Transcription

This is a mechanical calculator. It can add, subtract, multiply, and divide numbers up to six digits long. I want to show you how it works, how we built it, and explore the difficulties in mechanizing arithmetic. What it takes to turn mental activity, like adding two numbers, into a physical, mechanical process. Building this challenged my understanding of how technology changes through time. What digital means, and even how numbers work. Let's go back to square one, or zero. It's common to teach basic math using a number line, where it's obvious, for instance, that three more than five is eight. And this is already a sort of materialization of arithmetic. That is, the numbers are no longer in our mind. It's not difficult to see how devices like a sector or a slide rule could employ this method. But those are analog calculating devices. That is, they give proportional answers, not discrete, certain numbers. Another way to materialize numbers is to curve the number line into a circle, as in a clock face. And this is what the first digital adding machines did. They curved the numbers into a circle, or a wheel, taking peculiar advantage of a structure at the heart of our number system. See, inventors like Blaise Pascal and Wilhelm Schickard realized in the 17th century that the positional decimal number system we use is well suited to being expressed in wheels, because in counting up, each decimal place loops from zero to nine and back to zero continuously. They dedicated one wheel for each decimal place, so each wheel only needed to have zero through nine on it. Just like Schickard and Pascal, I'll number these wheels zero through nine and dedicate one to each decimal place. If we were trying to use Roman numerals, we couldn't get away with just having ten teeth, or using the same design of wheel to represent tens as that which represents ones. Before making this calculator, I had never really thought about how the position of a number encodes part of its value, and how this is very advantageous for mechanizing arithmetic. It means we only really need to deal with adding up to nine. See, these wheels only have zero through nine on them, because in a way, the number ten doesn't exist. Well, obviously it does, but we don't have a discrete symbol for it. We just have zero through nine, and then a tens place that a number can be in. If I put this cover on with these windows, it'll be more obvious. See, over here we have zero through nine in the ones place, and here, zero through nine in the tens place. So, to make the number twelve, for example, we want a one over here, and a two over here. Twelve. After arranging the number wheels so that each one represents another decimal place, Pascal and Schickard only needed a way of carrying the one, as we say. That is, when this one moves from nine to zero, the next wheel over needs to move from zero to one, so that we get ten. In theory, this is easily done with this little lever, or tooth, which attaches to this wheel, but could nudge the next one along. But as you see, this would turn this in the wrong direction, so we need an intermediary here. Now when I turn this one, it moves the other one just one tooth over. And now we could add numbers by rotating these wheels the desired amount. I could first add six, one, two, three, four, five, six. Then I could add another six by moving it sixtieth again. One, two, three, four, five, six. I've got twelve. So with these gears, I've taken the process of carrying the one out of my mind and into the matter. I can now add without thinking about the carrying procedure. It's been mechanized. But the carry isn't the only hard part about arithmetic. It can be hard to hold large numbers in our head and remember where we are in an operation, especially when we're doing a lot of addition and subtraction, which is the basis for multiplication and division. To alleviate our mind of those burdens, we need to abandon this method of directly driving the wheels because it forces us to remember how much we've turned them. Instead, we need to have some way of having one consistent motion that can add different values into the wheels as needed. As we are already working with gears, notice how I can add nine to this wheel if I drive it with a nine-tooth gear. Instead of counting the teeth like before, I let the gear do the counting. Resulting in nine. If I then wanted to add five to that nine, I could switch out this gear for one with five teeth and then do the same rotation, resulting in 14. So with these gears, we added nine and five and got 14. And instead of having to count the teeth out, we just always fully rotated the gear. That way, the math is much more material than mental. This is what the polymath of Wilhelm Gottfried Leibniz realized in the 17th century. And he came up with a device for easily switching out new gears to add new numbers. And it's really just a stack of these gears, one on top of each other, numbering in teeth nine, eight, seven, all the way down to one. It's called a stepped drum. And if you can imagine, it makes changing out the gears a matter of just shifting this drum up or down. If I change