Improving Euclid’s Fifth Postulate

Michael Chabon (author of The Amazing Adventures of Kavalier and Clay), writing in the New York Review of Books, discusses The New Annotated Sherlock Holmes. He quotes Holmes in The Sign of Four:

Holmes remarks to Watson, referring to A Study in Scarlet, and continues:

Honestly, I cannot congratulate you upon it. Detection is, or ought to be, an exact science, and should be treated in the same cold and unemotional manner. You have attempted to tinge it with romanticism, which produces much the same effect as if you worked a love-story or an elopement into the fifth proposition of Euclid.

Some of us feel, of course, that the fifth proposition of Euclid would only be improved by a nice juicy elopement.

What? I hope not too many people believe that. Juicy elopements are all well and good, but they would not improve Euclid’s Fifth Postulate:

If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

The postulate is already cumbersome enough! Mixing elopements into the act would only degrade an already-confusing situation. What would you want, the two indefinitely-extended straight lines to exchange secret vows of eternal love on the side on which the angles are less than right angles? Madness.

Of course, the Fifth Postulate (or Parallel Postulate) is not necessarily “true”; by relaxing this assumption, we are led to non-Euclidean geometry, which ultimately provides the basis for the geometric understanding of spacetime achieved by Einstein in general relativity. Everyone knew that the parallel postulate was suspiciously ugly, but it took quite a while to gather up the courage to simply discard it and see what happens.

Likewise, of course, I doubt that many elopements would be improved by Euclid’s Fifth Postulate. Like it or not, the beauty of mathematics springs from its rigorous austerity; it’s a different kind of beauty than we might find in a clandestine marriage, but no less compelling for that.

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