 I'm Bill Hammack, and this is a page by page commentary of the entire book Albert Michelson's Harmonic Analyzer. I'll share with you information that didn't make it anywhere else. I assume that you've watched the video series, and so have a sense about how the machine operates. So now we see the machine and a picture of me. Notice the labels. We try to make them appear as if they were printed on the thing they're labeling. Look for them throughout the book. The introduction focuses on Albert Michelson. Likely you've heard of him from the Michelson-Morley experiment, or his measurement of the speed of light. At the time he designed this analyzer, he was a professor of physics at the University of Chicago. Now, on the right here, you see the machine marked with dimensions. Well, this machine was most certainly built using inches as units, as was the standard of the time. We decided to label dimensions in metric units. This visual table of content shows you the main parts of the machine, and it lists the pages where that part is located in the book. These pages also serve as a visual table of contents. We link the parts of the analyzer to their mathematical function. Here we do that for synthesis. On the left, we go through term by term the synthesis equation and how to think about each term individually. There's a more detailed mathematics section of the book on page 98. On these pages, we do the same visual table of contents before analysis. Now, it's easy to understand what analysis does. That's shown visually on the left page. It takes a signal and tells you what sinusoids to add together to approximate that signal. But it's hard to understand exactly how it does this. This difficulty is exactly why this machine was built. This section is the meat of the book. We take you part by part through the machine. Turning the crank on this machine is really cool. It takes a full 80 cranks for the entire machine to make a complete cycle. When you crank a whole cycle, you can literally feel when the gears are in phase. When a rocker arm is pushing an amplitude bar upward, the crank requires more force, and when all the gears are in phase, the crank is much harder to turn. If you look carefully here, you see a small fiducial mark. It helps to orient the crank when reattaching it with a tapered pin. There are four different combinations of attaching the crank and the pin, and only one will fit properly. The cone gear set is probably the most distinctive part of the entire machine. It's not a single gear shaped like a cone, but rather it's a set of gears that form a cone. We've never seen this arrangement of gears in any other type of machinery. When we first got the machine, we expected the gears to make clicking or clunking sounds. Turns out it's very smooth and incredibly quiet. The crank does not rotate the cone gear set directly. This gear is attached to the crankshaft, and in turn engages the cone gear set. This is why it takes four turns of the crank to rotate the cone gear set by one revolution. Without this gear reduction, the crank would be more difficult to turn. Here you can really see the grime on the cone gear in many of these photos. Notice the shape of the teeth on the smallest gear on the cone. They're much pointier than the teeth on the other gears. The cylinder gear set is driven by the cone gear set, but unlike the cone gear set when operating, you have to often touch these gears because you adjust them by hand to set the machine for sines or cosines. We often get the question, do you have to set them all to be sines or cosines? And in principle, no. You could independently set a gear to be a sine or a cosine, or even a phase shifted sinusoid. In practice, though, it'd be hard to do because when you turn a gear, it tends to turn the one next to it. The cam we show here is a bit schematic. Without taking apart the analyzer, we could not determine exactly what it looked like. One reader wrote me that he thought it was more likely an eccentric hub. Each rocker arm has two holes on its left and right sides about an inch away from the pivot. We're not sure what they're there for. At first, we thought that a rod could be inserted through the holes in order to align the rocker arms. This is nearly impossible without binding the rod and rocker arms. Plus, aligning the arms is more easily done by aligning the notches on the cylinder gear set. So the purpose of these holes remains a mystery to us. While watching the synthesis video, you may have noticed that these trails coming off the tips of the rocker arms are not perfectly sinusoidal. This distortion is primarily due to the fact that the tips are not moving perfectly vertically. They're moving in an arc and therefore also have some horizontal movement, making the waves look a little sharper. Notice that you can also see sine waves on the tips of the rocker arms. The fact that the motions of the machine produce sine waves in two different directions hints at a deep concept in Fourier analysis, the concept of duality. This is an extremely important concept in modern mathematics. For this machine, it means you can transform back and forth between the function as represented on the rocker arms or as represented on the output at the front of the machine. They're both equally valid ways to describe that particular function. One reader asks what holds the amplitude bars in place. It's just friction. The springs at the top of the machine press them lightly against the rocker arms. In fact, you can sweep the amplitude bars out past the ends of the rocker arms. It's alarming when it happens, the amplitude bars crash into the base of the machine. The first time I did this, I thought I'd broken the analyzer, but the machine is resilient, just put the bar back on the rocker arm and it's fine. Notice that the amplitude bars are more rusted at the base and at the top. You generally touch the bottom to move the bars, so we guess that moisture from hands accelerated the corrosion. The measuring stick is not precisely a part of the machine, but it's an accessory tool that came with it. We believe it to be original because the style of the knob matches knobs in the magnifying lever. Also, it looks very similar to tools shown in pictures of under analyzers. The measuring stick is a very simple tool, but