Lobachevsky / Von Staudt, Steiner, Poncelet

0:00 by a point which is here E, d'accord, on this particular AF and then on tourne, on tourne AF sur this point AG here, d'accord, at the same time AB, this, it turns a AK, and there is an intersection here, because it is parallel, it is parallel here, we turn here, the intersection is black, ok? Now there is this point L in comparison to the page, the intersection, you see? And now we turn to the inverse, and then this L, it's not L', that means that this SE is actually A-B and it is said that it is parallel to A-B. It is to say that already we are talking about the point choice, because we have to prove that it doesn't depend on the choice. D'accord? where, finally, the thing that is transitive is a bit more difficult, I guess. But it's the same type, the same type, absolutely classic approach. Sauf that we are in another space and we make an image a little differently, etc. Well, that's the first part. It's to say that it's a relationship with equivalence in condition that we respect the sense. It's to say... Because in the second sense, it's a different thing. Well... The second point, also, on the geometry... It's just to show that if we have an arbre right, an arbre aigu, it is possible to always find a perpendicular here,

2:30 that it is parallel to the other side. This is something which is interesting and bizarre. I think it was known before the Batcheson, it was a legend, it was already that. You choose, let's say, you take a distance here, you take a perpendicular, and then you take a perpendicular here, and then here, like that, And we see that what we call the difference between 2 Pi, the somme of the angles, and 2 Pi. They accumulate and they are growing, strictly growing. It is to say that they cannot become negative. And to this reason, there is still something like the principle of 2 Pi. So it's to say that there is K-linik here and there is no more torsition here. D'accord? And the consequence directs is that if you have an angle aigu, you can always find a right, that it is parallel to the two sides. Something like that. That, I'll use this consequence after. I would say that at a certain moment, I would not like to modernize things, but to simplify things. Because there is a certain reasoning that is so complicated, and now we can do the same thing in the same way. But in fact, this is also remarkable, because there are some properties of the history of the anthropological theory that ressemblent to the Caioquedian. This is what we call the theorem, if we have three planes. And this is remarkable, because at the beginning, as I already noticed during the conference of Jean-Jacques, that for all this theory, it should already start by three dimensions. Otherwise, I don't know. I don't know. If we have three planes, and that the lines of the intersection are parallel, of course, in the Sarge Labatchevsky,

5:00 so these triangles here are all the same. They are the same as the same. I don't know, we're going to pass in dimension 3? We're going to pass in dimension 3? That's what I want to say. And if not, it should change everything. D'accord? Well, that I don't want to... I don't want to... I just want to replace it by another theory. It's after Chirot, like you said, we can replace it by another theory. If we plant here, we are always in the space thermally. D'accord? On plane ici, on appelle que 3A est parallèle à plane, c'est parallèle à sa projecture orthogonal, bon c'est vrai, je dois le dire, et après le théorème, et il est théorème encore absolu, c'est que 3A est parallèle au plan, il n'existe qu'une seule plan B, bon parallèle à plan alpha dans le sens disons normal qu'il ne coûte pas, il y a le seul. It's to say that it's an analog of the 5th century which is still valuable in the perforate position. And that the problem is also simple. The first thing is how to make this parallel parallel. This is a parallel parallel. And the definition of parallel parallel is the same. It doesn't change. I don't know, I don't know, of this plane-là, L, A, B, L'. L'', it's the orthogonal. And now, if we take the other plane gamma, we can show that this gamma is equal to alpha. d'accord et maintenant je prends L sur gamma en droite et j'ai fait cette construction AC projection octagonale DAB sur ce nouveau plan et BD projection octagonale de AC et maintenant on voit bien que And this angle BAC, it's more small than BAL, and it's in the same plan. Why it's more small? Well, it's not possible by construction, but it's more important that this angle BAL,

7:30 it's the angle which corresponds to this function, which is the limit. Because I said that it's parallel. So that's the geometry, and then we can see it. I can allow you to interrupt again once again. You are passing in dimension 3. Auparavant, all of your definitions were on the geometry hyperbolics in dimension 2. So what do you mean? Yes, yes, yes. There is a lot of things, but a lot of things. Just a little bit. There is also the geometry hyperbolics in dimension 3, right? Yes, yes. and it even begins, you see, it doesn't make dimension for dimension. It's not the case where it begins with the plenumetrie, after it starts with the styrumetrie. It's not like that, it doesn't start with the styrumetrie. All these constructions are only with the dimension square, even with the plenumetrie. And I'll finish the construction, I'll finish. D'accord? And we'll see why. because in fact they use this P, that's the base. That's the theorem that, if you want, you can replace it. I don't understand that there is a theorem that says that there is one single plane that is parallel to the alpha, and it contains a right-wing. That doesn't seem to be true in the case. Yes, yes, yes. Let me see. Because that's a little... On a right, L is parallel to the plane 1. Well, there is nothing. Because L is parallel to the plane, it is parallel to the plane. It is parallel to the projection of the plane. That is very false. You see? Parallel to the sense of the plane. So there is no other plane. Until that L is in this plane, and... D'accord ? D'ailleurs, c'est pareil avec la progestion entre les mains, c'est le sens... c'est évident, ça ? C'est, disons ça, ça c'est devient du Réme absolu, c'est les mêmes. Bon, maintenant on passe le troisième point de ce soir. Bon, on sait que, car il est bien, il y a trois types de faisceaux parallèles. Fissos de droite, fisso parallèle et fisso, je ne sais pas comment ça s'appelle, qui se croisent dans le même point.

10:00 Et dans le cas de Mathieu, il y a un troisième type en fait. Il y a trois types, ça c'est le même. Il y a droit divergent, comme je ne sais pas par le Jean-Jacques, et bon, divergent et tout perpendiculaire, par exemple, pour notre droite qui est appelée un axe. and, on the other hand, there is also the right to the parallel, like in the previous case, but this parallel is special. Now, we can look at the line in the space, if you want, the normal surface of this vessel. Yes, here it will be a circle or a sphere, like in the previous case. It doesn't change, it's important, because the geometry of the sphérique will be the same in the hyperbolic and in the california. Here, as I said, we have a right on the plane, but here there is a line where a special surface, which is not an analog in the california, which is called sphère, or cercle-limite, or surface-limite. This is the definition here. Do you want to mention a little bit your definition? But it's true that Balcharski, here, he considers this as the limit of the cercle. It is to say that if the rayon is infinite, then the cercle becomes the cercle. And now, yes, what he said, even if something is wrong, but in the sphere, in the sphere, the circle of the region. And now, the case, the principle is that in the sphere, they share this property of the right of the circle which, in fact, I see Lee found... Chez Lee, there is a term to say that the Eglise This surface is the same. We can say that it is constant. But at the time, we did not think about the curve, but it was something that you could put in a piece of this plant and apply n'importe where. And that is, we can do the measurements there. We can do exactly what it does. Well, here,

