Alright, so we're talking about decoding RM codes, okay? So in this context, it turns out we were looking at finite geometries, and what I'm going to do in this class is to begin with a simple example and talk about how lines and planes and all these things look in a very simple example so that you get a rough idea. And after that, we'll try and go into some more general things. Okay, so the example I'm going to take is EG4,2. So what is EG4,2? The points of this equation jump at 3. Basically, 0,1,4, right? And for some reason, this is not BW. The points are 0,1,4, so how many of them are there? There are 16 points, okay? So just to visualize, maybe we'll write these points down like this, okay? And maybe a little bit below here. So those are the 16 points in the geometry, okay? So you're definitely used to the real geometry where you have an infinite number of points and you always deal with equations. Here there are just 16 points, so this thing is pretty easy, okay? So for the subsets of the points, we're going to call them as lines and planes and all that. So one flat, two flat, three flats. The dimension is 4, right? So you'll have one flat, which would be the line, and then two flats, which you can think of as a plane, and three flat, which is called the hyper plane, one less than the dimension, and then whatever you think together is the last one, okay? So one point, which is all 0, is called the origin, just like the problem, okay? And it's easiest to think of these flats that go through the origin, okay? So because any flat that goes through the origin will be a subspace of 0, 1, m, okay? 0, 1, 4.  So once it's a subspace, you know, subspaces are defined by bases or they're dual bases, right? So either the bases itself or the set of orthogonal things describe the subspace. So it's very nice, okay? So you start with lines and planes and all that that pass through the origin, okay? So how do I define a line if I want one of the points to be the origin? So usually if a line passes through the origin, what is its equation in the 2D plane around? It's some y equals mx, right? So you have some constant m and then you multiply with x, okay? So if I say the line passes through the origin, right? So what you need is actually just one more point on the line. Any other point, okay? So you have the origin and then if I give you any other point, say x1, y1, okay? So let me do it that way, okay? So if I give you the origin and then I give you some other point, x1, y1, what do you do to get the line? You basically take this vector and then multiply that vector by all possible scalars. So that will give you all the entire line, right? So that's what you do, okay? So there is a similar argument here, okay? So I give you the origin. If I give you any other point, say for instance, this point, what should I do to get the line passing through the origin and that point? I have to take that point and multiply by all possible scalars. But for me in the binary case, the scalars are only 0 and 1. So if I multiply by 0, I'll get 0. Multiply by 1, I'll get that point itself, okay? So as it turns out, the origin and any other point will form a line, okay? So how many such lines will we have? The pass through the origin, 15 lines will have passing through the origin, okay? So now how do you get lines? So is that clear? So for instance, lines through the origin are something like 0, 0, 0, 0, 0, 0 and then 0, 1, 1, 0. Then we have other lines, 0, 0, 0, 1, 1, 1, 1. So 15 such lines I got. Okay, these are the lines that pass through the origin. So how do you get a line that does not pass through the origin? Okay, so if you have a line that does not pass through the origin, you can always shift it by some constant, translate it so that it passes through the origin, okay? It's the same principle you can use here also. So you take a line through the origin and then add something to it. Add a constant number to it, that number to it. You'll get some other line that passes through some other two points, okay? If you already do, so if you want a line, so if you want any other line, so you take this guy and say add something, so 0, 0, 0, 1. So take this line and then add 0, 0, 0, 1. I'll get another line. It will be 0, 0, 0, 1 and then 0, 1, 1, 1. So that will be another line. So if you think of this as a line, this girl is another line, okay? So that's the idea, okay? So lines can be described that way, okay? So lines are described by two points that pass through one, right? So in the binary case, when this guy is two, the scale-ups are really only two. So we'll have only two points in the line. You can't have anything more than two. So there will be only two points in every line, okay? So another thing that we'll have to use is this notion of an incidence vector. So what is an incidence vector? So I have 16 points, okay? So what I can do is I can think of, sorry, I have 16 points. I can think of putting these 16 points in a row, okay? So I'll put 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0. All of them in a row, okay? And then if I have a subset, okay? If I have a subset of these points, here the points appear up, okay? If I have a subset of these points, so the subset is the line, okay? So maybe it's the line, okay? 0, 0, 0, 0, 0, and then 0, 1, 1, 0. I can define an incidence vector for the subset. What is an incidence vector for the subset? If a particular point belongs to the subset, so that entry you put 1. If it doesn't belong, it would be, okay? So like this will be the 0, 1, 1, 0 from there, but the row will be 1. Everything else will be, okay? So what will be the length of the incidence vector? It will be 16, okay?  Yeah, this set and we have a lot of subsets. You can always think of incidence vectors, okay? So this is a very commonly used technique in common topics and all this. We can use incidence vectors to do a lot of counting. Yes, we can do that also here. So one nice thing with respect to each element codes is incidence vectors are vectors which look like they could be code ones, right? In a way. They are length, 2 param, length 16, they are the right length, and they have boundaries, 0 or 1. So incidence vectors of these subsets are, they belong in the proper space at least, okay? We will show eventually that incidence vectors of these one flats and two flats belong to each element codes. We will do that. But for now, these are just vectors of length 2 param, okay? So in general, let me hear in this case, vectors of length 16. So, okay? Eventually these incidence vectors will are the bridge between this finite geometry and read the rate miller codes, okay? So through these incidence vectors, we go from geometry to read miller codes. Okay? So that is lines. If I want planes, what should I do? Okay? So a plane is basically a two-dimensional subspace. Okay? So you have a two-dimensional subspace of these 16 points. You need two basis vectors. Okay? So maybe I take these two as some basis vectors. This guy, maybe this guy. Okay? It turns out once you have, I mean, any subspace will definitely have zero also in it, right? What will be the, what will be another one? See, I am defining a two-dimensional subspace with these two as the basis vectors. So what are the vectors in the subspace? Yeah. So it's the sum of those two, right? So it's going to be just 0, 0, 0, 1, which is this guy. Okay? So those four guys form a plane which passes through the origin. Okay? So it's a two-flag that goes through the origin. It's also a two-dimensional subspace. So it's nice. So if maybe you wanted to find some other plane that way. So all you have to come up with is two different points in this. Take that as the basis of your plane and then form the linear combinations. We'll get the plane. So how many points will that be in a plane? In a two-flag, you'll have four. Okay? So likewise in a three-flag, you'll have eight. Okay? So of course beyond three-flag, it's not really interesting in the three-dimensional thing, right? So four-flag is interacting and after that you won't get anything else. Okay? So okay, so for two-flag, there are four points. One easy thing is to take the basis and then form the four vectors which are there in the two-flag. It turns out you can also have planes that don't pass through the origin. How do you think of those planes? In R3 for instance, you can have a plane that does not pass through the origin. You can always translate it to make it pass through the origin. So in reverse, you can start with a plane that is in the, through the origin and then add something more. Same thing you can do here. So I'll take this plane and say add 10102. Okay? So I need another shape here. So I'll use the shape. Okay? I'll add 10102 each point of that plane. So what will I get? Okay? 1011. And then I'll get 0010. And then 0011, right? So that's it. So you can see where even in terms of there is a translation effect going on, I mean this is just the way to write down the points. Write down the points in any other way you like. But you can see it kind of makes sense to tell these things of planes and all that. It's not the point. So that would be another plane which does not pass through the origin. It's translated away from the origin. So it's not a subset. But still if you have four points. All right? So that's the, that's the idea. Okay? So you can have, for instance, incidence vector for both for a two-flag. Okay? So you have an incidence vector for the two-flag. How many ones will it have? For instance, if you take this squiggly ellipse incidence vector, this is going to be one. This is going to be one. And then you'll have a one at 10000. And then you'll have a one at 1001. So that would be the incidence vector for the two-flag. Okay? Already one nice thing you're observing is, if I have an n minus r-flag, what will be the rate of the incidence vector? Two-flag and minus r. Okay? So it's all the rates are powers of two. And that's something which is very nice where it will occur. Because we know powers of two is what really happens and occurs in the rate millipole definition. Okay? So all those things are nice connections which you can think of. Okay? All right? So this is the, this is kind of something that was, I was alluring to go to the end of last class how it works all day. Okay? So some interesting questions that we turn out that will be very important in final geometry at least is given a geometry like this, how many lines are there? So let me see. Take some time to think about it and answer this question. In this, 16.014, how many lines are there? Is that the correct answer? 16, 3, 3, 2? Is that okay? Will you not be 16, 3, 2, 2, correct? Everybody's happy with that? Okay? That would make sense? No? Maybe, yes. Okay? So, so, so there are various ways to do this counting. One easy method that I like is to do the, do the incidence vector kind of thing. So what you can do is, you can, you can make a, you can make a big, big matrix. Okay? You can make a big, big matrix. You put the points there. So how many talents will there be? 16 points. And then you put lines here. And you fill it up with incidence vectors for the lines. Okay? So maybe you have capital or lines. Okay? How many ones will there be in each line? Two. So there will be two ones. So where is this two? How many ones will there be in each column? 15? Is that okay? Everybody's happy? Is that correct? 15, the correct answer? Okay, so let's say 15. Okay? So if you say 15 ones. So two ones in each row. And then 15 ones in each column. So now this gives you a formula for L. Does it not? You count the total number of ones in the matrix, column wise and row wise. Okay? So you count it column wise. You have 16 times 15. That has to be equal to L times 2 at two and right. Okay? So L equals 16 choose two. Okay? So I wonder for lines it is really quite simple. I don't have to do it. So planes and all it can be a little bit more tricky. Okay? Not that difficult. But you can count the total number of planes. Okay? And then you might want to count things like, if I have a line, how many planes contain that line? Okay? So let's say this is the number of lines. What about the number of two flats? Number of two flats as well? 15 C2. 