Diagrams, Euclid's algebraic geometry / discussion / Davide Crippa: Diagrams, Euclid's plane geometry (last part) / discussion

0:00 I'd like to thank the organizers for convincing me that John Sowell was very persuasive in inviting me to do this conference. So I'm going to talk about, my talk really has two parts. One is Diagrams in Euclid, which is what I love particularly, and the second is at the request of the organizers, Diagrams in Algebra and Geometry, which I will attempt to find something. So we start with Euclid, and in order to understand the content of the slides I'm going to show you, I need to remind you of Euclid's book one proposition, 35, 36, and 44. So, 135 is the proposition that says that you have parallels. Could you pick another, speaking of material? If you have two parallelograms on the same base, so here's one parallelogram and here's another parallelogram, you have two parallelograms on the same base, so this one here and that on there, then they're equal. And by equal, he doesn't say what he means by equal. We understand equal area, but he doesn't define the notion of area. But from, you'll see from the proof what he means by equal. Oh, very good. Because black is better than green, you want to prove that these two parallelograms are equal. So what does he do? He says, take the left-hand parallelogram and add oh no we're going to subtract we're going to subtract this piece here and add this piece here so what do you get if you take away this and add that you get a triangle and this triangle is obviously congruent for this triangle over here they have all the same angles in the same size so you take this one you subtract that and add that and you get

2:30 this triangle you move it over you get this triangle now you take this away and that, you get the other one. So from this we can infer his notion of equality of figures means it's possible to decompose, to add and subtract equal things, and to get the other one. So that's 135. And then 136 says if you have parallels, if you have two parallelograms on equal basis, not the same base, but equal basis. So here's one base and here's the other base and you have a parallelogram over here and another parallelogram over there then he says these two are equal and the proof is simple you just draw two lines here and you refer to the previous proposition because now you have a third parallelogram and this green one is on the same base as that one so this one's equal to the green one and this one here up there is that one so that's equal to that one. Okay, so that's the review of those two propositions. Now, in preparation for this talk, I would not attend the previous talk in diagrams last year at Stanford, but I've looked at some of Ken Saito's papers, and Ken Saito has examined the diagrams in manuscript editions of Yussel, a number of different manuscripts, and he made several observations. So I'll show you. I've copied out what he has here, 136. And so the proposition up here, you have this, you have this parallelogram here, this parallelogram, you want to prove that they're equal, and the proof is by this one. So what's interesting about this diagram is that it is not general. It's what Saito calls hyper-specification, or super-specification The diagram represents a much, a very particular case of the theorem in question. It's sort of ironic in this case because you're supposed to prove this one before that one. Well, they're congruent in this diagram. They're both squares, so it's obvious. However, of course, he gives the proof, the same proof. So this is Saito's commentary, this hyper-specification in these diagrams.

5:00 And you see all through these different, all through, these are manuscripts. Codex P, V, V, V, and P. These are manuscripts that are here. They're all pretty much the same diagram, and they're all squares or maybe rectangles. And then he has some printed editions. He has Zamberti, which is pretty much the same. Grenaeus is the first edition of the Greek text, the 1533. Comandino has them tilted, but they're still congruent. Gregory is late, 1703, I think. He has it a little bit different, but not too much. Perard has discovered the manuscript that's called P, named after him in the Vatican. He has them slightly different. And August, that's an early 19th century edition. This is really general from our point of view. So his point was that after a while, in modern editions, of course, we've got to draw a picture like this. We'd never draw a picture like that in a modern textbook. So that was Saito's comment. And I thought it would be interesting, since he's discussed only manuscripts, to look at some printed editions of Euclid and see what happens. So my question was, when did the change from the hyperspecification to the modern form actually take place? it. Saito studied all of the propositions in Euclid's first book, but I'm going to illustrate only with two propositions, 136 and 144. So I need to tell you what 144 says, to 144, you're given a triangle, and you're given an angle, and the problem is to construct a parallelogram which contains this angle and whose area is equal to this triangle. The construction is a little bit complicated, but you can see in the diagram over here, here's Heiberg. So Heiberg gives us a general picture. Here's his triangle, his angle, and he puts the diagram. So Heiberg is pretty general. But in the manuscript, it's the same thing again. You've got rectangles instead of parallelograms, and you've got an angle here which does not appear at all in this parallelogram, and you've got a triangle whose area is nowhere near

7:30 construct so this is what he calls metrical inexactness in other words the diagram in the manuscript it's a symbolic representation of the proposition but it does not represent accurately the metrical quality so the angle is not the right angle and the triangle is not the right area this is the manuscript by the way. Codex B, the same thing. The angle doesn't appear here. And again, this one, here has the right angle. We've got the correct angle in this picture, but the area of the triangle doesn't seem to correspond to the area. So now let me show you the list of printed editions that I examined in preparing for this talk, and I'll review briefly what these I did not look at the first edition of 1482, that's the Rattel edition, or the Zamberti edition. I didn't have access to those. Zamberti is around 15.05. So those are the first two printed editions, being the Rattel edition with Campanus's version, from an 11th or 12th-century translation from Arabic, whereas the Zamberti was translated directly from the Greek, the first translation from the Greek. Pacioli reprinted the Campanus edition, and the 1516 Paris edition is interesting because it has both the versions of Campanus and Zamberti next to each other, so you can see the difference in the text and the difference in the diagram. Then here's the first Italian edition, Tartaglia, Phineas, all Phineas had a nice, very pretty edition. The Basel edition is essentially reprinted in the 1516 edition. Schäuble, I'll show you some examples here, uses no letters on the diagrams, which is interesting. Pelletier's Island is the first German one. Falkadel's first French edition. Condale, that's the podium. Billingsby's the first English edition. Comandino, of course, is a very well-known edition. Clavius is a very well-known edition. These are some minor editions, Dunod-Lotts, Romain de Léa, Michel, Aurélien, Michel, Giordano. Barros is pretty well-known. He was the teacher of Newton. And then I put a couple of later ones. Simpson made an important edition then. And then Perras, this is Perras.