12:30 there is a piece of parallel. here, there are several, let's say, for the circle, and it takes an hypothesis that this relationship between the arcs and the circle depends only on the distance, on the distance. And from where it turns out that this function is exponential? I don't know, I don't know, because it's a very valid argument, but the idea is very simple. functions, c'est que f de x plus x, c'est la même chose que f de x pour f de x, c'est immédiatement on en donne 1 plus 106. Et après, ici ce point, intéressant, il dit bon, on peut toujours choisir tous les langues de la même manière que f de 1, ça sera E, la base de logarithme naturel. Et ça, bien sûr, c'est où vient cette idée d'unité absolute lambda. Il y a discussion, mais pas ici, dans le nouveau élément d'une. C'est quelque chose que, bon, il y a cette unité absolue, par contre il y a une espèce de choix arbitraire. Maintenant il n'est pas arbitraire aussi quand même, mais eux c'est pas quelque chose d'arbitraire. Bon, aussi je saute ce point parce que même si je crois So it's very, very important to understand what is this absolute unity. Because there are still in this occult, even today, with all the chemistry non occult, I don't know, I don't know, I don't know, I don't know, this idea, I don't know, constant de plan comme quelque chose de naturel, ou, je ne sais pas, la velocité de lumière comme un truc naturel, ça paraît un peu bizarre. Dans ce cas-là, on voit que ce genre de choses sont tellement naturelles comme un V3A. Mais là, quand il retrouve cette unité absolue qui est en fait une constante négative, tout simplement, He does, in fact, these relations functionnelles that satisfy the function of F, the actions that are defined at the beginning, the fact that we can compare the length of these quadrilataires and if there is a dilation intrinsically, is it a consequence explicitly of the actions of the body at the beginning or is it an action supplémentaire implicit?

15:00 It is a bit more than I, of the way that we can make the measure. This property-là, when I said that each part... There is a property of homogeneity of the measure, which includes the free mobility. Yes, yes. And also the fact that there is this percent absolute, which is the case. It's a good thing. It's a good thing. A is a complete. But then you can always choose from a normal way. You can't say anything. I'll never forget it. I'll never finish it. I'll never forget it. Now the most important point of this story And, well, I cite, the triangles of the surface limit, that is, on the sphère, have between their angles and their sides the same relations that we demonstrate to exist. It's a bit hard, like the graph of Batchevsky, that is, it's a double. The same relations that we demonstrate to exist in the triangles rectilignes in geometry ordinary. It shows that we can talk about some models here, but of course, we already see this model of planclidium in the space, not clidium, but not at all. In the sense that we are used to do the square, etc. And now why? In fact, it's simple, I think it's important to verify all the actions, but I'm just going to verify the 5th postulate. It's immediate to this theory that I showed you earlier, because what are the doigts here? Yes, it's the gap between the planes. Because there is a parallel, with the parallel, we don't know if there is a plane, and for the planes, as I said, it's analog to the 5th postulate, and so on this surface, It's a tendency to plonger in something a little bit bizarre, to develop this new intuition, and we work on that. And then we fall on something which is, if you want,

17:30 more strong than Schoenberg, because we fall on something that we think about surface, limit, something bizarre. Then we say that this is the proprietor of the occult. It's something that we plant occult. It's the same genetically occult. And from that, we can do everything. Again, I simplify it a little because the bachelors it's not the easiest way and the idea, of course, is to this horosphere on a plane tangente, d'accord, on a point donné, c'est-à-dire qu'on sait que sur ce on risque c'est un peu la géométrie collettienne, c'est-à-dire qu'on peut faire une trigonométrie normale, et les angles sont partagés, ça c'est important. Si j'ai le plane tangente, je l'ai même dans un angle, je vais vous montrer tout à l'heure, et en gros c'est ça, et à partir de ça, on développe trigonométrie des plans, qui sera hyperbolique. Vous voyez, c'est inversé, c'est-à-dire on a tendance à penser hyperbolique comme courbe et comme, je ne sais pas, plat, mais ici il faut penser à l'inverse. C'est plan qui est plat, comme toujours, mais c'est le collédien qui devient courbe. Bon, et peut-être, je crois, ça aussi clair qu'on peut le faire, hein, je saute ou je développe. Well, this is a modernization, a little modernization, because what do we do? We have two perpendicular doigts. We take here this circle limit tangent, point O, d'accord? And we are interested in this distance to A, and how to A to B, which I have noted S, and this little thing V here. We recalculate. And then we use the formula from earlier, to compare the two A. And all of a sudden, we get to this kind of thing, this formula, that is a rapport between O and V, here and here.

20:00 These are two characteristics. And we are interested now, because here we know how it works, we already know how it works, but we are interested in this kind of thing. d'accord et c'est qu'on obtient bon ça c'est juste ça m'a dominé d'abord ça j'ai introduit une connotation que si nous c'est hyperbolique si nous c'est hyperbolique la base on peut couper ça e plus en x plus e plus en moins x sur 2 ici e plus en x moins e plus en moins x sur 2 So this is the triglyceride, the tangent triglyceride. And so here, we already have three dimensions, on the triangle rectangle here, the triangle rectangle, on the parallel, on the third dimension, the parallel, and here we have the horisphère, the sphere of the mu. and now we can calculate S2 and S3 and for S1 we can just go to the second one here because here we can only calculate two sides here because it touches this point-là so it's already simple and now, from this formula And here, for example, we can calculate each formula in its hyperbolic formula. For example, if you look at the theorem of Pythagore, it becomes this thing. It's not a substitution, it's not a problem. The theorem of Cosinius, it becomes this. But it's important that here, there are angles... Yes, of course, I wrote cosinus hyperbolic of A. And what does that mean? In the sense of Bachir, what he would say, is that this thing is calculated by its function. Because that, it gives a rapport between the triangles and the angles. So, as I said earlier, every triangle is defined by... And it is defined in terms of metric, by Cézanne. D'accord? Well, and we have to do the same thing,