15 C2? Not 16 C2? 15 C2. Okay? So you can take any two but one of them cannot be 16 C3. Why did it go to C3 now? Won't you get repetitions? Okay? So let's do this once again. Okay? So I'm just doing this the fun. It's not really that useful. So let's say how many two flats are there in this? Okay? So how do you go about doing that? How many two flats are there? See lines were easy. Any two points between any two points over the line. But how do you think of, how do you think of two flats? Right? How do you, how do you do that? Okay? What's the question? Right? What's the question? So you have to be careful about ever counting. Okay? So because you can count planes in multiple ways. See for instance, if I take this quickly plane, I could have taken the basis to be these two, or these two, or these two. Okay? Any which way I take the basis, I will get the same plane. Okay? So I cannot just count 15 choose two. There I'm ever counting. Right? If I say 15 choose two, it gives you all possible choices. Right? So it would be slightly careful there. And then you have to divide by something. What should you divide by? So let's say a number of two flats through the origin. Give me the number of two flats through the origin. How will you add that? See if we divide by three. Yeah. So you have to divide by three. Right? Is it clear? So you have to take 15 choose two, and then you have to divide by three. Why do you have to divide by three? Because every plane has three different bases. Okay? And you're counting all the possible bases. That's why we divide by three. We're overcounted by three. Okay? So that will be the number of two flats that pass through the origin. Okay? So likewise, you can do. So if the number of two flats passing through the origin is 15 choose two by three. What is the number of two flats passing through any other point? So again, the same thing. Okay? So once you answer that question, you put this matrix.  Right? Each plane has four points. There are 16 points. And how many planes are there through each point? 15. 15 choose two by three. So number of planes times four should be equal to 15 choose two by three times 16. You solve that and you'll get another place. Okay? So you can put the matrix like this. The critical question to answer is how many flats, how many two flats or three flats or whatever pass through a particular point? That you can do by this simple counting argument. Once you do that, you can answer the question of total number of planes, total number of two flats, three flats, etc. So it starts with counting. It will vary on overcounting itself. The only thing you have to be careful about is overcounting. Okay? We're okay. So these things are not too critical. Maybe there will be some assignment questions. But it's quite easy to do this calculation. Okay? They're okay. That's fine. So we can maybe check that. So number of two flats through the origin, number of two flats, two, two, zero, zero, zero, zero. It's going to be equal to, like you said, 15 choose two. That's correct. But then you'll be overcounting by three. So what is it? Seven times five, thirty-five. Okay? So this answer is thirty-five. So I don't know what this method is. Correct? Okay? So thirty-five planes that pass through every single point. Okay? So number of two flats, total number of two flats times four should be equal to 16 times thirty-five. So that gives you the number of two flats to be 140. Okay? That will be the total number of two flats that pass through the, that are in this geometry. Okay? So like I said, these are all finite geometry. Subjective will be finite. You can't have anything infinite. Okay? So number of planes in R2 is no, number of lines in R2 is no need to ask a question. Your answer is infinite. You know, immediately what it is. Here it's not like that. Okay?  So it's not too hard that you have to do it carefully. How do I get this three again? Three comes from the, the answer to the question, how many different bases are there in a two-slide? In one two-dimensional subspace that passes through the origin, how many different bases are there? That answer is three. Okay? That number can be calculated. It's a little bit more tricky as you go larger and larger, but it can be calculated. Okay? Think about it. So the binary case is very easy. The non-binary cases, it becomes a little bit more tricky, but it can be done again. Okay? All right. So that's done. Another question I want to ask is, suppose I fix a line. Let's say a line through the origin. Okay? So I fix a line through the origin. How many planes will contain that line? That is my next question. Fourteen, right?  Fourteen. Fourteen. Fourteen. Fourteen. What are we going to do? No. And then if these are fixed, then one more point is fixed. Because that's just above here. Some of these two is that itself. A fixing time. There's nothing more you got. So you take any other point. So suppose I fix this line 0000011, 00001. I have the liberty to pick one other point. So that gives me one more. So I can pick that plane in two different ways. I cannot pick every this point at that point. So it's going to be 14 by 2, so 7. 7 would be the correct answer. Okay? So you pick these two. You pick one point. So you pick 00001. That tells you that 0000 is also there. So you can't say all 14 will give you different planes. Two of the points will give you the same plane. So you divide by 2. You get 7. Okay? So these are simple calculations. But if you do it for the first time, it will be packed through the overcounting. Yes, watch out for the overcounting. Okay? So it's a simple question of dividing, but still you have to watch out for overcounting. Okay? So the number of planes that pass through a fixed plane is 7. Okay? So something like that. So those are the answers that you can do. So my point is, the main point I want to make is, in any finite geometry like this, all these questions have exact answers. Questions like how many planes, how many lines are contained in a plane, how many planes pass through a given line, how many planes pass through a point, how many free flats pass through a point, how many free flats are contained in how many free flats are contained in free flats. All these questions have fixed definite answers. Okay? So you can think about them, do the computation, and get the answer. Okay? So that's the point I wanted to make last class. I don't know if it was very clear to you that this is the way you count them. It's not very clear. Okay? So now, next comes the main result which I kind of informally stated for you. The main result which links these two things first. Incident vectors of M minus R flats in EGM comma 2 are probe worlds of... I don't know why I've been so precise last time because we probably didn't mention these incident vectors and all that. This is the first. So you take an M minus R flat in EGM comma 2. Okay? R is the same R, okay? It's not some other arbitrary thing. R flat in EGM comma 2. And then take its incidence vector. So its incidence vector will have a weight of 2 power M minus R. It's a M minus R flat. It's going to have a M minus R basis vector. So you pick 2 power M minus R for a different point step. So that will have a weight 2 power M minus R. So in fact, the non-linear code words, they are minimum weight code words. They are minimum weight code words. Since you do see it, they have a weight 2 power M minus R. And 2 power M minus R is the minimum distance of RMR comma M. Okay? So the incidence vectors of M minus R flats are the minimum weight code words of the RMR code R comma M. Okay? So that is the connection. And this gives you an easy way of coming up with the minimum weight code words of the RMR code. Okay? And like I said, the rules of RMR codes are also RMR codes.  Right? Because ultimately, if you're interested in orthogonal checks, you're going to be interested in lower weight guys. Right? You don't want very high weight vectors. Right? Because when orthogonality is going to be difficult, you want lower weight guys. Right? And minimum weight vectors of RMR code were simply incidence vectors of flats. And due to this, RMR code is again some other RMR code. So you only have to come up with incidence vectors of R flats to think about methodically coding it. That's the idea. Okay? Okay. So let me throw this to you. This is not very hard. It's probably very easy to code. Okay? So if you have an M minus R flat, okay? So far I've been talking about the basis vectors. M minus R basis vectors completely fit further. An equivalent picture is the dual space. Okay? So an M minus R flat, once you go to the dual space, it's defined by R equations. R equations of the term summation aij vj sbi the i is 0 so we'll go from 1 to m. aij i is from 1 to R aij is 0 at 1 not at 0. I mean, of course, I mean not at 0 is not the really big thing. It has to be linearly independent and all that. So this is this defines an I said this defines an M minus R flat so it cannot be dependent and all that. So it'd be perfect. And di also is 0. Okay? So any M minus R flat is defined by R equations. And remember, the degree of each equation is 1. Okay? So now I want to show this this belongs to R comma m. Okay? So what does R comma m have? It contains evaluations of of what? Evaluations of polynomials of degree less than or equal to R. Okay? So this is like a polynomial of degree 1. Okay? But my points are those which evaluate to which evaluate to 0 when I plug in both. Okay? So I have to move from 0 to 1. Right? These points that satisfy the problem where M minus R flat have to evaluate to 1 when I plug it in. Okay? And then I'll get the proper code load. Okay? Do you understand what I'm saying? Okay? So maybe I should write it down. Okay? So suppose I want to make an incidence vector for our M minus R flat. What would be the incidence vector? So I have my points on 0 and I'll go to R1. Okay? What do I do? I take each of this V1 and Vm. I plug it into the R equations. If all of them evaluate to 0 then I know I should put a 1 here. Okay? But I have to move from this world into a polynomial evaluation world. How does a polynomial evaluation world work? I have a polynomial. I first V1 to Vm. It's a Boolean polynomial. I simply plug in the V1 to Vm here and I put that value there. Okay? If I want the incidence vector as this way to belong to the real number 4 I should come up with a F of degree less than or equal to R which will evaluate to 1 whenever a point is in the M minus R flat which will evaluate to 0 otherwise. So the whole trick is to come up with that construction. How do you come up with an F of V1 to Vm which has degree R or less than or equal to R and then it should evaluate to 1 whenever I plug in a point of 2 flat M minus R flat it should evaluate to 0 otherwise. Is it okay? Okay? That trick is simple. It's not a very difficult trick. All you have to do is this. Okay? You take this and then you make a product of I equals 1 to R and these guys summation a i j v j plus b i but then you add a 1 to it. Okay? Don't just do it that way. You add a 1 to it. Okay? And then this j only goes from 1 to M. Okay? So that's the trick. Okay? Now I have a polynomial of degree what? Equal to R. Okay? So evaluation of this polynomial at all the points will definitely belong to the Read-Miller code, R M O F, R M I M. What is so nice about this construction is for every point in the M minus R flat what will happen to this expression for every i? It will be 0. So that will be 1. Overall the polynomial is lower than 1 and you'll get