Hilbert's Problems (last part) / discussion / C19th mathematical journals (1st part)

0:00 Can you look at the page number on top of the page? Another example of the introduction. It's around 65. There are a few pages and it's quite specific about the role of time. I want to say two things, both of which I think is tied to the work of Philip and one of which is to expect to be done. Sheldon has spent quite some time with Grünflagen, I think, on the screen, working on the relation of figures. But Grünflagen is quite heavily illustrated by him, although he might have been purely up-to-date, but I know that Grünflagen is still there. And I think, so, Grünflagen, what's the link between this? It's going to engender a long debate and break with that topic now, so I think I'll move on. There are, at this time in the mathematics world, very substantial debates about the nature of mathematical thinking, what is mathematical thinking, how does one do it, and this is the theme raised various times before and after the theory of Liberia, which is a famous microagent in the late 19th century, but the theme is already there.

2:30 Since this was defined on the fact that it's possible spontaneously to be invented mathematics. And this is contrasted by Klein with the way mathematics is written in Berlin by the Berlin panel. And that's what's called the representation of mathematics by Klein. It's portrayed by him as necessary but insufficient. There should also be intuitive, geometrical approach, and what I think Hilbert is, is very productive in terms of detecting the chemistry of formalism and language. And now for all speculative remark, I don't think it's time to talk at all. Within the Mathematics Museum at this time, there is a great deal of attention. The most popular topics are about mathematical languages, Tejano especially has been pushing an ideographic, formalized language which would eliminate natural languages altogether, and this is followed by Thierry and other formal artists who have been creating that style. It is completely unbelievable. So I think that this question of how you think about mathematics and how you write about mathematics is alive in mathematics and physics rather than a kind of oblique interest.

5:00 The theme of this is that it becomes so much later when we get the thinking of mathematics and mathematics and mathematics formalized quite immediately. That was 20 years ago. It's around and it means something very hard to pin down in Hilbert. Hilbert wants to advocate quantum sort of intuition in mathematics. And that may also somehow or other be coming up at this point. I would question general trends in mathematics pretty much. It's not that far in this text, it's an interesting remark, about rigor versus simplicity. And somehow that is also, in my mind, connected with mathematical language, mathematical symbolization. I could give you another Hilbert if he says any more of this. I can't believe he would be writing these kind of reflections about the nature of mathematics. He wasn't aware of the fine polemic in Berlin, and the leading Berlin poet died, as he said politely. And general discussions about being informative. Mathematics we are not very precise in formulating. There is no orthodox position in the world of mathematics we are subscribed to, but not very much there. So perhaps I think I will take a break. Ten minutes break and then we can begin to hear the last paper.

7:30 Thank you for your attention. I am very happy to be here with you today because I would like to talk to you about physics and I don't want to go back on the subject of physics. I am very happy to be here. So I will talk to you about the questions that we have for the women at the Faculty of Mathematics. I will try to speak quite slowly. Before we start, we will talk a little bit about the background. The mission that Karine has given me is 33. So, it was to explore a little bit this practice of problems in mathematical physics. So, to see a little bit how it works. So, I accepted this position. And I'm going to explain to you what I'm going to do. So... Oh, great! Oh, great! So, the type I chose is question-solution-problems in mathematical physics. So, the words question-solution-problems, behind, we go down a level, we are no longer at the level of Hilbert. These are the words used by the editors to qualify the rubrics, questions, solutions, problems that exist or do not exist in the mathematical space that we are going to produce. So I'm going to show you a little bit of the curriculum on the curatorial part. The curators will observe very carefully. But that's their problem, isn't it?

