Nombre et grandeur chez La Ramée

0:00 It's going to make it happen to Marco. So we have interverted the order because François had a train to take. And so we will start by the exposé of François Logey, who is at the UFN of Limoges, on the Ramus and the Ramus. How many times do you have? I can't count. There are certain numbers that you don't have to do. So, can you see the camera? Can you see the camera also? I'm going to go over there. Thank you.

2:30 I can't speak English, but I can't speak French in English. There is no English native speakers here. If you don't speak too quick, it's okay with me in French. There is no problem, because you can speak French and can speak Italian if you like. Okay, that's awesome. No problem. Good offer. Then I can walk and say Dutch. So I try to speak slowly and speak Italian. Thank you very much. On peut le corriger, l'amender, et commencer par là c'est une manière d'introduction. Nombre et grandeur dans les mathématiques de la Ramis et dans l'Emmanuel Ramis. Alors l'Emmanuel Ramis, on dirait seulement deux mots. On a évoqué ce matin avec un ramiste, un partisan de Ramus. Je vous ferai simplement rapidement deux auteurs en réclusion, deux autres ramistes, Salignac et Schoenert. Vous trouverez pour Salignac un extrait. L'ombre et grandeur, pour ajouter la pluriel, la pluriel s'impose pour l'ombre et grandeur. Les objets qu'il dit Pierre de Laramé dans ses traités de mathématiques sont des classes de nombres, des genres, des classes distinctes, distinctes les autres et presque étanches, on pourrait dire.

5:00 And more precisely, I would like to talk about the science of numbers and the science of grandeur. The arithmetic, the science of numbers, the geometry, the science of grandeur. It is in fact that through these doctrines, through these arts, arithmetic and geometry, these arts that he can constitute, put in order, that Aramé can make his lecteur to the objects of the arithmetics, the geometry, the number, the grandeur. And then, the last remark, here it is a little complement, to complete the precedence, These two doctrines, arithmetic and geometry, are defined, as Ars Béné-Mérandi, Ars Bénén-Métendi, Ars Béné-Métendi, Ars Béné-Métendi, Ars Béné-Métendi. But these two principles of doctrine fix the usage of mathematics, calculs and measures applied to the fins variées. Arameh, promoteur of mathematics and practical, It explains, in a very ambiguous way, that these continuators also commentent, complètent, etc. There are three remarks that allow me to engage in the reflection of this type. Science of number, science of grandeur, rather than number and grandeur. Nombre et grandeur ne sont présents dans l'œuvre de Laramé, je l'ai dit, qu'à travers la médiation des sciences qui en fixent les usages. Vous avez dans les premières pages de ce petit fascicule les tables des matières de son arithmétique, la dernière version de son arithmétique en 1534. It is used in two books, this arithmetic.

7:30 It talks successfully about the numération of the non-grantier. Numération simplex. This morning, we were talking about the word numération in a passage numération, in a passage de Snell. The numération is the art of the language of the algebra, namely the annotation and the art of posing the operations. Numération simplex, arithmetic of the numbers entier, and numération comparativa for the fraction. This table of materials, I don't recommend it. They just show everything that there is a classic in the arithmetics or in Aram. These are absolutely traditional materials, very close to what we find in the arithmetic manuals, J'ai une paris métique en l'occurrence, et le premier manuel en usage dans les universités de plus loin d'image. On trouve notamment ce qui implique les raisons, les catégories anciennes pour les raisons, les raisons suivies, les liquides, les chiffres particulières qui appartiennent, etc. Des choses absolument traditionnelles. So, there are classes of objects, so. For the geometry, you also have the table of materials. It's the same thing. We introduce the kinds of euclidians, figures, figures planes, we study successfully, and the order that impose the method. And then, by the figures planes, we introduce the different species the triangle, the rectangle, the rectangle, the rectangle, the rectangle, etc. and then the rectangle of the rectangle. So, a new order. There is a little bit of a new way in the geometry of Aramé. It is a completely classic. Some of the objects that are introduced in these traités, are made necessary by the order. So, if you look at the geometry of the chapters 20 and 21, the surface of the 12th, and the 21st, the surface of the 21st, the surface of the 12th, which is in Latin, excuse me, but also in the common sense.

10:00 Well, why are you talking about it? I prefer to speak in English. So, these chapters 20 and 21 correspond... The 20th century produced an object, the surface curve convexes, which is calculated on the line curve. The definition is not given, but it is defined as the surface curve as the surface which is inégal to its limits. And then, among these surfaces, he will study the sphere, like he studied the sphere, in the order of the plane, the sphere. The sphere is defined as a surface curved, which holds the distance of the center of the space. Well, I'm not going to go into detail. I'm not going to just survoling these traités-là, with a little bit of assistance on the algebra, but what I want to show you is how to compose these traités-là with an order, This is the characteristic of the Aramets, and also what is visible there. The objects that are visible, let's say that these are the idealities, which the doctrine envoie, and which are present through the radiation of a doctrine, which is constituted, We see, for example, in the definition of number, in the book 1 of the Arrhythmica, at the beginning, in the Arrhythmica 1569, the number is defined as that, according to which one counts each other. Thus, the one counts each other, the two counts each other, the two counts each other, and the three counts each other, and the other counts each other, and so on. Before we even thought about it, before we even had access to these objects, we had access to these objects.