10:00 Three languages, Greek, Latin, and French, based on the manuscript he discovered in the Vatican. And then I'll show you a couple of pictures And he's just the modern English translation, which is the standard English translation of the present. Okay. So here are some images from 135. Oh, these code letters refer to the list of revisions I just showed you, which I can't always remember. So let's look at the hyper-specification here. This is 135. So one of them is square, and the other ones are at a diagonal. And here, it's interesting in the Campanus, you see, in Euphrod's actual text, he usually treats just one case. For 135, he treats the case where the two things do not overlap. But there's really three cases to study. One should parallelograms overlap a bit, or the case where they overlap a great deal, like that, for example. So Pamparis puts the three of the two other cases, so this is the case that Newton does, and these are the two other cases, whereas Zamberti only gives us the one case. That's the translation . So, as you look down, there's still, mostly, one of them is a rectangle or even a square for quite a while. Now, Forcadell, this is the first French edition. This one's a little bit interesting because he puts the, oh, thank you, he puts the standard picture over here that he's copied from one of the earlier editions, but using little letters, he puts his own picture. So in his own picture, he has a general form. So this is 1564. And here, Fusate, 1566, he really has a completely general picture. So for 135, we get generality in the middle 16th century. Here's a few more later ones, the 135. Comandino is more general. so they begin to be they begin to be more general long system curiously

12:30 Giordano when we talk about in relation one 136 still has a rectangle so it's regular my impression is that most of the editors simply copied the diagrams in the previous guide and just use them over again without thinking about it very much now let's look at 136 See, that's Pacioli, which is a copy of the Campanus edition. Here's the Paris edition, showing Campanus this way, and San Verdon. So they're both still rectangles, and even though we began to get the second one looking a little bit different, the first one is still a square. And the first one I've been able to locate where it was truly a general diagram was Giordano here. This is a truly general diagram. This is my mistake. He didn't draw multiple lines here. And Giordano is actually, it's not an addition of Youth and the Gentleman so much. It's called . He writes new proofs and so on, but he follows the traditional order of the pieces. So that's 1680, so it took essentially 200 years to get the generality in the picture for 136. Here's some more 136. By the time you get to Simpson, you've got a perfect example of a picture. But these pictures persisted a long time. I mean, here's one that's slanted, but they're still congruent. Okay, here's 144. So in 144, we're interested in the metric exactness of the diagram. And already in Pacioli 1509, this one is metrically exact. You notice he's got the triangle here with its base twice the length of the corresponding parallelogram. and the height is the same as the parallel again, and that makes the area correct. So already, right away, without shielding, the area is correct, and that's pretty much following in all the later ones that I looked at. Sometimes the angle isn't quite right. Here, the angle is between the correspondents and here, but the area is more exact.

15:00 So it seems that the metric exactness problem was already pretty much solved by the time of the first printed edition, whereas the hyperspecifications have persisted for a long time. And where is the angle, Robin? Whoops. I'm sorry. The original? In that one slide? In this one here. Here's the triangle. Here's the angle. And here's the picture showing the same triangle in such a way that you can see that the area is correct. And the angle is A to L? I've just copied these free hands off of the text, but it's more or less, see here the angles is a right angle, that's correct. This one seems to be wrong, because that should have been a right angle. So the angles don't seem to come out so well. Here the angles seem to be pretty good. But the area of the triangle comes out pretty well in most of these diagrams. So that's sort of curious. You always wonder, you see, when somebody's making an addition of Euclid, Is he really thinking while he's doing it, or is he just copying earlier ones? So these are some more of 144, and again, they're mostly pretty accurate. That one's not so good, but this one's good. What's that? That's the 3rd edition of 3rd edition of 3rd edition. This may be the 1st edition of 3rd edition. Okay. Then, so those are the diagrams. you a few pages from some of these early editions because it's fun to look at. This is the title page of the Paris edition of 1516. See, it shows Euclid's Elements by Campanus here with his commentary and by Zamberti. They thought that Zomperny was giving Theon's version, and Campanus was considered his commentary. At those times, somebody mentioned already early in the conference that until the middle of the 16th century, most people believed that the demonstrations had been added by commentators, and that the statements were due to Eucharist. And it was only Comandino, I forget who it was, that first pointed out that actually

17:30 the statements and the proofs came from Euphos. So at that time, the Catholicism was called a commentary. And then these extra books, Euphos is the first 13 books, and they usually added 14 and 15 attributed to Hypsychles. So here's a page from the 15, 16 edition. They hadn't yet figured out how to put the diagrams in the text. They had the diagrams off to the side. These diagrams are for 1.7. That's the one, 1.7. That's the statement that you cannot have on the same base and with the same size. You cannot have two triangles having the same base and the same two sides. Now this is actually an interesting diagram because it's an impossible exist. So here we have, this is Campanus' version with the three different cases. This is the one, Zanderity shows us only this case, which is the one Euclid shows, but Campanus has the three cases. But while we have this picture on the board, I want to mention something else about it, because one of the defects of using diagrams in Euclid elements is that Part of the proof depends on something that's visual there in the diagram but is not justified by anything in the text. Let me explain that in the proof of 1-7. So 1-7 says that you cannot, if you have a base, A-B, and you have a triangle A-B-C like this, you cannot have another triangle that has the same base and the same two sides. and the way they prove that is they imagine that you have another triangle like this so this side is equal to this side and this side is equal to this side and you want to show that's impossible so how do you do that well first of all this is an impossible picture because you know it's not going to be possible but how do you do that so you would prove those like this you draw CD and And then you used a previous proposition, 1.5.

