How To Find A Logarithm Without Much Computing Power


I don’t yet have actual words committed to text editor for this year’s little A-to-Z yet. Soon, though. Rather than leave things completely silent around here, I’d like to re-share an old sequence about something which delighted me. A lon while ago I read Edmund Callis Berkeley’s Giant Brains: Or Machines That Think. It’s a book from 1949 about numerical computing. And it explained just how to really calculate logarithms.

Anyone who knows calculus knows, in principle, how to calculate a logarithm. I mean as in how to get a numerical approximation to whatever the log of 25 is. If you didn’t have a calculator that did logarithms, but you could reliably multiply and add numbers? There’s a polynomial, one of a class known as Taylor Series, that — if you add together infinitely many terms — gives the exact value of a logarithm. If you only add a finite number of terms together, you get an approximation.

That suffices, in principle. In practice, you might have to calculate so many terms and add so many things together you forget why you cared what the log of 25 was. What you want is how to calculate them swiftly. Ideally, with as few calculations as possible. So here’s a set of articles I wrote, based on Berkeley’s book, about how to do that.

Machines That Think About Logarithms sets out the question. It includes some talk about the kinds of logarithms and why we use each of them.

Machines That Do Something About Logarithms sets out principles. These are all things that are generically true about logarithms, including about calculating logarithms.

Machines That Give You Logarithms explains how to use those tools. And lays out how to get the base-ten logarithm for most numbers that you would like with a tiny bit of computing work. I showed off an example of getting the logarithm of 47.2286 using only three divisions, four additions, and a little bit of looking up stuff.

Without Machines That Think About Logarithms closes it out. One catch with the algorithm described is that you need to work out some logarithms ahead of time and have them on hand, ready to look up. They’re not ones that you care about particularly for any problem, but they make it easier to find the logarithm you do want. This essay talks about which logarithms to calculate, in order to get the most accurate results for the logarithm you want, using the least custom work possible.

And that’s the series! With that, in principle, you have a good foundation in case you need to reinvent numerical computing.

Bob Newhart interviews Herman Hollerith


Yesterday was the birthday of Herman Hollerith. His 40th since his birth in 1860. He’s renowned in computing circles. His work in automating the counting and of data made the United States’s 1890 Census possible. This is not the ordinary hyperbole: the 1880 Census’s data took eight years to fully collate. Hollerith’s tabulating machines took … well, six years for the full job, but they were keeping track of quite a bit of information. Hollerith’s system would go on to be used for other censuses, and also for general inventory and data-tracking purposes. His tabulating company would go on to be one of the original components of IBM. Cards, card readers, and card sorters with a clear lineage to this system would be used until fully electronic computers took over.

(It’s commonly assumed that the traditional 80-character width of a text terminal traces to the 80-hole punch cards which became the standard. Programmers particularly love to tell that tale, ignoring early computing screens that had different lengths, particularly 72 characters. More plausibly 80 characters owes to two things: it’s a nice round number, and it’s close to the number of characters you can type on a standard sheet of paper with a normal typewriter font. So it’s about the “right” length, one that we’ve been trained to accept as enough text to read at a glance.)

Well. In about 1970 IBM hired Bob Newhart to record a bit, for … fun, if that word applies to IBM. Part of the publicity for launching the famous System 370 machine. The structure echos the bit where Bob Newhart imagines being the first guy to hear of Sir Walter Raleigh’s importing of tobacco, and just how weird every bit of that is. In this bit, Newhart imagines talking on the phone with Herman Hollerith and hearing about just how this punched-card system is supposed to work. For decades, though, the film was reported lost.

What I did not know until mentioning to a friend two days ago is: the film was found! And a decade ago! In a Swedish bank vault because that’s the way this sort of thing always happens. Which is a neat bit of historical rhyming: the original fine data from the first Hollerith census of 1890 is lost, most likely destroyed in 1933 or 1934. So, please let me share with you Bob Newhart hearing about Herman Hollerith’s system. The end appears to be cut off, and there are Swedish subtitles that might just give away a couple jokes, if you can’t help paying attention to them.

Like a lot of comic work-for-hire it’s not Newhart’s best. It’s not going to displace the Voyage of the USS Codfish in my heart. There are a few spots to me where it seems like Newhart’s overlooked a good additional punch line, and I don’t know whether that reflects Newhart wanting to keep the piece from growing too long or too technical or what. It’s possible Newhart didn’t feel familiar enough with punch card technology to get too technical too. Newhart did work, briefly, as an accountant and might have had some reason to use the things. But I’m not aware of his telling any stories of doing so, and that seems a telling omission.

Still, it’s great to see this bit has been preserved, and is available. And is a Bob Newhart routine about early computer technologies, somehow.

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