# Dictionary of Mathematical Eponymy: Osborn’s Law

Today’s entry in the Dictionary of Mathematical Eponymy is, by some distance, the entry that’s been most useful to me since I learned about it. (The Elo rating is probably in second place.) It’s also a unique entry in that I have next to no information about its originator.

### What is Osborn’s Rule?

Osborn’s Rule, or Osborn’s Law, was introduced to me in sixth form when I was struggling to remember by trig and hyperbolic trig identities. It says:

If every product of two sines in a valid trigonometric identity is replaced by the corresponding product of hyperbolic sines, multiplied by -1, the result is a valid hyperbolic identity, and vice versa. (Cosines are replaced by hyperbolic cosines, and vice versa).

For example, if I know that $\cos^2(x) + \sin^2(x) \equiv 1$, then I also know that $\cosh^2(x) - \sinh^2(x) \equiv 1$. I’ve replaced the product of sines ($\sin^2(x)$) with the corresponding product of hyperbolic sines, multiplied by -1 (namely, $-\sinh^2(x)$).

For a more complicated example, $\cos(2x) - \cos(6x)\equiv 2\sin(2x)\sin(4x)$; the corresponding hyperbolic identity is $\cosh(2x) - \cosh(6x) = -2\sinh(2x)\sinh(4x)$.

### Why is it important?

I’m not sure it’s all that important - the part it played in my getting my A-level grades was minor at best - but I think it’s neat. It ties together circular and hyperbolic trigonometry in a way that appeals to me, and can be explained using the exponential definitions of the various functions involved.

That’s why it interests me, I think: it was one of my first exposures to a Proper Mathematical Oo, something at the right level for me that seemed a bizarre coincidence at first, but on closer inspection turned out to have some depth to it.

### Who was George Osborn?

It took me a good deal of digging to find out anything at all about George Osborn, not least because there are several much more famous people with very similar names.

Some things were immediately clear: Osborn’s Rule (a “Mnemonic for Hyperbolic Formulae”) was published in The Mathematical Gazette in 1902, and he wrote several algebra textbooks with C.F. French around the start of the 20th century.

So, time for some sleuthing! This logic article suggests that a George Osborn (1864-1932) corresponded with Charles Lutwidge Dodgson about logic, and this contribution to the Mathematical Gazette unlocked the whole puzzle: it gives his address as The Leys, Cambridge - a prestigious private school whose alumni include Andrew Wiles, Martin Bell, Jamie Murray and the brothers Hitchens.

Osborn was Senior Physics Master at The Leys from 1888 to 1926, an excellent chess player, keen on Spanish literature and the study of the New Testament - aside from being an excellent mathematician (he was 17th Wrangler in 1887). He died on June 20th, 1932.

It seems his grandfather was Rev. Dr. George Osborn, a Methodist Scholar; his father was also a George.

I’d love to know more if you have any more resources!

* I'm extremely grateful to Alison Lainchbury, archivist and librarian at The Leys, for her help in piecing together Osborn's career and helping to turn phrases from "seems to have" into more definitive language.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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