20:00 1.5 says in an isosceles triangle, the base angles are equal. So from A, you've got two equal lines, so the triangle ACB is isosceles, which means this base angle is equal to this base angle. Now from B, you have another isosceles triangle. So that means that this base angle is equal to this base angle over here, using one five. Now he says, the red angle here, this red angle, I'm sorry, I'll move more to the side. Yeah, the red angle here is bigger than the green angle, right? it's bigger than the green angle. But over here, the green angle is bigger than the red angle, strictly bigger. And therefore, the red angle is very much bigger than the red angle. That's impossible because the red angle is equal to the red angle. That's the contradiction. Now, in order to prove this, to do this proof, you have to see that the red angle is bigger than the green angle. In other words, you need to know that this line here, CB, actually falls in between the lines AC and AD, inside of this angle. And for the other part over here, you need to know this line here, AD, lies in between inside the angle CBB. That's obvious from the diagram. But there's nothing in Newton's elements for his postulates or his explanation that says why that should be so. What's missing is a notion of one thing being between another thing and drawing that as a consequence from the hypotheses. You see, these angles, the hypotheses had to do with the base and these sides. So to draw a conclusion about the relationship of these angles to each other, and which one is contained in the other, is already rather subtle. This problem wasn't solved until the late 19th century with Pache, who made some axioms And these have been incorporated in the Pilbert's series of axioms from his Wundlogen-de-Gablin experience from 1899, and they're standard now. But I'm not going to go into that so much. But if you do discuss the diagrams in Newton, you certainly want to say which aspects were not successful, and here's a clear case of using something that's visual on the page, but is not justified by any of the text.

22:30 Okay. here's another one this is another case where there's a question of metric exactness this is 122 I think the problem of making given three segments any two of which are greater than the third you want to construct a triangle so here's here's components in version and here's some version and in some of the I didn't show you the manuscript one, but some of the manuscripts, they draw three equal lines, and then they draw a triangle with three unequal triangles, so it's not metrically exact. But we're getting much closer. Here you've got C is short, B is middle, and A is big. More or less corresponding to the size of this triangle. And down here, we have three different lengths, and we've got three different sides. So these are more or less metrically exact. Here's the, this is still in the Paris 1516 edition. The top three pictures here are Campanus's pictures, and this one is Zamberti's. And then down here you have 136, and you see how it's, it's, one of these is Campanus and the other is Zamberti's. Let's see. This is Campanus. This is actually a typo, it should have been Zamberti here, this is Zamberti's picture. But you see that, again, they're congruent rectangles, whereas he's proving it in the case of general parallelograms. Now, I've never actually seen the Zamberti edition, but I'm assuming this is copied pretty well from the Zamberti's edition, similar to earlier. here's 144 and you see this is metrically exact the base of the triangle is drawn as twice the side of the parallelogram and the height is the same so the area is exact there and for some curious reason it doesn't give the angle and the triangle they're not shown but here in Zamberti's version it gives you the triangle and the angle and this is pretty good so that's that's fine Okay, then let's see, what's this one? Well, this is the Basel edition.

25:00 No, no, sorry, this is Schäuble. Ah, Schäuble was interesting, 1550. It's interesting because in his version of Utrezelos, this is a translation, he does not put any letters So, for example, for the proof of 1.5, 1.5, oops, you can't see that, 1.5 is the one that says if you have an isosceles triangle, the base angles are equal. So you probably remember, I'll show you the diagram that you find in most editions, the common diagram is like this. You have a triangle like this. here's your isosceles triangle ABC and if its proof runs like this you choose a point D down here you mark off an equal point over here E and then you draw these lines like that and then he proves that this big triangle is equal to that big triangle using side angle side and then he gets these angles are equal and then he shows that this triangle here is equal to that triangle and therefore these angles are equal in the base, and then by subtraction you find that these angles are equal. That's a very short version. Usually you put the inversion, every angle is named, and every triangle is named, referred in previous ones. But look what Scheichel does. He doesn't put any letters. So what is he going to do? How is he going to explain that proof? Well, he has to draw it many times over. And the result, see, here's the Euclid's diagram, to this one so you see it there inside the picture and then out of that you deduce that this triangle is coming to that triangle and that there's how they fit together in that picture and then down here you see the you see the final result and there is a justification why does not use letters simply does not use them well in the text if you read the text there's no letters because it does explain why why he doesn't use letters yeah I can't remember maybe maybe he said something in the beginning I think he probably thought it was easier if you start reading the text yeah he doesn't refer to the letters you know so it says the two parts in this proposition which the first one is trying to equal size

27:30 the angle of the base, the main angle of the third, are equal to each other. And it's all words. There's no letters. It makes it rather hard to follow because he has to talk about this angle and that angle, the angle you mentioned previously, and the angle of the base and the angle on the other side, all kinds of locutions like that, instead of using letters. There is one part that you can use in it, which is a normal thing to move from the to the That's right, that's missing. Because Yeah, because usually in the statement of Euclid, the first statement is very general. Yeah. A triangle with two equal sides had two angles at the base. In any case, let ABC be a triangle, and assume that side AB equals AC. Then the angle of B should be equal to A. Yeah, yeah. So that's missing here in a circle. Good point. I'll show you that one. Okay. Now, this is the front of the first German edition by Zylanger. Unfortunately, the color doesn't come out. I've got a color Xerox here. The title is written. It's very beautiful. Very beautiful. Beautiful title page. This is the first German edition, 1560 or something already. The reason I wanted to mention this one, this is not really a translation. He says, somewhere in the beginning, he says, well, the proofs are too complicated for the understand. And therefore, I'll just give explanations. But I wanted to show you particularly what he says for 1-7, because almost all of the printed editions have something like this, like some pair of keys. Or maybe they have it in a slight angle. They all have more or less this diagram. His is totally different, you see. He says, here's the one triangle. make another triangle with the same legs, because, look, you know, here's one of the same legs, and here's one of the same legs, and they don't meet. That's his explanation. It's not really a proof.