it works well. It is accurate and it's fairly repeatable to use. The springs and levers at the top are integral to the machine's operation because they are where the summation occurs. These springs were Michelson's great insight. He'd considered using fluid pressure and electrical currents to add sinusoids and decided that springs were, and I quote him here, the simplest in practice. Despite their importance, it's easy to forget about these parts because they are a foot or two above the head of the operator. The springs in this machine are in great shape, but they can wear out. I came across a report by a graduate student at the University of Chicago in the late 1920s used one of Michelson's analyzers, likely one built 20 years or more earlier. He reports that quote, by the time I got my hands on the machine, the springs had different force constants so that you got some startling results. Here's the underside of the summing lever. In the video, we call this a pivoted bar and then realize when writing this companion book that a pivoted bar is, of course, a lever. We use the word pivot because the most surprising aspect of this lever is the large knife edge pivot that it uses. This was some kind of very careful design choice. Likely it was to reduce friction. Now, keep in mind all of the terms we use, like amplitude bars, rocker arms, and so on, were invented by us. There was no user's manual or workshop plans for this machine. This counter spring counterbalances the pull of the 20 smaller springs. The curved tapered post that holds up the spring can be moved up and down to adjust the tension. The summing lever has a limited range of travel and needs to be kept around its middle point. If the pull of the counter spring is too much, that is, the post is too high, then the summing bar will tip too far downward and get stuck. And if the post is too low, the summing lever will drop and will not be held up on its pivot. Once the height of the counter spring was set, we did not need to adjust it further. This magnifying lever amplifies the motion from the summing lever. It's a simple brass rod with a threaded end. It can be unscrewed from the angled bracket that attaches it to the summing lever. When we brought the machine to our shop, it didn't have a writing mechanism, and so I took data by hand. I would measure with a ruler the motions of the tip of this arm and then plot its position against the number of cranks. I recall getting tired of measuring the small motion, and so I had the machine shop make a 3-foot long magnifying lever so it would sweep out a much larger, more easily measurable arc. And this long arm weighed too much and never worked. This vertical rod allows the operator to adjust the vertical position of the output on the paper. In electrical engineering, you would call this the DC offset. Adjusting this knob is very difficult because the position of the pen is very sensitive to small changes on the rod. Also, the magnifying lever which it's attached to can wobble up and down during adjustment. The magnifying wheel further amplifies the motion of the summing lever. We noticed that the metal in this wheel was not magnetic, and so we thought it might be aluminum. We learned later from a catalog by the analyzer's manufacturer that they often used an alloy of aluminum called magnallium. They describe it as very strong white and light non-crosive alloy, and so that's likely what this wheel is made from. The idea was to get a very lightweight wheel. You can see here how we wrap a wire around the hub of this inner wheel. Note there's no lip, and so sometimes during adjustments, the wire would slip off and unravel. This malfunction was particularly frustrating because the wire attached to the magnifying lever and the wire attached to the pin would get tangled. The platen has two brass clips that hold a piece of paper. We found we got the best results when we used glossy photo paper. Anything else seemed to have too much friction. The platen itself is surprisingly heavy. The translational gearing controls the horizontal scaling of the output. I find it fascinating that the scale is adjusted by changing the speed of the platen relative to the crank's motion. The next six pages detail the translational gearing. It is very compact and so it takes quite a few photos to really document here. You'll likely find the labels on the image is useful to keep you oriented. On this page, we're just showing you how the gears are exchanged. Notice that on the left the threaded nut has been removed, and on the right the crank has been removed. Careful study of these images will likely repay the time spent with them. The left is a view from the top and the right is a view from the back. The pen mechanism proved to be the most challenging aspect of the machine, mostly because when we discovered the machine, the writing mechanism was missing. To recreate it, we looked at photos, but these didn't help too much. First, we found several different types of writing mechanisms, none of which seemed to fit our machine. Or second, the photos often didn't have enough resolution to really see what was going on. So this is our best guess. The piece you're looking at was made by our machine shop. The holes in the brass block were drilled to reduce weight, but we're not sure now if we really need to do that. Notice that the ink pen is a sharpie spray-painted to look metallic. We don't know what kind of pen they would have used. I suspect it was a fountain pen of some sort that just barely touched the paper. We thought about that, but concluded it would be too messy. We also tried a pencil, but found it had too much friction on the paper. The pinion gear is not actively used while the machine is in operation, yet it is essential for setting up the machine. It is used to set the machine for sines or cosines. This lever is attached to cams that push the pinion gear into contact with the cylinder gear set. This brass spring pushes against the pinion and keeps it from contacting the cylinder gear set when the lever is disengaged. Notice this little post here. We've called it the mystery post because we're not entirely sure what it does. It seems like it may serve as a stop to prevent this lever from being pushed down too far. Interestingly, it has the same diameter as