12:30 And the method then requires that we separate the two angles, the two angles, the arithmetic, and the geometry, because each one has two orders of grandeur absolutely distincts. This is the famous law of Solon that Aramé applies in all these traits. On ne mélange pas les ordres et on doit enseigner la géométrie géométriquement, l'arithmétique arithmétiquement. On ne mélange pas et on doit introduire en bon ordre les différentes espèces de quantités qui sont les objets de chaque science. La doctrine ne nous renseigne pas sur la façon dont on accède à ces objets-là, à ces idéalités. they teach us their usage and, before all, their écriture. When you take the method of the Aramé, you always use it in the background. Numération, you always use the numération, the way to pose the operation, the way to note the numbers This is part of the first part of the arithmetic of 1539 on the notation, notation. For what is the name, I just give you three lines. We consider first the notation, then the numeration. The symbols for writing and writing, the numbers by écrit, in abaco, in the text latin, are the following, 1, 2, 3, 4, 5, etc. This note of the notation includes some lines, but what I'm interested in here is the conclusion. Here is a sort of alphabet of symbols for any number. But it is not an alphabet vocal, but manual and scriptural. This is not an alphabet vocal, the alphabet of the arithmetic, it is an alphabet manual and scriptural, it is necessary to calculate the writing of the math. So, the first point to discuss, if you will, the first affirmation,

15:00 in my development, for me to learn how to learn the math and math. The math and math, it's before all to learn how to write. The soin, I'll show you that what characterizes this method of math, is certainly not the content, but it's a particular soin that puts the operations in these manuals. This lecture of the mathematics of Aramay shows that, for him, the technique of the writing is part of an integral of the math skills. And this is also the name of Walter Ong, one of the commentators of the Aramé, Walter Ong, who has made a lumineuse study on the dialectic at the Renaissance, and the dialectic, the logic of the Aramé. What montre Walter Hang, is that the transformation, the work of the Aramé témoigne the transformation profonde of the culture of the Renaissance. The method that he promew, that he promew the Aramé, for, well entendu, Irei, to be able to do it in a vocabulary a little more tardive, but to be able to do it in a vocabulary a little more tardive, but to be able to do it in a way to do it, to be able to do it by writing. And the dialectic, like we call it Aramé, it's the core of the reform, Araméen, the dialectic, It's an instrument logique d'une pensée qui est désormais destinée à être couchée par écrit plutôt que délivrée oralement. S'il y a, dit Jung, une liaison intime entre la logique de l'aramée et la communication nuette et silencieuse du livre. The Arborecences which abondent in the books of the Aramay are simply the résumés

17:30 which allow the lecturers to appropriate the contents of the book, which allows us to intellectuals, who is before all a lecteur. But the content of the book is the reflection of the idea of the pensée. The place that he assigns to mathematics in his program of reform she is liée to her interest for the disciplines which show, with the geometry, the model of just reasoning. But to my opinion, there is not an indice in the text, but we can suppose that this place it also ties to the fact that l'Arabay recognizes in mathematics the sciences where the writing plays a particular role. And for the algebra, the writing of the algebra interests the most at the point. because there he sees an oeuvre, to reprendre his formule, an alphabet manual and scriptural particular. In the same time, his conception of mathematics, the one that I presented a little bit more, the concept of mathematics as a science of ideas, of objects, intentionnel, accessible only by intuition, is divided into two different classes, from two different branches, with the geometry geometry, Well, this concept, we're going to look at, it empires to give a place to the algebra in a mathematical program, in some mathematical program.

20:00 In fact, it appears impossible to recognize, behind the algebra, the objects that they let us know in the idea. Well, it's a series of hypotheses. A series of hypotheses, what I will do now is to try to illustrate what we can call the migration of algebra in the work of Aramé, And then, first, I'll show you very quickly the Algebra. The Algebra of Laramé, very quickly, because... I'm going to present it only through his model. The Ramay takes for model an algebra, the first published in France, the original film, in 1550, which was printed in Paris. It was printed in 1550, in Germany, and printed in the identically in 1551, at Paris, at Cavella, It's an imprimeur régulier des professeurs royaux. En 1550, elle est préface d'une édition des éléments du club. En tout cas, elle est là comme préface d'une édition des éléments du club en 1551. Elle est imprimée à Paris comme traité indépendant. On peut se pencher sur le contexte de cette publication. On peut faire l'hypothèse que c'est à l'initiative de Pierre de Laramé que ce traité a été imprimé à part.

22:30 I don't want to talk about it. There is a possibility for Pierre Delaramé or his entourage. There are Magnin, Pena, other people who are interested in mathematics. There are people who are close to the Royal Collège and who are interested in algebra at this point. In any case, the imprimeur signale that it is the initiative of a mathematician who gave this treaty for the first time to Paris, because he felt the need and he had recommended this treaty. I have compared the same features. Yes, I compared the same features. Is it not the same features? In any case, the same features. So, it's the same features. So, what do you call it? That's not what you call it. No, no, no, no. The features are very interesting. One of the characters popular, but... I don't know, the page... ...the planche... ...the planche... ...the planche et la crème... ...the planche et la crème... ...the planche et la crème... ...it's not a difference of the pagination ...but it's the same text, ...and exactly the same... ...the planche... ...it's strange... ...but it's strange, ...the fact that... I don't know it, it's going to consult the historians of the book on these questions, because, actually, the special ones for the Algebra are rare, I imagine, and they should be engraved. So, I don't know about which one or which one hasard the two-treaties are found in such a problem. Well, this is the first one to be imprimed in France, the first one to be imprimed in France,

25:00 perhaps at the initiative of Pierre de Laramé or of Son Courage. At a time, in any case, Pierre de Laramé s'intéressent de près aux mathématiques, who is teaching at this point, and he is interested in the algebra. He has already announced that in the course what he would expect for the students who could have under his férule, he would expect a sixth year to be consacred entirely to mathematics, without neglecting the skills of the algebra. He s'y intéresse et il enseignait, en tout cas il a prévu de l'intégrer à un curriculum pour les élèves. This is the model for the algebra of 1560, a model, a reference, but the parent It's not so obvious that the first report is to say that the treaty of Chebel is almost two times more long than the original Ramay. He is much more detailed, much more detailed. He has much more problems. He gives proofs of his operations. There is more details, more information in the book of Chebel than in his copy. I gave you one of the original parts of this book of Aramé. It's the short preface. It's the page 5 of our article. It is practically the only passage where Laramé expose his project in terms of general. He is defined as algebra, he introduces the objects of algebra and the way to the notation. He announces the organization of his practice.