a 1. For every point which is not in the M minus R flat what will happen? So at least one equation this will evaluate to 1 and that will make the entire polynomial 0 and you'll get the 0. Okay? So the evaluation of F is the same as the incidence vector. And it has degree R which means the incidence vector belongs to the Read-Miller code, R M O F. So it's a very simple course but not really that fancy. But it's a nice little construction. This trick was very interesting. How do you go from some equations that are equal to 0 to a polynomial whose evaluation is something that can be controlled? It is not a very difficult trick but it's an important trick to get this answer. Is that okay? All right. So if you have... So there are few other things that you should keep in mind now. Okay? So you have a lot of inclusions there which should be mind to laugh. Read-Miller code of R, M is contained in what? Read-Miller code of R plus 1 comma M. Right? It's fully contained in Read-Miller code of R plus 1 comma M. So if you have M minus R flats belonging to R M of R comma M will also belong to... The incidence vector of M minus R flats will also belong to R M of R plus 1 comma M. Okay? So once you fix M minus R R plus 1 all the way to M the incidence vectors will still belong. That's because of this inclusion. Okay? All right. So that's the first point. Another way to think about it is what about M minus R minus 1 flats? If I have an incidence vector of M minus R minus 1 flat that will be R M of R comma M. Is that okay? Is that correct? I didn't make a mistake there. Yeah, that's okay. All right? M minus R minus 1 into brackets this is M minus R plus 1. Okay? Because larger weight. Okay? So if you have M minus R plus 1 flat that will also belong to this scale. Okay? Is that okay? So let me write that down very carefully. Maybe with an example. Okay? So let's take an example. Let's take the 4 comma 2. So let's take R M of 2 comma 4. So both will contain incidence vectors of the robot. So first of all... Okay? So incidence vectors of M minus R right? So 2 flats in EG 4 comma 2. Is that okay? What else will it have? It will also have incidence vectors of 3 flats. It will also have incidence vectors of 4 flats but 4 flats are very trivial. This is the entire all ones. All ones is there. Is that okay? Is that clear? If you consider... If you contain the incidence vectors of 2 flats and 3 flats. So if you look at any other readmiller code, let's say readmiller code 3 comma 4. This will contain incidence vectors. I don't have to keep repeating the incidence vectors. When I say it contains 2 flats, I mean it's clear that it has to be the incidence vectors. So it will be 1 flat, 2 flats and okay, so that's how you have to think about it. So it will have all these things. Whatever the readmiller code is 1 comma 4 3 flats. Okay? So what is the dual? The dual of this case okay, right? Am I right? So if I want to do majority logic decoding of r m of 1 comma 4 okay, suppose that's what I want to do and I have to look for orthogonal parity checks in in the dual, right? I have to find the orthogonal parity checks for r m of 1 comma 4 which will be 3rd words of r m of 2 comma 4 okay? So what you do there the strategy is as follows okay, so like I said there was a main result what is the main result that I stated? r m of r comma m can be decoded by r plus 1 step majority logic decoding right?  Okay? What does that mean? So when I say r plus 1 step majority logic decoding remember these steps right? So the hamming code I illustrated that right? So you take the first you find orthogonal checks on a subset of the not just e0 but say e0 plus e1 e0 plus e2 and then you recursively repeat it repeat it repeat it until you get to just e0 alone and then you repeat the same process okay? So if I say half of 2 bar m minus r minus 1 in each step so what does that mean? j j has to be equal to 2 bar m minus r minus 1 okay? It has to be that right? Only then you will have you will have this many okay? So so in terms of what you have to do in r and r 2 and r 4 is to first if I am not wrong you have to start with two flats okay? right? Okay? So r m is 1.4 is two-step so you have to start with all the two flats okay? And then ask the question can I find so let me rephrase it okay? So you should start with trying to find orthogonal parity checks on a one-file okay? So that's the idea so let me write that down so here's an example m number is decodable m number is majority logic so let me write that down so two-step majority logic decodable okay good? j equals what? 7 right? No discussion number 7 is coming okay? So we have to answer the question 7 So what you do is you start with some one flat okay? So let me say the number 2 comma 4 and it contains contains what? two flats okay? So you start with any one flat okay? some one flat so that passes through the origin okay? And then ask the question then ask then what you do is you collect all all two flats passing through this okay? So you start with the one flat so that passes through the origin 0 0 0 0 0 0 0 0 0 0 1 something like that and then you collect all the two flats that not passing through I am sorry they say that contain this one flat right? getting confused but contain this specific one flat the answer to that is what? how many two flats will there be? we just do that calculation it's going to be 7 okay? you take the incidence vectors of all these two flats you know that they are parity checks for my code how do I know that how do I know that the parity checks for Rm1,4 what's the deal? Rm2,4 all the two flats belong to Rm2,4 the next thing is they will all be after the NL on this one flat okay? think about that it's a relatively easy thing to think about so you can go back and think about this geometry if you like okay? so if I take this line and then look at all the planes that contain that line they will be after the NL and that line okay? all of them will check that line well, all of them contain that line and no two of those planes can have anything in common it cannot happen okay? then it won't do after the plane because if they have something in common the other thing will also do the same it cannot be different right? because this line is fixed you have one other point you have two other points in this plane if there is some other plane which overlaps in one point what will happen? what is the other point? is it this overlapping in anything else? okay? think of this is true in okay? this is true in R3 also okay? if you have two planes that have a line in common they cannot have anything else in common because they are overlapping in that line they cannot overlap any models otherwise they will be on top of each other okay? think of R3 if you have two planes that are intersecting in a line okay? they cannot intersect anywhere else they will intersect anywhere else then the two planes okay? the same proof will be used here and show that these will form orthogonal part of each side the one-flat okay? that's fine reasonably clear okay? so so you take so these form an orthogonal part let's set up okay? on the one-flat is it okay? right? hope it's clear it's not okay too fast but that's all the two-flats that contain this one-flat there are seven of them so I can find seven that are orthogonal on the one-flat okay? so that's the first step so now what do I need? so that's the first step in my method logic decoder okay? now I can do two-step in the next step what do you do? you just repeat this process for several one-flats that pass through the origin okay? so this is one-flat whereas the same of them we just take any seven okay? no point in going beyond seven if you want you can go beyond seven also if you want to correct more than where a correcting capability you can go beyond seven so you take all the flats that approve origin for each of those flats you repeat the same exercise okay? so you take any other one-flat so that's the first step the first step will contain one-flat how many such one-flats do you need? you take seven such one-flats okay? repeat for seven such seven or more I'll say seven or more such one-flats okay? one-flats through the origin right? so all of these are one-flats to the origin what does that mean? these one-flats to the origin can overlap only with zero they cannot overlap anything else so all these one-flats to the origin form an orthogonal set for E0 for the origin itself okay? then in the second step we use both sets to be quite easy okay? okay? in the first step we have found the the the subset corresponding to the one-flat we have found okay? now we repeat that for every one-flat that you have say you have seven of them they all of them are passing through the origin remember that okay? all of them are passing through the origin so these one-flats now form an orthogonal no I can't say a parity check but they form an orthogonal subset for E0 why can't I say it's a parity check one-flats are not an RM2 come up they're not regular parity checks and one-step checks out okay? now in the second step use one-flats to find but recently clear okay? so this is for the origin what I did for the origin I can do for any other point okay? it doesn't matter origin is just some one point you just translate it you can just keep repeating that's the nice thing about geometry the origin is not successful simply translate repeat so we'll do the suitable adjustments in your translation okay? so maybe we'll do this explicitly for the RM1 come up so with one guy maybe you can just repeat the same process okay? so I don't have to do it for every single point I have to do it maybe for the origin and then do it for one other point then do cyclic shifts after that in a suitable notation okay? so maybe we'll do this explicitly with one guy maybe you'll like that a little bit okay? so let me just check this guy okay? so this thing is not going away okay? so this was our so let's take the one flat zero zero zero zero zero zero zero one okay? like I said there are seven two flats that contain this one flat okay? so this so what are the two flats? so I'm going to write the equations directly okay? I'm looking for E0 plus E1 so I'll go with B in the parity set what else will that be? so it's just a question of taking one note so suppose I take zero zero one zero what will be the fourth point? zero zero one one so one parity check is E3 equals zero then I have E0 plus E1 what else can I take? I cannot take zero zero one one again because it's already there so maybe I'll take zero one zero zero so what will I get? zero one zero one what else? zero plus E1 plus I cannot take E1 again I'll take E6 E6 plus E7 equals zero and then not zero I'm sorry what am I saying? sorry so we will all do the syndromes right? S1 S2 S3 so on okay so how do I find this S1?  Okay, so you repeat for E1 plus EI, E1 equals to E3 plus E2, let's say 7, just 7, you can do more if you like, but 7 is enough, okay, once you start at 7 then what do you do in the second step? Okay, in the second step I have E0 plus E1, E0 plus E2, so until, E0 plus E7, just simply take majority again, break, there will be no tie as such, majority will be, okay, so all these are caps, okay, so that's how you do the decode, okay, so in general if I go to RM as r comma m, okay, the general logic will follow as such, okay, so if you go to, so this is here, right, in this example, so if you have any code, any ridmila code reasonable size, you should be able to repeat the same process, start with a suitable flat and then you keep on doing one more, one more, one more. So in general what will this picture look like? If you want to decode RM of r comma m, the dual is what? Dual is RM of m minus r minus 1 comma m, right, that's the dual, okay, so what you do is, in the first step to start with an r flat, okay, you can find all the r plus 1 flats that pass through this r flat, that number will be exactly 2 power m minus r minus 1, so the same logic, it's not very hard to do that computation, okay, you will see the number of r plus 1 flats that pass through a fixed r flat or the container fixed r flat will be 2 power m minus r minus 1 and they will all be orthogonal on the particular r flat, okay, so after the first step you find r flats, so how many such r flats do you do?  You take some other specific r minus 1 flat which is contained in this r flat, okay, and you pick several r flats which are all contained in the r minus 1 flat, okay, so after the second step you will have found the r minus 1 flat, now you will repeat the second step for several r minus 2 flats inside the container in the r minus 1 flat, so on and so on and so forth till you come back to the final step where you have many and then you have the points, okay, so that's the step, when in every step you will have at least 2 power m minus r minus 1, the limiting side will be the last, first one, okay, you will have 2 power m minus r minus 1 checks which are after the next, you can do the computation, okay, so that is the majority logic decoding for an arbitrary read mode of code, and there are improvements to those, there are alkanes of refinements, usually people don't decode the read mode of code directly, they always puncture it and decode the punctured version, why is that nice? Because that's the clip, once you find something for the first bit, the same thing shifted around will work for all the bits, and as many steps as you take, and at least that will work, the same thing works, okay, so that is the nice thing to do, okay, so we will stop here for today, hopefully this point was reasonably across.