10:00 There are quite a few. There are quite a few books. Great! So, why did I decide to take this course? Well, first of all, I'm interested in the fact that it's a little bit of a three-part course per week. So, first of all, I'm a mathematics teacher, so the practice of problems, in a way, is based on that. So, for about twenty years, I've been teaching mathematics. So, I'm used to... You can also explore different journals or manuals to not get bored and do exercises that are different from each other. So it's an ordinary part, it's not necessarily in French, it's to try to see a little bit on the right and on the left what kind of problems I can apply to my students. So that's what I would call a classical course. I also have the habit of... Exploring some of the old problems with Mathematica. It's a way of teaching Mathematica. As long as I expose references in yellow, it's not complicated. In fact, I refer even more to the algebra part of the lecture. And so the last step, that's why I'm here today. I initiated for a few years a step in the history of Mathematical Press. So I'm finalizing a thesis on this on the direction of the registrar at the Université d'Orsay. So I'm doing a thesis and I'm also coordinating with several other people a small group of studies around the scientific and technical press in that sense, but more focused on the mathematical press. So throughout this exhibition, we're going to travel. So it's a fairly general lecture, trying to avoid being too generalistic, so we're going to go through a certain number of periodicals while staying more specifically in Lyme, Paris and Berlin, we'll see a little why Lyme, Paris, Berlin, knowing that there will be another point of passage where we will meet other movements.

12:30 So I will focus on the first two quarters of the 19th century and especially on the first half of the 20th century, with a little distortion in the 60s, and it will mostly be localized in the first half of the 20th century. So the corpus, there are a lot of newspapers that I have illegally studied. We're not going to go through all of them. The paper I'm working on the most specifically is the one at the bottom, the one in the middle. We're going to expose it in time. And we're going to see a lot of other things. The interest of the mathematical press is that it is now easily accessible. There are a lot of memorization programs. Most of the journals I'm going to talk about are partially miniaturized, in different programs, in different structures, so it's quite impressive. By the way, all the extracts I've cited are essentially miniaturized. The 19th is often considered as the emergence of the specialization of the 13th. Generally, we are certain to say that the analysis of Gergon, the analysis of pure and applied mathematics, is the first journal of mathematics. It was written in 1810 in Nîmes, which we will obviously manipulate a little later. This does not mean that before Gergon, there was no mathematics in the press. Just two remarks. For example, at the end of the 18th century in Leipzig, We find Elin Mourke with one of the Bernoulli who has created several newspapers with a lot of mathematics.

15:00 He will write one of the 20 mathematics. We also find this in the beginning of the 19th century in England with the Mathematical Compositorium. We will make a very small incursion on the side of the last to be mentioned. So, in terms of color methods, it is a journal that I have studied a lot. So the reference zone was studied in a study article, not in a journal, but in an interesting paper published in the fourth volume. What is in green, in fact, refers to the corpus in which my class is located. So page 1 to 3 show examples of how this journal is constituted. So there are... On page 2, for example, you will find a section on mathematical questions where, in general, there is a numbering of questions, what is the author of the question, and, eventually, what is its institution of belonging. So these are very varied things. If you have problems, it depends. So there are answers in the journal. You will find articles on the page H3, which is an article by Wallace, where he takes up things that appear by certain pieces in the mathematical network of physics. I will stop here on the operations of today, we will return to what we showed earlier. And now we are going to focus a little more on the journals of Gerdon, the journals of pure and applied mathematics. So this is a journal that was launched by Gergon and Laverneuve for the first two years. So this is a journal that existed between 1810 and 1832. So the references in the orange part, I will insert some examples so that you can understand the question of this journal.

17:30 So we will see some examples together and I will try to explain why I chose the examples that I chose. Before I explain, I'll give you an example of the original analysis of Gerdon, it's not too hard, it's very difficult to find, just to see a little bit the physical form of the experiment. So on page 5 of my class, you will find a number called proposed questions. Generally, they do not necessarily give the name of the person who will be the writer of Gerdon. I do not necessarily give the name of the person who asks the question. Sometimes he does it, sometimes he does not. So there is a question to be asked and then on the next page, page 6, you have the question resolved or not, it can be a million or less, or not resolved. So the way it is written, the part of the question resolved, it is a synthesis that is usually done by Gergoine. So here I'm going to insert some pages to answer the problems raised at page 5. I'll make a summary. Often he makes comments. There aren't too many, but in general there are often notes at the bottom of the page where he criticizes the solution, where he makes an allusion to an article in a newspaper. It's important that Gergoyne intervenes frequently in the columns of his lectures. On page 9, I would like to give you another example of a question about equations to know if they are imaginary or not roots. In theory, the proposed question is in this form. For some questions, I tried to know where these questions came from, how they are constituted.