27:30 So if you want a quick translation, the algebra is a part of the arithmetic, which is a part of the arithmetic, which is the number of numbers in this calculation propres to figure out in proportion continuous. These figures are the polygraphy, etc., number figures, because we are in the What do I do here? You will have maybe some comments to do. We have a nomenclature of the algebraic species. We have for first term the unit, for second term the coté, for third term the carré, for fourth term the cube, for fifth term the pi carré, solide, cubo carré, pi solide, pi carré. to find the symbol. The symbolism, there, the Aramee doesn't need a special case, it doesn't include the typography, it uses the letters minuscule for one which doesn't work. For another, the unit, we have a U, but the Aramee immediately that this symbol, we don't use it. on a besoin de plusieurs signes, qui sont... He doesn't note the entries by letters, but he doesn't note the entries by letters. This is a heritage of Chebel, who introduced a N to note the entries that he doesn't know. What is there again, compared to Chebel, apart from this little preface? Well, it's the structure of the rubric. In all the Aramé, you have the structure of the mer, two parties. The two branches here are, like in arithmetic, numeratio simplex and numeratio comparativa, comparata. The first part is the calculation of the number of algebra.

30:00 And the second part is the theory of the equation. The equation is the comparison between the numbers. This part is less precise, less detailed than the of Chebel. I will not wait for it. The first part, where the ramet concentrates all the numbers and the operations, is more interesting. not only for symbolism, but also for the way to pose the operation. The symbolism is not the one of Chevelle. Chevelle was quite original because he introduced the notations caucyste, but he didn't use it. He uses the abbreviations. He uses the primus for the first puissance, second, second. It uses the same symbol to note the first puissance, the first puissance, and the racine carré. There are two uses of the symbol L. The operation of the racine carrée is the only one. It's an inconnu. No, I don't know. Les espèces de nombre qu'il détaille, donc il distingue les opérations dans son premier livre, sont les sourds, avec différentes espèces, avec cube du carré, et les dinômes et résidus de l'indirationnel qu'il retient, par élection, c'est l'élection, qu'il est mis comme nombre particulier de l'algème.

32:30 What I would like to say in page 6 is just this way to noter, to resume an analysis of a binôme. Analyse des côtés d'un binôme, le texte de Chebel d'une part, le texte de l'un ramé de l'autre, pour montrer un petit peu la différence. Et ce qui me frappe chez l'un ramé, c'est cette manière de disposer les étapes successives du calcul. On reconnaît une procédure, là aussi, très éprouvée, avec la fille. It's simply the disposition of the operation that serves as a guide to the lecturers or to the machinaticians, to the usage that we expect for this library. That's what I'm going to do, this way to summarize at the maximum, to summarize at the maximum, the mathematical procedures, in giving the most possible details. But all happens in the typography, in a certain way, in the typography, on the page. The numbers that you have here are the numbers that you have here are the numbers that you have here. The rame is absolutely loyal to this model. I'll let you see the second one. This is not an originality. I think the mind is actually better What is the page 2 page? Yes page 6 That's right. It's 21. That's right. I think it's 23, the 1,8 ? That's right. I don't want it. No, it's 23 to 27. That's right.

35:00 Ah oui, en fait l'exemple, là c'est 13, c'est 13, moins racine, moins ou plus, c'est plus, plus racine de 448. C'est 13, plus racine de 448. C'est 23. Non, 23, regardez. Et après en bas c'est 11. Non, c'est moi qui l'as fait pas dire. 11, 12, racine 110, oui. Well, the plus, it's not indicated, but it's the rest. No, no, no, no. On prend la moitié de chaque élément du binôme, c'est un binôme 23 plus racine de 448. On prend la moitié de chaque, on les valait au carré, So, the moitié of 1,5, I don't know. 122,25. That's the square. And then we'll do the same thing as the other side. And then we'll do the fraction of the square. So the square is 4,8. Well, no, the moitié of the square, that's how much? It's the square. It's the square. It's the square. It's the square. It's the square. It's the square. Then we make the subtraction spinneret. We take the root of the result that we add to the half of the large segment. The large segment is what? The most large of the two terms. And then we do the subtraction. This is a procedure ultra-classique. It's just the procedure. It's precisely the notation and the way to dispose of the operation that you have noticed. So I move on to the second book. The originality of the treaty,

37:30 it lies in the organization, which is the difference between the model of the Chevelle, and in this way of posing the operations, of traiter the operations. The rest, it's a simple summary of Chebel, especially the examples that he gives for the equations. The examples that he brings are much less than Chebel, and they are all emprunted. Those who rest are due to Chebel to five exceptions, but those who are not emprunted to Chebel are also examples that we find in other ways. which is original, so it is the adoption of a notation more practical than the Chevelle, and Chevelle already had already romp with the notation of the Kossis, the traditional. And then, what is also interesting, perhaps, is to signify that the Aramee is on a model, which is not any one, but Chevelle, an algebra, an algebra which was very fast to traiter the elements of Clyde. In this case, Chevelle, in general, was the part of associating the algebra with the mathematics. And, undoubtedly, the Aramee was interested in this aspect. on a model on a treaty which rapprochait which allowed to make the link between algebra and the market, and the art of the practice, to make a model, to make a model, to make a connection with the mathematics. And the project of Aramé, when he envisage, in this case, to write an algebra in the curriculum, and when he envisage to write an algebra,

40:00 to push the algebra from the mathematics, to make a place, in sum, on the two disciplines. But it confere the dignity of mathematics savants, arithmetic and geometry. And, however, to attach the algebra to mathematics savants, it is not simple. Just because it is necessary to cohabiter a new discipline with the two branches. classique. Alors, une cohabitation ici simplifiée dans la mesure où l'aramée prétend présente l'algèbre comme une partie de l'arithmétique. Mais on va voir que son opinion sur ce sujet to vary the course of the time. I'm starting to describe this migration of the algebra in the language of Laramé. We're going to come back. In 1555, Pierre de Laramé is the first part, the first version of his arithmetic. 1555, une arithmetic en 3 livres, 2 livres consacrés comme dans celle de 1569 aux nombres entiers et aux nombres fractionnaires, et puis un troisième livre consacré aux nombres figurés. What are these numbers figurines ? For Pierre de Laramé, this is a theory that we find at the time. He would have been envoying to Goethe, but he did not. He would have been envoying to Nicola de Gérase, where we had an edition of Greek from 1538, but he did not. Il renvoie à Euclide. Les livres 1 et 2 de cet arithmetique de 1555 rassemblent tous les résultats d'Euclide portant sur les entiers fractionnaires. Le livre 3 rassemble tous les énoncés.

