Calculus and a Calculator

Earlier this year, [Dan Maloney] went inside mechanical calculators. Being the practical sort, [Dan] jumped right into the Pascaline invented by Blaise Pascal. It couldn’t multiply or divide. He then went into the arithmometer, which is arguably the first commercially successful mechanical calculator with four functions. That was around 1821 or so. But [Dan] mentions it used a Leibniz wheel. I thought, “Leibniz? He’s the calculus guy, right? He died in 1716.” So I knew there had to be at least a century of backstory to get to the arithmometer. Having a rainy day ahead, I decided to find out exactly where the Leibniz wheel came from and what it was doing for 100 years prior to 1821.

If you’ve taken calculus you’ve probably heard of Gottfried Wilhelm Leibniz (who would have been 372 years old on July 1st, by the way). He’s the guy that gave us the notation we use in modern calculus and oddly was one of two people who apparently figured out calculus, the other being Issac Newton. Both men, by the way, accused each other of stealing, although it is more likely they both built on the same prior work. When you are struggling to learn calculus, it is sometimes amazing that not only did someone think it up, but two people thought it up at one time. However, Leibniz also built what might be the first four function calculator in 1694. His “stepped reckoner” used a drum and some cranks and the underlying mechanism found inside of it lived on until the 1970s in other mechanical calculating devices. Oddly, Leibniz didn’t use the term stepped reckoner but called the machine Instrumentum Arithmeticum.

Many of us remember when a four function electronic calculator was a marvel and not even very inexpensive. Nowadays, you’d have to look hard to find one that only had four functions and simple calculators are cheap enough to give away like ink pens. But in 1694, you didn’t have electronics and integrated circuits necessary to pull that off.

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Fourier Machine Mimics Michelson Original in Plywood

It’s funny how creation and understanding interact. Sometimes the urge to create something comes from a new-found deep understanding of a concept, and sometimes the act of creation leads to that understanding. And sometimes creation and understanding are linked together in such a way as to lead in an entirely new direction, which is the story behind this plywood recreation of the Michelson Fourier analysis machine.

For those not familiar with this piece of computing history, it’s worth watching the videos in our article covering [Bill “The Engineer Guy” Hammack]’s discussion of this amazing early 20th-century analog computer. Those videos were shown to [nopvelthuizen] in a math class he took at the outset of degree work in physics education. The beauty of the sinusoids being created by the cam-operated rocker arms and summed to display the output waveforms captured his imagination and lead to an eight-channel copy of the 20-channel original.

Working with plywood and a CNC router, [nopvelthuizen]’s creation is faithful to the original if a bit limited by the smaller number of sinusoids that can be summed. A laser cutter or 3D printer would have allowed for a longer gear train, but we think the replica is great the way it is. What’s more, the real winners are [nopvelthuizen]’s eventual physics students, who will probably look with some awe at their teacher’s skills and enthusiasm.

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Retrotechtacular: Pascal Got Frustrated at Tax Time, Too

While necessity is frequently the mother of invention, annoyance often comes into play as well. This was the case with [Blaise Pascal], who as a teenager was tasked with helping his father calculate the taxes owed by the citizens of Rouen, France. [Pascal] tired of moving the beads back and forth on his abacus and was sure that there was some easier way of counting all those livres, sols, and deniers. In the early 1640s, he devised a mechanical calculator that would come to be known by various names: Pascal’s calculator, arithmetic machine, and eventually, Pascaline.

The instrument is made up of input dials that are connected to output drums through a series of gears. Each digit of a number is entered on its own input dial. This is done by inserting a stylus between two spokes and turning the dial clockwise toward a metal stop, a bit like dialing on a rotary phone. The output is shown in a row of small windows across the top of the machine. Pascal made some fifty different prototypes of the Pascaline before he turned his focus toward philosophy. Some have more dials and corresponding output wheels than others, but the operation and mechanics are largely the same throughout the variations.

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Harmonic Analyzer Mechanical Fourier Computer

If you’re into mechanical devices or Fourier series (or both!), you’ve got some serious YouTubing to do.

[The Engineer Guy] has posted up a series of four videos (Introduction, Synthesis, Analysis, and Operation) that demonstrate the operation and theory behind a 100-year-old machine that does Fourier analysis and synthesis with gears, cams, rocker-arms, and springs.