20:00 For this one, we have an example. If we look later in the proposed production, it turns out that the question emanates, so it is probably asked by Gergone, it emanates from a memory that has been published in the And this article, I just gave you the first page, is a list of problems concerning algebraic equations. In general, there is often a condition data on the coefficients that explains how many similar laws exist or not. So here, Gervon, who has written a report on the work of Ernest, extracted a question for himself. Gerdon's analysis, through this aspect, is a reflection of what can happen in some scientific academies. There is very little mathematics in general, but sometimes there is often the local approach of a geometer. There are a few pieces of mathematics, and there this question emanates from this piece of the academy. On the next page, page 11, there is another question to be solved. I'll show you how it's written in response to a question. So, when it's written by a subscriber, very often, it's Japan itself that writes the solution. Not always, but by a subscriber, it's a generic way to designate the act of the creation of Japan itself. At the end of the 11th solution, page 11 or 11.12, you will find what I was saying at the end, a typeface of jargon, where often they refer to another article, here in this case to an article published elsewhere, in the journal Krem, so we are going to... So the typeface of jargon is something that is very frequently used, therefore a typeface of jargon.

22:30 And to finish the little tour that we will do in the Journal of Jarlon, I put you a question composed, page 13, which comes from Steiner. Steiner asks a lot of questions that come from a very important author of the Journal of Jarlon. In this question composed there, I just put you the first page, in fact it is a list of theories that are to be demonstrated. Geometry-planned theorems will be found in the book. So here, Japan's page number is quite interesting because, I say, although we have not yet decided to designate the authors of the numerous questions proposed in the book, we feel that when these questions consist of theorems of great importance, they can be an act of justice, at least of compliance. So from time to time... The cycle between the forms of mathematics and physics. To complete this survey, in page 14, I will show you the end of the paper where he exposes the correspondence, he makes a balance, if you will, on the questions that are asked and how they will be resolved if they are. So for example, this page here... This is a small remark on the number of numerations that exist. The Anabes Ergon were numerated last year by Lindam, who was a member of the Commonwealth. In the numerization, you only have the articles.

25:00 There are no, for example, the geometrical fields of the pages, which are not numerated. And here, it is not the result of Lindam's numerization, it is the result of the Google Books numerization, which is often criticized in France. All these are book numerization, but we find this corresponding rubric which is quite interesting to see how it works, to have the object as a whole. There are some books of numerization that do not have the object as a whole. When you work on a journal of a writer without having the boards, it's sometimes... So on Google Books, there are not all the analogies I give, but some of them are interesting. So that's what I would say in a few words about the analysis of jargon. There is a question and answer section which is quite important. I don't work specifically on the analysis of jargon. I work with Christian Gérynouk who has done a thesis on the analysis of jargon without exploring the thematic that we are trying to see today. So there is a whole... There is a lot of work going on around this, around the analysis of the question-answer rubric to see a little bit how it is going in the field, so there is a lot of work going on. So the years of Gerdon started in 1810 and ended in 1832. Here I will continue with what I have called a first editorial push, located around the years of 1825 and 1826. It's not just a European phenomenon. In New York, Adrien is launching a journal that will last for a few years. There are a lot of questions, questions and answers that are very interesting, that I will not talk about here. In Belgium, in 1825, the mathematical and physics science was launched by Ketley and Garnier. Very soon, it will be Ketley who will take over. Garnet continues to work in these publications without being the main editor.

