Books like Euler

Leonhard Euler was one of those stratospheric geniuses that gets thrown up occasionally in any field. A pastor's son from Basel, he distinguished himself by the quality of his work – his name is attached to dozens of mathematical formulas and equations – as well as by its quantity, which is almost inconceivable. No one ever did more maths than Euler. And this against a background of personal disability: he started losing his sight in the 1730s and was functionally blind by 1771. Like Beethoven composing music that he couldn't hear, Euler was writing mathematics that he couldn't see.In 1775, into his sixties – blind, remember – he was still producing more than one academic paper a week. So huge was the backlog after he passed away that Euler managed to publish fully 228 papers posthumously – they were still coming out decades after his death.Yes. Euler published more papers dead than most mathematicians manage while alive.In 1911, the Swiss Academy of Sciences decided to publish Euler's complete works. They brought the first volume out that year, and they are still not done. Eighty volumes so far and counting.When he turned his attention to a problem, he blew it out of the water.A simple example is the so-called ‘amicable numbers’. Amicable numbers are pairs of numbers whose proper divisors sum to each other. What does that mean? Well, the classic example is 220 and 284. The number 220 can be divided into the following smaller numbers:1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110…and if you add all those together, you get 284. Coincidentally, 284 divides into the following smaller numbers:1, 2, 4, 71, 142…which, added together, make 220. 220 and 284 are therefore said to be ‘amicable’.This is a completely pointless property, but nevertheless intriguing if you're a mathematician. It is also very rare. The pair 220 and 284 was known to the Greeks, but after that no one in Europe could find another example until Fermat in 1636 – he managed to show that 17,296 and 18,416 are also amicable. At which point Descartes, Fermat's great rival, was determined to find a pair as well – he worked on it for two years and eventually came up with 9,363,584 and 9,437,056.Three examples in more than 1500 years.So a hundred years after Descartes, Euler decides to have a go. He publishes a paper in 1750 in which he finds 58 more pairs!This was the Euler way. ‘Let me just have a quick look at this problem you've all been worried about – OH NO HERE'S EVERYTHING THERE IS TO KNOW ABOUT IT. Also, I just invented a new branch of mathematics to deal with the matter. Bored now…’Euler is particularly associated with the number e, the base of the natural logarithm. Roughly equal to 2.718, e emerged from studying compound interest and is not difficult to understand in itself. But it's a very strange number all the same. It crops up everywhere in mathematics, though it's independent of any counting system; clearly, it is one of the fundamental values of how the universe fits together, indeed of how numbers work as a concept.It's this that makes Euler's most famous equation so extraordinary. Known as ‘Euler's identity’, it emerged from his work on complex analysis and is, first of all, breathtakingly simple:The reason mathematicians get so dewy-eyed over this is that it shows a beautiful and completely unexpected link between the worlds of pure maths and trigonometry – a bizarre, unintuitive relationship between the five most important numbers in mathematics (0, 1, i, e, and pi). There is no reason these numbers should be related, and no one really understands what it means that they are, except that it tells us something fundamental about reality.This equation regularly tops mathematical polls of the most beautiful result of all time.William Dunham is quite an Euler enthusiast and expert; if you're interested in the subject, I recommend his TED-like video tribute on YouTube. But this is a small book and can only touch on a few of the most impressive of Euler's accomplishments. Also, it is very very focused on the maths. I had been hoping for details of Euler's life in Switzerland, why he had a bust-up with Voltaire, and who he was sleeping with. In actual fact, every page of this book looks like this:A lot of it is way above my A-level understanding of maths. But in the absence of any other good biographies of Euler, it'll do – and if you have the skills to follow the proofs, it's likely to give you a lot of delighted, mind-blown moments.