Book 3 of the Nicomachean Ethics. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Jeffrey Edwards. The Nicomachean Ethics by Aristotle. Translated by Thomas Taylor. Book 3. Chapter 1. Since, therefore, virtue is conversant with passions and actions, and praise and blame accompany things of a voluntary nature, but pardon and sometimes pity, such as our involuntary, it is perhaps necessary that those who speculate concerning virtue should define the voluntary and involuntary. This will also be useful to legislators in conferring rewards and inflicting punishments. Those actions appear to be involuntary, which are done by force, or through ignorance. And the violent is that the principle of which is external, being of such a nature, that it contributes nothing to the advantage of him who acts, or of him who suffers, as if, for instance, the wind, or men who are the masters of someone, should bring him to a certain place. With respect to such things, however, those are performed through the fear of greater evils, or for the sake of something beautiful in conduct, as if a tyrant who is the Lord of parents and children should command one to do a certain base action, and on these conditions, that if he did it, his parents and children should be saved, but if he did not, they should die. With respect to such things as these, it is dubious whether they are voluntary or involuntary. Something of the like kind also happens in losses at sea. When, in a tempest, the goods of the ship are thrown overboard. For simply considered, no one throws them into the sea willingly, but everyone who is in due with intellect does so for his own safety, and that of the rest of the crew. Such like actions, therefore, are mixed, but they are more similar to voluntary actions, for they are then eligible when they are performed. But the end of the action is according to opportunity. A thing, therefore, must be said to be done voluntarily or involuntarily, then when it is done. But he threw his goods into the sea voluntarily, for the principle of moving the organic parts in such like actions is in the man himself, but those things of which the principle is in himself, he has the power to perform or not. Such things, therefore, are voluntary. Simply considered, however, they are perhaps involuntary, for no one would choose any one of these on its own account. But, in such like actions, men are sometimes praised when they endure something disgraceful or painful for the sake of great and beautiful circumstances, and if they do otherwise, they are blamed. For, to endure the most disgraceful things with a view to nothing beautiful or moderate is the part of a bad man. To other things, however, no praise is given, but pardon is granted to them when a man does what he ought not to do in consequence of being compelled by such things as surpass human nature and which no one can endure. And perhaps there are some things which we ought never to do by any compulsion, but we ought to suffer the most dreadful evils and die rather than do them.  This, however, sometimes difficult to judge what is to be chosen in preference to something else and whether this is to be endured instead of that. And it is still more difficult to persevere in our decisions since, for the most part, things which are expected are attended with molestation and things which we are compelled to do are base. Hence, both praise and blame are given to those that act from compulsion and to those who do not. What kind of things, therefore, are to be called violent? Shall we say that they are then simply to be called so when the cause is in things external and the agent contributes nothing to the action? But things which are of themselves involuntary, indeed, but are now eligible and are eligible instead of certain other things and the principle of which is in the agent, these are of themselves indeed involuntary, now and instead of certain other things, are voluntary. They are, however, more similar to voluntary actions for actions are conversant with particulars and these are voluntarily performed. It is not, however, easy to show what things are to be chosen in preference to others for there are many differences in particulars. But, if it should be said that things delectable and beautiful are violent for they compel us to act, being external, if this should be said all things will thus be violent for all men do everything for the sake of these and those, indeed, who act from violence and unwillingly act painfully but those who are influenced by the delectable act with pleasure. It is, therefore, ridiculous for a man to accuse external things and not himself when he is easily captivated by things of this kind and to consider himself as the cause of beautiful actions but delectable things as the causes of his base actions. Hence, the violent appears to be that the principle of which is external and to which the thing compelled contributes nothing. Everything, however, which is done from ignorance is not voluntary but that is involuntary which is attended with pain and repentance for he who does anything from ignorance and is not at all indignant with the action is not indeed performant willingly because he acts from ignorance nor yet again does he performant unwillingly in consequence of feeling no pain from the action of those, therefore, who act from ignorance he who repents of what he has done appears to have acted unwillingly but he who does not repent since he is a different character from the other may be said to have acted not willingly for since he is a different character it is better that he should have a proper name to act from ignorance likewise appears to be a different thing from acting ignorantly for he who is intoxicated or enraged does not appear to act from ignorance but from someone of the above mentioned circumstances yet not knowingly but ignorantly every depraved man, therefore, is ignorant what ought to be done and from what actions he should abstain and from error of this kind may become unjust and, in short, bad but an action ought to be called involuntary not if he who does it is ignorant of what is advantageous for ignorance in the deliberate choice of a thing is not the cause of involuntary conduct but of depravity nor is the ignorance of universal the cause of it for men are blamed on this account but it arises from the ignorance of particulars in which and about which every action is conversant for, in these, there is pity and pardon since he who is ignorant of any one of these acts involuntarily perhaps, however, it will not be amiss to define what and how many the particular circumstances are which render an action involuntary they are, therefore, the circumstance of the principal agent the circumstance of the instrumental agent the circumstance of the end and the circumstances of the action itself no one, therefore, will be ignorant of all these unless he is insane but it is evident that neither will he who acts be ignorant of them for how can he be ignorant of himself? a man, however, may be ignorant of what he does as is the case with those who say that they have spoken unawares or that they did not know what they said was arcane as escalus with respect to the mysteries or as when someone throws a catapulta not knowing what he throws a person also may fancy, like Merope that a son is an enemy and that a spear which has a sharp point is blunt, like a ball or that a stone is a pumice a man, likewise, striking another with a view to his safety may kill him and, wishing to show the mode of exercise and wrestling he may strike him whom he wished to instruct as there is ignorance, therefore, in all these particulars in which there is action he who is ignorant of some one of them appears to have acted involuntarily and especially in those things which are of principal importance but those appear to be of principal importance in which there is action and that for the sake of which action is undertaken since the involuntary, therefore, is denominated from an ignorance of this kind it is besides this necessary that the action should be painful and attended with repentance but, as the involuntary is that which is done from violence and through ignorance the voluntary will appear to be that of which the principal is in the agent who knows the particulars in which the action consists for, perhaps it is not well said that actions which are produced through anger or desire are involuntary for, in the first place, indeed if this were admitted no other animal would act voluntarily nor would children and, in the next place whether are any of the actions which we perform through the influence of desire or anger done by us voluntarily or shall we say that worthy actions are performed by us voluntarily but base actions involuntarily or would not this be ridiculous since there is one cause of both of these perhaps, too, it is absurd to call those things involuntary after which it is requisite to aspire but it is necessary to be angry with certain things and to desire others such as health and discipline it appears, however, that things involuntary are painful but that those which are the objects of desire are delectable again, what difference is there between the errors which are caused by reason or by anger with respect to their being involuntary for both are to be avoided the irrational passions also do not appear to be less human but the actions of man proceed both from anger and desire it would be absurd, therefore to consider these as involuntary having, therefore, defined the voluntary and involuntary it follows that we should discuss pre-election or deliberate choice for deliberate choice appears to be most aligned to virtue and by this, as a rule a judgment may be formed of manners more than by actions deliberate choice, therefore appears indeed to be a voluntary thing yet it is not the same with what is voluntary but the voluntary is more extended for, of the voluntary children and other animals partake but they do not partake of deliberate choice and we say, indeed the things which we do suddenly are done voluntarily but not according to deliberate choice but those who call it desire or anger or will or a certain