Conversations

0:00 Thank you for being invited to Paris 7. Until when? Until October 15th, right? Yes, until October 10th. October 10th, so you can take advantage of it. And so, for a day around Galois. So, I don't know if you have all received the message on Teut, but the program has been modified. Because Katia Alcela can't come, to her great regret, and so we're going to go back to the lectures. This morning, we'll have a lecture by Massimo Gallutri and Caroline Mérat, and this afternoon, Sylvia, from Jean-Jacques Gignac, who is not yet here. And so we start with Massimo Gallutri, who will present to us a memory of Gallois on continuous fractions. Disconnected is a non-Hausdorff protector. When you think about it, the first memory of a galois on a continuous equation is contained by the comparable measurements of a galois on a geological equation. But it contains interesting details. In fact, it depends on the range of the range of the continuous equation.

2:30 But it is very easy to reverse independence and to demonstrate the dynamics of the range from how it is done, for example, in Harvard. And so these results are remarkable. It's just a fun to see what are the irrational and quadratic equations that have a development that is immediately recognized. But in fact, it's a good result. Depends on the direction of the wave, so in the first transparent, I give the list of one or the other, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, I give the list of the vaccines, the applications, The problem is to find a machine to give an approximation of this real machine, of this phenomenon. If we want to give a continuous interpretation of each of these racines, we can assume that there is only one racine between a given interval, which is between the first and the consecutive interval, this is a very strong hypothesis, but it can be obtained either from the theorem of Lagrange himself on the difference of groups in terms of...

5:00 Or by the rule of Descartes, if we want to put it in a more intelligent way. In any case, in the memory of the Grange, we only have the equations of T2 and T2. So, in the case of the equations of T2, it is very easy to have a separation of the two. So, if we take this hypothesis, the idea of ​​the Grange is to generalize the development in the continuous fashion of the real man. For the whole part, we make a variable change, which is to simply take the whole part, C0, and add a C0. This new polynomial has two roots that are larger than the roots, while the other roots are negative. And if we continue this way, we manage to give the root. So there is only one session to transform the language. With these, we continue to change and at the end we arrive and we continue to be given a set of terms. And Grange, who has already started his automotive career, by giving a complete characterization of the irrational parameters,

7:30 he has a continuous development that is immediately periodic. And, as I said, Ganois deduces from Grange's terrain, but we can easily reverse the table. I wrote here that it should be noted that the MNUMN and Renoir's predictions of a young Renoir have a considerable importance from a point of view of history because Renoir's ideas have had a great impact on the following narrative that historical research has favored the analysis of development, but we do not have detailed information on the hours, on the specific sources. We know that Renoir... But if we try to go beyond this general information to see what exactly he has learned, it is not easy to see them. And so I would like to say that this memory, in my opinion, is given by the fact that we can arrive with a certain precision, it is a hypothesis, but it seems to me reasonable, to see what he has learned from the arithmetic research of Gauss. Exactly the theory of von Friedrich and the theory of von Friedrich and the theory of von Friedrich. And another thing is the demonstration of Calois in his memoirs, which is very formal.

10:00 In fact, Calois has an idea of the art, which is, as he says, partly formal. And if you see, for example, the books of Hardy on continuous fractions, Hardy gives a large part of the theory of continuous fractions, of formal fractions. There is only a form that can be given as a basis. And we don't need it to have a bit of... It is enough to interpret the three parts as one, it is enough to interpret them as one. And that is exactly what Camus did. And there are calculus. But if you look here, there are p squared minus a p cubed. And the first thing that we have to observe is that, obviously, we can represent these units when they have a value of zero. And if the determinants are equal to A, then the same unit is equal to T. If we choose T equal to A, then it is easy to see that if we take this type of A, we can evaluate B.

12:30 We have a quadratic form that depends only on A. And if in this quadratic form we give the same value that we have chosen, then we obtain effectively a unit that is very easy to... Just a second. So this is a sheet of Galois grouilleau that has been preserved? Yes, that has been preserved and that has not been considered as important in the prediction of Bohr and Adler, because they say that these are the calculations of analytical geometry. These are the calculations that show the nature of the theoretical theory of quantum mathematics. The typical problem of the theoretical theory is that we have a formula that has x2 plus xy plus d. The number of squares is equal to the number of m in the sequence of hands, i.e. the value of the variable x in the square, such that the quadratic source gives the value of m. And you see, if there is a quadratic source in the sequence of hands, what are the possibilities of having it in the line? We will determine later what are the other possibilities, which are different. So, tomorrow we will have to think about it. So, if you look here, to get the results of this, we have to assume that the number is the same as it is, while in the theory of Gauss, as you know, the central term is always given as the double term. So we can imagine here that there is more to the book of the first chapter of the legend than to the book of Lagos. But in my opinion, for those who follow, there is a tunnel that has also read the book of the legend, and that is quite clear, the second memory of Galois. This will give a contribution that wants to identify and improve a result that is given in the legend during the circumnavigation of the Atiyah Witten.

15:00 So it is very clear that Kabbalah knew the legend very well. He cites it precisely. If there is a result in the legend, I want to improve it. It is an article that I wrote a few years ago on the archive. And so he knew the people very well. But I think he also knows the wisdom of Négoz. There are indications, in my opinion, in the demonstrative structure of Négoz, which shows that he considered, in my opinion, the practice of quadratic form in Négoz. So, yes? You have just shown that we can, according to the form you give, the poetic form, make hypotheses about the lecture, but don't the notations you use also allow us to make hypotheses about the authors who read, because often we take up a notation of these readings when we work with something? The only thing I found that shows a lecture on quantum mechanics theory is... Yes, but what I mean is that I don't know much about science, about Gauss, about the genre... In the article on quantum mechanics, I have a hypothesis on the lecture on quantum mechanics which is given by the fact that the demonstration of Gauss does not make much sense in this case. All of these are related to equations. And the key is a bit of science, if we think about it, depending on the theoretical theory of quantum mathematics. But that's all I have to say. I don't know how to convince you. I would say that the theoretical theory of quantum mathematics has been initiated by Lagrange. He had a lot of important computations by Lagrange and then by Sotosi. And God certainly knows these definitions. And God could come back to this text.