the geometry a little, I can set up the same arrangement as before, like this, where this is our ten-tooth gear, similar to this one, and this is our stepped drum, like the one from before, with teeth numbering nine, eight, seven, all the way down to one. This arrangement makes the user input easier because selecting a different numbered gear just becomes a matter of sliding this gear along this square shaft. If I want to add nine, for instance, I shift it over to nine and fully rotate the stepped drum. Now nine is on top. On a side note, these square shafts are really key components. They allow freedom of parts horizontally while still translating rotational movement. I only seriously considered making a calculator once I realized I could easily source square shafts and tubes from Home Depot. Anyway, now we have a way to input a number, like five, by sliding this gear and add it by rotating the stepped drum. Notice that this action also stores the number that was inputted as the linear position of this gear. If I don't adjust anything, this setup still retains the information and mechanical configuration that is needed to add five. Similarly, the result of five is stored in the amount this gear is rotated. This kind of memory in the way inputted and outputted information persists will be important when we get to multiplying and dividing. Next, let's make the resulting number easier to see. Since we're inputting the number by sliding from above, we'd ideally want the result wheel facing up as well, something like this. To do this, I'll make a pair of bevel gears. I had a lot of trouble figuring out how to cut out bevel gears. I experimented with all sorts of techniques, but I settled on this method of tilting my scroll saw table 45 degrees and cutting into it like this. Now the number dial is in a better position. For now, let's put carrying the 1 off to one side. We know we can make some sort of mechanism to perform the carry. Instead, let's focus on subtraction. In theory, this would be easy to do. Just turn the step drum in the opposite direction. But that would be another thing we need to think about, and we're trying to mechanize all that thinking. So instead, we need some way to turn the step drum the same way, but have the dial turn in the opposite direction. By merely adding one bevel gear here and connecting the dial to that one instead of the other one, the dial turns in the opposite direction as before, subtracting instead of adding. Now with the bevel gears configured like this, I could add in a number by turning the step drum. There I added in 4. Remember, we're looking at the top number here. And I could just slide this part over, and by turning the drum in the same direction, I could subtract that number. So we're back to 0. I think that's a beautiful mechanical solution. If we added something like the mechanism we made before for carrying the 1, we would now have a machine for adding and subtracting numbers up to 9. But we want to be able to calculate with big numbers, because those are the especially hard ones to do mentally that a machine could really help with. This means adding more stepped drums. I'm not working off of any plans or blueprints, but basing my calculator off of the arithmometer invented in the mid-1800s by Thomas de Colmar. So I went with the same 6 inputs and 12 outputs that were on his standard model. This meant cutting a lot more gears. I cut all the gears by hand, in total 156 with around 1500 teeth, out of plywood on my scroll saw. Historically, these machines were made of brass or a similar metal, but my background is in woodworking, and when I started this machine, I didn't really know how to work with metal. So now we have these 6 columns of stepped drums, the 10-tooth counting gears they mesh into, and the sets of bevel gears that would connect to the result styles and allow me to switch between addition and subtraction. Now we need some way to link up all the drums so they turn at the same time. We could use gears, but then every other column would turn in the wrong direction. Thomas de Colmar's arithmometer used these sets of bevel gears, but I started to loathe cutting so many bevel gears, so I decided to use a chain and sprockets, which in my mind is slightly more elegant, plus I've never seen a mechanical calculator use a chain like this. The links in the chain have one hole that is slightly bigger, and one that is slightly smaller, so the chain rolls well while also staying together. In total, I cut out 152 axles and 304 links. Now to add a crank to the top, I had to make another set of bevel gears, and also a ratchet and click to stop anyone turning it backwards. I'm going to refrain from putting the output dials on for the moment. For now, I've just written the numbers on the bevel gears. We can begin to see how this calculator can add large numbers. Starting at all zeros, I can add a number, like 125, by sliding these gears to the corresponding number of teeth on their step drum. Remember, each column here corresponds to one decimal place. So for 125, we need a 5 in the ones place, a 2 in the tens place, and a 1 in the hundreds place. The rest will leave zero. We can turn the crank fully, and that number shows up in our output. So that was 0 plus 125, which is obviously 1, 2, 5. But now I can do 125 plus, say, 734, by sliding to the correct number, 7, 3, 4, and cranking to equal 859. That calculation had no carries. None of the decimal places went above 9. If one had, like adding 3 to this 9, notice how in that decimal place, the answer is correct, 2. 