opinion do not appear to speak rightly for, deliberate choice is not common to us irrational animals but desire and anger are and the incontinent man, indeed acts from the influence of desire but not from deliberate choice on the contrary the continent man acts from deliberate choice and not from the impulse of desire and desire, indeed is contrary to deliberate choice but desire is not contrary to desire desire likewise is conversant both with that which is delectable and that which is painful but deliberate choice is neither conversant with the painful nor the delectable much less is deliberate choice anger for, in the smallest degree do things which are affected through anger appear to be affected by deliberate choice nor yet is it will the will appears to be near to it for, deliberate choice, indeed is not among the number of things impossible and, if anyone should say that he deliberately chooses impossibilities he would appear to be stupid the will, however is directed to things which are impossible as, for instance, to immortality and the will, indeed is also conversant with things which can by no means be accomplished by him who wills as that a certain player or person engaged in athletic contests may be victorious no one, however, deliberately chooses things of this kind but only such as he thinks can be affected through himself farther still the will, indeed is more directed to the end but deliberate choice to things pertaining to the end thus we wish to be well but we deliberately choose those things through which we become well and we wish, indeed, to be happy and we say that this is our wish but it is not fit to say that we deliberately choose to be happy for, in short deliberate choice appears to be conversant with the things that are in our power neither, therefore, will deliberate choice be opinion for opinion, indeed, appears to be conversant with all things and no less with things eternal and impossible than with things in our power opinion, likewise, is divided into the false and the true and not into good and evil but deliberate choice is rather divided into the latter than into the former in short, therefore, perhaps no one will say that deliberate choice is either the same with opinion in general or with some particular opinion for, by deliberately choosing good or evil, we become affected with a certain quality but this does not happen to us through forming an opinion and we deliberately choose, indeed, to obtain or avoid or to do something of the like kind but we form an opinion of what it is or to what it is advantageous or in what manner we do not very much opine to obtain or avoid it and deliberate choice, indeed, is praised because it pertains to that of which it is necessary to partake more abundantly or with rectitude but opinion is praised for its truth we likewise deliberately choose those things which we especially know to be good but we form an opinion of things which are not very much known to us and the same persons do not appear to deliberately choose and opine the most excellent things but some indeed opine that which is better but from vice choose those things which ought not to be the objects of choice it is, however, of no consequence whether opinion precedes or follows deliberate choice for our attention is not directed to this but to the consideration whether deliberate choice is the same with a certain opinion what then or what kind of a thing is deliberate choice it is no one of the above mentioned particulars it appears therefore to be a voluntary thing not everything, however which is voluntary is the object of deliberate choice but that which has been the subject of previous deliberation for deliberate choice is accompanied with reason and the discursive energy of reason and this the name appears to signify the object of deliberate choice being that which is eligible in preference to other things Chapter 3 but whether do men consult about all things and is everything a subject of consultation or about certain things is there no consultation perhaps, however that must be called a subject of consultation not about which some stupid or insane person consults but which is an object of consultation to a man endued with intellect concerning eternal things, however no one consults such as concerning the world or the diagonal inside of a square because they are incommensurable nor does anyone consult about things which are in motion but which are always passing into existence after the same manner whether from necessity or naturally or from some other cause such as conversions and risings nor does anyone consult about things which subsist differently at different times such as about drought and rain nor about fortuitous events such as the discovery of a treasure nor yet about all human concerns for no alas a demonian consults how the polity of the Scythians may be governed in the best manner since none of these things can be affected by us but we consult about things which can be performed by us and these are the rest of things which we have not mentioned for nature, necessity and fortune appear to be causes and besides these intellect and everything which energizes through man the individuals however of the human species consult about things which may be performed by them and indeed in those sciences which are accurate and sufficient to themselves there is no consultation as for instance there is no consultation about letters for there is no contention how we should write things as are affected by us yet not always after the same manner about these we consult as about things pertaining to medicine and the art of procuring money and about the art of the pilot more than about the gymnastic art because the former is much less accurate than the latter in a similar manner also we consult about the rest but we consult more in the arts than in the sciences for we dissent more about them consultation however takes place in things which have a frequency of subsistence but of which the event is immanifest and in things in which there is the indefinite in things also which are of great importance we employ counselors distressing our own judgment as not sufficient we consult however not about ends but about things pertaining to ends for neither does a physician consult whether he shall heal the sick nor a rhetorician or whether he shall persuade nor the politician whether he shall establish equitable legislation nor does anyone of the remaining characters consult about the end but proposing a certain end they consider how and by what means it may be obtained if also it appears that this end is to be obtained through many media they consider through which of them it may be obtained in the easiest and best manner but if through one medium through this and through what likewise this may be obtained until they arrive at the first cause which is discovered in the last place for he who consults appears to investigate and analyze in the above mentioned manner as if he were investigating and analyzing a diagram it appears however that not every investigation is a consultation for mathematical inquiries are not consultations but every consultation is an investigation and that which is last in analysis is first in generation and if indeed in consulting we meet with an impossibility we desist from consultation as if there should be occasion for money and this cannot be procured but if that about which we consult appears to be possible then we endeavor to obtain it those things however are possible which may be accomplished through ourselves which are accomplished through our friends are in a certain respect affected through ourselves since the principle is in us but at one time instruments are explored and at another time the use of them and in a similar manner in other things at one time indeed that being investigated through which the end may be obtained and at another time the manner man therefore as we have said appears to be the principle of actions that consultation is about things which may be performed by man and actions are for the sake of other things hence the end will not be the object of consultation but things which pertain to ends neither therefore will particulars be the objects of consultation as whether this thing is bred or as well baked or as made as it ought to be for these things pertain to sense but if a man always consults there will be a procession to infinity the object of consultation however and the pre-eligible or object of deliberate choice are the same except that the object of pre-election or deliberate choice is something which is now definite for the pre-eligible is that which is preferred from consultation for everyone ceases to investigate how he shall act when he has reduced the principle to himself and to that part of himself which ranks as the leader this part is that which he deliberately chooses but this also is evident from the ancient polities which Homer has imitated for the kings of these polities announced to the people what they had deliberately chosen to do since however that which is pre-eligible is an object of consultation appetable of things which are in our power pre-election also or deliberate choice will be an appetite of or tendency to things accompanied with consultation for forming a judgment in consequence of having consulted we desire conformably to consultation we have therefore adumbrated what pre-election is and what the things are with which it is conversant and have shown that it belongs to things which have reference to ends chapter 4 that will however pertains to the end we have shown this end to some persons appears to be the good and to others apparent good but it happens to those who say that the object of the will is the good that what he wills who does not choose rightly is not an object of will for if it were an object of will it would also be good it may however happen to be bad and it happens to those who say that the object of the will is apparent good that the object of the will is not a natural subsistence but is what appears to anyone to be eligible a different thing however appears to be eligible to a different person and if it should so happen contraries appear to be eligible if therefore these things are not approved we must say that simply and in reality the good is indeed the object of the will but that apparent good is the object of the will to everyone to the worthy man therefore real good is the object of the will but to the bad man casual good just as in bodies to such as are well disposed those things are salubrious which are in reality so but other things to such as are diseased and the light takes place in things that are bitter sweet, hot, heavy and each of the rest for the worthy man judges of everything rightly and in everything the truth presents itself to his view for according to every habit there are things beautiful and delectable which are peculiar to that habit and perhaps the worthy man very much excels others in this that he sees the truth in everything being as it were the rule and measure of things but with the multitude deception is present on account of pleasure for pleasure though not good appears to be so good therefore choose the delectable as good but fly from pain as an evil chapter 5 since the end therefore is the object of the will but things pertaining to the end are the objects of consultation and deliberate choice the actions which are conversant with these will be actions of deliberate choice and voluntary but with these the energies of the virtues are conversant virtue therefore also is in our power and in a similar manner vice for in those things in which to act is in our power not to act is also in our power and in those things in which we have the power not to act we have likewise the power to act hence if to act worthily is in our power not to act basically will likewise be in our power and if we have