30:00 But I thought that was interesting, because it shows at least some intelligence there. He's thinking about what he's talking about, not just copying the previous ones. Now, here's the book of Giordano, Oitri di Restituto, with a beautiful title page. And again, you miss the color. It's red and black in that beautiful, beautiful one. And the reason I mentioned this one is because I wanted to show you what he does with 135 and 136. It's curious, his 135 there is still special. It's still in a special position because he set a rectangle. So he did not make it general for 135, but when you get to 136, He does have a completely general figure for 136. So I don't know why that is, it seems illogical to me. If he's making a general figure for that one, why didn't he make a general figure for the other one? Now, then for a typographical point of view, I want to show you the, if you haven't seen seen this before, the additional Oliver Byrne, 1847, and again, the Xerox machine wouldn't do the color. So he's got a picture here. And his use of diagrams, he decided that it would be easier to understand if you used color in the diagrams for the proofs. So instead of using words, he gives a proof using colors. And I'm sorry, the copy here doesn't show of the colors, but here's the Xerox in the original color. Hypographically, it's a fantastic volume. Book lovers love it because of the colors. There's no mathematical content of it. There's nothing interesting in mathematics. But look how he proves it. You see, he's proving 136. So here you have two parallelograms. And he says, draw this line and this line, giving the colors. And then he says, well, you can't see the color. This is a black one. That's a blue one. because of the previous proposition and therefore the black one is equal to and parallel to the red one right so this one's equal and parallel to that one therefore this one is equal and parallel yeah so then then this yellow one and this dotted one are equal and parallel these ones here and therefore this guy over here that funny one is a parallelogram but this one's equal to that by the previous proposition 135 that's equal to that therefore they're

32:30 So there are very few words, and it's all done in pictures with these colors. Well, it's still flat on the page, but with the colors, it's a totally different way of doing diagrams. Online in color? Oh, wonderful. Okay. So please have a look at it. It's a fantastic version. Okay, now before I go on to algebraic geometry, I just want to mention the problem of diagrams in non-Euclidean geometry, and the difference is that one of the things about a diagram in Euclidean geometry is that the diagram is the object. If you draw a right triangle, if I draw a diagram, if I draw a right triangle, assuming I've drawn it accurately or exactly. This picture is not only a picture of a right triangle, the way a photograph is a photograph of a person. It not only represents the right triangle, it is the right triangle. So in Euclidean geometry, the diagram you draw is the object you're talking about. If you draw a square, it is a square. And one result of that is, if you draw some complicated diagram in Euclidean geometry accurately, and you notice a relationship that you haven't seen before it's probably true and then of course you need to prove it abstractly but so one of the things about the Euclidean diagram is the diagram is the object itself now in non-euclidean geometry we can't do that because non-euclidean geometry takes place on a curved surface and when the paper sheet of paper is not curved so how are you going to draw a non-euclidean diagram well you have two choices one choice is to use the the model. Poincare model of a Euclidean geometry, you take a circle in Euclidean geometry and you take circles, arcs of circles inside here that are orthogonal to this outside one and those represent lines. So the points of the non-Euclidean geometry are the points inside this circle gamma. These things here are lines in the non-Euclidean geometry even though they appear to be curved. This is the picture that's used for the proof that the three altitudes one point. So here's a triangle ABC, and then here's one altitude, and there's another altitude, and then you want to show the third altitude passes through there. So this is one choice. You draw the picture in Euclidean geometry, but using curves of Euclidean geometry

35:00 to represent lines. The other choice is to pretend that you're in the hyperbolic geometry and draw lines, but you make them a little bit curved because they aren't straight. So Tobert's proof that whenever you have two parallel, two non-intercepting lines, they have a common perpendicular. So the common perpendicular is going to show up in here. So in this picture, if you drew straight lines, it wouldn't make any sense because you have these limiting parallels. These guys are limiting parallel, meaning they're closer and closer and closer because they never touch. So this is the other choice, is to draw slightly curved lines but have them more or less straight. And here's another picture. This picture is having to do with what I'm talking about. Oh, given an angle, there is some other line that's limiting parallel on both sides. So there's another line in here that's limiting parallel there and limiting parallel there. So in this case, the diagram is not the object itself anymore. It's merely a figurative representation of it. that's not exact in either direction. Okay, so now we've come to diagrams and algebraic geometry, and I had some difficulty trying to decide exactly what that was, so I've found three different kinds of diagrams. One kind of diagram is merely a table. You might call it a table. I don't know if you'd call it a diagram or not. So here's a table, and what does a table do? It's simply a graphical way of gathering together a fair amount of information in one place, and it's visual. It's easy to see, and these are the possible degree curves in between three states so these exist and these are the inductors that exist so does that count as a diagonal yes sure it is oh okay this is from my book on algebraic function when I was preparing my talk I was in New Hampshire my summer place I didn't have access to a library I just had access to my own book so I'm sorry to be showing you all come pictures from my own book So your claim is, I use diagrams?

