Introducing a Very Small Number

Last time I talked mathematics I introduced the idea of using some little tolerated difference between quantities. This tolerated difference has an immediately obvious and useful real-world interpretation: if we measure two things and they differ by less than that amount, we’d say they’re equal, or close enough to equal for whatever it is we’re doing. And it has great use in the nice exact proofs of some sophisticated mathematical concepts, most of which I think I can get to without introducing equations, which will make everyone happy. Readers like reading things that don’t have equations (folklore has it that every equation, other than E = mc2, cuts book sales in half, although I don’t remember seeing that long-established folklore before Stephen Hawking claimed it in A Brief History Of Time, which sold a hundred million billion trillion copies). Writers like not putting in equations because web standards have evolved so that there’s not only no good ways of putting in equations, but there aren’t even ways that rate as only lousy. But we can make do.

The tolerated difference is usually written as ε, the Greek lower-case e, at least if we are working on calculus or analysis at least, and it’s typically taken to mean some small number. The use seems to go back to Augustin-Louis Cauchy, who lived from 1789 to 1857, who paired it with the symbol δ to talk about small quantities. He seems to have meant δ the Greek lowercase d, to be a small number representing a difference, and ε as a small number representing an error, and the symbols have been with us ever since.

Cauchy’s an interesting person, although it seems sometimes that every mathematician who lived in France anytime around the Revolution and the era of Napoleon was interesting. He was certainly prolific: the MacTutor biography credits him with 789 published papers, and they covered a wide swath of mathematics: solid geometry, polygonal numbers, waves, inelastic shocks, astronomy, differential equations, matrices, and a powerful tool called the Fourier transform. This is why mathematics majors spend about two years running across all sorts of new things named after Cauchy — the Cauchy-Schwarz inequality, Cauchy sequences, Cauchy convergence, Cauchy-Reimann equations, Cauchy-Kovalevskaya existence, Cauchy integrals, and more — until they almost get interested enough to look up something about who he was. For a while Cauchy was tutor to the grandson of France’s King Charles X, but apparently Cauchy had a tendency to get annoyed and start screaming at the uninterested prince. He has two lunar features (a crater and an escarpment) named for him, indicating, I suppose, that Charles X wasn’t asked for a reference.

Did King George III pay too little for astronomers or too much for tea?

In the opening pages of his 1998 biography George III: A Personal History, Christopher Hibbert tosses a remarkable statement into a footnote just after describing the allowance of Frederick, Prince of Wales, at George III’s birth:

Because of the fluctuating rate of inflation and other reasons it is not really practicable to translate eighteen-century sums into present-day equivalents. Multiplying the figures in this book by about sixty should give a very rough guide for the years before 1793. For the years of war between 1793 and 1815 the reader should multiply by about thirty, and thereafter by about forty.

“Not really practical” is wonderful understatement: it’s barely possible to compare the prices of things today to those of a half-century ago, and the modern economy at least existed in cartoon back then. I could conceivably have been paid for programming computers back then, but it would be harder for me to get into the field. To go back 250 years — before electricity, mass markets, public education, mass production, general incorporation laws, and nearly every form of transportation not muscle or wind-powered — and try to compare prices is nonsense. We may as well ask how many haikus it would take to tell Homer’s Odyssey, or how many limericks Ovid’s Metamorphoses would be.
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