17:30 I think it seems, however, that the best organization is the systematic organization of the first law, after the best part of the first law. He knows the mathematics research very well, it's quite clear. So I think he has improved this text. I think the difficulty of recognizing the phantographic theory here, which is the contradiction of Bohr and Haas, is a difficulty that I accept because it is a solidity. After the work of Pipichlet, the theoretical theory of quantum mechanics was replaced by the theoretical theory of number of days, and then by the theoretical theory of ideals. So, as of today, there are not many mathematicians who understand the theoretical theory of quantum mechanics, and therefore it is a subject that we see in Guillaume as a farce. Thank you. In the 5th session, we have a graph that has this structure, and for which, as you can see, the central coefficient is the quantity, which must always be the denominator. It uses the ABC recommendation, and the determinant of the ABC equation is denoted by B. They give the definition of the reduced form, which will correspond, in my opinion, to the idea of reduced equations in the case of this problem. A reduced form is the definition of Gauss. We will obtain the reduced form, the reduced form, in which A, then positively, that is to say the absolute value of A, The ratio of D plus D is the ratio of D minus D, which is positive, and minus the ratio of D, and the determinant of D is positive, and non-squared. Gauss will eliminate the interesting case in which we can share the square root of a product of the form. And we can easily see that if we associate the form 1... The definition of gas consists in the demand that the equation 2, where we have to suppose that B is equal to 2 real roots and B is equal to 2, is such that the root modules are larger than the unit, while the module of the other is smaller than the unit.

20:00 And this symmetry, where there is a root larger than the unit and a root smaller than the unit, is negative. There is a minor that is smaller than minus one, or the other sign that is between zero and one. Natural asymmetry of mathematical material is not practical because if we change the signs to a variable, we have to represent the same number simply by changing the signs. But this asymmetry is not very natural in terms of anticipation. If we talk about anticipation, it means that we change the variable of the decomposed sound. In terms of documentation, this symmetry does not appear in the books, but also in the theory of Fonko and Raddick, this symmetry does not appear in the books. And where there is a demonstration of the law, who maintains this symmetry, there is a demonstration of which all the symmetries are the same. In fact, there is an article in which it is not counted. And who comes from this symmetry? Is it clear to you? Well, then, let's move on quickly to the topic of mathematics. Another important definition is that of contiguous forms. If the form ABC has the principle of a terminal number that has a certain condition on the paper, we say that it is contiguous. And when a designation of two axes is necessary, we say that the first is contiguous to the second, to the first part, and that the second is contiguous to the first, to the last part. The four contributions are equivalent, i.e. one passes in the form of the other with a modular matrix, and the matrix is determinable in time, and if the determinants are equal, one says that the form is equally balanced, and the other says that the form is equally balanced, and the other says that the form is equally balanced, and the other says that the form is equally balanced, and the other says that the form is equally balanced, and the other says that the form is equally balanced.

22:30 A reduced form, A, B, C. You can recall the reduced form that has this combination of signs. We can find a reduced environment that is not contributed by one or the other party, but we can find the contribution. That is to say, we have the necessity to find a reduced form that contributes to A. In a fortified form, the important point is here, you see, in a positive form, the ratio of the program of the equation, you see here the equation of the second aspect, if we take the ratio z, and if we take the entire part of this ratio, where we have to choose the same ratio of z, if we put z prime equal to 1 over d minus z, The second-order polynomials that integrate the radicals Z1 and Z2 correspond to the continuous form. So you see that we pass in a continuous form with the same method of lagging through which we pass a polynomial to the next polynomial, in search of the development of a continuous concentration. I will not go into the context of the presentation, it may be necessary. You can see the results here. Here is a good example. If you take a reduct form, for example this one, given by the number 4, 6, minus 1, to which you can associate an equation,

25:00 you can take the double of the second term, you can have an equation, and you can have another equation. Transforming the branch, if you think about it in terms of the patient and you transform the branch, you have the following problem. During the period of the form, that is, if we move on to the contributory form, we have, as you can see, we have a similarity that is remarkable. The third is exactly the same. There is still some difference in the second and the third, because you can see that there are two equal forms. And so, the calculation of the absolute value of the quantity of x gives 1 to 1 to the a, and we have exactly the same thing as we said about the patient, a form of a continuous patient, which is immediately continuous. So we have the idea that a square wave of the reduced core gives the patients There is a development, a continuous expression, which is immediately periodic, and we can see it by taking the period of the form of the quadratics. So, it seems obvious that if Bernoulli had read this text, and I have not read this text, and I have not said anything about it, he understood immediately that the period of the quadratic form was exactly the development of a continuous expression of the quadratic irrational associated with it. I don't want to go into too much detail here because the general idea is clear and perhaps I will give you the texts if I want to go into more detail. So, I come to the results of my law, for example, the demonstrative structure. And now, in particular, the demonstrative structure, in my opinion, is not correct. It reflects the idea that I had on the photographic theory, which was projected on the theory of the equation, but which makes no sense at all.

27:30 In the Law of Man, in an explicit way, this theorem says that if one of the roots of a particular equation is a continuously periodic equation, this equation will necessarily have another equally periodic equation that one will obtain by dividing the negative units by this same continuously periodic equation written in a diverse order. If there is a development of a quantum fraction given, for example, by A, B, C, D, which gives a ratio, the other ratio is given by minus 1 divided by B, C, B, D, B, A. A continuous fraction where the terms are written in a more complex way. Galois, on the other hand, considers a continuous fraction that is anterior to the four terms because he says that the uniform step is a curve, which would be the same if we put it in a bigger number. And it's quite obvious, in the end I gave a modern demonstration, the Galois' New Demonstration, apart from the Galois' New Demonstration. That is to say, it seems to me, if you want, it seems to me, if I can say the same thing for your argument, that this Galois' New Demonstration text is stronger, by the way, than the way you present it, because here, what he says... All of these are developments between two roots of an equation and a computer data and not, as you presented, the development of an analytic and a quadratic equation. So it seems to me that this context is closer to that of Lagrange than the one you gave. Well, the demonstration for me is simple and simple. The equation of the second step, which is the term of the second step, we have a continuous attraction which is a medium and a priori.

30:00 Then we have ABCD, we have to repeat ABCD, ABCD, ABCD, etc. Then we have the equation of the second step. This results in the difference between the first step and the second step, The idea is to bring the term to Atman and to reverse it. So you see immediately that this emotion is going to be written in the form of 1 divided by 1. And by proceeding in the same way, we arrive at minus x equals 1. Minus x divided by 1. It seems to reflect the evidence. It is not quite the same as the expression of Galois and Cerf. In the previous lecture, we assumed that the proposed root was greater than the unit, and we considered a root of a form smaller than the unit, and therefore the demonstration of the fact that this equation that has this root must also have a root of the form minus dCVA. Well, obviously, what is given in this form does not have a development, a continuous fraction, immediately perpendicular, because our first term is zero. So a term with a continuous fraction comes to zero. So, it doesn't make sense to say that there is a sign of the form and I will give all this. And what I think is that here there is this symmetry that comes from the theory of quadratic forms. In my theory of quadratic forms, I actually consider an equation that has two varsities that are smaller than the positive unity and a varsity that is smaller than the negative and larger than the positive unity. This part of the demonstration comes from the fact that he projected, in my opinion, in a considered way, that there is a root that is smaller than or equal to 0.1 and that there is another root that is smaller than or equal to 0.1.