22:30 but what we observe very quickly, it is that, that is very similar to the case sphérique known since, I don't know, Ptolemy, or almost. D'accord? That if, if we place... On pense que notre rayon de sphère où on fait gémitry spirite, on prend racine de moins 1, on en vient à notre formule, c'est comme bien tout de suite. Et c'est étonnant que chez Lambert, on trouve cette remarque, mais bon, c'est dans cette story, quand même, je ne le dis pas sérieusement, mais c'est vraiment prophétique, because he says something like the third hypothesis, as George explained, the satirical, the case hyperbolic, is perhaps verified on a certain sphere imaginary. That's exactly what he said, Lambert, without knowing what he said. I don't know. He had written the line. Well, I don't know, but it's difficult to imagine that he really had a little bit of this thing. Ah, but he did Lambert. L'autre remarque historique, en fait, c'est que la même année 1840, quand Zobaczewski publiait son travail, c'était une autre publication en journal d'écriture, par Minton, peut-être que vous connaissez, où il observait en gros la même chose, mais il n'est pas travaillé avec géométrie non-politienne, avec axiomatique, il est travaillé avec with the communicative constant. And he noticed the same thing. And in fact, there is a book where he analyzes what Batshevsky did, what he learned in his Biblio in Kazakh. And they had this journal, with the work of Mindy, but apparently Batshevsky didn't touch it. That's it. If we can imagine that Batshevsky was touched, of course, he could already have an identification with a negative constant. And it's something that Batchesky doesn't ever talk about. Now, it's just two words. What is Batchesky he-même? Because that's a simplification. He's doing something a bit bizarre. Maybe he is guided by this idea of, I don't know, analogies, dualities. And in fact, perhaps, we can clarify after, with all this work that said Jean-Jacques,

25:00 what is the sense of this rapport between the scales sphériques and the scales anthropological that Batchesk has never very well understood. Well, there's a small sphere here, and there's a transformation of the triangle here in the spherical sphere, in fact. And then, there's a small sphere. Well, I don't know. It's the same thing, but it's a lot more complicated, maybe more interesting, and maybe there's a little bit of protection. I hope I'll explain it. Here, we use, I think, what, in France, it's called the patron. It's a little bit more complicated. And finally, at the end of the theory of the sign of the cause Pythagore, he cite 15 equations. Well, it's a sort of 4 generic, but I think it's 15. and with its function, and if it actually marks this duality with the case sphérique, it is remarkable that the equation of the geometry plate, which is hyperbolic, for the plate it is hyperbolic, and the last citation that I have done is the work of the fundamental to answer these questions. Is there a argument for consistency, etc. and I believe that it is the most... Well, it is not the only thing I have done. Yes, but you see, it is bizarre because in fact, if the theory is consistent, I would like to say consistent, but I would like to say consistent. But complexification, it's kind of... It's kind of... It's kind of... But for me, it's absolutely formal. D'accord. Well, you can say it's not much like that. But it's an idea, yes. In fact, there's a view... Analogies... Or maybe it's more than analogies, yes. On peut peut-être préciser... Dialogies in a sort of... It's a complexification. Complexification. It's a variation. You replace the variables in the variables complex. So, it's an analytical. The series is already covered. It's a different variable. It's a different variable. It's a simple parallelism of a formula.

27:30 And it's for that it's enough. It's like you saw the duality as simply an asymmetry between the annonces. But that's an event. It gives a sense of math. But for Walshiewski, like you said, it's a simple parallelism. And in the way I can... I have... the theoretical theoretical is consistent. The theoretical that I make, is consistent. But of course, because you can perfectly have a theoretical theoretical consistent. Well, I think the first geometry is that it is really historical, very special. However, all these assembles of work, there is still a difference between French and English, and also I think there is a lot of comments because the most important comments that there are in the edition of US, It's not on the interpretation, etc. Without this commentaire, it's very difficult to read. It's a great job, I think. If we can reduce this commentaire, and if we take this example, it's something like 1000 pages, or something like that, and I think it's necessary to do it, because otherwise everyone knows the name, and he cite certain things, but it's still a great field of historical work. Well, thank you. So you need to go to the room. It's about 12 hours. How about ? I'm going to go to the room. It's like God is Yeah, it's like God is Yeah, it's good to have two hours The last time, the 11th match

30:00 It was a little secret to be exposed Yeah, but also First time, it's good to have It's amazing It's the Wichat that you have to come with. I'm going to find the Pancaribu. You can find the Sto-Triman. The first one at the beginning. Yeah, the Sto-Triman. Just as conclusion, I think that, in fact, this idea of interpreting, in theory, in theory, that's more important than what we found in the HBERT, what we found in HBERT is just a model possible. You can go through the bureau before you get it. I'm on the bureau, you can go through it. I'm with a... But I'm on the bureau, it's on the bureau. I can go through it. I can go through it. I can change my computer. I want to open your logiciels with Aperçu. You can open it with Aperçu. You can see it here. You can see it here. There, there. There. There's another. There's a picture of the PDF. There's a picture of it. There's a picture of it. Super! It's difficult to see it. Yes, it's very difficult. So, we are going to change two genres of geometry. We are going to pass from the geometry of the Lovatshevsky to the geometry projective. It is a work that I have done, to make a story of the geometry projective from the point of view synthetic, not from the point of view analytical, but from the point of view synthetic for the 19th century, between Poncelet and Watey, an English-American team, Redland. And one of my surprises in this story, it's a thing I didn't expect,

32:30 because of the time there are surprises in the history, and there are results. It is the importance of Von Stoet, not only by its results, but by its conception of mathematics. I'm going to discuss with you today about what I call the radicalism synthétique, I could say the radicalism geométric of Von Stoet. Well, I'll start a conference of history in Germany, which is about three quarters of biography, and then one quarter of mathematics. And quickly, you can use it historically. So, he is born in 1718. What is important about his formation, he is working with Gauss. and so it's also an encouragement to work on the side of the geometry projectiles he did all his career at Erlangen and he disappeared in 1867 one other surprise is that when we do not have research well then the books that will interest us it's in 1847 he diffused a geometry of Erlangen which I will talk about and the supplements, which is between 1856 and 1860, in which we will find, in addition to his general theory of the geometry projective, a theory called the theory DG, which is a way to introduce the algebra in a procedure purely geometric, which we will try to convince you, and a theory, and I think it's essential to his ambition with the Bay-Trigger, a theory of the imaginative elements that at the time posed problems in geometry. Just to remind you, if you click on Google, it depends on the day, but Stout, eh bien, on ne tombe pas trop sur le côté géomètre, on tombe sur l'archiméticien, parce qu'il a fait un théorème de Stout-Clausen sur les nombres de la logique, et il est beaucoup plus référencié dans les moteurs de recherche pour ce théorème que pour ses travaux en géométrie. Alors, quels sont les objectifs ?