42:30 What's your question ? What's your question ? What's your question ? Libre 1, 6. Non, non, pardon. Les livres arithmétiques. Pris dans les livres arithmétiques. C'est 18, non ? Oui, oui. Enfin, je n'ai pas le détail. Mais... D'accord. Le livre III rassemble tous les énoncés qui témoignent de ce qu'il appelle l'ancienne théorie ptégorifienne des nombres figurés. Alors là vous avez le détail dans le document que vous donnez, page 3. Les définitions 7, 7 à 20 et 22, les définitions 5, 11 et 12 et 14 et 17. Du livre 9, il reprend les propositions 1, 6 et 8, 10, 8 à 10, et puis quelques résultats qui auraient du livre 2, des propositions 1 à 10, et puis deux lèmes du livre 10. He justifies the displacement of propositions of the book 1 and of the book 2, which relates to the geometry. He justifies the displacement of a particular arithmetic in saying that in the elements, Certains things are established in the grand plan, which is the common number of numbers. Communion numerum. On the report here, on the explains as if they were proposed for the numbers. The second part of the algorithm is still divided in three books. but there is a 3rd edition of 1532, which is divided for the first time in 2 livres, so the part on the numbers figurate is different. This is a edition that has something in common with the edition of the Algebra. It is anonyme. It is not original, but it is anonyme.

45:00 There are even a dozen. There are the grammars anonymes. Most of them are the re-editions under the name of the author. There is only the algebra which has not been re-edited with the name of the author. So, there is only anonyme. Well, in 1562, it was anonyme for the first time divided in two books, which is the same imprimeur that the Algebra. From the point of view typographic, it is pretty close to the Algebra, that is to say that the Algebra is a typographic trait of a different quality, Contrairement notamment à l'arithmétique de 1555 qui est une vision très soignée. L'arithmétique de 1562 est également assez bâclée, si l'on veut. There are other common points between these two traits. First, it's the appearance of the practical problems. The arithmetic of 1565, it's a compilation of elements of truth. It's a composition of the arithmetic elements of truth. The synopsis are accompanied by the Greek text, and they are just illustrated by the examples. In the Arrhythmia Theta of 1572, the presence of the key is very discreet, and there are also examples of practical problems. On résout, on illustre les dénoncés, les propositions, avec des exemples d'arithmétiques commerciales. Parenté entre les deux ouvrages, j'ai l'impression qu'ils appartiennent à un même ensemble. On exclut le troisième livre des figurés, on publie un traité d'algèbre qui reprend la théorie sous une nouvelle forme, the theory of the figure, which is on the numbers figure. So we have the impression that the arithmetic and the algebra, the arithmetic of 1562 and the algebra of 1560, are part of the same ensemble, a science of numbers, in which the algebra would be on the point of gaining

47:30 a place to pay, an autonomy. And yet, in 1560, we are there, at the end of the definition of 1560, Well, the end of the year 1560, 1569, it's the year where, on voyage in France, in Germany, the Ramay published its last mathematical traitors. The Scolae Mathematicae, the Arithmétique and the Geométrie. So you have the table of mathematics. So, this is the Nouvelle Réglise qu'on porte sous le livre. La Science des Nombres Figurés n'est plus. Par contre, dans la Géométrie de 1539, vous trouverez les traces de cette Science des Nombres Figurés. Et vous avez donc l'élément 4.9. Dans l'élément 4.9, il définit la figure rationnelle, like the which is comprised in an author and a rational basis between them. And to justify the introduction of this element, of this proposition, he envoies to a tour of the definition of 2-1, to the proposition of 18-19, to the definition of 17-18. It was an effort in 1579 to justify this improvement and to justify the attribution of the geometry of results on the numbers figurines. There is a little demonstration in a corollaire to the element 4-9, I believe, page 7-G, page 7, column to the left. On appelle donc le nombre de la figure rationnelle figurée et le nombre d'où il est produit côté du figuré, etc. On produit ici le nom général figuré à partir des noms spéciaux blancs, carrés, cubes, dont une Euclide. It would have been called by analogy the number of factors of the figurative, the iterative, but it would be called only the coté and it is not necessary to change the figure.

50:00 Certes, la géométrie est, pour la plus grande part, si générique ou si juriste, mais elle ne peut pas être traitée autrement que géométriquement. Cependant, on associe à une certaine partie de la géométrie les nombres, et on l'explique par les nombres, et les commentateurs attribuent aux nombres qui ont des dispositions géométriques des non-géométriques. Well, this habit, taken by the commentators of a certain number of dispositions géométriques and non-géométriques, has caused Proclus to conclude abusively that there was analogies between arithmetic and geometry, which is abusive. This analogy has abused all the mathematicians, for all the Greek or Latin, as well as they have attributed the doctrine of the number of the number of the number of the number of the number of the number of the number of the number. The comment is that the reason why this analogy is valid. A part of that, there is no study, although there was a science figurative in the 3rd of the arithmetic of 1555, although here in the geometry, this science figurative doesn't exist anymore. It passes very little time. And of course, he doesn't talk about figurative, he doesn't talk about figurative, he doesn't talk about figurative. and he says that the figure is only the size of an ombre of an ombre. If you study on the mathematics class, on the mathematics of Pierre de Laramé, he said that he has transferred his algebra in his geometry. He has not renounced to the algebra, but he has transferred his algebra in the geometry, which is also a little bit in the school of mathematics. He has not transferred, because it is abusive to say, he has not transferred,