9 plus 3 is 12, or 2 in the ones place. We're just missing a 1 in the tens place. This is what I was trying to explain earlier by saying we never need to add more than 9 in any one decimal place. So we need to reintroduce a mechanism to carry the 1 between columns when any one column moves between 9 and 0. So first we need a method of detecting when that happens. Remember that these aren't the real result dials. The actual ones will live above, like this. So we need to detect when this moves between 9 and 0. By attaching this disc with a bump on it on top of the bevel gear and aligning that bump between 4 and 5, which is opposite 9 and 0, the bump can signal a carry to happen. Now when the dial rotates between 9 and 0 and that bump comes around, it needs to cause something to happen, which will add 1 in the next column over. We already know how to add a 1 by introducing just one tooth of the stepped drum into the 10-tooth counting gear, like this. We can basically copy that mechanism by adding another 10-tooth counting gear onto the same shaft, and below it, in line with the stepped drum, we'll put this piece that has just one tooth of the stepped drum, so that then it will just add one tooth. Ignore this other disc for now. When this piece, with this one extra tooth, is moved in line with this 10-tooth gear, it will add 1. But if we move it out of the way, when we don't want to perform a carry, it will bypass that gear. So now we need a method of moving this piece in place when this dial turns from 9 to 0 and that bump comes around. To do that, I made this lever, which will stretch diagonally from one column to the next. So when this bump comes around, it pushes this lever, which acts on this fork and moves the assembly in place. To reset this operation, I cut this spiral ramp at the end of the assembly with the extra tooth. When we continue rotating, this post down here will interfere with the ramp and move the tooth so that it bypasses the gear. Now it will bypass it. Here's a better view of that. You may be wondering now what these brackets down here are for. And the answer is that they are a first of a series of mechanisms in the calculator a quality which is critical to making a digital machine work, as I'll discuss later. The bracket holds this spring that attaches to these arms in what's called an over-center arrangement. This means the arms always want to be either all the way forward or all the way back. The arms hold this shuttle that is attached to the extra tooth piece. This means that the assembly as a whole is sprung to want to be either all the way forward and under the gear or all the way back and bypassing the gear. This gives the mechanism a self-policing tendency towards either definitely adding an extra one or definitely not adding an extra one. It is binary, spring-driven to have no middle ground. Let's put all those pieces in place for all six of the drums and two more columns that will only perform the care operation to allow larger resulting numbers. With those spring-driven brackets, all the carry levers are now very clicky and certain. It's starting to really come together now. The next part to add is something small, but its effects make this machine much more powerful. It's merely a device for keeping track of how many times we turn the crank. It's starting to really come together now. The next part to add is something small, but its effects make this machine much more powerful. It's merely a device for keeping track of how many times we turn the crank. I'm not going to get into the details, but this method of counting the crank turns took me a while to design. With this train of gears, I can make a dial up here turn by a tenth every time I turn the crank once. Part of what makes this so useful is that this dial will live on the assembly that carries all the results dials for each column, and each column will have its own dial that counts the crank rotations. You'll see how important this is later when we start multiplying and dividing. Speaking of the results assembly, it's got to hold a lot of parts very precisely. It needs to hold all of the result bevel gears with their discs with the bumps for signaling a carry, then also all the number wheels, and it needs to hold seven bevel gears for counting the crank turns. As you can see, all of these axles also have this gear that's missing its tenth tooth. This is for resetting the dials to zero. We'll get to that in a moment. First we need some structure to hold all these axles in place. This base plate holds the bevel gears and discs with bumps. In truth, the bumps act on this little arm, and it's this arm that hits the diagonal carry levers. Now one layer up, we have this slot that this rack goes in. When we run this