37:30 I guess so. You are claiming that? Okay. I guess so, yeah. So if this is a diagram, this is one kind. It's a table. It's just a simple way of putting together a lot of information in a visually compact form. Now, another kind of diagram in algebraic geometry is more what I would call a picture. So it's an attempt to suggest some of the ideas that are taking place by means of something or other. So this is a picture of the speck of the following two variables, and the point is that it's like an ordinary Cartesian plane. So you've got axes x and y, and there are points, and here's a curve. But in addition to those, in algebraic geometry, we have something called generic points. So if you have a curve, the curve has something called a generic point, which is sort of all around. It's not localized anywhere. It's sort of along the curve, but it does belong to the curve. So in this case, the artist who was a former student of mine had artistic ability. He drew the generic point by making a fat thing with little things next to it. Now that doesn't mean there's a lot of points there. It's simply an artistic way of giving the idea of something that's not anywhere exactly And this one over here is supposed to be the generic point of the plane. So this is, again, something that's not localized anywhere, but sort of all over. And this is a purely artistic device. You know, a painter might do that. It doesn't have any particular meaning. So in this case, this is a picture that has some artistic reason to its existence. Here's the picture of blowing up a curve in the plane. you imagine the plane over here and here's this curve and the operation of blowing up this point here gets blown up and separated out so this is a sort of trying to imagine what the surface up here looks like this this whole line is mapped down to this single point and every everywhere else it's one-to-one so you can imagine the surface everywhere except at that point and what happens to curve, it gets split out, so it's no longer singular. Here it has a singular point, and there it gets spread out. So again, this is an artist's idea of how to represent something. It's supposed to give you some visual imagination, even though we don't use the diagram to prove anything. It's simply to have some kind of picturing model. So I've got a number of these.

40:00 I'll show you some more diagrams which are using sort of artistic ideas. This is supposed to represent a family of curves, with that one here, the original curve, the space curve, going to a plane curve. So you imagine the family of curves and it's spread out here and then it sort of gets collapsed until the end, it lands in the plane, that's this one here. So you see the curve, it was a non-singular curve, but it ends up crossing itself. And again, in the picture, the artistic idea, these little thin ones, sort of give you the idea of motion. You know, the way they do in in in cartoon strips sometimes, they show you something moving along. So I was lucky to have this student of mine who has artistic ability. And here's a picture of a ruled surface. So you imagine the surface here together with that being down to a curve, and these are the rulings, and those are a couple of curves going on the surface. And here's the real, well, here's another one. Now this is interesting because, again, it's a fairly good representation of something that happens. If you have a real surface, you want to transform the real surface into another one, and what you do is you blow up this point P, and when you blow up P, what happens is it turns into a whole curve E, And this line here becomes that line over there. So now you have two. And then what you do is you blow down L, which becomes Q. And again, you have a line. So you get one rule surface is transformed into the other, but it's a different rule surface. It's called an elementary transformation of the rule surface. And here's the tour de force picture. This is Kiranaka's example of a complete non-projective variety. You start with a smooth variety here, a three-dimensional variety. Imagine a three-dimensional variety of two curves that intersect each other in two points. And in the neighborhood of the point P, you blow up C first, like this, and then you blow up D, and you move that together. But in the neighborhood of the point Q, you blow up B first, and then you blow up C. So you get something up here, which has been patched together by these two different procedures, and the result, you prove that it has no invading in the projection space.

42:30 So that was from hearing off this thesis in 1960. So all these ones I've shown you are sort of pictures, or just trying to give something to, but you don't use it to argue with. Now, the last kind of thing I wanted to show you was diagrams that are actually used in proofs. And here's a couple. This is the so-called octahedral axiom of a triangulated category. So what happens is here, you're working on a category. So a category is something that has objects and arrows. They correspond to, well, it could be anything. It could be groups and homomorphisms. So in yesterday's talk, we heard about groups and homomorphisms or it could have been modules and morphisms or it could be algebraic varieties and morphisms between them in the case you have objects and arrows and it's very hard to resist the temptation of drawing them graphically like this so A is an object B is an object and this this represents an arrow going from one to the other so there's something called a triangular category and this is the things called triangles So, x, y, z is a triangle, and x prime, y prime, z prime is a triangle, and then, oh, no, I guess we start, what is it? We start with x, z, and then there's an axiom that says you can complete it to a triangle. So, you can complete it to a triangle x, y, z. And then, I forget exactly what it says. There's another triangle you complete it, and then you draw some maps, and then you have to commute with each other. And commuting, we heard about that the other day. commuting in the diagram means that whenever you take take a sequence of these arrows and there's different roots to get to the other place you have to have the same result so starting from X you could go to Z and then Y prime or you could have gone to Y and then Z prime and then Y prime so that's an example of the commutative diagram here's a slightly worse one this is part of a proof and what happens here is you start with u and you have a representation of

45:00 u by i and y over here and then you have another representation of u by j and q over here and you want to compare the two of them to each other so you dominate those by another one r and another one s you get a big diagram like this and you have to show that a whole thing bunch of things is the chart of the proof. The chart of the proof is how you compute them, how you compare them, because there's various previous propositions that say this one could be transformed, this one by proposition four, and this one could be transformed by that by proposition two. So it gets pretty complicated sometimes. And you're not going to write that in words. There's no way you want to write that in words. You write the diagram, and then the reader says, okay I have to check all of these things oh and then I brought along the largest diagram I ever drew in the course that I was teaching No, in the other sense. Other sense. This way? No, no. Another one again? On the third dimension. Not this way? Still one, okay. No, still one, no. Apparently not. Yes, yes, still one. no no no no no no no no no no no no no no wait a minute i have to look at i have to look at this one so i know which one is on top okay those little things are on top but they're backwards this way okay now it's okay okay um this is the spectral sequence of a composite function well partly you see I got it I made this out of a photograph this is a photograph after I made the diagram well the students went out to get the department photographer saying you have to record this recorded it you're a functor F and a function G so GF is the top of the function and then you have some you long exact sequences so you get long exact sequences here and there's various arrows going between them and it gets very complicated it takes about half an hour to explain that's certainly a diagram