52:30 There's no algebra, certainly not in the geometry. In the Scolae Mathematicae, we'll see. Algebra, there's no algebra. What's in the Scolae Mathematicae, published in the same year? and it's what we talked about this morning, it's in the Scholar M.S. University, which is a famous critique of the book of the Glyde, which is what we found in the book of the Glyde, it's a classification, it's a rational, this classification is totally ill. What we can retenir from the book is simply the distinction between rational and irrational. It is a book totally redondant and totally utile. This critique of the book is developed on 4 books, between 21 and 23. It depends on 4 livres. Yes, because the Scholarly Mathematical is a commenter. So, on prend les propositions de Clive une à une, et si elles sont inutiles, eh bien, on signale qu'elles sont inutiles. Donc, ils détaillent l'ensemble du contenu du livre. Et ce qui est intéressant, c'est le dernier, because, as you said, he gives a lot of pain to do this demonstration. And not only does he give a lot of pain, but he does it in two times. In the first time, he details the whole of 115 propositions. And in the second time, he exposes the book X, using his notation, his notation algebraic. And I will give you the beginning of this book 24, So, we had to expose with so much the 10th book of the elements, we had to fear that the time of obscurity has not been shown enough to be shown to the novices and to the incultes. We have instituted all the matter of the 10th book of the 10th book, according to our reasons and our method, as if we wanted to take a look at it for us to see clearly how

55:00 at which point this book is very strong. So, we add a couple. I had first thought that in this cavern were hidden some treasures of sagesse, but having all found out and found out, having all checked with calm and delicacy, I realized that never the subtleties of the lignes as rationnelles had no use in their own life. I didn't realize that That seems new, etc. But what I'm interested in is also the end. For that the demonstration of the paradoxes and the whole of the book will be made as clear as possible, it will be done again the exposé through the subject of algebra. Let's take care of what there is of technique. This is the last phrase, I think, in the passage that you have given. the first paragraph here, Verbo tanquam technologia quae dami atangatur. Faites-on donc attention à ce qu'il y a de technique ici. Le mot technologia, ici, nous retrouvons surprenant, intéressant. Il y a une technique de l'algèbre, mais ce que retient Pierre Delaramé de l'algèbre dans ce dernier ouvrage où il l'évoque, It is an aspect technique, a language that will allow you to make a quick summary of the book of the book of the book. I will not examine in detail this book. What I've noticed is that it doesn't take the time to explain the same connotations that in the exam of 1560. It reminds the techniques operative. And then, of course, it benefits an exam very attentively. I suppose qu'il reprend les résultats de l'Irodite et qu'il réexpose certains, au moins, avec son langage angébré. Well, in this case, it seems clear to my eyes that we are in the scolae, in the scolae,

57:30 in an oeuvre that is not the creation of the Aramé, it is a simple commentaire of the elements. It is not in the scolae that he understands his method. So the language of the Algebra is simply utilized here as a particular technique, as a particular language. And in the Algebra science, there is no place really in place. How many times do you have to do this? Five minutes, maximum. That would be all right. When he started to study the Algebra, at the beginning of the year 450, L'aramé devait être frappé par la parenté entre la littérature algébrique à laquelle il avait accès et les mathématiques pratiques. Alors certainement ça pouvait l'intéresser que ça. Mais en même temps, son projet c'est quand même d'élever l'algèbre au rang d'art savant. Lever l'algèbre au grand art savant, lui faire une place aux côtés des disciplines mathématiques nobles, arithmétiques et géométriques, telles qu'elles sont enseignées dans l'université. the model, the model of Chevelle, the model of Algebra and Compandioda, was chosen for this proposal. It was a treaty that established the link between the elements of Euclide and Algebra. It's a trait, by ailleurs, which could be interested in the Algebra of Chevelle and other Algebra of Algebra, it's the fact that we developed the study of the binomes and residues,

1:00:00 so the classes irrationnelles, which were used to create a link supplementary with the elements of the guide. and then the science of the form figure which was developed in 1855 he could also expect to give a new form in an algebra so his project was not not to promote an art practical mais d'élever l'algèbre au rang de l'arithmétique et de la géométrie savants, de faire de l'algèbre une discipline. Alors, faire une discipline, c'est rassembler tous les éléments de la science algébrique, jusqu'alors réparpillés dans les divers livres de guide, constituer un corps de doctrine successive de faire l'objet d'un enseignement et donner l'outil de cet enseignement, un auxiliaire, c'est-à-dire un manuel conforme au principe de la méthode. Il fallait faire un livre. Or, ces sciences, cet algèbre, These sciences, algebra, arithmetic, geometry, are construed as porting on the idealities, which are the objects that the species are placed in the genre in the number of other. non seulement la méthode de l'aramée maintient cette caractéristique, cette forme d'ontologie, mais en plus elle accentue la démarcation entre les deux sciences avec médité jouée. Alors de quel genre sont les objets de l'algèbre ? Jusqu'à la fin du XVIe siècle, on se posait question, l'aramée n'est pas le seul à poser cette question, So it's not a question that the discussion has allowed an avance notable, but it's also a cause. The third element, if you want, it's a complement. The reform pedagogic of Aramé conduit to give a place in the book and written as a vector of the science savante. Sa méthode, c'est non seulement l'art d'ordonner la pensée, mais surtout l'art de coucher par écrit cette pensée.

