## Wednesday, December 26, 2012

### Logarithms and the end of the world

Hardly a day goes by when I am not asked "Hey Math Guy!  What do logarithms have to do with the end of the world?"  Now that the end of the world has come and gone, I will share an interesting coincidence from the annals of math history.

When the natives are restless, they use log rhythms to communicate over long distances

Prosthaphaeresis

I am not talking about prosthetic devices, like artificial limbs and fake noses. At least not yet. I predict that I will, however, be talking about fake noses soon. And I am definitely not talking about electrophoresis or paresthesia[1]. Prosthapheresi Prosthespaeoius Prosthaphaeresis is a technique used to simplify multiplication.

Here is the simple approach. When you have two numbers to multiply together, you first compute arccos and arcsin of both. Then you add and subtract to get some other numbers. Then you compute the sine or maybe cosine of these and add them together. Or maybe subtract. Then divide by two.

Easy enough, right? Except for the silly fact that all these trig functions will probably involve multiplying, this simple method turns multiplying into the easy process of addition and subtraction. The trig functions may be seen as an impediment, but if you happen to have a trig table handy (or a scientific calculator app for your smart phone) then you can just look them up. And lemme tell ya, 16th century astronomers always had their trig tables handy.

Required reading for all 16th century astronomers

So, when some unknown person came up with the idea that a trig identity could be used to make the multiplication process faster, it got a lot of airplay. The phrase "(sin x + sin y)/2 = sin ((x+y)/2) cos ((x-y)/2)" was on everyone's lips.

I have a similar formula that turns multiplication into addition, subtraction and a quick look up in a table of squares: ((x+y)/2)2 - ((x-y)/2)2 = x y. Let's say we wish to multiply 9 X 17. The sum and difference are 26 and 8, which become 13 and 4 when you divide by 2. I happen to know that 13 squared is 169, and also that 4 squared is 16. The difference (169 - 16) is 153. I'm so confident in this method, that I am not even gonna check that with my calculator.

This formula has become known as John the Math Guy's Prosthaphaerisoid. I have not found any evidence that this formula was known back in the 16th century. I have looked in the index of all my math history books (there are 127 of them on my shelf), and have not found a single mention of the formula under that name. It's too bad. If they had known the formula earlier, Pluto would have been discovered so much earlier and the whole "is Pluto a planet" thing would have been settled already.

Invention of logarithms

Enter John Napier (1550 - 1617), who is remembered today as the inventor of logarithms. In 1614, he published a book of logarithms called Mirifici Logarithmorum Canonis Descriptio. I am sure most of my readers have a well worn copy of this book sitting on their night stand.

The father of modern-day ciphering

Napier took inspiration from several sources for this boon to cipherophiles [2]. Primarily, he drew from the idea of Prosthaphaeresis which was indirectly introduced to him by Tycho Brahe, an eminent astronomer of his time [3]. Probably at a dinner party of 16th century math geeks. Napier also drew on the work of Michael Stifel (1487 - 1567). While Stifel most certainly did not invent the formula that makes logarithms possible, he was likely the first one to state that logarithms turn multiplication into addition, and division into subtraction.

am an = a(m+n)

The formula that made it all possible

Doomsday prophecies

It would perhaps be to Napier's dismay that he has not been remembered for his book A Plain Discovery of the Whole Revelation of Saint John (1593). This book was a bestseller during his day, and even after his death, was printed in twenty-one editions, and was translated into Dutch, French and German. Both the words and the numbers were translated, by the way [4].

Napier was a vehement Protestant. In his book he viciously attacked the Catholic Church, claiming that the pope was the Antichrist. Napier also offered his prediction that the Day of Judgment would come between 1688 and 1700.

Michael Stifel entered an Augustinian monastery and was ordained in 1511. After much dissatisfaction with the Catholic Church, he sought the help of no less a personage than Martin Luther in setting him up in Protestant pulpits. Unfortunately, Stifel's career as a minister was tumultuous, having to move from several parishes due to anti-Lutheran sentiment, and leaving another due to war. Stifel predicted that the world would end on October 3, 1533, and many believed him. As a result he was arrested and forced to leave yet another post. There is not much profit in being a prophet.