32:30 So, we are talking about a development of continuous, immediately periodic pressure, or, if you like, to give a broader definition of a development of continuous, immediately periodic pressure in people who are smaller than or equal to 0.1. It should be a continuous patient, immediately periodic, otherwise one forgets the part in which it belongs. At the end of this demonstration, we see that either A is a continuous patient, immediately periodic, or B is a continuous patient, which one forgets by reversing the previous one. We see that if one of the roots of the patient is x, it will necessarily have another root x minus 1 over b. If there is a non-positive program of an idea, even if it is negative, as it is between 0 and 1, it is an inverse. An inverse normally makes no sense, it is a continuous equation. If there is a non-positive program of an idea, even if it is between 0 and 1, even if it is between 0 and 1, it is a non-positive program. So, you see here, it's a test, of course. An opening of the theory. There has certainly been an opening of the theory of the elephant, which is quite clear. But I also think that there has been the partiality. It is important to note that the period of the quadratic form is exactly equivalent to the immediate periodic development of a continuous fraction, but the period is a kind of simmetry and it also gave this simmetry in its demonstration. And Gallois also gives the result of this theorem, in terms of the quadratic form, which makes no sense. And this result, this circle, this result...

35:00 And in the form of a form of a form of a form of a form of a form of a form of a form of And we have a positive case that has the shape of a first step P, and then this case is given by P plus 1. And this is the first case of Lagrange. This case produces an equation that has the root A. And then in the third case, the other case is given by minus 1 plus the first step P, P is the first step A. So the equation at the beginning has the root that is P plus, and P even B is included between 0 and minus 1. We can repeat the reasoning if we have a period of two terms. Gallois does the same thing. He shows that after two passes, two transformations of the range,

37:30 we arrive at an immediate development. But if we have an immediate development of the other root, and this is only possible if we have a student, it is a real result. And, based on this example, in an even more explicit way, if you will, the method of the branch can also be transformed into an example of an equation of a degree of 2, of a degree of 2, which has a development which is immediately periodic and also symmetric. What happens if... So, if A is this one... Yes? Do you mean that the example is of this nature or is it you who puts A in the place of the root? No, it's exactly the same. I consider the case where there is a development between a continuous sample which is periodic and symmetric. And he says what is the general structure of the patients around them, etc. And then he gives a numerical example. It's me who does that. Okay, so these are the general considerations and the American examples. So, there is either A, a root, since A is a development, a continuous equation, which is a symmetric equation,

40:00 and the other equation is B, which is A, and the two roots are A and minus one is A, and the equation that has these roots is therefore A x squared minus A squared minus one is x minus A. These equations belong to the general theory in the case of Nix-Bedard, but in the case of Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Sepphorne, Also, the woman is an equation of the three. Who speaks to her? In this form, she is a living being in this form. The equation is a simple symbol of the world, a map, and therefore speaks to her almost the same way as the branch, that is, we have a pre-transformation of the branch. She says it explicitly, but also in the pre-transformation of the branch, in a very precise way, in the text of the branch. And then, the machine transforms the branch into a living being in this form. They all start with the transform equation and we see that it is used in the last equation, in the same coefficient of the equation given here. And then in the root of the equation 14, as you can see here, since we have transformed the formulas with 1, 2, 1,

42:30 and in the first equation we have done the root, and in the third equation we have done 1, 2, 1, And you see that in the text of Galois, we do not have the consideration of quadratic irrationalities, but we only have the consideration of polynomials. What he uses is to transform from the branch the entire part of the ratio, of the ratio greater than or equal, and With only these things, he manages to give the development of the continuous fraction. So he does not consider the quadratic irrationals, nor the developments of quadratic irrationals, but only the polynomials, the polynomials of the whole part. The idea is that there is a certain structure, and with a certain structure of racines, by using only the whole part, and only the transformed parts of the branch, he can give all the returns. This is a real reading of the text of the branch and of the theorem. There is no need to mess around with the quadratic equations, the square roots, etc. You don't need any square roots, any considerations on the square roots, etc. And so, yes, it is the work of a student who simply has the will to do it. Maybe it was the work of a student, in fact, he had the will to do it. It is quite obvious, but he did it in a completely modern and brilliant way. A modern demonstration, an exhibition, a remarkable fact is that it has a very modern form. It is part of the theory.

45:00 In modern democracy, if we have, as we are used to, a convergent, a continuous democracy, we have pi h over pi h divided by six zeroes over six h, and it means immediately that if we reverse the order, if h is equal to six zeroes over six h, then this continuous democracy is given by pi h over pi h squared. And, before we think about CE, QH and QH-E, as a recent example, we can give, as you can see, the structure of a continuous fraction of a matrix product, a matrix, it's important, and if we take the transposé of the product, obviously the transposé of the same degree, that is to say, through the matrix, we have CH, CH1, CH2, CH0. And if we take the transposé here, for example, when I go to Piaget, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, According to the formula given in the Transparency Order, the function also contains a periodic variable which is given by reversing the order, an inverse of the equation obtained from the but the substitution of the variables x and minus z over x exchanges a result of the equation

47:30 We simply make a change from one equation to the other, simply with a change of variables and with a part of the theory of continuous equations that is entirely given in terms of variables. Then, of course, we can interpret the equations of the second degree in a linear way, but to go from one equation to the other, it is not necessary to have a numerical interpretation in terms of variables. And this part of the equation is very clear. The reason is that it is simply the opening of the grid. We have many of the same in the world. And the reciprocal theorem for the equation of roots I do not have it here. I know exactly the law. If we say that the root equation that is understood is minus zero, then we take the second root here, we will say that it is understood. And we come to the conclusion that all the parts are absorbed in the development of the world. And, if we want, there is a mathematician, who is remarkable, of course, who has given the results of the law in direct use of the transformers of lagrangians. This demonstration is done, but you immediately understand that it is well given on the field, this demonstration, without any difficulty. With this demonstration, you can see that the result of the math can be put in the theorem of Gallois' theorem, and we demonstrate the theorem of Grange's theorem, in continuous proportion, from here. There is also a theorem, which is very strong, but which, once again, in the case of the equations of degree 2, can be seen.