1:02:30 Et pourtant, en dépit du soin qu'il accorde à mettre au point une écriture algébrique épurée, il considère finalement que cette écriture est une simple notation pour des objets qu'il ne parvient pas à identifier. But it's not, in a way, it doesn't give the algebra a value or an autonomy suffisant. I think it is limited, in the case, by the oncology, by which the algebra and the science which, in general, portent toujours sur les objets intentionnels que l'on vit, qui sont là, dans les diversités, les classes, les objets bien identifiés, et le langage en lui-même, comme technique, comme... technique autonome, pouvant permettre de résoudre des problèmes de tout ordre, but he doesn't have his autonomy here, Mr Pierre de Laramée. Well, I'm going to stop, unfortunately, the epigones. There were two algebraic traités, which is inspired by the Aramée. One is an author, the other is less known, but not less interesting, Salignac. The two have been formed in a series of academies which have been quickly conquered by the pédagogues of Oramus. Elizabeth Briggs said earlier that Snell was passed by Marbourg, and Saliniens and Schoenert were also passed by Marbourg. They were formed in the same milieu of Oramus. So two algèbres, two traits d'algèbres, one in 1580, the other in 1586, I mis par erreur 479 for the traité of Solignac.

1:05:00 Two traits who regretted the Algebra in Aramé, and I modified it a little bit, and some modifications are interesting, It's an extrait of this one that we can see. The chair is open. But there is one behind me. There is one behind me. There is one behind me. There is one behind me. I have a question about the difference between the 1555 and the 1562 And, simultaneously, in the same period, it's also the key of the Algebra, so if I have well understood, you see a parent, an evolution between the 7 parts of the number, the number figurated, which figure in the 1555 and the key of the Algebra of 1560. My question is, do you find the elements of structure between the one and the other, or is It's rather a hypothesis. The concomitant, it's the disparity of the novel. However, the parent is more between the little doctrine on the non-refigurations that we found in 1569, and the novel on the non-refigurations that we found in 1555. It's there, in the 1569, that we find these elements. there are no elements of common? No, the algebra 1560 is a copy of the algebra of Chevelle with simply notations different, with an organization different, a different structure. I ask that question, because as you have described it, we have seen that with a 6-6, we have seen that this part is related to the algebra, but it's not that you wanted to say? This part on the figure had a rapport... It's the number figuré as an object of the study in a branch of mathematics. In the first case, it's attached to the arithmetic,

1:07:30 and in the second case, it's a part of the arithmetic, but in a separate case. But these are the same objects that are listed, But it's not just the same content, it's the same form. There are some other elements communs. Marco? Yes, now I have always the same problem when we talk about algebra at the end of the 17th century. I have always the same problem when we talk about algebra at this time. So there, it's okay, it's fine. I'm going to talk more quickly. It's quite extraordinary. It's not very difficult, but I can talk more quickly. I have always the same problem. But there is a context, I suppose it's very horrible. Because it seems to me that It's all this project, and in conclusion, it's about a passion, an ambiguity. Not at all, but at all. And it seems that you have already shown it. And I read in page 5 of the commandment, algebra, F, part of the medical, which is what you do, by the way. And on the other hand, when you talked about this, you said something that I have not seen, but I think you said something about it. There is no real expression of the genre that you let yourself know, as there is a word, which is what you do. There is no object of the representation of the gender that is supposed to be like idealism. You can see, it's very complicated. We are not able to do this in French. We are not able to do this in French. So you have to do this in French. And from the other side, you don't know, but the conclusion was in a sense that it was It's clear that there is an effort to make an algebraic discipline. You said to bring everything there in the algebraic discipline, so that I think it's very problematic, because everything there is in the algebraic discipline,

1:10:00 I don't think so much about it. Yes, I do. Well, it's another question, but it's the second part of the question. I can't concentrate on the two of them, to elevate the algebra to the level of the algebra But what is the algebra the algebra is a notation for the two objects that they can't identify The question that I have is that I have the impression that for us the things are pretty easy to be regained because the algebra they are in theory On projette la division moderne, on la projette sur l'époque, etc. Mais au fait, cette manière de voir les choses ne nous rend pas absolument pas compte de la situation de l'époque. Je ne crois pas qu'on puisse dire qu'il y a trois théories ou trois disciplines en science. Ça dépend du sens qu'on donne à ce terme. And, by the way, it is very difficult to understand why the algebra and the algebra are different if the difference is made by the objects that they are concerned. The objects that they are concerned, at least in the Hamel, are the same. Well, it's not the same. It's not the same. Yes, the same. The same. The same. Yes, of course. Yes, of course. So it's not the same. But then, for what? Because for Pierre Delaramay, the numbers are not the same. We have all the numbers, we have all the irrationnels, and then we have, So, what? The irrational, the binomes, the residues, which are the classes? Well, I believe that the difficulty that we have to do with things, and that for us, when we talk about the arithmetic, the arithmetic has to be a number. But in a sense, like we, we have to be a number, in which all this, there is a number. In our conception of the arithmetic, there is that. Well, if you read the book of Euclid, there is something else to say. There is something else to say. And I have a little impression, because I would like to have an affirmation, that what you're saying is that the algebra is all the numbers that we can't read

1:12:30 if we read the book of Euclid, the book of Euclid. In particular, it was a problem. It's an animal that you could use to work with the binombs, it's the solution of the equation of the binombs, it's the way to treat the binombs, it's the way to treat the binombs, but it's not an other thing with other objects. I don't know anything, I don't know anything, I don't know anything. What I'm trying to see is what it could be before that it was devenu something else. What is it for Pierre Delaramette? Lui, l'ambiguïté que tu soulignes, elle est bien là, elle est entre tout ce qui le concerne de plus classique dans la mathématique universitaire. La mathématique universitaire, c'est parisphétique géométrique. And then, a tentative of renovating the curriculum, a tentative of introducing new content, but with a tension which is huge, because normally the content of science is fixed, and the content of science, for the mathematics, well, it's Euclid. and then, in the geometry, he will introduce a little bit of stuff on the iconics, a little bit of this, a little bit of that, but the content is completely fixed. So, what happens to the algebra here is to bring the algebra to the traditional algebra. I think it's useful. How to do it? Well, there, he's a bagarre with this story. So I'm trying to tell you, if I understand, that the way to tell you what to do is simply a way to avoid the difficulty, but if there is something new, but we call it a part of it to be able to put it in the colicolons? C'est-à-dire qu'on doit mettre quelque chose de nouveau, c'est ça que je comprends. On a quelque chose de nouveau, mais c'est quelque chose de nouveau où on n'a pas le droit de mettre parce qu'il faut qu'on reste au classique. Donc on appelle ces nouveaux parties de la numérique et on a résolu le problème.

