclidean geometry. However, this seems to be circular reasoning, as he is proving that the acute-angled case contradicts the right-angled case, before even proving the right-angled case.

Alex: Do you want some Pringles?

Charlie: Yes please! Oh, wait! If I draw some lines on this to make a triangle it is less than 180O. So a Pringle is hyperbolic geometry. It seems that triangles can’t get very big. In fact, (Charlie draws a triangle) I think I’ve found one with an angle sum of 0O! It seems that this is about as big a triangle as I can get. There are many ways to define absolute distance. One is the square root of the area of the largest triangle. This is similar to elliptic geometry, where there are many absolute units. One of the most common ones is the radius. Another is 2r√π. You can’t get bigger triangles than the whole sphere minus a point, and the root of this area is this length.

Alex: Now elliptic geometry makes sense to me. The lines are great circles, the largest possible. Of course, small triangles have an angle sum close to 180O but bigger triangles can have angle sums up to 900O, if you allow a triangle to be bigger than half the sphere.. All lines cross, so parallel lines do not exist. It seems that elliptic geometry was much more practical than Euclidean geometry as sailors were travelling on a giant sphere. But the only person who knows everything about this is your Castleran. Let me get out Castleran Pocket Edition. It seems that it is almost out of battery.

Alex: TELL ME ABOUT NON-EUCLIDEAN GEOMETRY.

Castleran: There are many constructs of non-Euclidean geometry. The sphere is obviously a construct of elliptic geometry. A finite hyperbolic geometry is formed by a saddle shape. An infinite but bounded one can also be formed by a pseudosphere, which is the surface of revolution of a tractrix. It has negative curvature everywhere apart from its edge. One of the first books on non-Euclidean geometry was by Girolamo Saccheri. It was his last work, published in 1733, and titled “Euclid Freed of Every...

(The screen goes blank)

Tom: It’s out of battery already. Have a look at this short book. Let’s see... This chapter is called “Euclid Was Wrong”!

Alex: You don’t realise that if you are late in handing these back in the fees will be tiny.

Charlie: You normally take so few books! And such small ones too!

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