35:00 Donc, il y a une petite intro, et puis, quand on lit la géométrie derrière la gueule, L'ambition de von Stott, on peut penser, c'est reprendre un projet, et on va voir que c'est le même projet que Steiner en particulier, mais fonder la géométrie sur la seule considération des relations projectives entre formes, formes fondamentales, formes au sens de la géométrie projective, c'est-à-dire les faisceaux, les ponctuels, les gerbes, des choses comme ça. The second trait, when we open it, there is no need to read it, there is not a figure. It is to say that you have Géomy Scheler-Waigel, Leibail-Trigger, there is not a figure. One revendication dès le début, to say that they constate that the geometry synthetic, the polysynthetic has made a progress at its time, but that the people are not all at the end of the analysis and research, because it is constantly, the work that they would like to do, those of Steiner and maybe those of Concelet, they do call to the measure. There are measures of angles and segments in the works of Poncelet and in the works of Steinem. And so, we can read, and I think it's not only an effect historiography, it's really part of the works and ambitions of Von Schott to place his work to radicalize the point of view of Steinem. So we'll have to discuss a little bit. But so, I'll remind you, I'll remind you, Steiner, it's a little before, it's 1830, 1835, or something like that, it's in my own room, he worked a lot in geométrie, he worked a lot in geométrie, he worked a lot in geométrie, and his ambition is to expose and organize, and so on, it's almost a citation, in a way systemically, all the corpus of the ancient and modern geometry.

37:30 It's the ambition to structure completely his exposure and his work around a certain number of properties that will refer to the germ of all the theory the problem, the problem, the problem and the problem of the geometry. And here it is extremely important, because we see that there is a new way to do the geometry, to do the mathematics, it is to do the theory, to organize the theory, and not to be attached to the problem. It is to say that we pass from a geometry of a problem to a geometry of a theory. So, to find a field director and an origin common to all the geometry, which are separated from each other, also my job. And, of course, the last one is important because there is a reference to the nature which will allow us to see the point of view of Steiner between geometry synthetics and geometry analytics. So, very quickly, on what Steiner fond of his work? Well, he introduced a certain number of objects, which are already known, which are banal objects, but it's from there that he will work. These are the base objects. The right, but as a support point, the vessels, so that those vessels of planes, they say rayons, So, it's a vessel of planes or a vessel of rayons in space, which is called in French, at least in the French, 1920-1930, long gerb. It's a word that has a little tendency to describe. And the main notion, it's how to build a correspondance between these forms. That's it. And, so, Steiner will introduce a notion of relationship projective between forms. So when I talk about form, it's a reference to this notion of form fundamental, not a concept of form. For me, a form, it's like that they call it. Then we talk about the idea that the right and the right and the right are in situation perspective. I have the figures just after. You take your face and you make a section by the right. There, the right, the points of the right, the intersection of the right and the right of the vessel,

40:00 form a perspective. So the right and the right are the perspectives. Then imagine that without moving the internal structure of the right and the right, and the right and the vessel are in a public situation. This is how it is defined. For example, here, it is clear that the vessel right here, is in perspective with the points of this or of this point, Then, by analysis of consistency, it is clear that the points which are in perspective, the points that are in perspective, which are in perspective compared to a same field of right, will be viewed in perspective. We can, between guillemets, dualize situations. The right can be in perspective when the right correspondents are on a punctual point. Here you have a right and a right in perspective, then you place... I didn't have to turn my right, Imagine qu'elle est tournée. Rotation refuse de marcher dès qu'on met des lettres. Et le faisceau soit en perspective oblique. And the intérêt and the idea of the position oblique is that we can come back to a position in perspective. It's to say that one right and a faisceau are in position oblique if, by a movement of the two forms, we can bring them to a position perspective. This is the first introduction. Then, Steiner introduced the notion of bi-rapport. So, as I mentioned, bi-rapport between the point of the right, or bi-rapport between the right and the right. And, of course, there is a character between the right and the right and the right

42:30 étant en perspective nous avons cette relation d'égalité des bi-rapports, donc il y a une conservation par perspective et donc puisque dans le déplacement on ne bouge pas du tout les situations internes des droites dans le faisceau et des points dans la ponctuelle, et bien ces rapports-là restent constants et donc Steiner conclut au caractère projectif du bi-rapport. That was already... After that, it was already... ... the art? No, not at all. That art uses this kind of thing, yes. But that... That's what I think... I think that you can serve to define a certain number of transformations from which you work. It's the objects that we serve, so that it's the objects on which we are working to fund our theory. It's the length of segments? Yes, it's the length of segments. So there was a measure? Yes, I'm talking about the Steiner. The difference is that Van Stoet was refused this part. It's super, it's the question that it had to be posed. And so, definition of the notion of relation projective from the bi-rapport and the generalization of this notion to all the others, in particular, so on the plane, etc. Now, let's continue a little bit, because what did geometry, at this time, is a theory icon. C'est-à-dire que la théorie déconique, on va dire que c'est la mesure à partir de laquelle on va juger la qualité d'une théorie. Alors comment Steiner arrive naturellement à une théorie déconique, et à une théorie déconique qui est réellement innovante, puisque les déconiques vont apparaître dans la constitution, dirons-nous, de la théorie de la géométrie projective, ce qu'on peut qualifier de géométrie projective, alors qu'avant, les déconiques apparaissaient comme sections d'un cône. from the measures, from the foyers or something like that. The question that was asked Steiner, is that if I take the right which joins the points correspondents of the two right which are in perspective,

45:00 they will define a rayon, that's to say that they are in a point. If I look at the question in a dual, if I look at the rayons in which they are in perspective the rayons correspondents of the two rayons which are in perspective, and it's like that Steiner amen. So when I take a right, two rightes in situation oblique, which is the right, the right, the right, the right, the right, corresponding to two rightes in situation oblique, and you look at the projection what that will define, what that will envelop, In the same way, if you take the rayons in situation oblique, when you look at the points of intersection of the rayons corresponding, what is the place of these points of intersection? Alors, la réponse de Steiner, c'est justement de montrer que ces lieux-là sont déconiques, au sens, et donc en 1832, dans son premier traité, il reprend la définition classique déconique to show that it is from these curves that we can find and find solutions to the question that they pose. All right after, and in a way retrospective, he would say that he had already the ambition to consider that it was in fact the definition of the conic that it was posed, but that in 1832, we would say that he was still not quite a bit audacious for the pose and that he had made, I don't know, all of a part of his treaty to show that he had found the classical theory of God. Well, is it that part there is a reconstruction in... Steiner, he said that 20 years later. From the other side, there are the articles of Steiner where he goes directly, saying that we are only interested in this propriety. And we will see that there is a citation, when it is the most important. There are two theories classiques, which are, the most simple thing is for the figure, is that if you take a circle, you put the center of the faisceau as a point of the circle,