27:30 In Berlin, you have Krenn launching a journal called Reinhardt and the Bantam Mathematics. We will talk about it later. And in Vienna, in 1926, you have Teichrich on physics and mathematics. which was launched by two professors from my studies, Dan Gartner and Tim Sauer. So in this part of the editorial I'm going to go to Berlin, but before going to Berlin we're going to stop a little bit in Brussels with the Ketley conference. So I almost brought you an original from Ketley, but I had no place, so I'll come back to what I'm telling you in the polycopy. This journal was studied by KDEM in 1978, but it is a rather editorial study, I would say, on the content of the journal, how it is done, how it is written. Financially, it is a study on the economic conditions of the creation of the journal for mathematics, mathematics and physics. So in this, I have just put three pages, including an optical page, so that means we're going to go fast, we're not going to stay long. On page 16, you have an excerpt from the question to be solved rubric. This is the first volume of the first volume of 1825. There is an elementary mathematical rubric. In another two-page note, we specify that in the future, the solutions of these questions that we will take in the elementary part of science These will be used to fill in elementary mathematical types. Mathematical and physical correspondence has developed quite quickly over time. In terms of rubrics, it is not a closed form, it is a form in co-creation. There is this rubric question-answer.

30:00 So there is a kind of rubric for everything where we put this rubric, question-answer, question-answer. We also put a lot of extracts of problems posed in exams, so in different universities. It's a way to measure what was going on in the universities of the Kingdom of the Netherlands. So that's what I'll say in a few words about Kepler. So a quick summary of Vienne. Vienne is very active on physics and mathematics. I'll also give you an example, which is the year of the beginning of the year. It's very difficult to find, I don't think it's in the book. They are not found in the French library of mathematics. So there is not really a rubric question-answer, there are rather memoirs, which often emanate from the analysis of jargon and from different French or German publications. But there is still, at the end of the class, the beginning of a creation of exercises, which are called analysis of the world. So these are rather exercises based on analysis. It turned out that it did not hold. We find it on the first two or three pages of the Itachi and then it disappears. The journal itself does not hold for very long. It holds for a few years, from 1826 to 1832, five or six years with publications that are more or less regular. Now we will stop a little longer on the great journal of Krehler. So Krehler... His journal was published in Berlin in 1826. It's a journal about mathematics. What I've written here is an excerpt from the preface that leads us to a possible rubric, knowing that this rubric is not important enough for all of us. So, very soon before we do this...

32:30 He mentions the Gergon equation, which still exists when he publishes a journal. He alludes to Keckley's correspondence. He has two French-language journals. He says that a German publication is missing. He explains that the Germans are also friends with the mathematics of the French alphabet. He launches a publication intended for... They are intended to be discussed in German, but it will not be possible, so there is a lot that goes beyond this framework. They also refer to the analysis of mathematics, of the metatones, for the rubric in Howard-Garvin, Exercises, where we have, in the polycopier, I have put some examples between pages 20 and 25. So, to give you an idea of how it's structured, it would only happen if it was out-of-garden or out-of-the-box, but it changes over time, sometimes it's out-of-garden, sometimes it's out-of-the-box, so it's not fixed. So, on page 21, you have an intervention by Rabe, who talks a lot about it in the journal of Frey. I took you page 22, so this is an example that is quite interesting. On page 22, you have the answer to a question that was asked in the gergones analysis, which I mentioned earlier. So this analysis of the gergones, we will cross it later. So you have several exchanges between what is happening in France in gergones and what is happening in Berlin. There are articles or sometimes questions like that. These are taken from elsewhere. So here you have the writing of the draft. On page 23, you also have another example of the Venetians,