Actual unretouched photo from Oct 3, 1533

It would be to Stifle's dismay that he has not become known for his own bestseller, A Book of Arithmetic about the AntiChrist. A Revelation in the Revelation. [5]
Not only do Stifel and Napier share the discovery of logarithms and prediction of the end of the world, but both men also accused the pope of being the Antichrist.

Much as he would have liked, Isaac Newton himself could not lay claim to the invention of logarithms, but he  had something to do with logarithms. He came up with an estimate of the sum of the harmonic series that involved logarithms. And, much like Stifel and Napier, Newton would have been dismayed that he has not been remembered for his endless religious writings. And, needless to say, he had a few things to say about the end of the world. He declared that is could not end until after 2060, when his pants would be done at the cleaners.

"This I mention not to assert when the time of the end shall be, but to put a stop to the rash conjectures of fanciful men who are frequently predicting the time of the end, and by doing so bring the sacred prophesies into discredit as often as their predictions fail."

Now there's a clever guy. If his prediction were to fail (the world were to end before 2060) who would be left to call him on it? And if 2060 comes and goes without incident, then it's well past his lifetime. Why worry?

One last mathematician's prophecy

Some of you are no doubt familiar with the greatest living mathematicomedian, Tom Lehrer. Well, maybe he is one of the greatest. There is at least one other vying for that title. Lehrer also had his predictions about the end of the world.

Oh we will all fry together when we fry.
We'll be french fried potatoes by and by.
There will be no more misery
When the world is our rotisserie,
Yes, we will all fry together when we fry.

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[1] Just to be clear, electrophoresis is not a way to remove unsightly, embarrassing hair. Paresthesia is not waking up one morning thinking that you are Paris Hilton.

[2] Cipherophile - One who loves cipherin'. I coined this word just for this blog. Just look for it to be used on the Jerry Springer show in the next week or two. My own prediction.

[3] Tycho Brahe was an interesting fellow. His chief contribution to science lay in providing his collaborator and rival, Johannes Kepler with data on the positions of the planets. Kepler used this data to do a little curve fitting that showed that the planets run their courses in elliptical paths. Newton in turn drew from this to establish the inverse square law for gravity.

Aside from having an odd sounding name that I don't know how to pronounce, Brahe also had a prosthetic nose made of gold and silver. His original nose was of the standard design with flesh and cartilage, but he lost part of it in a sword fight with a cousin about a mathematical formula. Ah! to be back in the days when scientists were real men, rather than sarcastic little blog-writing cipherophiles! Brahe owned a pet elk, which met it's demise when it was fed too much beer at a party. It toppled to it's death on a set of stairs.

[4] I know it is unlike me, but this is a silly statement. Oddly enough, the symbols for numbers were standardized across Europe at this time. This was before ISO.

[5] I personally am a bit dismayed that this title has already been taken. I had been planning to use that very title for my memoirs.

1. I've been a long-time fan of Tom Lehrer, but had no idea that he was also a mathematician (I initially assumed you were joking about that). I checked Wikipedia and learned that he taught at MIT.

2. Dear John, I thought Gauss invented curve-fitting to
extrapolate the orbit of Ceres. What did Kepler do
re curev-fitting? [P.S. about curve-fitting, see
some of the examples at www.civilized.com [and see "amusements"
there.] gary knott, garyknott@gmail.com

3. Steve, Would I joke about something as important as Tom Lehrer's true avocation?

Gary, I was sloppy in my words use. I intended for "curve fitting" to apply generally to any method where you make a curve of some sort kinda go through a bunch of data points.

As for the least-squares method of curve fitting, there is a bit of a dispute as to who first came up with the idea. Legendre was the first to publish (in 1805). Gauss followed up in 1809, but claimed he had been using it since 1795.