47:30 and that you look at the rayons that join, that intersects on the circle, Well, these two vessels are projectiles. And since this morning, I search for an angle that intersects the same arc on a circle. It's an angle capable. Yes, it's an angle capable. So, it's just an angle capable. You can say what you just said. You have a vessel. You take a vessel that is in the center of the circle, a vessel that is in the center of the circle. You have the rayons that are cut on the points of the circle. All the rayons of the vessel? Each rayon, it's called the A, the other one is called the A, etc. And for the same paramètres, it's the two... Which paramètres ? Because the Fesceau, it's an infinite family of Thomas, is it right ? Yes, it's the four, it's already a lot, right ? D'accord, d'accord. For the same paramètres, it's supposed to be... But I don't see which paramètres. Now, if you want to move on... Chacun des Fesceaux... The two fessions are parallel, they are projective. How do you do it for reading? In the picture, in the picture and in the projection. How do you do it? There are two centers on the affichage. You put the fiches on the one? We'll read the annoncer. Ah oui, on ne peut pas avoir à la fois l'énoncé et les figures, c'est dommage. Les deux points quelconques, ça c'est les deux points qui étaient sur mon cercle, les deux centres des faisceaux, d'un cercle sont les centres de deux faisceaux de rayons projectifs dont les rayons correspondants se coupent en les autres points du cercle. Alors ça c'est des trucs qui sont connus depuis longtemps. because it's the theory of the theory of the angles capable, which is the term, which intersects the same arc or the additional arcs. You have clearly the conservation of the piracy. So, this is a theory that has been known for quite a long time.

50:00 Maybe. I don't know, I don't know. In the same way, you have the dual. If you take two tangents to the circle, and you consider that the rayons of projection are the other tangents to the circle, each of the tangents to the circle is associated here, one point of this right to one point of the other right, and this correspondance is also projective, to be in the sense of the conservator. Well, the... Yes, exactly. But it's a theorem, I tell you, there's enough mathématiques that we find quite easily in the book, even intermédiaires. The only theorem that we have to show is that two angles which have their summit on the circumference of a circle and intersect the same angle, the same arc, which is very difficult. It's a very difficult way to do math. And from there, you see that you preserve the sinus. Eventuellement, you have a little difficulty if you pass from the other side, to know that you will intersect a arc which will be supplementary. But, with the sinus, it doesn't make a problem. So you have a conservation of the bi-rapport. And as Steiner has taken care of to define the notion of the projective both by the idea of projection of the position oblique but also from the two reports. Here, the measure serves as well. And so, he generalizes this theorem to the second degree of the cone and the other sections of the cone. And so, he arrives. It is so important to be able to see a little bit of the figures. And again, when you take two rightes tangentes and uniconiques, these rightes are in relation projective by the whole rayons of projection. The other tangents would be the rayons of projection of the relation projective which exists between the two rightes, the rouge here. In the same way, you have here, between the two vessels, and so, he obtains, as a theorem, that, by reciprocity, and there, if we look at it, the demonstrations, they are, from the time in the time, it's a little...

52:30 It's to say that Steiner insistes very much on the direct side, the theory that I just showed you. And at a moment, there is, I don't know if there is a certain lightest in the argumentation, in all the rigueur, on we say, on the right, on the right, because the grand theoretical, the definition and the engendrement of the technique, from Steiner's point of view, the points of intersection of the rayons correspondents of the two projectiles engendrent a conic. The same, the rayons of projection of the two conctuels projectiles enveloped a conic. These are the two theorem. The theorem, as I defined here, is the point of view, and that it is the good point of view puisqu'il qualifie ces théorèmes comme les plus importants pour l'étude de ces figures, que tous les théorèmes connus auparavant sur celles-ci, car ils sont les théorèmes véritablement authentiques, puisqu'ils sont en effet si universels que presque toutes les autres propriétés de ces figures se déduisent de ceci, de la manière la plus simple, la plus claire, etc. Et il termine son exposé systématique des propriétés, etc. que je l'écris, justement en démontrant tous les théorèmes possibles et imaginables sur les courbes néponiques et sur les surfaces du second degré. and he expose also the theory of polarity. So, where we are? And here is the moment where, let's say, von Staubt, a young mathematician form by Gauss, he says, hey, I have a little projectile. He has given a definition of the form of Steiner, which means that Steiner has a definition of the use of Steiner. Yes, yes, yes. but there is just the transparent which arrives. We're going to try to see a little bit. At the end of my first exposé, this is my first thing. On va voir what are the... Well, with Steiner, the theorem fondamentals

55:00 concern the correspondence between forms. That's the point. The theory of the conic becomes a chapter of the theory of the correspondence between forms. I don't know that the study of the lie, where the elements correspondents intersect, if we look at a vessel, or liées par, si on regarde, les ponctuels. La définition de la correspondance des formes, et en particulier le théorème fondamental, qui dit que dès que j'ai trois couples d'éléments correspondant relation, ma correspondance projective est définie, dépendent de la théorie du bi-rapport, et donc de la notion de mesure. Et, comme chez Concelet, et comme chez la plupart des... the theory of polarity, and a theory that comes after the theory of the phonics. It's a theory subordonnée, it's to say, on first of all the phonics, and on deduce the theory of the polarity. Then, where comes Von Stott? So Fausthaut reprend l'idée de travailler avec des formes fondamentales, comme tu disais, et tout simplement il entre un peu plus dans le détail. Il va distinguer les formes fondamentales de première espèce, qui sont donc des déploiements d'un seul objet géométrique, à savoir donc ponctuel pour le point, droite coplanaire pour les droites dans le plan, et plan coaxio pour NCC. Des formes fondamentales de seconde espèce, which corresponds to the deployment of two objects, so the systems of plants, in which there are points and the droids, and the gerbes of plants and the droids. Excuse me, the gerbes are all the plants? The gerbes, it's the ensemble of all the droids in Dimanche 3. It's the translation that was used in the 20s, 30, when people traduise the books of geometry of this time, who are talking about the sap. It's false, but it doesn't mean that it's a gerb. It's a gerb, a plant, it's all... Yes, yes, yes, yes, yes, yes. And so, there is a very beautiful citation of Frank Schlourke,

