James Tam · Transcript · · Updated
CPSC 409 Sept 28 – Transcript
Video by James Tam · Watch the original on YouTube
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- Babbage designed the difference engine to reduce calculation errors using the mathematical property of constant differences.
- Babbage and Clement eventually experienced a major falling out, leading to a split and the loss of Babbage's designs and tools.
- Babbage subsequently conceived of a completely different programmable machine called the analytical engine.
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to do it really to to have a fair job considering those odd edge cases rather than just okay it's outside that narrow bound of what I considered the bottom line that I want to recap is the tables are very useful I described that but despite the extraordinary measures that people took in general despite the extra extraordinary measures that people like Babage took there were
still errors as long as you have a person involved. Uh you just can't help it. So this is a little bit more describing 40 errors on a sync how extreme it can be.
Uh 1,000 errors in one I assume that's one text and the only foolproof method. So there's more examples of where you can't avoid the errors is just automate the process assuming that process is correct. But they actually did find something correct although it was limited using differences. Oh actually I'll go maximize. I'll build this a part of their time.
Essentially what they wanted to do was uh no this actually wasn't too bad. What they wanted to do was um simplify the design of machine of course. So the fewer math operations it had to implement the simpler the hardware. But when we look at Babbage the machine, it was actually fairly complicated. What it did was the difference is when we were trying to compute the value of function.
So I have a simple one here, a linear function. So you have different X values. X is 1 2 3. There's value in determining the different values of the determining for different values of X. What does the function evaluate to? What does it work out to? So 2 * 1 + 3 is 5.
uh 2 * 2 + 3 that's 4 + 3 7 computing this is possible with hardware but the thing is it adds more complexity more things can go wrong it's mechanical so it wears out more so the idea is they tried to have something simple and they use the mathematical property of differences if you comput compute a function that's raised to the first power
and you have the results of that function Here when you subtract a later result from an earlier result you have a difference two. And notice when we get to f ofx is equal to 3 we subtract this value from its immediate predecessor. It's two.
That's exactly oops waiting room here. That's exactly the idea of oh he left before I could let him in. of the differences. It's a constant difference. And this folks, this is a pretty important part here.
It's called the difference engine or region uh reason. You see, you'd start off with this base value here, right? You could compute this by hand.
or some other mechanism. Look it up or whatever. Make sure it was correct. And then from there, all the machine had to do successively was add two to it. Oh, another person's in the waiting room. It's a property of constant differences, that constant sum.
this math principle Babage didn't come up with. And in fact, he wasn't the only one that worked on a difference engine or even was the first person that designed a difference engine, but he was uh almost certainly the most famous.
Now, with the Whoops. Clear this up. Is there any questions about the first difference? Fairly straightforward. I think I went through it slowly. What if you have something where the exponent is greater than one?
Well, then u do this a piece at a time. Oops. Reason that fill wasn't set. There you go.
So we have x squar when we compute the value of f ofx. So squared value you should be able to do in your head basically square these x values. We have 3 5 7 it's not a constant difference. We can't add three to the base case here.
We can't add five. However because it is now a power exponent that is greater than one. We have to compute it to a further level. We then subtract this versus this. So we take the the uh let's see so here's the the first difference here we take the later value immediate predecessor value we a difference of three we take then another later value it's immediate pre
predecessor value difference between 9 - 4 is 5 then at this first difference subtract these ones 5 - 3 is 2 and if we go a bit further so we have n uh 16 - 9 is subtract these ones. We end up with a constant difference.
So for the power if it's the power of one we just have to take it to the first difference. If it's squared we take it the seance second difference. Again we compute the base value and then based on the base value after a certain point after what is it after a certain point then we can just keep adding a constant difference. So
over here this constant difference can be added on with the cubic we have to compute the third difference until we get a constant difference. The idea is we just keep adding that con. Oops, someone's in the waiting room. Constant difference. So that's how the difference engine works.
It's based on the prop mathematical property of differences to compute a series a sequence of numbers because it's not the sum just to be accurate. The sequence is that line at 1 2 3 4. The the series would actually technically be the sum of that value.
Okay. So, there were other other people who actually conceived of the difference engine. One person who wrote it was Clipstein. Clipstein um he this is where it gets a little bit on the hairy side. And it doesn't actually it isn't clear enough to attribute credit as I remember because things were lost in the he did
write about it in this treatise. It includes one of the first references to such a device. However, Clipstein wrote this publication described the computing device the um difference engine. However, in the appendix, now that had a bit of complexity, Hessen, that's his um ethnicity,
Mueller actually did invent one, and he described one that was even more ambitious than the one he described. Now, if you're kind of trying to sort through this, well, who actually published first? Well, this guy produced a book, but then it referenced to someone else who who had some writings, although this writing wasn't published.
Yeah. So I suppose officially I wouldn't put this thing on exam in this case official publication this one I suppose and this guy's first because it's unofficial publication. So it's kind of yeah the guy published the guy wrote something but he didn't publish it and then this guy who published kind of referred to this one but the point is Babage wasn't first at all.
The first person who referred to it, they described such a device, the published one was Kipstein. Whether or not you get Mueller, well, yeah, I kind of agree with Williams that it's kind of ambiguous.
They're trying to attri if you're trying to attribute credit, you need enough proof to be able to say, "Yes, this guy did it." I mean, these days it's very easy. You have a conference, you have a journal, there's the publication. Is there anybody else who actually if you're trying to say who came up this concept or who was cited as coming up this term, you have that as proof. This is precise. So you can actually go to this paper and look through it. This is precisely but in a
more complicated form what I'm looking for in the exam details where you can actually describe not a paper but what I've said in class about a concept. So that shows that you know this is the source but you also understand why this is proof as opposed to just say pointing right here to this and then I don't know trying to use this as a copy paste. So it has to be some understanding and in this case because it's so on the
hairy side I wouldn't quite push it on to exam because they're not there's not enough evidence from the historians who've looked at the data. So motivations, why did Babage actually um design such a device? Well, some of it's the obvious part that I discussed earlier, the need to reduce the errors in the calculations,
accuracy. Regardless of how much care they took, errors would show up and sometimes the correction of the errors produce more errors. The idea, when did Babage first conceive of it? sometime 1812 1813 when he was still a student at Cambridge.
The story goes that he was in the Cambridge Library and his colleague saw this saw saw Babage sleeping and then Babage awoke and he said, "Well, what were you what were you dreaming about? What were you sleeping? What were you thinking about?" He said, "I wasn't sleeping. I was actually thinking about this machine that could compute logarithms automatically.
I wasn't resting my eyeballs. I was actually doing work. Power nap I guess they might call it. Okay. However, it wasn't enough obviously 1812 1813 for him to be working on an advice. So he tried it the way everybody else tried it and saw there were still errors and problems and he thought well we need to do something