1:15:00 Donc c'est en fait un tour de passage d'appeler l'objet de parties de la numérique. Mais je ne crois pas, je ne crois pas qu'est-ce que c'est... La ranée... Et d'ailleurs, toutes les conclusions sont de contraires, But it's not because it's not that intelligent, it's not because it can't be able to live, because there are no objects. Because for having objects, it's a different way. The way that people think about it, it's not that they can't be able to live. So we can continue to think that in modern age there is arithmetic, arithmetic, arithmetic and algebra, but this is not the way we can describe the situation. There is a theory of numbers and a theory of grandeur, and then there is something that is very difficult to define, So it's a problem historically to define it, but that it's not another theory that's a good idea. So it's not a difficult question. And that I'm not a critique of your exposé. I think that, what you're showing is that your exposé, it's these difficulties-là. And I'm happy that you're showing it. Yes, yes. Like you said, I'm pretty sure. Yes, yes. There's nothing to answer. And now, my second question is, I don't have seen something that I haven't seen before, but then what do we do? So the question that I'm going to ask is that if we, as an arithmetist, we think, as an arithmetist, as an arithmetist, as an arithmetist, and something else, but not even more. If we think about that, we can't say that, so, we need to put something else that we consider the number? For example, the algorithm for calculating the racine, because I know there is no algorithm for calculating the racine. It's the case, it's the case, but it's the case. So, it's something else, that we need to put something else that we consider the number? Yes, of course. So these things are called algebra, but in fact it's important to work on the numbers. It's just a way to work on the numbers.

1:17:30 Algebra, it's what we want to say about the numbers, that we can't bring it to the structure of the system. Is this a way to do it? You always put the comment on the version after. I'm in the Avant. What is that after the Aramee? It's all different than the Aramee. What is the Algev? Algev is the Algev. The Algev is the Algev. The elements that we find in the Euclide are the things that we find in the Euclide, and then it's the traits of the Arrhythmia Contraty. Just simply the traits of the Arrhythmia Contraty, they are not apt to make an object of an introduction in the field of mathematics. It's a savant. Well, that's what I do with the path, the path, the path, that's not my problem. Well, it's the Smith. What is the figure? Well, it's the part of the arithmetic which is on the number figurative, so on a particular class of the number. I don't know what the codes are. C'est ce qu'il écrit. Écoute, là ça ne parle pas de nom figuré là. Et si, binoméridus sont des noms refigurés, les irrationnels sont des noms refigurés, et toute sa nomenclature... Ah oui, oui là, le nom figuré, c'est ça. Ca permet d'expliquer en quoi on voit des noms figurés chez Zidane ? Je vois ça que c'est des noms figurés. Oui, alors comment... Voilà, comment est-ce qu'il importe cette terminologie-là dans Euclid ? S'il y a des définitions... It's not a square or cube. Yes, but it's not that. It's the only object in the book 3 of the arithmetic that he does the theory. The theory is the cube. The figure is not the sense of Nicomac. It's not that he doesn't cite Nicomac. It's even the sense of Nicomac. Nicomac, the Greek edition, which is 1538, is not Greek. Yes, it's not that you can't say anything. Because there are things in Boes, and there are things in Boes. Yes, Boes, he can't ignore it. Boes, he can't ignore it. But, why is he never...

1:20:00 I tell you that there is a lot of information that depends on the non-figuration and that the non-figuration of the non-figuration has the possibility of reading the non-figuration of Boes. Yes, but I think that's not part of these lectures. Yes, absolutely. I don't know, but it's not what they recognize as... I think it's really important to emphasize that they employ an expression with a sense... Well, if it's true, I don't know why my definition is very genomic, that is the part of this algorithm that isnotic. It's entirely a algorithmic. It's very, with the algorithmics. Second, with the algorithmic. It's not clear if you don't have… But no, no, that is the first one. That the 7 periods, the arrays are not lit, maybe the 4c… And therefore, we can operate a little limous rule. No, because in the Eclipse you have numbers in terms of the car, in terms of the cube, and it's not the algebra. I have the impression that it's the thing that you do. I'm not sure what you do, because it's not that we don't have any numbers. It's not a theory of the operation numérique. Because the algebra is the theory of the operation numérique. No, I'm saying that. What I'm saying is why I think there's algebra in Eclipse. It's to understand. This is a question. It's true, but it's not a problem. No, no, no, no, no, no, no, no. Wait a second, wait a second, wait a second, wait a second. If the non-figure is the square or the cube, if the algebra is the part of the algebra, then there is the algebra, it's the number 8, which is the square or the cube. So I have the impression that if you want, it's in this way that you need to go. And not in the sense of the algebra. I don't do that. There's something to say, right? Yes, but the example, the table of mathematics... The arithmetic? The arithmetic. I'm trying to figure out how to do it. Yes, but it's an exclusion. What? The arithmetic. So, in the algorithm, there is the number of numbers pervers and ampères, which is the operation of the addition of the division, the proportion of the number,