the vertical rod extending from the magnifying lever. On the base plate, there's a ring of wear that matches the knob from that vertical rod. Perhaps it was used to store an extra knob. We really don't know. Providence is a term used for a historical object. It's the documented story of who owned it and where it was kept so the object can be traced through time. We don't have anything like that for this machine. I visited the university archives and looked for any hint of the machine and turned up nothing. The nameplate clearly tells us who built it, William Gartner and Company. The Gartner Company still exists today as Gartner Scientific Corporation. They make ellipsometers, microscopes, and other optical instruments. A larger version of this machine was also built. It calculates with 80 sinusoids instead of 20. We found two distinct models of the 80-gear machine. Note the difference in the cone gear of each. In this model, the cone gear is very much like the cone set on our analyzer, but four times the size. It truly is humongous. In this, we assume later model, the cone gear is different. It appears to be made of two separate cone gears each driven at a different rate. Thus allowing for more compact gearing was still generating the same motion as a single giant cone gear set. On the right-hand page, you see a photo of this analyzer in its glass case. If you want to see it in person, just visit the University of Illinois and go to Altgeld Hall. You can find this machine on the second floor, not too far from the mathematics library. The next 16 pages show output from the machine. For each output, we use an input schematic to indicate how the amplitude bars were set. These are cosines, each of a single frequency ranging from cosine of 1x to cosine of 20x. If you look closely, you'll notice that all the amplitudes of each cosine are not exact. This could be caused by the springs wearing out and having different spring constants, but a much more likely cause is different friction between the pen and the paper. If the pen is pushed too far onto the paper, there will be increased friction that resists the movement from the summing lever. This will result in a smaller apparent amplitude. Note the squash section in cosine 19x. What happened here was the paper was not flat on the platen, but bulged out slightly. When the pen reached the bulge, there was a sharp increase in friction and the pen could not move up and down as freely. In the next two spreads, we move a square wave across the rocker arms. It produces some very cool-looking patterns. On this spread in the one following, we've sampled a cosine of varying frequencies. We use cosines of two, one, one-half, and one-quarter periods. Notice as the period decreases, the peaks in the output move closer together. On the left, we've input a square wave to produce a sink, a sink to produce a square wave, and some frequencies to produce beats. On the right-hand page, the top three inputs are vessel functions of various parameters. Keep in mind that we didn't use the same translational gearing for each calculation. You can determine what gearing we used by just counting the periods. For example, the plot on the top of the left-hand page was made with a medium and medium gear combination, shown on page 57, because it has one period. While the graph below it, the square wave, was made with a large small combo, because it has two periods. On the left are examples of some arbitrary input and output pairs. The output on the right was produced with the machine set for signs. Everything else in this output section was done with the machine set up to calculate with cosines. This is the key paper about the analyzer. It was written by Michelson and Samuel Stratton. Often the machine is called a Michelson-Stratton analyzer, but we don't use this term, because Michelson clearly had the idea before he worked with Stratton, and it even built a 20-wheel version of the machine as a prototype. Stratton, though, was no doubt important in getting the larger version built. Stratton was head of electrical engineering in Illinois, before moving to Chicago to work with Michelson. Stratton had a long association with William Gartner, whose company built this analyzer. We suspect, but could never prove it, that Stratton was seminal in getting the analyzer described in this book to the University of Illinois. This is a schematic of the key components of the analyzer. This paper contains a lot of data produced by this 80-sinusoid machine. Because it had more sinusoids, the output it produced is sharper than an ordinary gear machine. The graphs on the left-hand side show you how to analyze a function that is neither even nor odd. In this book, we mostly considered even functions that require only cosigns to be approximated. We could just as easily have used only odd functions, and then use signs throughout. In these two pages, we go deeper into the mathematics. Our goal is to lay out the math in a way it would be discussed in a class on Fourier analysis, so that someone who has had such a class can understand the analyzer. Another thing to note here is that the derivations of the equations show, especially for analysis, the simplifications and approximations made by the machine when calculating. These spreads show the analyzer at eight different angles. It's a great way to imagine walking all the way around the machine. It feels like the closest we could get to letting you be there in person. We've worked with this machine for two years. We spent the first months of each of us pouring over the machine to discover all the parts and understand how they work together. We had many debates about the best way to show the machine's operation for both synthesis and analysis. Eventually we realized the videos weren't enough to capture all of the magnificent mechanics, so we set up on photographing and assembling the book. It's hard to believe it's been two years since we first discovered the machine. We will certainly miss working with the analyzer, but we hope with the videos and the book that you can enjoy it as much as we have. Thanks for going through the book with me. I'm Bill Hammack, the engineer guy. Click here to learn where to purchase the book we just went through. There's also info on the wall posters that are available.