35:00 who for themselves, in fact, have two proposals of answers to the question asked by the Venetians. So you have two contributions, with a complement brought by the editor, in the case of L. The initial demonstration, page 23, is written by a subscriber of the present newspaper, it can be Gergol Mignani, who was a subscriber of the French newspaper. He intervened sometimes in this field, with the intention of passing by the name of a subscriber. He also intervenes in the correspondence of experts. A little more lighting, it would be nice to see a little bit of it. Do you want to see how much or do you want to see how much? We don't see that much. It looks like I'm going to turn it off. Earlier we turned on the light a little too much over there, but it was pretty good. Yes, pretty good. Yes, that's it. You turned on the light at the right time. No, wait. I'm going to turn it off, it doesn't work over there. The balls are not in the same order, they are in a straight line. In page 23, we have several interventions on the same question. On the other side, we have an allusion to the demonstration that was published in the 2013 journals. So there are other variables between the two journals. So, to finish this little passage in Berlin, in page 24, I have on this side an excerpt from 1855, in the volume 50 of the Journal of Ecclesiastes. You have several dozens of pages that summarize all the compositions that were read in the Journal of Ecclesiastes,

37:30 by the author and by the classification of the articles. So, this gives me an idea of the question of the rubric in Haute-Garenne, which is given at the bottom of page 24, in the 24th generation of mathematics and physics. So, there is a rubric in Haute-Garenne called No problem is theoretical, and it continues on the next page. So, there are quite a few in relation to the set of minerals that constitute the common scale. In general, the questions are not numbered, as we had seen at Gergol. It's from time to time, there are questions that appear, so there are some people who don't know them. And at the bottom of page 25, there is this analysis of the 50 first volumes of Threl, which are written in German and in French. It's Threl who wrote the first volume of Threl. In the second column, at the bottom of the two small columns, you have the section called Mathematicians and Physicists. The column on the right explains why some of these subjects are anonymous. It is to avoid having to rely on names. In particular, in the section Out of the Eye, there are many questions that are simply signed anonymously. By the way, you can also see the hand of Crenn in the mirror. It doesn't remember. It doesn't remember. There are other major pages in Crenn's book that don't remember. So now we're going to the end of the 19th century. We have the second editorial section.

40:00 So, in 1935 and 1937, there are several years of research. We have also published some books in the way of the Academy of Sciences, in 1935. You have a journal called The Geometer, which I will talk about a little later. What is interesting is that many of the books, even if we are not going to go into detail, exist on the first few months of the year 1836. You have the Mathematical Journal of the United States, which was published in 1938 and in 1936, and in 1937, you have the Quindim Mathematical Journal of the United States. So we're going to stop for a while on the Geometers. So it's a journal that was created by Dugard in 1936. Guillaume is a professor at the University of Le Grand. His father was already a professor at the University of Le Grand. Guillaume launched a journal in 1936. He explains his goal a little bit. What does he mean by his journal? This is a preface essay. In all of this, he uses technical terms such as belt, sea, forest, and so on. So, in all of this, you have some notes, some black notes on some pages, and there are also a lot of questions and answers to these programs. So, I put in the chat box, in just a few pages, to have a more general idea of how to explain this program. So we're not necessarily going to go through all the discourses, I've put a couple of pages, so it's not going to be too complicated.

42:30 So, these are questions proposed by Kerr-Rack, they are not classified, so in page 26 you have a question on continuous fractions, at the bottom, taking a look at the structure of the journal, you have the list of subscribers, to give an idea that the journal does not last very long, but still has 250 subscribers. I hope that you will be able to understand everything. With the professors who are also in Paris, in the province, we see in particular many students. Among the 243 students, there are approximately a lot of students. These are students who are either in graduate schools or in private institutions. In Paris at this time, there are many private institutions, in addition to the classical courses in the Royal Colleges, which prepare the courses that we can do at the time of the pandemic. This journal is somewhat a marker of what is happening in this world of institutions that we find in Paris, but not only, in each of the provinces, there are private institutions that prepare the oral courses of the public schools. So, we continue a little bit with Penrose, which is a study, page 217, which is a study which is always a little bit, which gives us a little bit of instructions, it's a little bit like a book. There are a few questions, question 43, for example, of the Nobel Prize, it's a question that we already find in the book of Kepler. So it's not mentioned, but there are a lot of things that we find similar. I'm just going to stop on two questions.