57:30 who says, no, it's not that there, he says that the two geometries, that's the system plant, that's the gerb of plants, are two ways to see the same geometry. It's not the ideas that are just liées to models, it's just that I have two theories, and they are the same, with, and eventually, the principles of translation, but we have the two-fold methods. Alors, l'intérêt, et on voit bien qu'il y a un point de vue de volonté en même temps que d'organiser des champs théoriques, Il y a aussi la volonté de trouver les bons énoncés, en particulier des énoncés qui soient, entre guillemets, compacts, qui réunissent un certain nombre de propriétés, et en particulier toute l'analyse avec les points à l'infini. It's to say that I don't have too much time, I pass. But Frank Stoltz insist a lot for, in a first time, to analyze, for example, the faisceau of points that pass into a point infinity, then he analyzes the faisceau of the plane, the faisceau of the plane, which are parallel, then he will show that in introducing the point infinity, And without metaphysics, without saying what is this point, it's not to worry about it, it's simply that if, instead of saying, of course, a vessel of points, a vessel of parallel, I say a vessel of right passing by the point of infinity, I'm able to unify all these enunciations, and all these enunciations, with one single form, And so, there is no, at least at the point, at the point, there is no plan projective in the sense of the plan that will integrate the infinite elements of an infinite, or an infinite space projective, which intègre the infinite elements as normal. But there is already a reflection on how the ways of saying

1:00:00 would allow to organize a whole corpus of annonces. Just a question. And for him, is the projectile, like, in the chapter special of the geometry, or rather, he thinks... Well, that's... It's a... It's a... It's a... What is the geometry projectile in the 19th century? Well, in a way, the term, in 1870, The first book of geométrie projective, it was Crémamma, with his traité. In fact, all these people, the geométrie projective, they don't exist. They need a geométrie general. And in particular, before... No, after what I tell you about, before you're interested in the theory of the phonics, what will do Fanchon? He will demonstrate the theorem of the air. It's surprising. Oh no, that's why the technological revolution is not a book of the geopolitics. It gives, perhaps, is the first good demonstration of the theory, at least the demonstration, when I say good, in the sense of modernity, of the theory of the theory. What are you doing with these stories? For us. For us, it's natural. That's what we see. It's... It doesn't matter how much it is? Yes, yes, I know. Yes, yes, I know. Well, let's continue. Remember, Steiner, as we asked the question earlier, he defined the notion of projective using the mis-rapport, so the measures. How did you do? He started with a notion, the notion of the harmonics. And he introduced this notion from the theorem of the quadrant of the complete theorem. I remember, if we take three points, here, for example, on the right, aligning, I can take any right which will pass to the right. I can build this configuration, this is this, this.

1:02:30 So what I do is this, this, and this, and then I join the right. So I finish this quadrant, the quadrant is there, and I finish it. This is the 6th corner of the quadrant, or the 2nd diagonal if you prefer. point D ne dépend absolument pas, ne dépend que de A, de B et de C. C'est-à-dire que vous refaites la construction avec d'autres droites, comme ça, vous le voyez, etc. Ce que vous allez obtenir c'est le point D. Donc le point D est défini et le dépend en fait, c'est assez intéressant parce que c'est des endroits where the dimensions are fixed. We have, in fact, a theory that concerns the points aligned, but that is still happening in the plane. It is not a property of the right, it is a property of the right in the plane. So ABCD is called the fourth harmonic of ABC and ABCD, the points ABCD are also formed in form harmonics or formed ponctuels harmonics. And of course, the notion of the harmonics is exported to all forms of the first species. So you have the harmonics of rayons, you have the harmonics of rayons. So this is the theorem, the introduction of the theorem as we define it. If three points and the right are given, and they say, they don't consume a quadrant of a certain way, etc. So that's what I'm going to explain. It's called the quatrième harmonique of this. Von Stott va introduire a notion of a correspondance geometry in terms of... but I'm not going to do it because it's an acronym. And so, he defines a correspondance entre formes géométriques, à savoir que tel élément a un élément de la première forme, j'associe un et un seul élément de la seconde forme, quelque chose comme ça. C'est entre guillemets ensembleiste, mais bon, il ne faut quand même pas trop élevé. Et une fois, parmi ces correspondances géométriques, il y en a des particulières, donc ce sont des correspondances entre formes de première espèce ou uniformes, qui vont être qualifiées de projectives, c'est celles qui conservent

1:05:00 l'amendicité. you have two forms of the first species, so at each element of the first you associate an element and one of the second. If you have four elements of a form harmonica in the first form, it makes a lot of forms, their images will also be harmonica. So the first It's the form of the fundamental perspective. Frank Schott introduced a sign to say that the form is projective. It's the sign that we use today. You know, it's the cap with the bar on the top. The good thing about Steiner is that there is no definition isometric which is supposed to be on the right side of the rapport harmonique for 4x, but it goes by the definition of the quadrants hugo. It's purely geometric. It's purely geometric. It's equal to minus 1, in this case. What? It's-is equal to minus 1, in this case-là? Uh... It's... In this case-là, but it's not all right. It's true, because at Steiner, the segments are not signed, so you don't have the "-1". It's only longer. You have to talk about dropper. At Steiner, you have only longer segments. And so it's on... In fact, there are three bi-rapports to really characterize a point on one's right, from one's right. So that's one of the subtleties of the bi-rapports. So the one who introduced the sign in the bi-rapport, it's Charles. What is the role of the bi-rapport? Yes, in theory modern, a form of bi-rapport is equal to minus 1. But at von Stott, there is no bi-rapport. Alors, une fois défini ça, von Stott montre, entre guillemets, une propriété qu'il qualifie d'importante, à savoir que si deux formes fondamentales et une forme projective ont trois éléments correspondants communs, alors tous leurs points sont correspondants communs. So if you have two forms fondamentales, most of all, here I've given it with the points correspondents communs,

1:07:30 it's on the same form of the first species, the same form uniform. You look at the correspondence between the elements of this form, so if I take a right, I take a projective between the points of the right on the same. S'il y a trois éléments invariants, correspondants communs, c'est invariant. Alors tous les points sont invariants. En gros, c'est l'identité. Donc dès que j'ai trois points qui sont invariants, la transformation projective et l'identité, c'est ce qui s'appelle le théorème fondamental de la métrie projective. Autrement dit, il dit que dès qu'on a trois paires d'éléments correspondants entre deux formes uniformes, It's all the transformation that is defined. Now, it's the beginning. It's really the beginning, because we put a little bit of time on it. The demonstration of Fanchot is false. Unfortunately, all the book and all the... all the... all the geometry of Fanchot repose on this theory. I will not have time so I will be able to be invited and after all I will be part of the NR so even one day I will not be invited and I will tell you that we can save Von Stout and it is not me who does it, it is Leblen who will in the first time look and say oh well we don't have it, we don't have it we don't have it like theory, like axioms and then we develop and we show that all the rest is It's very good, etc. Then, it shows that, under certain conditions, when is it that we have it? In particular, there are two actions of continuity. In fact, when I say Von Stott II, it's not true. It's simply that for Von Stott, a punctual, between guillemets, it's a punctual, he continues. And he didn't use it. At the time, people didn't use it. At the time, people didn't use it. It's not a problem. It's not a problem. At this time, if I take a mobile from left to right, and a mobile from left to right, and at the moment, they meet. At this time, they didn't use it. It would have to attend the dating, and it will have to attend to the next one. So there's a whole story, extremely complicated.