the power of not acting worthily, we have also the power of acting basely. But if to act, and in a similar manner, not to act worthily and basely, are in our power, and this is to be good or bad, it will be in our power to be worthy or depraved characters. And to say, with a certain tragic poet that, quote, no one is willingly depraved, nor unwillingly blessed, close quote, seems to be partly false and partly true, for no one is unwillingly blessed, but depravity is voluntary. Or, unless this is admitted, what we have just now asserted must be controverted, and it must not be said that man is the principal and generator of actions in the same manner as he is of children. But, if these things are admitted, and we cannot refer them to any other principles than those which are in our power, it follows that those things are in our power, and are voluntary, of which also the principles are in our power. The truth of this appears to be tested both privately by individuals and publicly by legislators themselves, for they castigate and punish those who act depravly, if they do not act from compulsion or from ignorance of which they were not the causes. But, they honour those who act well in order that they may excite these and impede those. No one, however, exhorts others to the performance of such actions as are neither in our power nor are voluntary, because no advantage can be derived from persuading us not to be hot, or be in pain, or be hungry, or anything else of the like kind. For, notwithstanding the persuasion, we shall no less suffer these things. For, legislators also punish a man for his ignorance, if he appears to be the cause of his ignorance. Thus, double punishments are ordained for those that are intoxicated. For, the principal is in themselves, because they have the power of not becoming intoxicated, and this, i.e. intoxication, is the cause of their ignorance.  For, it is in our power to pay attention to what ought to be known. Perhaps, however, it may be said that a man is a person of such a character that he cannot pay attention, but such persons are themselves the causes of their characteristic qualities, in consequence of living negligently. The causes likewise of men being unjust or intemperate are in themselves, in consequence of the former acting wickedly, and of the latter spending their time in drinking, and things of the like kind. For, energies in everything render those who employ them similar to such energies. This, however, is evident from those who exercise themselves in any contest or action, for they persevere in energizing. To be ignorant, therefore, that in everything from energizing about the thing, habits are produced, is the province of a very insensate man. Again, it is absurd to suppose that he who acts unjustly is unwilling to be unjust, or that he who acts intemperately is unwilling to be intemperate. But, if anyone does those things from which he will be unjust, not ignorantly, he will be unjust willingly. Nevertheless, though he should wish, he will not cease to be unjust, and become just, for neither does he who is diseased become well by wishing to be so, even though it should happen that he is voluntarily diseased by living intemperately and disobeying his physicians. Prior, therefore, to his living intemperately, it was in his power not to be diseased, but after having abandoned himself to intemperance, it was no longer possible, as neither is it possible for him who has thrown a stone to resume it. At the same time, it was in his power to emit from his hand and hurl the stone, for he contained the principle of action in himself. Thus also, to the unjust and intemperate man, it was possible from the beginning not to be unjust and intemperate, on which account they are voluntarily so, but when they are become such characters, it is no longer possible for them not to be so. Not only, however, the vices of the soul are voluntary, but in some persons also the vices of the body, which likewise we reprehended. For no one reprehends those who are naturally deformed, but we blame those who are so through the want of exercise, and from negligence. The like also takes place in imbecility and mutilation, for no one would reproach a man who is blind from nature, or a disease, or a blow, but would rather pity him, but everyone would reprove him who is blind from drinking wine to excess, or from any other species of intemperance. Of the vices, therefore, pertaining to the body, those indeed that are in our power are blamed, but those which are not are not reprehended. But, if this be the case in other things also, the vices which are reprehended will be in our power. If, however, someone should say that all men aspire after a parent could, but that we have no authority over the fantasy, and that such as everyone is, such also does the end appear to him to be. If, indeed, everyone is to himself in a certain respect the cause of habit, he will also be, in a certain respect, the cause to himself of the fantasy, i.e., of the conception which he forms of a thing in his imagination. But, if no one is the cause to himself of bad conduct, but he acts evilly from an ignorance of the end, fancying that by so acting he shall obtain the greatest good. And, if the desire of the end is not spontaneous, but it is requisite that everyone should be born endued as it were with sight, by which he may judge rightly, and may choose real good. And if, also, he is naturally of a good disposition, in whom this is well implanted by nature, for that which is greatest and most beautiful, and which can neither be obtained nor learned from another person, but which such as a man is naturally, such he possesses, and to be naturally inclined to this well and beautifully will be a perfect and true natural goodness of disposition. If these things are true, why will virtue more than vice be voluntary? For the end appears, and is similarly posited both to the good and the bad man, either by nature or in some other way. But, referring other things to this, they act in any man or whatever. Whether, therefore, the end, whatever it may be, is not apparent to everyone from nature, but there is also something with him who acts, or whether the end is natural. Yet, because a worthy man performs other things voluntarily, and therefore virtue is voluntary, vice also will be no less voluntary. For, in a bad, as well as in a good man, there is similarly a power of acting, from himself in what he does, though the intention of the end is not in our power. If, therefore, as we have said, the virtues are voluntary, for we ourselves, in a certain respect, are the concauses of habits, and in consequence of being disposed in a certain way, we propose to ourselves a certain end. If this be the case, the vices also will be voluntary, for a similar reason. We have, therefore, spoken in common concerning the virtues, have adumbrated the genus of them, and have shown that they are media and habits. We have likewise unfolded what the things are from which they are produced, and have shown that they are caused by energies, and are the principles of energies, similar to those by which they are generated, that they are likewise in our power, and are voluntary things, and this in such a way as right reason shall ordain. Actions, however, and habits, are not similarly voluntary, for of actions we are the lords from the beginning to the end, since we have a knowledge of particulars, but of habits we are only lords of the principle. The accession, however, of particulars is not known as it is in diseases, but because it is in our power thus to use, or not to use particulars, on this account our habits are voluntary. Resuming, therefore, the discussion of each of the virtues, let us show what they are, what the quality of the things is, with which they are conversant, and how they subsist, but at the same time it will be manifest how many there are, and in the first place let us consider fortitude. Chapter 6. That fortitude, therefore, is indeed a medium which is conversant with fear and audacity, has been already observed by us, but we evidently fear things of a terrible nature, and these are, in short, evils. Hence also fear is defined to be the expectation of evil. We fear, therefore, all things that are evil, such as infamy, poverty, disease, the want of friends, and death. The brave man, however, does not appear to be conversant with all evils, for it is necessary and beautiful to be afraid of some things, and not to be afraid of them is base, as, for instance, not to be afraid of infamy. For he who is afraid of this is a worthy and modest man, but he who is not afraid of it is impudent. He is, however, metaphorically called by some a brave man, for he has something similar to the brave man, since the brave man also is fearless. But perhaps it is not proper to fear poverty or disease, or, in short, such things as neither proceed from vice nor from ourselves, yet neither is he who is fearless with respect to these a brave man. We denominate him, whoever brave, from similitude, for some men who in the dangers of war are timid, are liberal, and possess a proper confidence in the loss of money. Neither, therefore, is he timid who dreads insolent conduct towards his children and wife, or envy, or anything of the like kind, nor is he a brave man, if he is confident, when he is about to be whipped. With what kind of dreadful things, therefore, is the brave man conversant? Shall we say with such as are the greatest? For no one endures dreadful things better, but death is the most dreadful of things, for it is the end of life, and nothing farther appears to remain for him who is dead, either good or bad. But neither does the brave man appear to be conversant with every kind of death, as, for instance, death in the sea, or from disease. With what kinds of death, therefore, is he conversant? Shall we not say with those that are most beautiful? But these are the deaths which happen in war, for such a death is attended with the greatest and most beautiful danger. And the truth of this is confirmed by the honours, which cities and monarchs confer on those who conduct themselves bravely in war. He, therefore, may properly be called a brave man who is intrepid with respect to a beautiful death, and such things as are the causes of death when they are near. But things of this kind are especially such as happen in war. Nevertheless, in the sea and in diseases, the brave man is intrepid, yet not in the same manner as sailors are. For brave men, when they despair of their safety, indignantly bear a death of this kind, but sailors have good hope of escaping from their experience. At the same time, brave men act with fortitude in those things in which strength of mind is requisite. Or it is beautiful to die, but neither of these exists in such like destructions as we have mentioned. The same thing, however, is not terrible to all men. But we say that there is also something which is above man. This, therefore, is indeed terrible to everyone endued with intellect. But the terrible things which do not exceed the endurance of human nature differ in magnitude and in the more and the less, and the like takes place in things pertaining to confidence. The brave man, however, is unterrified as a man. He will therefore indeed dread things of this kind, yet in such a manner as is proper and as reason prescribes for the sake of the beautiful in conduct, for this is the end of virtue. But it is possible to be terrified at these in a greater and less degree. And it is also possible to dread things which are not dreadful, as if they were so. Of the errors, however, in the endurance of things terrible, one consists in dreading what it is not proper to dread. Another in dreading not as is proper, but another in not dreading when it is proper, or something of this kind. And in a similar manner in what pertains to