45:00 I have a question, so the question is very clear, and I am on the screen, it is in there, but it is on the screen. So this question is a question that was proposed by Michael, precisely Michael is a young man originally from Albi who graduated from the institution of Orbeck. The institution of Orbeck is one of the important institutions of the capital compared to the rest. There is a classic question that we will address later, so I do not want to go into too much detail, it is a question of the plane geometry, so if we have a pentagon and if we move it to the sides, so I put in my hand the picture that goes with it, so you can see it well, if we go to page 34, which is from the plan of the geometry of the museum, the plan of the geometry of the plan, so page 34, On page 35 of my book, you can see page 67, which illustrates this proposal. We have a pentagon in the middle, A, B, C, D, E. If we extend the sides of the pentagon, if we trace the five circles, which are circled in different places, R, A, M, A, B, etc. So we have these five circles that intersect, we have a certain number of points that are on the same circle. So that's what we call a nickel and a teren. We're going to see a little bit where we're going to find the teren. So at the beginning it's a question proposed by a student, my name is Nickel. He asks other questions of geometry, so some are, we cut into the questions of the resolute in jargon, proposed by Steinman. We can think that, in the world of institutions, the analysis jargon was very much worked by the students. The theorems, here is the strong version of the nickel theorem, you have a pre-version that we find in a mathematical repository by the artist de Wallach that I mentioned at the beginning,

47:30 and you have the questions that we have here. So here, it is a question asked by the nickel theorems. There are a lot of questions about bigars at the time, and we find them in the books at the time of this lecture. So here it is quite simple to explain orally. There is a bigar, it is not mentioned, but it is in the book. The teacher who sleeps in me. So we have a bigar. So I have my ball. What I want to know is what the journey is for the four of us to return to what we said in the beginning, because it doesn't work. So this is a question that was also asked by Michael. So Michael is an important player in this journal, the Geometer. He asks questions, we answer them. It's not a question for anyone. So this is what we find a little bit in the questions that you put in there. He is going to answer question 28, so I invite you to read my answer. So question 28 starts at page 33, where it holds a few pages. It shows us what we call today the nickel theorem. We don't have time to go into the details here. In the nickel writing, there is a point that was tested by the editor, that is, by Villard. So it's the plane geometry. When the question is answered, it will be the subject of the question, because there is a lot of discussion about this question proposed initially by Witten. And here, it's the same, Guillaume, in general, does not mention the name of the author of the question, because he says it's very easy to ask questions, so there is no reason to give the name of the author of the question. But on the other hand, for Witten, it is an exception, he mentions it sometimes in his questions, he also explains it in his debates.

50:00 These are people who have a lot of knowledge, and I ask them a lot of questions, and they also answer some of them partially, so we often talk about them in the video. I also put you an extra answer to the question 30 minutes. So this, I put you this, so it's actually page 28 of my article, where the structure, the answer to the question, is generally the same, it's a synthesis. Most of them are written by the editor, with various contributions he could have made. So here, page 8 is just a part of the answer, the answer to Dugard's problem. So we find people who are rather students. We find Michael, of course. And the answer that is given, which is less complicated by the way. So this one is quite classic, the one of Mikkel and Karkaradek. So Karkaradek is a student of the institution in Brest, who will integrate the Polytechnic School the day after, and he is someone who will become a pretty important engineer because he participates in the construction of Saint-Nazaire. These are just some examples of mathematical publications of the 19th century, which we will no longer find later as the author of the article on mathematics for revisions. So here, we found him as a student at Institution, an institution that was awarded to the Lycée Amnesty. And Mikkel and Mikkel Karajek, by the way. If we want to solve the problem, we can trace the diagonals of the bivouac. The other symmetries are parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation, parallel to this delineation,