47:30 okay now now to finish i wanted to spend a little bit of time talking about um community diagrams and this is the case where we actually use the diagram in proof and the proof the diagram is an essential part in the proof we actually work on the diagram to carry out a brief so I still have 15 minutes I think I'm going to give you you have more like five five minutes oh okay I'll do a little faster then I'll I'll speed up talking faster. We can do 10. We can do 10. Yeah, okay. So I want to give you a brief, a little bit of review of modules over a ring and category modules so you can understand what I'm talking about. So I'll suppose A is a ring, and M will be a module, and then we have homomorphism between modules, which I write with an arrow, F. And there's the notion of kernel and co-kernel, so the kernel of F is all the collection of elements in N, which go to zero. F of N equals zero. And then there's the image of F, which is the collection of elements in N, such that there exists an element in N with F of N equal to N. so the kernel here goes to zero and the image is everything you get so I think we've seen we've seen these notions before and then whenever you have then there's also the notion of injective and surjective I have to recall injective and that thing is injective if you never have two elements going to the same thing so that's the same thing as saying the kernel is zero nothing goes to zero in groups so that's injective and surjective means everything is in the image so the image of f is equal to the whole thing so these are these are the terminologies I need now the way this is recorded in homological algebra is by saying by the notion of exact sequence so if you have a bunch of modules let's say m n l and so on which could be long or short you say it's exact at a certain point we say it's exact at the point n if you have these mappings f of g if the

50:00 image of f is equal to the kernel of g so what does that mean image of f equals kernel of g means first of all if you take something in n that goes to zero and N. Well, if you take something in F, it goes to N, and then it's going to be in the kernel of G, so it's going to go to 0 here. So the composition is 0. So it tells you more. It says if you have something in the middle, which goes to 0 here, so it's in the kernel of G, then it actually came from something over here. So that's the notion of exact sequence. In a particular application, this is a simple application, but if you write, if n, if the mapping from n to m is injective, then we think of n as being a sub-module of m. We identify it with its image, which is a sub-module, and we write that in the following way. We say 0 to n to m is exact. You see, what does it mean for this map to be exact? It means the kernel from n to n is equal to n to 0, which is 0. So that says the kernel is 0. And surjective, n to n surjective, if and only if, when you write a 0 over here, this is exact. Because what does that say? The kernel of n to 0 is everything, because everything is in the base of 0. So that has to be in the image of n. So that says it's surjective. So instead of saying injective or surjective, we can represent it by this little piece of the diagram or this little piece of the diagram. Then, this is the notion of a short exact sequence, which is written 0 to M to N to L, 0. So, what does that tell you? It tells you that M is essentially a submodule of N, and that the mapping N to L is surjective, with kernel equal to M, which means that L is isomorph to the quotient of N mod M. So it says basically that M is contained in N and that L is isomorphic as a quotient in N mod M. Now I want to give you one example of a proof using this terminology in notation, which is called diagram chasing. Someone mentioned diagram chasing yesterday. So I'm going to give you a proof by diagram chasing, which is a proof that takes place on the diagram. So, this is the so-called 5-lima, and it says, supposing I have a short exact sequence,

52:30 let's say, m prime to n to n double prime 0, and another short exact sequence, n prime to n to n double prime 0, and supposing I have mappings from n prime to n prime, n prime and call an alpha beta gamma. And suppose that alpha and gamma are isomorphisms. Isomorphism means both injective and surjective. One-to-one correspondence if you like. Conclusion is then beta is an isomorphism. An isomorphism means injective and surjective. So I'll prove this by diagram shape. So I will show that first of all beta is injective, and secondly, beta is surjective. So what do we do? To show it's injective, I suppose I take an element x here, suppose it goes to 0 in n. Well, then it goes to 0 over here in n prime. I think I forgot to say this was a commutative diagram. It's supposed to be a commutative diagram. okay on the other hand x goes over here to some x prime in our double-prime and double-prime and this goes to zero because it's commutative but gamma was an isomorphism so if x prime goes to zero that means it's zero so therefore x went to zero over here but this is exact so it came from something over here so there's no y over here that goes to x yeah sorry there's a one x goes to zero so it's in the kernel so it comes from some y and this y goes to a y prime over here and this y prime goes to zero because it's commutative again but if y prime goes to zero this injector that means y prime is zero but alpha is an isomorphism so if this is zero then this was zero so zero so therefore x is zero and that's what we wanted to prove because we wanted to show that if x went to zero then it was already zero so that proves injectivity now to prove surge activity uh take away these This is called diagram chasing, because I use the diagram, and I chase around the diagram, using its various properties as I go along. Now I want to show the surjective, so take a Y and N. Okay, it goes to something or other, Y prime and double prime, and this is an isomorphism. So that came from something Z prime over here, maybe.