50:00 A very strong field is formed here. If we take an algebraic equation, which has only a ratio greater than the unit, we can make a subsaxon equation, and we arrive at a certain point where there is only one ratio, which is greater than the unit, and all the other ratios are as they are when the entire milimeter is in the interval of less than 1.0. We can have real ratios, we can have complex ratios, but all the real ratios have been understood in a very simple way. The real part of the complex root is also a complex root. But you see, in any case, that in the case of this gradient equation, there are two roots, one real root and the other, and in a finite space, we get to have one root that is more or less a root that is complete. We don't need to give a general answer to get these results. In any case, I don't know, I don't know, no, I think that... This theorem, which is the theorem, was given to Vincent and his brothers, and it seems to me that the theorem is published in the appendix, but I think, I don't know for sure, but I think that this edition appeared after the death of the king, so he could no longer use Vincent's theorem, but he could, obviously, for the equations of the degree of two, the thing that he did there. One must not end up with the living situation of the career, but there is still an idea of the career, including a problem. With this premise, we consider a polynomial and we consider it as an effect of a transformation of variables that must be done in the development of the Taylor coefficient of the polynomial of the following polynomial, and therefore the quantity of the alpha, beta and gamma are given by itself.

52:30 And this is the polynomial. We can start from the polynomial of the client. X and Y, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X, X and X. And so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, and so on, And so, if with a finite number of steps we arrive at this situation, this situation must be given to us, and therefore, if these conditions are obtained according to the finite number transformed by Lagrange, they are maintained once they are obtained. The conditions are as follows for this exam. Consider a fraction, like this one, which has a positive ratio of large units and a negative ratio, including a large number of zeros. And here, there are results that correspond to the quadratic force theory, which tells us that if we pass from a quadratic force to the next quadratic force, and the form is reduced, we also have the reverse of the procedure, which is to say that we pass from the next force to the end of the procedure. And so, this means that we have a positive ratio in front of the unit and a negative ratio in front of the unit.

55:00 Evidently, all the successive transformations given by the momentum of the Maldivian average, and since they are an infinite number, because the term of degree zero must be an infinite number, so all the successive transformations of the Maldivian average, and since they are infinite, we arrive at a transformation, a transformation that equals the transformation of the Maldivian average. The answer to the previous formula is either one of the known terms in the bioreaction transformation, or the F'x polynomial, which is the polynomial given by our president. If we take the values of minus zero and minus zero, we have a prime polynomial, which has a negative sign that is greater than zero, f of zero is greater than zero, but according to the previous formula, So we have f of n plus 1 in a greater way than 0, and c is equal to f of n plus 10 to the power of 0, and so the polynomial has only one positive root, and gamma has a different root, and that is that the inner part of this root is n, so if we consider that the inner part of this root is n, we have to reverse the situation and arrive at the final result. And it is very important that from the equations that have been applied here, the equations that have been applied here, if not that the polynomial is repeated, the same thing happens for the Polynomian Polynomian Polynomian Polynomian Polynomian Polynomian Polynomian Polynomian The results of the law are very interesting, either because it allows us to see the range, or because it gives us a strong hypothesis in the field of quantum research and in the field of quadratic theory.

57:30 I would like to come back to what you presented by saying that the demonstration of reasoning that is also presented in your example is not conform to the modern model of Rieggemann. I was touched by the fact that there is a moment in this lecture where Penrose relies on the uniform step of the calculations, the small argument of the uniform step of the calculations to make a conclusion. If we compare the formal demonstration that you have... In the context of the example, we can say that from the point of view of the calculation path, it is the same calculation path. In fact, there is an expression that uses All of this makes him realize how he can look at his example, namely that in the context of his example, he presents the step of the calculation which will be uniform. So what I would like you to try to deduce is what exactly is missing in his presentation in the context of the example.

1:00:00 What is missing and what is the ingredient that is missing and that is added by another presentation? That is to say, if you want, we have two ways of presenting. One that would be in the pretext of an example and the other that would be... Or does the formal presentation feel like something that really adds something that escapes the way of presenting in a paradigmatic way? I don't know what to say. I don't think I'm going to be able to answer that question. Where is the difference between the Galois demonstration and the demonstration that we have in practice in the French school? There is no general matrix theory yet, but we can do something that is comparable to the matrix theory. So here we have ABCD, we do the preposition, B, B, B, C, B, A. We see what happens here. What do we not do in the point of view of the model? We would not know ABCD, but we go to ABCD, point by point. If you want, to put it in another way, to put it in another angle, I think there must be a way to show that Gallois did not choose his example in any way, but that he chose his example so that his example allows him to develop the uniform approach of calculations in a systematic way. That is to say that we can imagine other examples where the phenomenon, the calculation would not appear in its entirety if we develop the example.

1:02:30 While probably there must be a way to show that the example is chosen in a way to be too long. And so there is probably something to add in the analysis. I'm talking about the choice of the example, the calculation path that can be uniform. Are you talking about the example of America? Yes, I was talking about the example of America. That is to say, if you want, he didn't choose any example. He chose the example that shows symmetry, that shows development, that is neither too long nor too short. A patient from which he said that he wanted a periodic equation, so he took a patient such that we arrive at a periodic equation called a passage. He could, of course, do it after two, three, four passages. The rule is to add a piece of paper that is not to be used, against another that is to be used, etc. It is a very easy technique. But one passage alone is enough to give the general idea. After the first passage, we arrive at the equation that has the formula. He took the simplest possible number and showed it on this example. The example is sufficient to have the symmetry and just once the symmetry, so I think that probably the example is not chosen by chance but that the example is chosen to be general, that is to say to show what it wants to show and that's why I would say that the example expresses something general

1:05:00 It is not a numerical example, it is a very specific example, but it is not a numerical example, it is a very specific example, but it is not a numerical example, it is a very specific example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is not a numerical example, but it is This is an example that includes all the steps, that is to say, it is a complete example, if you will, it is an example that includes all the steps of any other example. So I think that if you want, maybe there is something to add when you say that it does not satisfy the example of Heidegger. I think there is something to add to the nature of the example and the fact that the example is complete on all possible reasoning. This is another thing that establishes a link to Gauss, in your opinion, and I think that it is something to be added in your article on the analysis of the example. Yes, in fact, in relation to the same thing. Just in terms of rigor, I have the impression that this way of doing things was something that was considered at the beginning of the 19th century as typically a reasoning that was considered valid and rigorous. That is to say that in fact, even if its theoretical demonstration, the first part of its demonstration, in fact, it can lack bits, it is not necessarily considered a problem. On the other hand, what he does afterwards is that he has a very general demonstration, possibly a bit bankable, but what he does afterwards is typical of the 19th century, typical of what is done in Cauchy or in other languages that make this type of reasoning, is that afterwards he takes a particular case, which is still a bit theoretical, but he adds hypotheses that make it work better.