In Synthesis, [The Engineer Guy] explains how the machine creates an arbitrary waveform from its twenty Fourier components. In retrospect, if you’re up on your Fourier synthesis, it’s pretty obvious. Gears turn at precise ratios to each other to create the relative frequencies, and circles turning trace out sine or cosine waves easily enough. But the mechanical spring-weighted summation mechanism blew our mind, and watching the machine do its thing is mesmerizing.

In Analysis everything runs in reverse. [The Engineer Guy] sets some sample points — a square wave — into the machine and it spits out the Fourier coefficients. If you don’t have a good intuitive feel for the duality implied by Fourier analysis and synthesis, go through the video from 1:50 to 2:20 again. For good measure, [The Engineer Guy] then puts the resulting coefficient estimates back into the machine, and you get to watch a bunch of gears and springs churn out a pretty good square wave. Truly amazing.

The fact that the machine was designed by [Albert Michelson], of Michelson-Morley experiment fame, adds some star power. [The Engineer Guy] is selling a book documenting the machine, and his video about the book is probably worth your time as well. And if you still haven’t gotten enough sine-wavey goodness, watch the bonus track where he runs the machine in slow-mo: pure mechano-mathematical hotness!

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Retrotechtacular: The CURTA Mechanical Calculator

The CURTA mechanical calculator literally saved its inventor’s life. [Curt Herzstark] had been working on the calculator in the 1930s until the Nazis forced him to focus on building other tools for the German army. He was taken by the Nazis in 1943 and ended up in Buchenwald concentration camp. There, he told the officers about his plans for the CURTA. They were impressed and interested enough to let him continue work on it so they could present it as a gift to the Führer.

This four-banger pepper mill can also perform square root calculation with some finessing. To add two numbers together, each must be entered on the digit setting sliders and sent to the result counter around the top by moving the crank clockwise for one full rotation. Subtraction is as easy as pulling out the crank until the red indicator appears. The CURTA performs subtraction using nine’s complement arithmetic. Multiplication and division are possible through successive additions and subtractions and use of the powers of ten carriage, which is the top knurled portion.

Operation of the CURTA is based on [Gottfried Leibniz]’s stepped cylinder design. A cylinder with cogs of increasing lengths drives a toothed gear up and down a shaft. [Herzstark]’s design interleaves a normal set of cogs for addition with a nine’s complement set. When the crank is pulled out to reveal the red subtraction indicator, the drum is switching between the two sets.

Several helper mechanisms are in place to enhance the interface. The user is prevented from ever turning the crank counter-clockwise. The crank mechanism provides tactile feedback at the end of each full rotation. There is also a lock that disallows switching between addition and subtraction while turning the crank—switching is only possible with the crank in the home position. There is a turns counter on the top which can be set to increment or decrement.

You may recall seeing Hackaday alum [Jeremy Cook]’s 2012 post about the CURTA which we linked to. A great deal of information about the CURTA and a couple of different simulators are available at curta.org. Make the jump to see an in-depth demonstration of the inner workings of a CURTA Type I using the YACS CURTA simulator.

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Hackaday Links May 9th 2012

Homecut – CNC Cutting Directory

homecut

So you have a CNC machine that you use as a hobby, but would like to do some actual work on the side? Or maybe you have an idea you’d like made. Homecut is a map directory where you can maybe hook up with the right person.

The Curta Mechanical Calculator

curta calculator

As [leehart] mentioned in our comments section, the Curta mechanical calculator is a truly ingenious piece of engineering. A quick Google search should find all kinds of information on it, but this article could be a good place to start for some mechanical hacking inspiration!

Luxman Amplifier DAC Upgrade

nand-dac

[R. Barrios] wasn’t happy with using the sound card for his HTPC setup, so decided to add a DAC module onto his reciever. The resulting audio quality was very good, and the build came out quite clean.  Check it out if you’re thinking of a hack-upgrade to your stereo equipment.

3D Printable Tilt-Shift Adapter

tilt-shift-lens

A tilt-shift lens a neat piece of equipment that is used to make a large scene look like they were miniatures. It’s a cool effect, but professional lenses to do this can cost thousands of dollars. This Instructable tells you how to go about printing your own. For more info on the technique itself, check out this Wikipedia article.

New 3D Printer on the Block

3d-printer

If you would like to take the plunge into 3D printing, but are looking for somewhere to get a parts kit, the [ORD Bot Hadron 3D Printer] may be worth a look.  The build quality looks great, and the price for the mechanical components is quite reasonable at $399. You’ll need to provide the electronics and extruder. Thanks [comptechgeek]!