1:10:00 In all the trials that you talk about, at the same time, at the same time, at the same time, it's evident that everything is an analytic. So the problem is, it only comes to the hypothesis of regularity. So it's not fair. Well, it depends on what you call it. It's going to be... It's going to be a discussion. Yes, it's true because, in a way, it would be interpreted as... When I say... You will notice that I call the geometry from Schroo de geometry pure, geometry synthétique, etc. What's interesting about it is that the science and the order is to take the axiomatic characteristics of Hilbert. Well, you can't demonstrate the theory of the projective, the theory of the projective, only with the axioms of incidence and the axioms of probability. It's a good thing. But so, there's a lot of work. That's why the history of the projective of the 19th century is fundamental. And at the end, I believe that for the construction and the conceptions actuals that we have of the geometry, it is perhaps more fundamental than the history of the geometry of the geometry. And it's where it happens, at the level of the axiomatic, it's where it happens, because the time is going to pass, right? Yes, I'm going to go back in 10 minutes, so I haven't done the third of my exposé. so everything that happens at the level of reflection on the axiomatic, on the relationship with the topology, and then in addition, it's from there that are going to elaborate the models pertinents for the geometry of the nuclear system. So, there is a series of theories that we can do in particular. projectivement, two forms fondamentales uniformes. It's just to correspond to three elements of the first and three elements of the other. And then, Van Schloot finds, between the end, because there I do again, historically, I don't think about it, but I'm going to say, the definition, on va dire, des correspondances projectives chez Poncelet,

1:12:30 Chez Poncelet, il n'y a pas de correspondance, et il y a encore moins de correspondance projective. Mais Poncelet regarde beaucoup des combinaisons de perspectives. Et donc là, si vous regardez la droite rouge ABC, la droite rouge A2, B2, C2, U et U2, If you want to build a projective relationship with A, A2, B, B2 and C, C2, you need to trace a right pass by A, to take a projection point, to project A, B2, C2 on this right, so A will be fixed because the right pass by A, and then after we take the projection. And when you combine the perspective from S and the perspective from this point, you have, of course, a relationship projective which is associated A2 to A, B2 to B, and C2 to C. So we have... In addition, there is a big advantage about the projective, I know that it's a lot more simple than you have done before. There are cases where there is no perspective. Yes, when it's in perspective, when the two right are in perspective. And there are three when the two right are confused. Ah, it's O plus 3. Yes, it's O plus 3. If they are in perspective, there is one. If U and U2 are different, there are two. And if you apply U2 on U, you have to add three. perspective pour envoyer ailleurs, et puis je reprends les deux là. Alors, maintenant, quand je sois de continu, souvenons-nous, on avait les formes de seconde espèce et les formes de troisième espèce. Donc, il va généraliser la notion de relation projective aux formes fondamentales de seconde et troisième espèce. Donc, il y a deux types de formes. Il y a and the forms are reciproc. So, if you have two elements... So, remember, there are two sorts of elements. So, we're going to take it in the system Plan, so that it's more simple. If you take a relationship with a point, so if it's a point, and it's a point, and it's a point, it's a point. You have a transformation collineaire.

1:15:00 And if you have... And if you have, of course, everything is a silence. It's-is that if you have a point, which is the point of intersection of the two right, Well, the image will be the point of intersection between the two images, etc. And, reciprocally, and it's the case of saying reciprocally, the transformation will be called reciproc if, at a point, you associate a right, at a point, you associate a point, and you see the type of relation that you have given. At a point which belongs to a right, what will you associate? Well, you will associate a right which contains the point. You see the genre? It's those who will interest us. Is this what we call a transformation for polar? But why? You call it why you call it polar? One consequence of the definition, is that these transformations that I just defined, they are evidently projective, because I have a respect of everything in silence. So the structure of the quadrangle complete will be transported, eventually inversed in the transformations reciproc, but we see that if I have in a system plan a quadrangle complete, the image that I have by a transformation collinear is a quadrangle complete, And if I do it by a correspondance reciproc, I will obtain a complete quadrilatère. It's to say with six points and four right. But in any way, we see that the theoretical fundamental of the harmonicity can be preserved. And so, these transformations are projective in the sense of von Stott, which is to conserve the harmoniousness. So, we don't have to do that. We arrive at the notion of evolution. The notion of evolution is very clear. It is to say that the points correspond double. It's to say that if it's on a form with the same support, you associate an element and a second element, then you associate an element, but you come to the difference. It's the modern definition that we have of the evolution. It's to say that when you think about the transformation, you get the identity.

1:17:30 It's not obvious, because there are all the theories around these elements. If there are two elements that transform, if there are forms of first species, if there are one element fixed, two elements that transform, then the form is in situation involutive. What is the doublement? The doublement, it's this. A is in A1, and A1 is in A1. So if you have A1, it's enough. If you have all the other ones, it's the same. The illustration is in dimension 2 and dimension 3. The most of the illustrations of this general theory are in dimension 2 and dimension 3. What is the general theory? The Bollstone. Bollstone is in dimension 3. Well, no, it depends. It's-à-dire, it started with things that were not really dimensions. It's-à-dire, it's the form of the first species. There are only elements that are associated. So if you look at the right, it's dimension 1. But if you look at the right, it's in the plane. It's not the case. It's not the case. It's not the dimension like that. And it will also work on a plane which will be in space. And there, for the moment, we are in the systems of 2e species or 3e species. 2e species which have two elements. The plane with the points and the 3e species, with the points, the points and the planes. D'accord ? Si tu regardes une autre... Prenons-le, une définition, si c'est ce qui m'intéresse. Je prends un système plan. Et à l'intérieur du système plan, je regarde une transformation qui, à un point, m'associe une droite. Et à une droite, m'associe un point. Et imaginons que cette transformation-là soit involutive. this transformation on the same and I get an identity. It's what I get, it's what I call a system polar. So, the technique technique which will become a fundamental,