1:07:30 And then, in the third stage, what we do is that we take an example on which we turn the machine and we see how it works. And this type of reasoning by stages like that, it's something that doesn't work in our memory, that's why. This type of reasoning is something that we find at other times at the Academy of Sciences or that we find in a certain memory of Cauchy, which is, for us it's not necessarily rigorous, but for the time... It is something that is totally accepted and that is part of the demonstrative and argumentative practices. I think you are right. I spoke of the language of rigor from the point of view of the authors who gave the theory of depression. The march is not strong or not. We have always tried to do it. But that is a certain question. But the other question is that it is mostly demonstrative. It can be given in a very simple way with the same approach, which is not considered as a theme in this lecture. There are, for example, many others who do not speak of the field of law, who do not speak of the field of law, but when these authors speak of the field of law, in one way or another, it is another demonstration. I do not have a complete reason for this, because it is also founded, as I said, on a consideration of the theory of penetration, a theory that at the beginning... We consider the number of variables indeterminately. Of course, these variables are constructed as formulas. And then, if we want, we can interpret them. It depends on the complex, on the number.

1:10:00 But if this formula can be given from before, it reverses the situation. There is one equation on one side, one equation on the other side, and we go from there with a variable change. There is something I would like to understand better, it is the work on the whole length, the work on the root of the equations, that is to say, when you talked about the work of Gauss. When Gauss develops on quadratic shapes, it is not related to the representation of whole numbers by shapes. In fact, he approaches quadratic shapes in a context of work with whole numbers. This is a whole coefficient. These are arbitrary equations. Arbitrary. So it is in fact another context and with a work on the roots of any real equations. So in fact, in the parallelization of the two contexts and in the articulation of the methods that come from the two contexts, There is a reading of Gauss, if I understand correctly, which reads the object in quadratic form in a different way from the way in which Gauss treats it, if he treats it in relation to the representation of numbers, or Gauss develops in a general way...

1:12:30 These treatments are then specified in relation to the representation of the whole world. In the field of engineering, there are two aspects of the research. There is what we have already said about the treatment of the scientific universe, which is a treatment of the resolution of the patient's life. So, there is a continuous development of the patient. But the equations that have been given so far are artificial equations. As long as we take the whole part of the ratio that we are looking for, we have a continuous development of the equation. And this is an aspect of Lagrange. But the same Lagrange also considered the case where the equations are artificial and gave the German equation on the development of the equation. This is the periodical debut of the students of the National College of Mathematics. This is the second stage. No, I would say that we are in the middle of it. While Gauss was interested in mathematics, so we have, how to say, a voice of the students, from the students who are mathematicians. And Gallois places himself, as to say, in the middle of Hilbert space and the connection between them.

1:15:00 You see, the question I ask is a bit about the constitution of the domain theory, its relations with other domains, and the circulations, both the separation but at the same time the circulations between different domains, I think that's what's at stake in this question. The algebraic theory of the number of paths is also in the intersection of the two sides because there is a period of transition between the old theory of the quadratic form of Gauss, the theory of the quadratic form of Dirichlet, the theory of the algebraic path, etc. There is still, in fact, at the intersection, at the time when they separate, I have the impression that this is quite interesting in this example, is that it is still between the two at a time when they separate. On the one hand, there is the quadratic form theory, on the other hand, there is the algebraic theory, and in any case, they are not the algebraic theory of man, the algebraic theory of the two, and it is a passage. And it is also for this reason, it seems to me, that it is a bit of a coincidence. And perhaps the disappearance of quadratic space theory has made those who have forgotten it a mess, because they considered it a mess and never considered it a goal. I would have never considered it a goal if it had nothing to do with it. Thank you very much.

1:17:30 We will resume with an exposed speaker. I'm happy. Thank you very much. But not the last one, because you will come back on the 19th of January. Carolina, welcome back! With Massimo, by the way. Both of us will come back on the 19th of January. Really? Yes, yes. I thought that Massimo, it was February that you came back. Ah, in February. No, February. Ah, I forgot. No, it's January, February. No, it's the 19th of January. Oh no, it's not the same day. Geometric projective, but it's part of the curriculum. I'll come back at some point. This is an exhibition in which I will not talk about the work of Gallois, but rather about what my mathematicians have made of Gallois' work, let's say, from the 1950s onwards, and more specifically, I will be interested in ... The transposition of Gallic words into books intended for teaching, i.e. into academic books, i.e. into academic books to be studied. So, as you all know, Maryse Gallois died at the age of 21 in 1852. You all also know that her research had not been approved by the Academy of Sciences and that they were published in fact in 1846 in the Journal of the Journal. These are well-known episodes. What is interesting is that, from the moment they were published in the Journal of L'UBI, they attracted the attention of many mathematicians all over Europe. And we often say, in classical reading, that these mathematicians only undertook to take Gallic works, to complete and arrange what was not in the Gallic works, which eventually led to the Gallic theory. It seems to me that when we address the question of the posterity of Gallo, we realize that the path that leads from these manuscripts edited by Lumine to the mathematical theory of Gallo, including in the form that it has at the beginning of the 20th century, is not at all linear.

1:20:00 And above all, that it is not guided by what could be called a intrinsic need for mathematics. A priori, it could very well have been different from what it has become. Of course, each of the readers of Gallois, of course, respects the rules and norms of mathematical reasoning. Each of them produces evidence in a matter of freedom. Nevertheless, the questions they ask, the approaches they put in place, the aspects they retain or ignore are elements that do not rely on mathematical criteria but that actually depend on the environments in which the readers evolve and also depend on the audiences to which these readers address. So from there, so this is a series of readings in fact which is quite largely inspired by the memoirs of Kathleen Goldstein on the re-readings of Fermat. And from there, it seems to me that we can open two paths for the analysis of the posterity of Galois. So a first way to see it is to place oneself in fact ahead of the re-readings. It is important to understand what are the mathematical practices that lead to these re-readings and in what epistemological cultures they are part of. And on the other hand, to analyze how these different factors intervene in the human mathematical text. And a second point of view is to place the re-readings this time around. That is to say, to question what they transmit, what each entry transmits from the Gallois writings, and from this point of view, it leads quite naturally to favor re-readings that have a relatively important dissemination because they are written to be disseminated, and in particular academic books, let's say university books, which are therefore addressed to students who have been published in the form of synthesis or manuals.

1:22:30 Since the double interest is that the type of source contributes to the expansion of the community that has access to theory, it contributes to the creation of a community around a shared knowledge. And on the other hand, these are works that characterize themselves as re-editions, which mark theory for a long time and then which make sustainable representation systems on a theory. This is another interest. So the question I want to address here is in fact the question of the re-reading, let's say, of the Gallois works, to try to show that they do not insist simply on importing knowledge that was already in the Gallois works, that they would only have transcribed, but on the contrary to show that they play a role in the elaboration of the theory itself and that they actually add things to the Gallois writings. So, to start, perhaps some precision, I will just give the points on which I will dwell in particular in the memory of Gallois on the conditions of paramatic resolubility. So, it is not, I want to emphasize, to do a re-reading that would be better than those made by mathematicians of the 19th century or which would be more objective. In fact, the big lines that I have been able to clear are rather work hypotheses. These are the main lines that I wrote later on, after having compiled and looked at relays that I would call the first generation. That is, I looked at what the first mathematicians who worked on Galois had done, I looked at which tracks they had followed. This allows us to characterize some major lines from which we can study this memory. What we lose sight of these days is that what Galois did at the beginning was something on the resolution of equations, above all, it is not something else than on the resolution of equations, the objective is very clear on this.