rack along, the axles will turn to that missing tooth in the gear. When we add the number dials, we'll align the zero on them with the missing tooth. And we'll cover the rack with another layer. Now we can add the wheels that keep track of the crank turns. They also have a rack to zero them. And we can add the main results dials. Then another layer, and we can put the cover on. As a whole, this was the trickiest assembly to get right. There's just so many parts that need to be held in close alignment. These are all the layers before I assembled them and put the gears in. This took me the better part of a summer to design and make. Before we can set that assembly in place, we need to link up all the sets of bevel gears to move together when we switch between addition and subtraction. This square shaft will control this plate that stretches under all the bevel gears, holding them in the same position. And now we'll connect this to a lever so the user can easily switch between addition and subtraction. Then we can set the results assembly in place. Now we can properly see when we add in a number, like 5 for example. For now, ignore the lower row of windows. Now if we add, say, 8, the carry mechanism comes into play, and we get the correct answer, 13. Let's try to see that carry mechanism working more closely. Here's the bump hitting the arm and causing the lever to move. That lever then pushes the extra tooth in place, and the 1 is added. Finally, the spiral ramp pushes the assembly back to default, as well as the carry lever and the arm. And now by flipping this switch, we can subtract a number, like 6, resulting in 7. Now we're almost done. To make it easier for the user to slide these gears, I'm going to suspend this fork above them. Now it's easier to slide the gears along the shafts. Now it's easier to slide the gears along the shafts. Now it's easier to slide the gears along the shafts. This beam also holds these leaf springs that are important for another form of certainty mechanisms, like the brackets I discussed earlier. like the brackets I discussed earlier. They have a ball stuck on the end of them that will ride on the bumps of this disc, biasing the number wheels to always be in one of 10 positions. biasing the number wheels to always be in one of 10 positions. Although in theory this mechanism is not essential, in reality it is crucial. It keeps the machine on a 1 or a 2 instead of between a 1 or a 2. We count to 10 because we have 10 fingers, or 10 digits, and in that scheme there's no half a finger, no half a digit. There can be 1 or 0 or 9, but not 2 and a half. These springs, because they prevent 2 and a half, keep the machine digital. Furthermore, this is what it means for a machine to be digital, that operates off of a discrete, certain set of values. Things aren't continuous, but jump by digits. I actually glossed over another mechanism which keeps the machine digital inside this assembly. See, all the dials also have bumps that spring-loaded ball bearings ride against, ensuring they turn a full number at a time, maintaining digital certainty and keeping everything discrete. It's not immediately obvious why a phone or a laptop is digital. Sometimes we might think it just means electronic, but what it really means is that it operates on discrete events. In this case, the teeth of gears, or in computers, the binary states of transistors. With those springs, we can now rely on everything to stay in line when we add large numbers and have many carries. To demonstrate this, I'll add in 999999, 999999, and then, to force the machine to perform many carries, I'll just add one. So many parts need to work perfectly for that to work. Let's watch it carry over to a million again. Now we're basically done. We just need to throw a case over the whole thing and see what happens. As Leibniz wrote, Calculamus, let us calculate. Adding numbers is simple. 762 plus 549. When I turn this crank, the machine will add the number set in these sliders into the number displayed in this upper row of windows. For now, we're going to ignore this lower row of windows. So for 762 plus 549, we first need to add in 762. To do this, we just set the sliders to that number. Remember that each column here represents a different decimal place. So for 762, we need a 7 in the hundreds, 6 in the tens, and 1 in the ones. Now we crank to transfer that number into the results. So up here, it reads 762. Now to add in 549, we just set the sliders to 549. 549. And crank again. So up here, our answer is displayed. 1,311. Now to prepare for the next calculation, we can zero the machine by adding in the results, sticking a stylus in this, and dragging back and forth to zero the windows. Let's try a subtraction. How about something difficult like 25,072 minus 19,871. Just like an addition, we begin by entering our first number into the results section. So we put 25,072 and then crank. So up here, it reads 25,072. Now to subtract 19,871 from that, we set the sliders to that number. 