52:30 So, to finish our very partial lecture of the journal, which lasted a few months, it was launched in January, it stopped in June, so 6 months. So there are still a lot of questions to be asked, so if we do a little summary, there are about a hundred questions to be asked, 40 of them were resolved within the journal. There are 56 unresolved questions. I will give you an example of a non-resolved question. It is not a non-resolved question because it was solved last night by a member of a math forum called Mathematik.net. This is a complement to the orange question. So, there is... Michael asked Russell a dozen questions. There are three or four unresolved questions and there are a few unresolved ones. And I invite you to this slide, there is the solution of this problem, I did not mention this question here, it was answered by Mr. Goult, an amateur of relative geometry. So I can tell you that the five questions have been answered by Mr. Goult since last night. It was just a little mathematical interception, so it's a pretty complicated problem to understand. So now we're going to leave the geometry and we're going to get to the journal of mathematics that we're going to apply, the journal of Ludwig, which is my main subject of study. I'm not going to talk about it here because there's no need to. So it was launched in 1936 by Ludwig. So I'm not going to spend too much time on the creation of the journal of Ludwig.

55:00 He had already been studied by Ludwig Leibniz in the late 14th century. I also mentioned him at the time of the interview because I had already mentioned him at the end of my thesis, which is not quite a reality. I still have a few pages to go, but I have no resistance. So that's it, I'm going to leave it up to you. I just put you an excerpt from the preface. So there is no allusion at all to a question-and-answer question-and-answer. I'm going to do a few remarks on Ludwig's journal. So they are addressed to another public. Geometry is more about the preparation of the textbook. The journal of the public is something else. It's more about writing. So I've put in my notebook two or three things. So this page, I wrote it in Libby, so between page 41 and 47. So page 41, we're not going to talk too much about it, it's the little part of the literature page. We don't see anything else. Page 42 is the most interesting. It's an article of our academic year. Page 42 is an article of a few pages. It's a cultural history of our academic year. And it's the resumption of what was in our time. So this time, it's to open a memory, a cultural history. There is no reference to genetics. Since this is a copy that came out, it's not quite clear because I gave you what we announced earlier. There was a part of the demonstration of which was published by Duyard. This part was not written in the version that was published in the book. It's page 4043, so it's related to the era of the genotypes of Pantagone.

57:30 So this article, this memoir, started with 10 questions. Geometry, Paine-des-Aubonnes, Anteorennes, an apparent memory in an academic journal, the journal of Lévy, and this in a practical sense. Now, when we refer to the Lévy article, most of the time we refer to the article in Lévy. It is rare to see a reference to the source of Lévy and to know the game. So, on page 44, I have written a start of the article, the teaser. If I put this arcturus here, we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 1842, so we are already in 18 So, in the 42-45th century, we see the arrival of the first former students of normal school in a significant way. And I stress this example of black, because in my knowledge, there are very few examples of black that are represented in this way. So, you have problems. Find the curve where the curve and the curve are constant. So, in general, it doesn't start like that, there are no problems, it's not a problem anyway. Another example, which is listed on page 45-46, is an article by Serret. Serret is... so here we are in 1951, so it's been 30 years since he wrote his book.

1:00:00 So, this article is interesting to me because it refers to... In the manual, Louisville published a publication on the mathematics of the world in 1950-1971, I don't know who, and then there was a series of annexes, and this article is one of the annexes of the publication by Louisville on the mathematics of the world, and it is also published in the Journal of the World, and in this article, at the end of 1946, we see something in the middle of the page that is very interesting. There are a lot of waltz notes in Libby, but it's not as polemic as in Gerdone. In Libby, it's forbidden to use polemic notes, but it's not really a problem. So the notes are generally quite polemic. In this note, I mention that Mr. Gerdon, a student of the Stéorènes community, proposed an exercise for normal school students. So one of them, Mr. Giraudet, gave the following genetic demonstration. And we are finally, in the last 40 years, we are already seeing the arrival of the first teachers of the Marmal School. And from the 1950s, it has become, in a way, it has become quite important, especially by the time of Bertrand, the teacher of the Marmal School. And it raises the concern that many of these students, that many of these students are not in the middle school. In particular, Giraudet, who will not say anything else in Boudin, but who is perfectly mentioned in his lectures. Thus, there is a whole tradition of entering the school of mathematics in Boudin through Bertrand's activities at the Normand school. In particular, we often find an article by Bertrand with a complement brought by one of his students.