confidence. He therefore who endures and fears things which it is requisite to endure and fear, and for the sake of that for which it is requisite, and in such a way as and when it is requisite. And in a similar manner, he who thus confides is a brave man, for the brave man suffers and acts according to the importance of the thing, and conformably to reason. But the end of every energy is the end according to habit, i.e. the beautiful in conduct. And the brave man fortitude is beautiful. The end also is a thing of this kind, for everything is defined by the end. For the sake of the beautiful in conduct, therefore, the brave man endures and performs all that pertains to fortitude. Of the characters, however, which exceed, he indeed who exceeds in fearlessness is anonymous. But it has been before observed by us that many things are anonymous. He, however, who fears nothing, neither earthquakes nor inundations, as it is said of the celte, will be an insane person, or one who has no sense of pain. But he who exceeds in confidence, respecting things of a terrible nature, will be audacious. The audacious man also appears to be arrogant, and a pretender to fortitude. Such therefore, as the brave man is with respect to things of a terrible nature, such does the audacious man wish to appear. And hence, in those things in which he is able, he imitates him. On this account also, many audacious persons have timidity, united with audacity. For in consequence of their audacity, when danger is not imminent, they do not endure things of a dreadful nature when they occur. But he who exceeds in fearing is timid, for he fears what he ought not, and in such a manner as he ought not to fear. And all such things are consequent to him. But he is deficient in confiding. As he exceeds, however in pains he is more apparent. The timid man, therefore, is hopeless, for he fears all things. But the brave man is the contrary. For confidence is the province of the man who hopes for the best. The timid, the audacious, and the brave man, therefore, are conversant with the same things. But they are differently affected towards them. For the timid and the audacious man exceed and are deficient. But the brave man is disposed towards things dreadful in the middle way. And in such a manner as is proper. And audacious men, indeed, are precipitate, and wish to encounter dangers before they arrive. But when they arrive, they are deficient in fortitude. Brave men, however, are ardent in encountering danger. And before it arrives, they are quiet. As we have said, therefore, fortitude is a medium conversant with those things of a dreadful nature. And such as pertain to confidence, which we have mentioned. And it chooses and endures them, because it is beautiful to do so. Or not to do so is base. But to die in order to avoid poverty, or on account of love, or something painful is not the province of a brave, but rather of a timid man. For it is effeminate to fly from things laborious. And such do not endure death because it is beautiful to endure it. But in order to fly from evil, fortitude, therefore, is a certain thing of this kind. Chapter eight. Other kinds of fortitude also are denominated according to four modes. And in the first place indeed, political fortitude, since this most resembles fortitude truly so called. For citizens appear to endure dangers on account of the punishments, and disgrace inflicted by the laws, and also on account of the honors they confer. Hence, the most brave men appear to be found, among those with whom the timid are disgraced. And the brave are honored. Homer likewise introduces such persons as, for instance, Diomed and Hector. Quote, shall proud polydemus before the gate proclaim, his councils are obeyed too late, which timely followed but the former night, what numbers had been saved by Hector's flight. Close quote. And Diomed, quote, but ah, what grief should haughty Hector boast, I fled inglorious to the guarded coast. Close quote. This species of fortitude, however, is especially similar to the before mentioned, because it is produced from virtue, for it is generated through shame, and the desire of the beautiful in conduct, for it is through a desire of honor, and a flight from disgrace, which is dishonorable. Those also may be ranked among brave men, who are compelled to be brave by their rulers, but they are inferior to the former, i.e., the politically brave, because their conduct is not produced through shame, but through fear, and is not the consequence of flying from what is base, but from what is painful, for they are compelled by their masters. Thus Hector, quote, on rushed bold Hector, gloomy as the night, forbids to plunder, animates the fight, points to the fleet, for by the gods who flies, who dares but linger, by this hand he dies, no weeping sister, his cold eye shall close, no friendly hand, his funeral pile compose, who stops to plunder at this signal hour, the birds shall tear him, and the dogs devour. Close quote. And the generals scorch the soldiers if they desert their ranks. The same thing also is done by those who dispose their troops before faxes, and adopt other methods of the like kind for all these employ force. It is necessary, however, not to be brave from necessity, but because it is beautiful to be so. But experience about particulars appears to be a certain fortitude. Wentz also Socrates thought that fortitude was a science, and indeed there are other such persons in other things. But soldiers are such in warlike affairs, for it seems that there are many vain terrors in war, of which soldiers are especially aware. Soldiers therefore appear to be brave because other persons do not understand the nature of these alarms. In the next place they are especially able, from their experience, to attack their enemies without receiving any injury themselves. They also know how to guard against and strike their enemies in consequence of being able to use their arms, and having armor of such a kind as is most excellent for the purpose of attacking, without being injured by their adversaries. They fight therefore like armed with unarmed men, and like athletes with those that are unskilled in athletic exercises, for in such like contests, not the most brave are the most adapted to fight, but those who are most strong, and those whose bodies are in the most excellent condition. But soldiers become timid when the danger is excessive, and they are deficient in numbers and warlike apparatus.  But the soldiers, in this battle at Hermaeus, encountered the danger at first, as thinking themselves superior to their enemies, but when they saw the full extent of the danger, they fled, draining death more than disgrace. The brave man, however, is not a person of this description. Anger also is referred to fortitude, for men likewise appear to be brave on account of anger, just as wild beasts rush on those that moon them, because brave men also are irascible.  For everything of this kind appears to signify the energy and impulse of anger. Brave men, therefore, act on account of the beautiful in conduct, but anger cooperates with them, and savage animals act through the influence of pain, for they act because they are wounded or terrified, since if they are in a wood, or in a marsh, they do not attack anyone. Hence, those persons are not brave or impelled to danger by pain and anger, for seeing nothing that is dreadful, since thus asses also would be brave when they are hungry, for they cannot even by blows be driven from their pasture. Adulterers, likewise, perform many audacious deeds through their lustful desire. Those, therefore, are not brave or impelled to danger through pain or anger. The fortitude, however, appears to be most natural, which subsists on account of anger, and which assumes deliberate choice, and that for the sake of which a thing is done, or the final cause. Men also, when they are angry, are pained, but are delighted when they take vengeance on the authors of their anger. Those, however, who act under the influence of these causes are indeed pugnacious, but not brave. For they do not act with a view to the beautiful in conduct, nor from the dictates of reason, but from the influence of passion. But they possess something similar to fortitude. Nor yet are those who are full of good hope brave, for in consequence of having frequently conquered, and conquered many, they are confident in dangers. But they are similar to brave men, because both these characters are confident. Brave men, however, are indeed confident for the reasons we have already assigned, but these, because they are fancy, they are superior to others, and that they shall suffer no evil from their opponents.  It is, however, the promise of a brave man to endure things which are, and appear to be dreadful to men, because it is beautiful to do so, and base not to endure them. Hence also it appears to be the part of a more brave man to be fearless and without perturbation in sudden tears, rather than in such as we're foreseen. For this rather proceeds from habit, and in a less degree from preparation, for things indeed which we're foreseen may be chosen from deliberation and reason. But in things which suddenly happen, a man can only conduct himself fearlessly, from the habit of fortitude.  They are, however, inferior to them, because they have no preconceived opinion of vanquishing the evil, but the former have. Hence the fortitude of those who are full of good hope continues for a certain time, but the fortitude of those who are ignorant of danger ceases as soon as the deception is apparent, as was the case with the archives, when they met with the Lacedaemonians, and thought them to be Sisyonians, and thus we have shown what kind of men the brave are, and those who appear to be brave. Since, however, fortitude is conversant with confidence and fear, yet it is not similarly conversant with both, but in a greater degree with things of a terrible nature, for he who is without perturbation in these, and who conducts himself in them as he ought, is more brave than he who does so in things pertaining to confidence. Brave men, therefore, as we have before observed, are called brave, from enduring things of a painful nature. Hence also fortitude is unaccompanied with pain, and is justly praised, for it is more difficult to endure pain than to abstain from pleasure. Nevertheless, the end according to fortitude may appear to be pleasant, but to be obscured and obliterated by surrounding circumstances, just as it happens in gymnastic contests, for to pugilists, indeed, the end for the sake of which they contend is pleasing, viz, a crown, and honors, but to be beat, since this pertains to the flesh, is painful, as is likewise every labor, because, however the circumstances which produce pain are many, and that for the sake of which they contend is small, it appears to possess nothing delectable. If, therefore, a thing of this kind also pertains to fortitude, death, indeed, and wounds will be painful to a brave man, and to one who is unwilling to endure them. The brave man, however, endures them because it is beautiful so to do, or because it is base not to endure them. And, by how much, the more he possesses every virtue, and is more happy, by so much the more will he be pained by death, where such a man most eminently deserves to live, and he is knowingly deprived by death of the greatest goods, but this is painful. He is, however, no less brave, and perhaps he is more brave because he chooses that conduct in battle which is beautiful, in preference to these goods. To energize, therefore, delectably, does not pertain to all the virtues, except so far as they come into contact with the end, but perhaps nothing prevents not only those from being most excellent soldiers who are most brave, but also those who are less brave and possess no other good, for these are prepared