19,871. And we flip this switch to put the calculator into subtract mode. And now we just crank the same way. And our answer reads up here. 5,201. And again, we can zero that by disengaging the results and moving this stylus back and forth up here. Now we're ready for our next calculation. Notice how whenever I turned the crank, this window increased by one. Since we began calculating, I've turned the crank four times. If we were adding together prices, for example, this could be used to tally how many items were purchased. But it also unlocks the secret to multiplication and division. Let me clear the crank turn windows. See, if I input 125, for example, and then crank once, I've added 125 once. If I crank again, I've added it twice. Or in other words, I've multiplied by 2. If I crank again, I've multiplied it by 3. And now 4. 125 times 4 is 500. So crank turns recorded in these windows are the multiplier. This is where the fact that the calculator stores the inputted number in the position of these sliders becomes so important. It allows us to easily repeat additions and subtractions. You're probably thinking that a multi-digit multiplication is going to take a lot of crank turns, then. And it's going to take some. But this calculator has a really clever way of dealing with big multiplications that I've been hiding from you. I can shift the whole results assembly over going into a new decimal place. This is possible because I mounted the results assembly on a drawer slide that can move horizontally. This means if I want to multiply something by 10, I don't need to turn the crank 10 times. Instead, I can just shift the assembly over from the ones place to the tens place. For example, 45 times 10 would look like this. Input 45, shift this over into the tens place, and then crank once. 45 times 10 is 450. So if I want to do a whole big multiplication like 652 times 213, first I input 652 down here, and then I build up the 213 in these lower windows. Since we're in the ones place, we first crank three times, multiplying 652 times 3, and then we shift over into the tens place. And then we'll crank once, multiplying 652 times 10, but also simultaneously adding it on to the previous times 3. So at this point, we've effectively multiplied by 13. Now I shift over into the hundreds place and crank twice, multiplying by 200 and adding it on to the previous times 3. So by only turning the crank six times, we've multiplied 652 times 213 to result in 138,876. Let's try division. Like multiplication, it's repeated subtraction with the divisor being the number of crank turns. So for something like 21 divided by 7, we first need to get 21 up here. So we'll add in 21 with one crank turn. And because our answer is going to show up in these lower windows, we need to clear this one out of here. Now we input 7 down here and set the machine to subtract mode. And now we crank until 7 no longer goes into 21. We know that 7 goes in at least once, so we can crank once. 7 is still smaller than 14, so we can crank again. And 7 goes into 7 one more time. So 21 divided by 7 is 3, with a remainder of 0. But that was an easy division. Let's do a more difficult one, like 355 divided by 113. To get the most decimal places possible, I'll start by shifting the assembly all the way over. Now we're going to act like there's a decimal point right here, so inputting 355 would look like this. 3, 5, 5. And I'll crank to add that into the results. Now like before, we'll want to zero our divisor. And then I'll input the number we're dividing by, 113, 3, and set the machine to subtract or division mode. Now it's a game of cranking until this number no longer goes into this number. So we know 113 does go into 355, so we'll crank. It goes into 242, so we'll crank again. And it still goes into 129, so we'll crank again. Now 113 does not go into 16, so we'll shift the assembly over once. Now 113 does go into 160, so we'll crank. It does not go into 47, so we'll shift over. And now it goes into 470, so back to cranking. Until now it doesn't go into 18, so we'll shift over again. And back to cranking. It doesn't go into 67, so we'll shift over. Now we have 113 going into 670. If we wanted, we could try to predict how many crank turns this will take. I think about 5. This is analogous to doing the guess-and-check method if you were working this problem by hand on paper. Let's see. 1, 2, 3, 4, 5. Yeah, 5, because 113 now doesn't go into 105. So we need to shift over. And crank. It looks like a lot of cranks. Now it doesn't go into 33, so we'll shift over. And crank again. Now it doesn't go into 104. And we'll shift. But we can't shift over anymore. So that's the farthest we can take this calculation So 355 divided by 113 equals 3.141592. I thought it would be fun to do an approximation of pi. And we have a remainder of 0.000000104. Shifting the assembly over takes great advantage of that underlying feature of our number system, that decimal position encodes part of the value of numbers. And the machine cleverly combines that fact with the way the dials themselves act as a memory, storing the value within each decimal place. Well, that's how you make a calculator. Except, not really. I haven't given you the dimensions of anything, and I'm sure I've left out some important details. You'd be hard-pressed