55:00 But this one's surjective, so there exists a mapping of something L of X, which goes to Z prime. Now what about this X? Is that going to give me Y? call it X prime over here but in any case I know that X prime this is commutative so X prime goes to Y prime therefore X prime minus Y goes to zero in n double prime so therefore X prime minus Y comes from something over here let's call it X circle minus Y circle or call it Z are you okay so this difference comes from something W but this was an isomorphism so there's a W And what happens to this one? This w prime goes to a u over here, whose image down here is x prime minus y. Now if I correct x by taking x, I guess I should take x minus u. I'm going to replace y by a different one. We use x minus u. When happens to x minus u, x minus u goes down here to x prime minus y plus x prime. Sorry. it goes to x prime minus the image of u which is x prime minus y and that's why so therefore it's subject yeah so this is an example of a proof by diagram chasing and just before I stop that's that's basically the end but before I stop I want to say a remarkable thing and that is that in order to do this diagram chasing you have to have tell us to talk about these these objects in in this case, modules over a ring, have to add elements that you can chase. The notion of a category of modules is more generalized by the notion of abelian category. An abelian category is a category that acts sort of like a category of modules. It has maps, it has kernels and co-kernels and exact sequences, but there's no elements. The objects don't have any elements. And there's a wonderful theorem of pride which says that any abelian category can be embedded in the category of modules, and therefore, you can actually carry out even though there aren't any elements there. They're sort of virtual elements. And if the proof works by diagram, they think it's still true. So thank you. Could you put the slide before the last slide? The last slide before the last slide? i i just want to i i'm going to make this one this one here yeah i mean to make a fun color

57:30 yeah in order to make you understand what i want to say about you okay okay you see the triangle U-S-R? U-S-R, yeah. You know, you drew it isosceles. U-S-R? You drew it as if it were isosceles, you know? Yeah, you know. Okay. Which has no meaning at all, of course. Okay. Yeah. So, what I want to point out by this remark is the fact that a diagram is also, depends also on the way in which you view it. Mm-hmm. Yeah. So, if you read Euclid diagrams that you think are special, as what they show, perhaps you read them in the same way as what you just did on this slide. What I mean is that the point is to understand how people were using or reading this diagram and not the way in which we read this diagram. And what I want to say about this is that I think that a diagram in general depends on the way in which we use the diagram. I see. So you would argue that those diagrams were for 136 were general? Exactly. Even though they looked special to us? Exactly. I see. Well, you have to argue that with Saito. No, no. I already argued with Saito. Oh, you did? Yes. What happened? And I think he agreed with it. And I think, I mean, just a point that I think is very important, at least to recover the way in which people were using and reading diagrams, and not use our own way of using and reading them. Ah, I see. That's the subject. Well, let me just comment about this, that someone earlier mentioned that when we use a diagram, we only see the part that's important to us. We don't see the other part. So I don't see the fact that that's an isosceles triangle. All I see is the letters and the arrows and their relations, and it could have been drawn differently. I could have put these down here. I'm sure, I'm sure, but I just wanted to point out that with a diagram in which you're familiar, that you read in a certain way, that there are ways in which you do not read, and perhaps that's what you do when we read a particular diagram, as if it was particular, instead of reading it as general. That was my point.

1:00:00 Thank you. And in this time, there are two unbalanced parts, because the first one is more general, it's more innovative, because it doesn't use the model properties of people, it's the only thing, it's only the kernel and the nature, and all that works very well, without assuming that . So it's the problem is, it's the third part. So the injectivity is more elementary and the subjectivity is more complicated? More diagrammatical. Well, diagrammatics, but the first part, the second part, the basic model is the model is the difference of zero. Yeah. That's all right. That's easy. That's it. Sorry. I'd like to query your claim that in the Euclidean practice, the diagram is the object, and here are three reasons against it. I knew I'd raise some tackle by that. Go ahead, Jeff. First of all, diagrams are always in exact, whereas the theorem don't claim that the square on the hypotenuse is approximately equal to a little clearer than that. First of all, the diagrams are always exact, but the theorems are not just approximately. We refer to the Platonic ideal diagram rather than the actual diagram we draw. Yeah. I agree with that. And secondly, as you say, there are these impossible diagrams. The theorem isn't about the diagram, it can't be because... And thirdly, the diagrams are in physical space, and some physicists tell us that physical space is known clearly in the presence of massive observance. So, I mean, of those of us who believe that there is some kind of truth in video geometry, and I'm one of those, don't, can't really accept that the diagrams we draw,

1:02:30 that we believe on the page, really are the object. Well, I can't object on anything you say, except to say that it's pretty, pretty close compared to all the other diagrams I was talking about. Oh, yeah. Okay. In contrast to the others. And, and I was, uh, this idea of a comparative account, it was not my idea, that's Marco's idea, he gave me that title. So, that's the comparative part, I'm comparing it, uh, in the algebraic. It could be an exact account. Yeah. Yes, my question was just to ask to Robin to elaborate this claim that the diagram is the object, but I see that it is not a so strong claim, because I immediately agree with Marcus, and in fact I do not completely agree with Marcus. I think that first of all your third reason is a very bad reason, the first and the second are of course true, but they are good reason to argue that we cannot intend the ease in a very literal sense, still I think that what Robin was willing to say was not really that is the object, but the point is that there are essential properties of the object, there are essential properties of the diagram, and there are only that, the diagram is not simply a representation of, it's a property of the object that cannot be explained in other way that as being properties of the diagram. Of course not all, because otherwise the ease would be ease, really. But I think that there is something important in this claim. I don't know if Robin agree on that. I actually have a comment on that, because when I teach geometry, I insisted my students do ruler and compass construction. True, sorry. Ruler and compass constructions. Not just theoretically, and not with cabri or geometry sketch bed. They have to have an actual paper, a compass, and two of them draw the paper. So what happens? I have to explain to them the difference between a mathematically correct construction