1:25:00 It is, on the other hand, a memory that poses problems of applicability, that is to say that Galois himself says that he does not know very well why it will be useful. He himself says that if we give him an equation, he will be unable to say with that if we can solve it or not. So the concrete application poses problems, we will say that this is not our point of improvement, and it is important in the context of the lecture. Then there are principles that he defines. A major principle is that of substitution. There are discussions on the rigor that he could put, the clarity of this type of definition, but to give a little context. So, the use of substitutions is something relatively classic, even very classic, since Lagrange's work, so since the end of the 18th century. It is not surprising when we talk about the theory of equations to talk about substitutions, it is quite normal. It is a notion that has been addressed and that Cauchy began to theorize around 1815 in articles published in the Journal of the Polytechnic School. This said, the articles published in the journal of the Polytechnic School have not been extremely used, it was not something that was completely accomplished. From the time of Gallois, the function of substitution was still relatively vague. On the other hand, this is something that Gauchy has greatly clarified precisely at the time when Gallois's writings were published in the journal of Louis VIII, that is to say in the 1844-1846. At the time when Gallois' writings were published, the readers had, on the other hand, a theory of substitutions that was much more successful than Gallois' theory. Then, the second notion on which Gallois relied quite heavily is the notion of adjunction. This notion did not come from Gallois either. Traces are found in Abel's works. It should be noted that this notion is not completely problematic in 1830. And we will see that this will not be the case only very very gradually during the 19th century.

1:27:30 It is something in particular, we will see that the French intellectuals of the works of Galois have completely left aside. This notion was finally approached rather by De Kynes, almost only by De Kynes in the 19th century. It seems to me that it is in fact very linked in any case to a question that was asked to us, it is a question of the nature of the numbers on which we work. At this time, they are still largely conceived as magnitudes and not as theoretical objects, which means that the notion of a direction does not really make sense. So then, to go a little faster, the three main steps of this memoir. In the first step, we take an equation and find a way to associate what Gallois calls a group. The second step, we realize that as we add quantities, we manage to decompose the group little by little, making it smaller and smaller. And the third step, from there, we can deduce a criterion for the resolvability of the equations. So, Pégalois first formulas in the language of groups, which is important, his reasoning is really based on... The dialogue between the adjunction and the repetition of the group, and once he has done that, he goes back to the language of equations. But Gallic reasoning is not a reasoning that is located, let's say, within algebraic calculation. So now, if we look, what I propose is to look a little bit from these big lines. Two quite detailed examples from the first manual that began to talk about Gallic writing, which is Joseph Serret's superior algebra course. Before moving on to this course, I would like to give you some context elements. The first version of this course was published in 1849. And it is directly derived from a lecture proposed to Sorbonne, because two years before, Serré replaced the suppliant Francker who had a lecture on algebra superior to Sorbonne. So Serré did this course and which is really published in its first version as a course. That is to say, we see that it is not organized as a treaty, it is organized as a course with lessons that follow one after the other.

1:30:00 After that, Céline occupied another chair at the Sorbonne, he occupied the chair of differential and integral calculus. And this course was modified little by little, so that it evolved into a second version that we will look at later, of 1854. And then there is a version that we can consider as definitive because it has almost touched after. The 1866 version, however, is not really a course, it is really a synthesis, organized completely differently, in chapters, in parts, not at all the same organization as the first version. So it's also a period in the 1850s when there are a lot of new algebra manuals that appear, that come out, and Ceres is not the only one to do that, but on the other hand, it is the only thing to do with Galois. He is also very well known as a teacher because he has written a lot of school manuals, but he is still a very good person who has published in the Journal of Liouville and in particular who, at the same time, published an article on the number of values ​​that functions take when permuting their variables, which was already very interesting to me. I studied at the University of Galway for a very long time and I found the last edition, there was a re-edition of this algebra course until 1928, it is something that has remained all along the 19th century. So this first edition of 1849, in which the synonyms concern, does not contain results directly related to the resolubility of the equations. However, the name of Gallois is mentioned in the 12th lesson, as you can see in the pages I gave you.

1:32:30 The first three are an extract of Gallois' memoirs that I wrote as a reference, because what will interest me in the first time is the letter 3 on page 2, page 420. And then, after these three pages, you have a page with a recto-verso extract. I cheated a little bit, I wrote this in the edition on Galois, which is not exactly the same, but I checked on the original, it's the same, the edition is not the same, but the text is the same. So if we look at where Galois intervenes in this hole, it is in the 12th lesson very exactly. The 12th lesson is an in-depth analysis of the 11th lesson, as I said, it is a lecture on mathematics, it is really organized by ETA. And it deals with the number of values that a function takes precisely when we have lost the letters that it returns. This is a question that Serré has already addressed in the research of mathematics. So he starts by recalling one more result of Lagrange. And so the result of the farm is here, which is reproduced on the photo here. So the second lesson is actually a numerical application of this result. And so it is the effect of the numerical application, which I did not put because it was a little long, but on the other hand what it explains in fact is that the calculations are really of an absolutely overwhelming length and that it is a result that is extremely painful if we want to use it. The main idea is really that. On top of that, what does he do? Well, precisely, we arrive at the point 502 that you have there, and he gives a theorem, in fact, which is the Galois theorem. What is also interesting is what he calls theorem. In fact, for Galois, it was only a lemma. For Galois, it was a result from the advantage of theory.

1:35:00 It was an intermediate result. And this one, if you look at it from the point of view of the enunciate, if you look at the photocopies I gave you, what is going to change is that the V-resolution is defined, while Galois, which you have already done in the M2, does not recall what the function V is. So there is another thing that changes. What is quite important is that we see that Serret has changed the connotations, that is to say that it is exactly like what we discussed earlier, that is to say that Galois works with A, B, C, D, etc. Serret works with X, Y, Z, etc. So he adds the notations which are more modern, clearer and which allow, in fact, to give the impression of a sort of corridor. If we now look at the proof of this result that you have in the Galois version and the series version on the sheets, there is a first thing, for example, if you look at the first step of the Galois lemma, it says that in fact it leaves the letter A which is fixed and it permutes the variables which are behind the Galois lemma. We see for example that Gallois permutes things like that, and then there is his reasoning of these terms, while Serré does something that is not very different or equal to what he does. So he allows variables and then he says that we will therefore have an equation of the form f of va equal to zero. And he says nothing else. So if we look at what makes it tight at this level, on the other page it is page 442 of the superior algebra.