1:22:30 and the proportion of the genetics. So, in fact, the proportion of the number is the number of the operation of the division and subtraction of the multiplication of the number. And then, in our study, there is a theory of Carvel, a theory of studies, a theory of one genome, and all that? Is that what you mean? Yes, in algebra. There are two things in Aramé. There is an ontology. this ensemble of objects that visent the mathematics, but that visent through a doctrine, a doctrine which is a book, essentially. The doctrine, from Pierre de Laramay, is the form is written. That is also the novelty. It gives an essential importance to the written as a vector of science. So, the algebra is always that, and we give him his manual, he gives him his hand. There's this ontology, and then there's the form written. The problem is to make it correspond to the two. For the arithmetic, it doesn't happen too bad. We take the materials classiques, we mix them in a new order, and it's fine. How do you think? I think in the book, the theory of the number of the functionnaires, for the algebra, it's even more complicated. Because already, division binary, theory of numbers simple, with their different classes, rationnelles, irrational, and then theory of numbers comparer, equations, because why an equation is a number and a comparison of two numbers? The problem is there. The problem is there. The problem is the syntax. You have to give a syntax to the algebra. What is interesting is that in the extract Salignac that you have, what you have said is that in the algebra of Salignac, Salignac is a part of the order of the order of the algebra. What he introduced, it's the notion of... ... in his context.

1:25:00 There's no binomes, no residues, no categories distinct, but there's no binomes. So all the expressions that we can form with the symbols of the algebra, will be binomes. On aura des expressions, on a un langage unifié qui permet de... On n'a plus des classes distinctes, on a un langage, on a une syntaxe. Les buts sont... Après, il pose la syntaxe. Et là, on s'achemine vers quelque chose que tendait, que disait l'aramique, qui était donner son autonomie au langage. Alors, on reste. Est-ce que je peux justement bondir là-dessus ? Puisque ma question est portée. Si tu veux, je pense que tu utilises le terme écrit d'une façon qui mérite à être décidée. Tu dis, en gros, le travailleur intellectuel, c'est un lecteur. Et tu parles de l'importance de l'écrit. But, if you want to show the page of the algebra, like the page 6, you have the impression that the intellectual has to use to use a notation and a disposition to work. And at this point, it's not so the reader who is in the game, it's the person who is in the game. So, when you talk about books, if you want, like the director of the culture, all of that, is it really the book in time of the movie or is it rather the instructions? And in relation to that, if you want, what I have said earlier in your exposé on this theme, it is when you said that he talks about notations by écrit, that is in abaco. Yes, it's a translation. Yes, it's a translation, but if you want, you can see that, precisely, in abaco, it's something, it's an inscription that you do, which can be temporary or inscribed in the book, but it's more an inscription at work than an inscription on the book. Yes, but then, there-dessus... So, if you want, I'm going to look at these two things, which are disjoints, and see what relates to the one and what relates to the other.

1:27:30 Yes, well, on the tradition inadako, I've been hesitated, but there are two occurrences in the passage that I've given. And I think that on these two occurrences, it colles. Well, now... It's a definition, by writing... You can tell me about two things, right? Yes, yes. After I have another question, it's a little bit more and more, it's... First I remind you of the definition that we've also cited. When we present the notation for the rhythm, we say it's an alphabet, but not an alphabet of the voice, but also, an alphabet of the hand and of... I have two important things. Yes, I will continue. The book published anonymously, 1560, what we can suppose, because the ramé has practiced this kind of exercise, is that it was a sort of support. I didn't want to go there, because we don't know if he taught this. We imagine it well, of course. But in any case, it's always something that should be accompanied by the oral comment. So, parenthesis, I would like to say that there is a project on the co-registral, with an enormous accent on the co-registral of the CCC, which is actually a new idea. And they have found a lot of documents that would be a type of publication and the way in which the students worked with their students. Yes, there was a little inventive that had been done by Randbrouille and a little bit of a publication where we found traces of these letters who were published and published, at the demand of Professor Royaux, because it was them who practiced this the most intensively, for that they were following before their course. On n'avait pas le droit de prendre des notes, on n'avait pas le droit d'écrire pendant longtemps, mais en même temps, c'était sans cesse de transgresser ces règles-là. Il y a des notes de cours sur des leçons de Laramé, sur des textes de groupe qui ont été imprimés à sa demande pour son public.

1:30:00 Ce type d'écrit là, un petit peu bâclé, ce type d'imprimer un petit peu bâclé d'Alger, il arrive peut-être qu'un 162. I consult the historians of the book, is there a support of the course? Where we would put the two kinds of books that you distinguish? It is written in vivo, the manuscript, the practice of the hand, and then the form of fix, the form of fixer of the training. And this is why we don't have to do it. Yes, because there we have to do it. It's not that it's not that it's that it is frappant. Absolutely, it's why I found, if you want, that there was a problem with the utilisation of different methods. It's the way to read this thing, it's easy to read. Yes, we have to do a comment. We have to do an aid. And it's even worse than that. What you write, it's not to read it, but it's to work with it. It's to work with it. It's an écriture to work with it. But in the same time, I have the feeling that when the Aramee publishes, it's to inflige the enseignments. So that's it. There it is, it's a form of intermédiaire, but there's no other... What you want me to say, if you want to continue, is that we would have here an illustration. It's an illustration. It's an illustration in the book of the notation with which we are going to work. And the statue of this thing, which would be an illustration? In the meantime, I can't pronounce it on the usage. It's to say that we are in the hypothesis. The simple hypothesis of French. that is possible. But there the historian peine to know the context, the usage. I think if you want to look at it, I think you could perhaps, in creating the reports between the text, in creating the exhibition, etc., in analyzing, you could certainly To absolument avancer dans aujourd'hui pour ta1500 conversation.

1:32:30 Je te propose ça en raison de l'accent que tu mets sur les cris, la lecture. Ça dépend du thème sur lequel tu m'oisliqué, mais je pense que ce medien de préciser de plus On va peut-être s'arrêter là on a appris une pause. It's at 4 o'clock. There's a cold coffee for the amateurs. And we're going to take a cold coffee for the amateurs. I'm going to take a cold coffee for the amateurs.