1:20:00 it's the evolution. And from there, von Stort defines what is a system polar as an area on which I have a system or a system, or a system, or a system of second species, or a system of three species, on which I have a polarity, a reciprocal, a reciprocal, a reciprocal, an involutive, so that it is a system polar, and then, he can define, from this point, just by looking, well, the image of a point, is a right, and this image is the polarity of this point. L'image d'une droite, c'est un point, et bien ça c'est le pôle de la droite, définition. Et il commence à développer la théorie des polaires, nous n'avons toujours pas vu, de conique. C'est-à-dire que c'est le truc qui les coiffe dans la théorie à l'époque, c'est qu'à l'époque, tout ce qui était polarité c'était par rapport à une conique. the point. You can remember how we do. If we take a point, we take a point at the exterior of the ellipse, we trace the two tangents, the right which joins, etc. That's the polarity of the point, and the reciprocal point, etc. So, we have to do this object. He construed completely the polarity of the polarity, which is the answer to your question, why it's called the polarity of the reciprocal, etc. Even if the vocabulary was introduced by Concelet The first time, there, we have a structure purely geométrica of what is a system of polarity. And the cones, they appear how? Well, like the double curves of the system of polarity. It's to say that there are two elements in the system of polarity, and in this case-là, the whole of these elements doubles, Autrement dit, l'ensemble des points qui appartiennent à leur polaire ou l'ensemble des polaires, les droits qui contiennent leur pôle, de toute manière, ça, ça décrit, si j'ai regardé les points, ou ça enveloppe, si j'ai regardé les droites, des courbes, des courbes du second degré, qui sont justement les coniques. Alors, tu ne sais pas, c'est-à-dire la définition de polarité qui prend le franchissement ? If you take a correspondance, for example, from a system plan to a system plan,

1:22:30 which has a point associated with a droite, a point, in respect of the incidence, all the terms of incidence that we want. If you take two rightes, you have two rightes which are in one point, You will associate two points, which is the image of the intersection of the two points. That's what we call a system reciprocal. Imagine that this transformation is involutive. So at one point, I'm associating to the right, and at this point, I'm associating to the right, but it's the same point. That's what constitutes a system polar. So it's abstract like that? No, it's not abstract, it's geometrical. It's totally geometrical. Because it's all built from the silence. It's in the sense that it's not abstract at all. No, because it's a very ascendant. It's not a problem, it's like this. But it's a transformation on a form. It's to say that I'm given a certain number of objects of base geometry, and I'm doing the correspondence, and I'm studying the properties of the correspondence there. So, what is the correlation of incidence that will be preserved, that's what gives us the importance of the correspondence? Well, it's the condition for that it is projective. Exactly. And so, after, in this theory-là, the conics will appear comme les courbes doubles de systèmes polaires. Et bien entendu, un des premiers théorèmes que von Stott démontre après avoir introduit sa notion des coniques, c'est ce théorème qui est justement la définition de Steiner des coniques. Et donc, avec von Stott, qu'est-ce qu'on a ? On a toujours les théorèmes fondamentaux concernant les correspondences entre formes, ça c'est ce qui est, dirons-nous, le point de vue qui va devenir majoritaire et qui va devenir la manière d'exposer la géométrie projective à partir de Steiner, ça ne signifie pas de bon changement. Les coniques ne sont même plus un chapitre de la théorie des correspondances projectives,

1:25:00 c'est simplement un objet de la théorie des correspondances projectives, puisque ce sont les courbes doubles d'un certain nombre de certaines correspondances projectives. La définition de la correspondance projective par harmonique, excusez-moi, c'est par harmonique, c'est projectif, des formes, And so, in particular, the theorem fundamental depends on the notion of the harmonicity which is even defined geometrically from the theorem of the theorem. And the theory of the polarity becomes, she, a chapter of the theory of the correspondence projectiles. So there is a whole organization that is done. Well, then, in three minutes, there is a question of the coordinates. So there is a recommendation for Fonsley for that the pure geometry is completely independent of the geometry analytically, but at the same time I have the same objects, the same principles demonstratives, and this recommendation passes by the history of the continuous continuity Staliner, on the coordinates, it's a bit ambivalent, because Staliner will be compensated to parties, to contributions, which are in geometry analytics. Simplement, there is no balance. If I do the geometry pure, I do the geometry pure, in the sense of Staliner, with the measure. If I do the geometry with the coordinates, I do the geometry with the coordinates, but I do not. It's quite clear. And so, it's what I call it, independent of the approach analytic and synthetics. But for Steiner, he said explicitly that the two approaches are complemented, because of all, it's about to study the same objects that are given by nature. There is a way that I use the numeral, the analytic or synthetics. In any case, at a moment, I can obtain the same theory, because they are the same objects. For Von Stort, who has another ambition, which is not simply to say that the geometry analytics has a great success, it is not the same ambition that von Schley, he doesn't satisfy, at this point of view, Well, there are two theories. We can use the one, we can use the other.

1:27:30 In any case, it's not very bad. For von Stott, it's going to be really organized the theory. And so, we have to hierarchize the theories. And he says explicitly in his introduction of the geometry of the value, that we have to subordinate the approach analytical, and a certain number of habits of calculus, to the approach synthetic. he gives an example of the duality. He says that we would teach the duality. I don't know where he enseigne, if it's the end of the second year, I don't know. I don't know where he wants to teach it, but he says that it can only do that because there is this symphony between the enonces and things like that. And so it's like there is also a way to see it, to see it, to see it, to see it. And so, what he showed me is that from the point of view synthétique, from the point of view geométric pure, we can go very, very far, and particularly introduce the notions of operation, of addition and multiplication, on an object that he can define, namely the G, and once he has defined that the G is the algebra of the G, Well, when I say algebra, it's in the sense of a generic, it's not a algebra, because it's rather a nano, but a algebra of the G. There, at this moment-là, but in a supplement, in an angle, as he said, he says, if I now admit the measure, I can give a number of numbers to my G, and once I have a number of numbers to my G, I can introduce the notion of coordonnées. But so, what he will show is that inside the geometry projective, the geometry pure, we can introduce all the notions of operations on objects. And there again, this is something that is quite innovative, because just now we knew that there were operations on the numbers, mathématiques, il y avait des objets, il y avait des opérations sur, on additionnait des longueurs, il y avait des opérations sur les angles, mais je ne connais pas beaucoup, alors donc, si, on peut regarder un petit peu, il y a Boole qui commence à additionner avoir des opérations sur des trucs qui ne sont pas clairs, ni des noms, on ne sait pas trop ce que c'est, mais toute l'école anglaise, mais là on a quelqu'un qui nous

1:30:00 I say, I take 4 points on a form, I call it a G. With my theory of evolution, I can define an addition and a multiplication on my objects. I'm a bit sorry, there are still 20 transparents to have the GPDG, but I really don't know. Thank you. Thank you.