Joost Bürgi And Logarithms

Logarithms are a common idea today, even though we don’t use them as often as we used to. After all, one of the major uses of logarithms is to simplify computations, and computers do that just fine (although they might use logs internally). But 400 years ago, doing math was painful. Enter Joost Bürgi. According to [Welch Labs], his book of mathematical tables should have changed math forever. But it didn’t.

If you know how a slide rule works, you’ll find you already know much of what the video shows. The clockmaker was one of the people who worked out how logs could simplify many difficult equations. He created a table of 23,030 “red and black” numbers to nine digits. Essentially, this was a table of logarithms to a very unusual base: 1.0001.

Why such a strange base? Because it allowed interpolation to a higher accuracy than using a larger base. Red numbers are, of course, the logarithms, and the black numbers are antilogs. The real tables are a bit hard to read because he omitted digits that didn’t change and scaled parts of it by ten (which was changed in the video below to simplify things). It doesn’t help, either, that decimal points hadn’t been invented yet.

What was really impressive, though, was the disk-like construct on the cover of the book. Although it wasn’t mentioned in the text, it is clear this was meant to allow you to build a circular slide rule, which [Welch Labs] does and demonstrates in the video.

Unfortunately, the book was not widely known and Napier gets the credit for inventing and popularizing logarithms. Napier published in 1614 while Joost published in 1620. However, both men likely had their tables in some form much earlier. However, Kepler knew of the Bürgi tables as early as 1610 and was dismayed that they were not published.

While we enjoy all kinds of retrocomputers, the slide rule may be the original. Want to make your own circular version? You don’t need to find a copy of this book.

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Calculation Before We Went Digital

We have to like [Nicola Marras]. First, he wrote a great mini-book about analog computers. Then he translated it into English. Finally, he opened with a picture of Mr. Spock using an E6-B flight slide rule. What’s not to like? We suggest you settle in when you want to read it — there are almost 60 pages of text, photos, and old ads for things like slide rules and adding machines.

There is a lot of research here. We couldn’t think of anything missed. There’s a Pascalina, Ishango’s bone, a Babylonian spreadsheet, an abacus, and even Quipu. Toward the end, he gets to nomographs, adding machines, and the early calculators.

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Pocket Calculator Isn’t A Brain Or Magic

If you predate the pocket calculator, you may remember slide rules. But slide rules take a a little skill to use. There was a market for other devices that were simpler or, in some cases, cheaper. One common one was the “magic brain” or Addiator which was a little metal box with some slots that could add numbers. However, using clever tricks it could also subtract and — in a fashion — multiply. [Our Own Devices] has a teardown of the device you can see in the video below. It is deceptively simple, and the description of how it works is at least as interesting as the peek inside.

We remember these on the market and, honestly, always thought they were simple tally mechanisms. It turns out they are both less and more than that. Internally, the device is a few serrated sheet metal strips in a plastic channel. The subtraction uses a complement addition similar to how you do binary subtraction using 2’s complement math. Multiplication is just repetitive addition, which is fine for simple problems.

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Laser-Engraving Hairlines: When A Line Isn’t A Line

When is a line not a line? When it’s a series of tiny dots, of course!

The line is actually tiny, laser-etched craters, 0.25 mm center-to-center.

That’s the technique [Ed Nisley] used to create a super-fine, colored hairline in a piece of clear plastic — all part of his project to re-create a classic Tektronix analog calculator from the 1960s, but more on that in a moment.

[Ed] tried a variety of methods and techniques, including laser engraving a solid line, and milling a line with an extremely tiny v-tool. Results were serviceable, but what really did the trick was a series of tiny laser-etched craters filled in with a red marker. That resulted in what appears — to the naked eye — as an extremely fine hairline. But when magnified, as shown here, one can see it is really a series of small craters. The color comes from coloring in the line with a red marker, then wiping the excess off with some alcohol. The remaining pigment sitting in the craters gives just the right amount of color.

This is all part of [Ed]’s efforts to re-create the Tektronix Circuit Computer, a circular slide rule capable of calculating all kinds of useful electrical engineering-related things. And if you find yourself looking to design and build your own circular slide rule from scratch? We have you covered.

Genaille’s Rods: When Paint Sticks Do Math

What is a hacker, if not somebody who comes up with solutions that other just don’t see? All the pieces may be in place, but it takes that one special person to view the pieces as greater than the sum of their parts. As [Chris Staecker] explains in the video below the break, Henri Genaille was one such person.

When French mathematician Edouard Lucas (himself well known for calculating the longest prime number found by hand) posed a mathematical problem at the French Academy, a French railway engineer named Henri Genaille developed the rods we’re discussing now.

Genaille’s Rods are designed to perform multiplication. But rather than require computation by the user, the rods would simply need to be laid out in the correct order. The solution could readily be found by just following the lines in the correct pattern. This might sound a lot like cheating, and that’s exactly what it is. No manual math needed to be done. Genaille also created rods for doing long division, which we’re sure were every bit as enthralling as the multiplication rods. Demonstrations of both are included in the video below.

While Genaille’s Rods have gone the way of the slide rule, we can’t help but wonder how many engineers and scientists carried around a set of marked up wooden sticks in their pocket protector.

If designing and building manual mathematical machines is something that you think really adds up to a good time, check out this post on how to design and build your own circular slide rule!

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Rubber Band “Slide Rule” Doesn’t Slide, But Rotates

Around here we mostly enjoy slide rules. We even have our own collections including some cylindrical and circular ones. But [Mathologer] discusses a recent Reddit post that explains a circular slide rule-like device using a wheel and a stretchable rubber band. While it probably would be difficult to build the actual device using a rubber band, it can do wonders for your understanding of logarithms which still show up in our lives when, for example, you are calculating decibels. [Dimitri] did simulate the rubber band for you in software.

The idea is that a perfect rubber band has numbers from 0 to 10 evenly marked on it. As you rotate a wheel attached at the 10 mark, the rubber band stretches more and more. So the 10 and the 9 have relatively little space between them, but the 1 and the 2 are much further apart. The wheel’s circumference is set so that the 1 will exactly overlay the 10. What this means is that each spot on the wheel can represent any number that differs only by a decimal point. So you could have 3 mean 0.03, 300, or — of course — 3. Of course, you don’t need to build the wheel with a rubber band — you could just mark the wheel like a regular circular slide rule.

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Circuit VR: The Wheatstone Bridge Analog Computer

We are always impressed with something so simple can actually be so complex. For example, what would you think goes into an analog computer? Of course, a “real” analog computer has opamps that can do logarithms, square roots, multiply, and divide. But would it surprise you that you can make an analog device like a slide rule using a Wheatstone bridge — essentially two voltage dividers. You don’t even need any active devices at all. It is an old idea and one that used to show up in electronic magazines now and again. I’ll show you how they work and simulate the device so you don’t have to build it unless you just want to.

A voltage divider is one of the easiest circuits in the world to analyze. Consider two resistors Ra and Rb in series. Voltage comes in at the top of Ra and the bottom of Rb is grounded. The node connecting Ra and Rb — let’s call it Z — is what we’ll consider the output.

Let’s say we have a 10 V battery feeding A and a perfect voltmeter that doesn’t load the circuit connected to Z. By Kirchoff’s current law we know the current through Ra and Rb must be the same. After all, there’s nowhere else for it to go. We also know the voltage drop across Ra plus the voltage drop across Rb must equal to 10 V. Kirchoff, conservation of energy, whatever you want to call it.  Let’s call these quantities I, Va, and Vb. Continue reading “Circuit VR: The Wheatstone Bridge Analog Computer”