One of the personality traits which my Dearly Beloved most often tolerates in me is my tendency toward hyperbole, a rhetorical device employed successfully on the Internet by almost four people and recognized as such as recently as 1998. I’m not satisfied saying there was an enormous, slow-moving line for a roller coaster we rode last August; I have to say that fourteen months later we’re still on that line.
I mention this because I need to discuss one of those rare people who can be discussed accurately only in hyperbole: Leonhard Euler, 1703 – 1783. He wrote about essentially every field of mathematics it was possible to write about: calculus and geometry and physics and algebra and number theory and graph theory and logic, on music and the motions of the moon, on optics and the finding of longitude, on fluid dynamics and the frequency of prime numbers. After his death the Saint Petersburg Academy needed nearly fifty years to finish publishing his remaining work. If you ever need to fake being a mathematician, let someone else introduce the topic and then speak of how Euler’s Theorem is fundamental to it. There are several thousand Euler’s Theorems, although some of them share billing with another worthy, and most of them are fundamental to at least sixteen fields of mathematics each. I exaggerate; I must, but I note that a search for “Euler” on Wolfram Mathworld turns up 681 matches, as of this moment, out of 13,081 entries. It’s difficult to imagine other names taking up more than five percent of known mathematics. Even Karl Friedrich Gauss only matches 272 entries, and Isaac Newton a paltry 138.
Besides knowing everything and writing abundantly of everything, Euler was also a fantastic symbol-maker. It isn’t appreciated until one tries doing without one how valuable a good notation for a concept is. When one has it, the hard parts of an idea seem to vanish, and new aspects of what could be done present themselves. Euler was among the best symbol-makers. He was not the first to use π (pi) for that number, but after he did, it was the popular choice. It is from him that we derive i as the imaginary-number unit, and Σ as a shorthand symbol for “take the sum of these quantities”, and the notation f(x) for a function of the variable x.
His skill in notation ran beyond pure mathematics. His work put classical mechanics, as Newton famously developed, into the analytic form that is still relied upon, along the way extending it so that rather than dealing with the point particles of fixed mass which had been studyable before, they could be objects, with bulk and distributions of mass, which occupied space and rotated as they moved. Physicists and mathematicians still study Newtonian mechanics and likely always will; but they study them through the light Euler cast and likely always will that.
Now I get back to e. For his physics textbook Mechanica, published 1736, Euler used e for that number, a little over 2.7 with some importance to exponential growth problems and serving some role in logarithms. Euler had used the symbol earlier in some of his manuscripts, but it was Mechanica which made it famous. Mechanica was a major book, one which would be foundational to understanding mechanics for a century-plus. And Euler’s use of e caught on from there. And why did he pick e?
No one is certain. Before e‘s appearance in the manuscript, though, he has already used a, b, c, and d; possibly it was just the first available letter. e is the first letter of “exponential”, the kind of function one comes naturally to after studying logarithms; possibly it was that association. One interesting hypothesis (proposed by Etienne Delacroix de La Valette) is that the e stands for either “ein” or “Einheit”, one or unity. Since Euler introduces the symbol in a sentence fragment others (I don’t read Latin well enough to try) may translate as, “for the number whose logarithm is unity, let e be written”, the mnemonic connection is clear. Possibly it was a combination of many influences; good notations are often like that.
After this there was not much choice. A few people attempted to use c for this base, and Benjamin Pierce in 1859 proposed a kind of open-paperclip symbol, but they’ve gotten nowhere.
That’s the name. Left to discuss is the meaning.