to make this exact same calculator yourself. But you could make your own version, now that you understand the basic principles. And I'm sure you could improve some things if you're handy. I know this because that's essentially what I did. I didn't work off any plans, I just studied pictures of the arithmometer and other calculators and made my own. Not copying dimensions or small details, but understanding the principles and playing around until I got it to work. Along the way, I made some improvements. This chain for driving it is one, those sprung brackets are another. The other big one is my method of springs on this bar to keep it digital. I can also loosen and slide them horizontally to fine-tune exactly where along its rotation the teeth stop. I found this could be really helpful in getting the carry to work well. Professor Matthew L. Jones describes the process by which designs for mechanical calculators emerged as a culture of emulation. That is, inventors and thinkers would observe other machines and then try to make their own. And in the process, they would find new ways of doing things. I read this book after making my calculator, but I think I fit into that culture of emulation. Particularly because Jones also articulates the interplay between theoretical design and material production as crucial. Makers like Pascal and Leibniz had to negotiate between how something worked in their mind and how it would work materially. Especially because they did not actually make their calculators themselves, but collaborated with craftspeople. Often the successive designs hinge on the success of the relationship between the initial designer and the skilled craftspeople. The point being that making a calculator requires not just a handy mind, but also a mindful hand. For example, I developed a feeling for springs while making this calculator that proved very valuable when I was making this mechanism. I had to sensitively cut away the springs to give just the right amount of force As Jones describes, often the physical realities forced the theoretical designs to change. And I can certainly attest to this. This culture of emulation and designs born through a dialogue between theory and practice challenges our notion of lone geniuses coming up with brilliant designs, then simply making them. That's not how arithmetic came to be mechanized. Furthermore, the history of calculating machines illustrates how we should not view the process of technological change as driven just by new materials or techniques. As I've shown, you can make a mechanical calculator out of merely wood and simple metal rods. And you don't need any fancy machines to cut gears. The accuracy of the hands suffices. It did not take some great breakthrough in manufacturing processes or in measuring devices. The most I used was a good ruler. But instead, some clever minds and hands with the proper motivation. And proper motivation is the key. More than new technology, what caused arithmetic to be mechanized was new interest in the problem. As much as it takes a certain arrangement of matter to make a calculator, it also takes a certain arrangement of society to incentivize the production of a calculator. Enlightenment culture prized intellectual pursuits and thinkers increasingly needed a figure with large numbers. Moreover, as Jones points out, these machines were status symbols in the 17th and 18th centuries, gaining their inventors fame and renown. As much as Leibniz dreamed of kings solving conflicts not on battlefields but on calculators, he also very much used his machine to tout his abilities to patrons and raise his acclaim. Even in the 19th century, Thomas de Colmar gained repute through his machine, being given the highest order of merit to see his machine become a commercial success. The arithmometer was the first mass-produced mechanical calculator. This is where what it takes to mechanize arithmetic becomes a question of not just proper intellectual interest but also sufficient market demand. As global commerce grew, these machines reached their heyday in the early 20th century, eclipsed by electronic calculators in the 1970s. I want to end by thinking about what it has taken to mechanize and replicate a thought. The way our numbers work, that 9 loops back to 0 and position encodes value, subtracting is a programmatic process, lends the thought of arithmetic towards mechanization. But as I've touched on a little at the end here, it takes more than that to produce a mechanical calculator. Clever minds engaged in the problem of materializing thought and clever hands sensitive to how to work with materials refined designs over centuries. It took many people, socially engaged in an endeavor shaped by a certain culture of emulation, motivated variously by necessity, fame, politics, and money to make this machine. Personally, I'm fascinated by how this one object manages to embody all that history, take advantage of and reveal structures in our number system, and orchestrate hundreds of mechanical parts, all towards the singular purpose of calculating numbers. That's why I made it. And in case you were wondering... Thanks for watching!