1:37:30 So already, he takes the trouble to avoid any form of confusion, to call instead of keeping the letter v. He takes the letters V1, V2, etc., so there is a form of indexation, there is an effort on it for the reader to follow, so it is also difficult to expose the didactic. And then, his reasoning is much more detailed, I will not go into the details. We see that there is a form of indexation that is made to make it clearer and there are really all the intermediate stages, even those which are relatively elementary, which appear in the proof. That said, the stage, on the other hand, the global reasoning is that of Gallois. Then, if we continue, there is a second stage which is still, which consists of Gallois as for Serré. To show that once we have done that, we can draw the value of the first variable, so the one that Gallois calls a and the one that Serré calls x1, from this equation, x0. So now, in fact, when he arrives there, Gallois estimates, he has demonstrated his letter. That is, Gallois stops there. For him, the letter is demonstrated, we move on. So, what is extremely interesting and what is the main difference between this closed gallery and the Galois Café, the Galois Café, and this is where we really see the effect of the work of scholarly rereading, is that if we look at the sequence of the manual of Serret, we see that he added a third stage. It is a method by which we can actually calculate the root.

1:40:00 That is, they do not intend to say that to calculate the root you would have to do this and that. They really add the steps of the calculation of the root. And if we look at the detail a little bit, it's just at the bottom of the page, at the bottom left. In fact, we see that it has nothing to do with what we see. It has nothing to do with what Fégalois did, it has nothing to do with Neuf, because in fact, we manage to see the relation between the effects of... Yes, okay. I hadn't seen, if you want, that 442 and 443 were... Code by code. No, but mostly shifted by a good amount of the part. That's what I hadn't seen. Now I understand. I think we're not trying to read it here, it's challenging the media. We have it here, we have it here. So that's it, and what's very interesting about it is that if you look at it, it has absolutely nothing to do with what Galois did, it has absolutely nothing to do with Neuf. It's extremely classic, in fact. He explains, we will look for the greatest common divisor, we will push the operation until we obtain the rest of the first line, etc., etc. It's quite common for the students to know how to use these results, and that's what is completely paradoxical. At the beginning, the work of Gawai was something quite abstract, quite theoretical, but which has a really practical purpose. And when we continue to look, we see that when he inserts this result in the class, he explains... This result constitutes an improvement in terms of calculation, which is the result that the branch formulated, and therefore the interest is that we can from there obtain a digital application, which is actually done, if you turn your sheet, I have not finished page 445, in fact there is a digital application to an example of the theorem, and it explains precisely...

1:42:30 It is very clear that it is much easier to use Galois' lemma than to use Abraham's theorem, because the interest is there. So paradoxically, I don't know if it's a bad story, but the Academy of Sciences had essentially reproached Galois' work, on the one hand for not being clear, but above all for being useless and not being applicable. This is something that Galois has done where typically it is presented from a radically different angle since it is presented as something that can be numerically useful and that has a practical interest. So if you want, in two words, Serret has completely extracted the Galois letter from its initial context because in fact in Serret's manual there is only this letter, there is nothing else, there is not what is around it. But in the initial context, he made it a method of calculation, so this book has become a method of calculation. If we look at another example, always from this serial manual, but this time from the edition just after, dating from 1870, we will see another way of reading, our interpretation of the book. So here too, before we move on to this thing, what I would like to emphasize is that in the closed interpretation, a way of thinking about equations in terms of calculations and solutions will appear in the background. which clearly correspond to the practices and expectations of the French mathematicians of the time, for which the dialogue between theory and application is absolutely fundamental. And it is an interpretation of Gallois' works that will be preserved in all successive editions, including the most recent ones.

1:45:00 So, in fact, what characterizes the Galois theory in France in the 19th century is not its architecture, as we can say later, it is not the general and abstract point of view that allows us to adopt, it is ultimately what characterizes the Galois theory is that it is part of the theory of equations, and the theory of equations is a story of calculation of roots. What French mathematicians insisted on is precisely this French tradition around the number of values that can take a function when we permute these variables. This is something that was studied by Cauchy quite heavily, which was studied by Serret, which was studied by Bertrand, which was studied a little later by Mathieu. There was a math grand prix competition in 1860 based on this question. It's really a research tradition, I think it's good, and it's a tradition that leads, indeed, to be interested in substitutions. That's where Jordan came in. And what is quite striking is that if we look precisely in the second edition of the Serret manual, for the first time, we have the Nougat theorem. When I say the theorem, it is the theorem to which the tool is its memory. In the case of quadratic equations, since the roots of any of them are negative, the ones of the first two are the ones of the first two, the others are the roots of the first two. So this is the result that Gallois came up with. And this is a result that Gallois demonstrated, as we saw earlier, by the principle of equivalence, adjunctions, groups, etc. So, in the manual of 1854, for various reasons, He does not publish something directly from the works of Gallo, but he publishes a demonstration that was given to him by Hermite. Hermite is normally someone who also knows the works of Gallo quite well. We see in his writings the same period that he read them.

1:47:30 Contrary to what many other mathematicians will do, what we see in there, you have the proof that is better, is that in fact they don't try at all to fill the gaps of the Galois demonstration. They don't go back to the principles that Galois had proposed. They don't try to do the same thing or to complete what Galois had done. In fact, they do something fundamentally different. It has nothing to do with the principles on which Gallois worked. In fact, I'm not going to go into details because it's quite long, but he has three sheets. He begins by assembling three sheets. These are sheets that, as you can see, there is absolutely no question of height in them. There is absolutely no question of adjunction in them. Apart from these questions, on the other hand, it is about permutation, the number of values that the functions take. It is relatively calculative, although without going into the details of the demonstration, we can see it quite well, it is still something that relies quite heavily on the practices of calculation, but on the other hand, it does not rely at all on group notions, on adjunction notions. Demonstrate these three terms once again. They combine all this. It is a process that is relatively calculative and it comes to the results of Galois and finally by a completely different way that reminds perhaps more, I would say, a demonstration that would be closer to what Abel would have done, for example, than Galois. Now I'm going to go. I would now like to talk about a second generation of re-reading of Galois' works. It should be noted that from the 1880s, Galois' works ended up appearing in many manuals, which have two common characteristics. First, they present themselves as a synthesis of the theory,

1:50:00 Practically, I don't know, I don't know, never, in fact, on the original writings of Gallois. These are manuals that are made from first-generation re-readings. These are manuals that are made from what was done by Serret, what was done by Jordan, what was done by Detti, or Detti quite rarely. And in this context, they could expect a relatively uniform image to be diffused. However, what we see when we compare these manuals is that, in fact, in order to construct a uniform image, they actually construct several Gallic theories that cohabit and that are ultimately quite different in their finality and in their method. I will not, of course, address all the possible and imaginable manuals, but I have selected, let's say, two German books, German languages. For those who have known a very important dissemination, an international dissemination, which has been translated into several languages, who have known several editions also, so who have been influential on a fairly long period, and who have been cited as references afterwards by other authors, and in particular by the authors themselves that we will see later. Excuse me, what is the date of the current date of the Netto's manual? It's written here, but it's not possible to read it. For my part, it's 1882. Eight hundred and eighty-two. All right, all right. Neto is from 1882. It is a book on substitution theory. The other book I will talk about is a very well-known book, which is the Agel's Manual of Bélair, published in 1895.