for danger and to lose their life for a small gain, and thus much concerning fortitude, and it is not difficult from what has been said to adumbrate what it is. Chapter 10 In the next place, let us speak concerning temperance. For these, i.e., fortitude and temperance, appear to be the virtues of the irrational parts. The temperance, therefore, is a medium conversant with pleasures has been already observed by us, for it is conversant in a less degree, and not similarly with pains, but about pleasures and pains in temperance also is employed. What the pleasures, therefore, are, with which temperance is conversant, we must now explain. Let pleasures, however, be divided into those pertaining to the soul and those pertaining to the body. Thus, for instance, the pleasures pertaining to the soul are ambition and the love of learning, for each of these is delighted with that which is the object of its desire. The body, not being at all affected, but rather the rational part, and those who are conversant with such like pleasures are neither denominated temperate nor intemperate. Thus, too, with respect to such other pleasures as are not corporeal, for we call those who are lovers of fables and narrations and who consume the day in such casual circumstances as present themselves, triflers, but not intemperate. Nor do we call those intemperate who are pained by the loss of riches or friends. Temperance, however, will be conversant with corporeal pleasures, yet neither will it be conversant with all such pleasures. For those persons are not called either temperate or intemperate who are delighted with objects of sight, such as colors and figures and pictures, though it would seem that there is also a proper manner of being delighted with these, and that it is possible to be pleased with them according to excess and defect. Thus, too, in things pertaining to the hearing, for no one calls those intemperate who are excessively delighted with melodies or players, nor those temperate who are delighted with them in a proper manner. Nor are those denominated temperate or intemperate who are delighted with odors, except from accident. For we do not call those persons intemperate who are delighted with the smell of apples or roses or odouriferous fumigations. But we rather denominate those persons so nor do they do with the smell of ointments and food. For intemperate persons are pleased with these, because through these the recollection of the objects of their desires is produced. Others also may be seen who, when they are hungry, are delighted with the smell of food. But to be delighted with things of this kind is the promise of an intemperate man. For, to such a one, these things are objects of desire. Nor do other animals receive pleasure from these senses, except by accidents, for neither are dogs delighted with the smell, but with the eating of hairs. The smell producing the scents, i.e., causing them to perceive food present, nor is the lion delighted with the voice of the ox, but with eating him, but he perceives through the voice of the ox that he is near, and is seen to be delighted with this perception. In length manner, neither is the lion delighted with seeing or finding a stag, or a wild goat, but he is pleased on seeing that from which he shall obtain food. Temperance and intemperance, therefore, are conversant with pleasures of this kind, of which also irrational animals partake. Hence, these pleasures appear to be servile and savage, and they are the pleasures pertaining to the touch and the taste. Temperance and intemperance, however, appear to use the taste, but in a small degree, or not at all, for the judgment of sappers is the province of the taste, which those persons employ who make trial of wines and season food. The intemperate, however, are not very much delighted with these sappers, but with the enjoyment of the food, the whole of which is affected through the touch, in meats and drinks, and in what are called venereal concerns. Hence, a certain person named Philoxenus, the son of Ericx, who was most voracious in eating, wished that he had a neck longer than that of a crane, as being one who was delighted with the touch. The touch, therefore, with which intemperance is conversant, is the most common of all the senses, and will appear to be justly disgraceful, because it exists in us, not so far as we are men, but so far as we are animals. To be delighted, therefore, with, and especially enamored, of such pleasures is beastly, for the most liberal of the pleasures which are perceived through the touch are not to be numerated with these, such, for instance, as the pleasures in gymnastic exercises produce through friction and heat, since the touch of the intemperate man does not pertain to the whole body, but to certain parts of it. Chapter 11 Of desires, however, so appear to be common, but others peculiar and advantageous. Thus, for instance, the desire of food is indeed natural, for everyone, when in want, desires either dry or moist nutriment, and sometimes both. And, as Homer says, both the young man, and he who is in the vigor of his age, desire the joys of love, but everyone does not desire this or that food, nor the same food. Hence, this desire appears to be properly ours, and it possesses also something natural, for different things are pleasing to different persons, and the same thing is more agreeable to some persons than to others. Few, therefore, err in natural desires, and they err in these in one way, viz in excess, for to eat or drink, what casually presents itself, till an excessive fullness is produced, is to surpass, in multitude, what is conformable to nature, since natural desire is the replenishing of indigence. Hence, such persons are called gluttons, as replenishing the indigence of nature beyond what is becoming, and those who are very servile become men of this description. But in those pleasures, which are peculiar or proper, many persons err, and in many ways, for they are denominated lovers of things of this kind, either from being delighted with things which are not proper, or being pleased with them more than is proper, as is the case with the multitude, or not in such a way as is proper, or not in that respect, in which it is proper. The intemperate, however, exceed in all things, for they are delighted with some things, with which it is not proper to be delighted, since they are odious, and if it is requisite to be delighted with some of such things, they are delighted with them more than is proper, and after the manner of the multitude. That excess, therefore, in pleasures is intemperate, and that it is blamable is evident. In pains, however, a man is not said to be temperate by enduring them, as in fortitude, nor intemperate by not enduring them, but he indeed is intemperate, who is pained more than is requisite, because he does not partake of pleasures, so that the pleasure gives him pain, in consequence of being desired by him above measure, and he is said to be a temperate man, who is not pained by the absence of pleasure, and by abstaining from it. The intemperate man, therefore, desires all pleasant things, or those which are most eminently pleasant, and is led by desire, so as to choose what is most pleasant in preference to other things. Hence also, he is pained, both when he is frustrated of pleasure, and when he desires it. For desire is accompanied with pain, though it seems to be absurd that a man should be pained on account of pleasure. Those, however, who are deficient in pleasures, and are delighted with them less than is proper, are not very frequent. For an insensibility of this kind is not human, since other animals also distinguish food, and are delighted with some kinds of it, and not with others. But he to whom nothing is delectable, and with whom, one thing does not differ from another, is very remote from human nature. Such a one also is without a name, because he does not very frequently exist. The temperate man, however, with respect to these things, subsists in a middle condition. For neither is he delighted with those things with which the intemperate man is especially delighted, but he is rather indignant with them. Nor, in short, does he rejoice in things in which he ought not. Nor is he very much delighted with anything of this kind. Nor is he pained if it is absent. Nor does he desire it, except moderately. Nor more than is proper. Nor at a time when he ought not. Nor, in short, anything of this kind. But such things as being delectable, contribute to health, or to a good habit of body, these he desires moderately, and in such a way as is proper. He also desires other delectable things, which are not an impediment to these, or which are not adverse to the beautiful inconduct, or above his income, for he who is thus affected loves such pleasures beyond their desert. The temperate man, however, is not a person of this description, but is one who acts conformably to right reason. Chapter 12 In temperance, however, appears to be more similar to the voluntary than to midity, for the former subsists on account of pleasure, but the latter on account of pain, of which the one indeed is eligible, but the other is to be avoided, and pain indeed astounds and disturbs the nature of its possessor, but pleasure produces nothing of this kind. It is therefore more voluntary, and on this account also it is more disgraceful, for it is more disgraceful, and it is more easy to be accustomed to these things, since there are many such in life, and the being accustomed to them is unattended with danger, but the contrary takes place in things of a dreadful nature. Timidity, likewise, may appear not to be similarly voluntary with particulars, for timidity indeed is without pain, but particulars so astound men through pain that they throw away their arms and act in other things in decorously, and on this account they appear to be violent. The contrary, however, takes place with the intemperate man, for particulars with him are voluntary, since he desires them, and his appetite is directed to them, but the whole of an intemperate life is less voluntary, for no one desires to be intemperate. We transfer also the name of intemperates to purile errors, for they possess a certain similitude, but which of these is denominated from the other is of no consequence to the present discussion. It is, however, evident that the latter is denominated from the former, nor does the transition appear to be badly made. For that which desires what is base is to be punished, and which has an abundant increase, but desires in a child are especially a thing of this kind, for children live according to desire, and in these the appetite of the delectable especially flourishes. If, therefore, this appetite is not obedient, and subject to the governor reason, it increases abundantly, for the appetite of the delectable is insatiable, and in the stupid man is every way diffused, and the energy of desire increases that which has lied to it, so that if the desires are great and vehement, they expel the reasoning power. Hence it is necessary that they should be moderate and few, and in no respect adverse to reason. But we call a thing of this kind obedient, and reformed by correction, for as it is necessary that a child should live conformably to the mandate of his preceptor. Thus also it is requisite that the part of the soul which energizes according to desire should live conformably to reason. Hence it is necessary that this part of the soul in the temperate man should accord with reason, for the end proposed by both, i.e. by a reason and desire in the temperate man, is the beautiful in conduct, and the temperate man desires those things which it is proper to desire, and as, and when it is proper, but reason likewise thus ordains, and thus much concerning temperance. End of book three recording in memory of Mitchell Edwards