Some weeks ago a reader sent me a note about an old article of mine, written back in 1989. The PDF, he reported, was missing a page. So I went scrounging through some untidy heaps of papers, found a copy, scanned it, and posted a new PDF. But along the way I stopped to reread my own words from 25 years ago.

The piece was a review of the first version of Mathematica, which debuted that summer of 1989. I began my review:

Sometime in the 1850′s William Shanks, a British schoolkeeper and amateur mathematician, began calculating the digits of the decimal expansion of π. Twenty years later, when he published his results in the Proceedings of the Royal Society, he had gotten as far as 707 digits. Today, with the aid of the computer program Mathematica, performing the same calculation requires no more effort than typing the expression: N[Pi, 707] . In response to this command, the labor that occupied Shanks for two decades is completed in less than eight seconds. Furthermore, Mathematica produces the correct 707 digits. Shanks made an error in the 528th decimal place that spoiled all the rest of his work.

I remember being wowed by the sub-eight-second running time of the π computation. (My notes say the actual time was 7.97 seconds.) That was with Mathematica 1.0 on a Macintosh SE/30 (16 MHz 32-bit processor, 5 MB of memory). How long would it take with more recent hardware and software? Naturally I had to try the experiment. Wolfram Research has just released version 10.0 of Mathematica. I ran it on the modern equivalent of the SE/30: a middle-of-the-road Mac laptop (2.6 GHz 64-bit processor, 8 GB of memory). The result: 190 microseconds.

I’m wowed again.

We don’t know how long Shanks actually spent noodling away at the digits of π. He did not—despite what I wrote in 1989—work continuously for two full decades. One known data point is that he extended his result from 530 decimal places to 607 places during March and April of 1853. Suppose he gave the task 60 hours a week during those two months. That’s roughly 500 hours for 77 digits, which we might extrapolate to 5,000 hours for 707. The average computation speed is thus 0.15 digits per hour, or 0.000043 digits per second. Compare this performance with later results:

Computer Year Speed Shanks with a pencil 1873 0.000043 ENIAC with punch cards 1949 0.0079 16 MHz Mathematica 1.0 1989 89 2.6 GHz Mathematica 10.0 2014 3.7 × 106

What I find surprising in these numbers is that the ENIAC was only a few hundred times faster than a Victorian English gentleman wielding pencil and paper, whereas my current laptop is a billion times faster than the ENIAC. Much of this recent speedup is technological—a blessing of Moore’s Law. But algorithmics surely plays a role too.

As I mused about these developments, I realized that I had no idea how Mathematica computes the numerical value of π, or whether the method might have changed between version 1.0 and version 10.0. I also had only a vague notion of how Shanks did his computation. It turns out that both he and the ENIAC programmers used something called Machin’s formula:

\[\frac{\pi}{4} = 4\arctan\left(\frac{1}{5}\right) - \arctan\left(\frac{1}{239}\right)\]

Thus they found the value of π by first finding the value of two arctangents. That’s good to know, but it raises a further question: How do you evaluate the arctangent of 1/5 or 1/239? You can’t just look it up in a table if you need 700+ digits of precision. The solution, I learned, is to evaluate some terms of an infinite series, and then sum them up:

\[\arctan x = \frac{x^{1}}{1} - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \cdots = \sum_{k=0}^\infty \frac{(-1)^{k}}{2k+1}x^{2k+1}\]

Mathematically, this is all you really need to know to understand what Shanks was doing in his computation of π. The whole job reduces to elementary arithmetic: Adding and subtracting, multiplying and dividing. But this was arithmetic at scale. I was having a hard time imagining the logistical and mechanical details of working with hundreds of numbers that run to hundreds of decimal places. “More is different” is a slogan from physics, but surely it applies to arithmetic too.

The last time I mentioned Shanks in these pages, I heard from Erwin Engert, who has done some thoughtful analysis of Shanks’s methods (and his errors). The documents on Engert’s web site include records of his own pencil-and-paper computations of π to 20 and 40 decimal places. I’m not brave enough to try that myself, but I thought I might be able to learn something useful by writing programs that follow some of the same steps a human solver might choose. The project would take a day or two, I supposed. Then one thing led to another (as it has a habit of doing).

For many of us, William Shanks is a figure to be pitied or ridiculed. What a sad waste of human potential to spend a major chunk of your lifespan churning out digits that no one would have any use for even if they were correct. What I’ve been doing over the past several weeks may be just as foolish and futile: I have expended a nontrivial chunk of my own lifespan tracking down all things Shanks—reading his publications, casting about for scraps of biographical detail, and most of all trying to reconstruct his work and figure out where his calculations went awry.

The main results of this dubious effort are in my new American Scientist column (HTML, PDF). I have also made available some supplementary material, including an annotated reading list and an IPython notebook with code you are welcome to explore in case you too would like to devote a portion of your life to following the Pied Piper.

In this posting I merely want to tie up a couple of loose ends: one item that wouldn’t fit in the magazine (it would have required a multipage foldout), and another development I stumbled upon after the article was sent to the printer.

In one phase of his computation, Shanks faced the task of adding up 506 numbers, each with 709 decimal places. Half of these terms were positive and half were negative, so the problem would naturally decompose into two parts. But half of the addition problem is still a tableau of 253 × 709 = 179,377 digits. What would such an array of numbers look like, if you wrote them all down on one sheet of paper? I don’t have a sheet big enough to answer that question directly. But here’s a hint—in very tiny type, with scroll bars. These are the 253 positive terms in the series for arctan 1/5. Feel free to add up the columns and write the sum at the bottom.

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