1:52:30 The advantage of this work is that we can see what people considered to be the works of Gallo at that time. We can consider that they embody in a way a kind of disciplinary matrix that goes far beyond the borders of Germany. And then I will show you French manuals that have not had the same opportunity. If we look at the introduction of Neto's manual, it is a book that, a priori, does not have a very ambitious point of view. Its goal, unlike other books, unlike Weber for example, is not to reform algebra and present a new way of seeing algebra. Concretely, what he wants to do, he explains, he wants to give a concrete foundation to the theory of substitutions in order to make the theory of substitutions easy to access. So, in fact, the idea is a little bit that the manual of Jordan, not the manual, precisely, the Treaty of Substitutions of Jordan, which at the time had three references on substitutions, is still a bit complicated for students. In addition, it is written in French, and we see it in other testimonies, but the way the French see it is not quite the same as the Germans see things, so it is not very comprehensible for the students, so we have to make an abordable substitution treaty. So, when it comes to equations, which is not a good part of the treaty, the sources used by Netto These are never written in Gallic, but they are re-readings of Gallic. They are without explicit inspiration. These are the manual of Serret, the Treaty of Jordan, and the works of Kronecker. Note that Netto was a student of the Christian Kronecker. Another thing to note is that, as we can see quite clearly on the table,

1:55:00 Netto does not develop a theory of Gallic. And if we look at the detail, in fact, we realize that Gallois hardly ever appears in it. There may be, in the table of contents, there may be two places where we see the name of Gallois. The same number, 223. There is another one. That's it, 188. But in fact, these are the only places where Gallois appears. So the name of Gallois is barely mentioned. If we also look at the content of the book, in fact, the name of Gallois is barely mentioned, it is mentioned only on these two items, elsewhere we do not see it appear. And the structure of the memory of Gallois is not preserved, it is completely exhausted. Jordan's work. Jordan affirmed himself that he was heavily inspired by the work of Gallois. If we believe Jordan, he didn't do anything, it's Gallois who did everything. And despite that, if we look at the book of Newton, we realize that the practices and the uses he seeks to transmit do not go away at all. For example, if we look at an equation, we realize that... The group of an equation as defined here has lost the fundamental link that Gallois had established with the resolution. For Gallois, the group of an equation was something that was linked to the resolution, whereas here, in fact, there is no more resolution. On the other hand, what is also striking about this manual is that, while Gallois made of the group the fundamental element of the resolution, the key of the group,

1:57:30 Here, the notion of group does not appear as a necessary principle, it appears more as an alternative to the use of an algebraic relationship between the races. In fact, the theorem that Neto presents on this subject is a theorem that is borrowed from Kronecker, and in which we see that it is the point of view of Kronecker that appears, that is, the notion of group that is on it. And so to say, to empty the principles that guided the Gallic approach. It is no longer about substitutions, it is no longer about resolution. On the other hand, it was quite heavily based on algebraic calculations. And there is another element that confirmed this impression. If we look at chapter 13, in fact, which is devoted to the resolution of equations, If you look in detail, it precedes chapter 14, and chapter 14 is the one that deals with the law of equations. In fact, concretely, it means that when we use the book, we can solve an equation without using the notion of group, it is not necessary, it is a plus, but it is not an obligation. And when we also go into the details of the demonstrations, we see that in fact the demonstrative method of Netto is systematically the algebraic calculation. It's the expression of the relations between the roots, it's the algebraic calculation, it's never tools inspired by work. The book that was published 13 years later is Weber's manual. On the other hand, we have a book that is clearly very ambitious.

2:00:00 It is something that, I think, has set up a modern algebra. So, Weber's agenda is this. He wants to create, to present a new algebra. And this new algebra, what he explains in the introduction, is that it must rely on the theory of numbers and on the theory of groups. It's an agenda that is clearly modern, at least that's what it seems to us. So, what we need to know, despite all this, is that at the beginning, what he did, the place where we see it appear, is volume 1. Volume 1, however, is relatively classic, it is not entirely on the table, since it is a book, volume 1 of the book of Weber, which is entirely devoted to the theory of equations. And finally, in these contents, there are relatively few ways to narrow it down. And in particular, while Weber wanted to present a concept of a group, to base algebra on the concept of a group, in this first volume, the concept of a group is not presented from a general point of view. It is presented only as a tool for the resolution of equations. The general introduction and what he actually did is still important. That said, this book has an interest for the re-readings from the point of view of Gallic posterity. It is the first manual that gives importance to the notion of body. That is to say that the other re-readings of Gallic, as I told you earlier, The adjunction did not work at all when we solved the equation in different forms, something that was not at all addressed. Weber's book is precisely the first to put this idea in the center, in the center of the book. Obviously, it's because Weber himself says that in fact he used, he was inspired by, that Delkine was right.

2:02:30 I had access to the notes that had been taken from Detkin's course in Göttingen in the years 1856-1858, and precisely this course by Detkin is the only one that puts forward the notion of adjunction very, very heavily. It is the only first reader of Galois to have addressed the notion of adjunction, and Weber, when he passes behind, he takes up this signal. So, if we look at the chapter 13, it is based on the definition of the notion of the body, then we have the notion of adjunction, and then we address the question... Decors obtained by adsorption of many elements and then the introduction of the resolution of the decors. So if we look, if we continue to go into the details of the chapter, it is only after, precisely, that there are the substitutions. And what you need to know is that if we go into the details that you have in the dictionary, if we go into the details of what Weber did on substitutions, It is very striking to see that he has developed a point of view which is relatively succinct already, and then it is completely autonomous of what can be found for example in Gordon's work. And finally, Weber kept substitutions that he strictly needed to develop his geometry theory.

2:05:00 So the general theory of substitutions that can appear around us has not been of great interest. And what is also quite remarkable, if we look at where this chapter ends, in terms of representation of what the Galois theory is, it is quite striking, because finally this chapter ends before the criteria for resolution and the dynamics of physics. That is to say that for Weber, the Galois theory is his original manual.