An Overused Intermediacy

I had wanted to talk about the Intermediate Value Theorem, since it’s one of those little utility theorems that doesn’t draw a lot of attention by itself but does have some wonderful results that depend on it. My context was in explaining just what Chiaroscuro had done when he figured out the fifth root of 1/6000th by guessing at it. I mean, he figured he was guessing at it, but there’s good reasons why this guessing would pay off and why he’d get to an answer near enough the right one.

And I wanted to talk about one of my favorite results of the Intermediate Value Theorem, at least as I remembered it: that at any time of the day or night, there must be at minimum a pair of antipodal sites — locations directly opposite the center of the Earth from one another — which have exactly the same temperature. Or the same humidity. Or the same of any meteorological measurement. I had read this, I was sure, in Richard Courant and Herbert Robbins’s masterpiece of mathematics writing, What Is Mathematics? and went digging about to find it precisely stated, particularly since as I remembered it was possible to get any pair of measurements — say, temperature and humidity together — exactly equal at antipodal sites.

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