1:05:00 and an accurate construction. Supposing you try to construct the pentagon. Well, it takes 14 steps. And by the time you've drawn 14 lines in circles and intercepted, there's always a little kind of bit of experimental error. And when you finally come to draw your pentagon around here, it doesn't quite come together. So they say it's not a pentagon. I say, well, you see, it's mathematically correct. not totally exact and of course you are the opposite situation there's a construction of the heptagon if you know this one there's a way to construct a heptagon which goes like this you start with a heptagon described in the circle you start with a circle here you take a diameter and using the radius of the circle you mark off a point here so that actually makes an equilateral triangle and then you drop a perpendicular here and you use this distance here as the side of your heptagon and you mark it off you know and it goes around seven times and you get a very nice heptagon so the interesting thing about this construction is if you do it accurately with your pencil and paper you've got a perfect heptagon however this is not a mathematically correct construction it's off by about one-fourth of one percent if you make the calculation Here's one used by artists. You find it in the book of, I forget names now. I think you find it in Durer. Durer in some of his books has a construction of a pentagon, which is also an approximate one. He maybe has this one, but I found it in a different book whose name I forgot. French the Frenchman who was basically known as an artist and a beautiful beautiful copper plates and he has a mathematics book that has 84 copper plates I'm talking about geometrical yeah yeah both these geometrical figures with little landscapes down on down the bottom so yes this construction but I forgot his name and we could say just as a comment I mean the diagrams and it makes sense to are to a real thing so that medically yeah but if some of these other diagrams like in that you put in geometry it doesn't make sense so we have several different well I saw a whole bunch of him you want me to hello yes I want you to see him you want me to call him first no no no no no

1:07:30 There was one of the, you know, showing the work with this kind of proof, showing that Yeah. It sometimes makes sense, because when you talk about , of course it's something that we see that we see in the dialogue that we see. But it's the same thing, the same thing could be said for society that's not sustained in the garden. So it looks like we have to decide how far our space in our garden goes. Because of course, I mean, in fact, it's the same, kind of the same thing for society. and I can see the special size, the size they're not safe. You can see that. Well, you know, this is famous complaints about Euclid, that he proves things that are obvious. For example, he has a proposition that says if you have a triangle, the sum of two sides is bigger than the third side. The commentators say, well, that's totally stupid because if you put an ass in one corner and a hay in the other corner, he'll go directly there. It's obviously shorter. No idea how to prove it. the next talk is about this is that right okay you can actually come up with rules to use the diagram in a way where you actually specify exactly what's your fate in your visual face inside you can quantify that right you look at the diagram is an actual symbol. You can actually make it very sharp and that sharp distinction connects pretty well with the vision. So rules of vision, what you're allowed to see and what you're not allowed to see or something? You can specify symbols which are diagrams and then you can specify rules and symbols of that, which match up pretty well with what you're seeing. I think you have it? Yes, I understand. But you have said rightly that in the region of East, for example, diagrams are more general than in the market, and you can find the height of the region, for example.

1:10:00 One thing which is strange is that if you say, for example, in the model edition for the more recent editions, the diagrams are still . Oh. But where did those diagrams come from? Did you take it from an earlier printed edition of Fröcklis or from a manuscript? I don't know. The Mauritius edition, we can buy. Yeah. You're talking about Morrow's edition in Princeton, yeah. Yeah, for example. But where did he get those diagrams? Did he get them from some manuscript? I think so. But the question is, why the need of rewrite the diagram arose to say, and not, for example, this prophet's commentary? Well, my guess would be that Profez's commentary and its re-edition is basically in a historical study. It's there for historical reasons. Whereas the reason why there's so many editions of Euclid, with more than 1,000 printed editions, is it was used as a textbook. In Italy, it was used after the reunification. They reinstated Euclid to be used in Italy after 1870 or something. And in England, it was used up to close to 1900. And Charles Dodgson, in a famous book called Euclid and his Modern Rivals, has the ghost of Euclid come back and defend his text against all of the current texts in England. He says that Euclid's order is much better. So it's been used as a textbook and a teaching book. And I think, therefore, it's more adapted to the modern way of Euclid. Can I come in there? Yes, go ahead. I was actually the last generation in Britain to learn from the condition of Euclid. Oh, really? Which edition did you use? We used an edition by Durell. Durell? I don't know that one. Who was a textbook writer. Was it an actual translation? No, no, no. This is written from an earlier edition. Ah, I think. So it's a textbook based on you. Yeah. But I wanted to come back to what I think were the rhetorical questions that you asked. are they just copying earlier ones or are they thinking yeah and there's a tradition and or there was a tradition of expectation of expecting reproduction

1:12:30 This was broken away from by a very famous school, Winchester College, in 1837. Up until then, an edition of Bucha had been studied as a classical text. diverged from it, then you were distorting the purpose of it, you were wrong, it was heresy. And the same thing happened with Trinfox, the famous Trinfox, and then schools, other schools in Winchester, and Winchester was leading this way, other schools in England, until the 1860s, and the time of Lewis Cowell's protest with the foundation of the Association for the Improvement of Geometrical Teaching in schools and universities, which later became a type of association. And there, really, you were expected to reproduce what you had read, otherwise it was marked wrong. There were no marks in the tribals. What I'm wondering is, in your studies, how do you find where the idea of thinking about you as opposed to reproducing, thinking about you as the content as opposed to treating the classical things, where that comes in in different countries? Oh, that's a very interesting question. And Klein, you know, wrote quite a long treatise on education in the teaching of theology, in the end of one of his books. I mean, I immediately think of someone like Clair Holt, for example. When you speak of Klein, you don't mean Felix. Felix Klein, yeah. Oh, you do mean Felix. Yeah, yeah. In one of his books at the end, he has a long study of teaching methods in different countries. And it's funny, because one country is abandoning you, but another one is reinstating you. So what I think of is, I think right away of someone like Clearhold, who published in 1741, I think it was Terry, and that's a totally different thing. He says, you know, the rules of studies are usually much more complicated, and people are breaking their brains over,

1:15:00 and why should we go through all these complicated definitions and difficult propositions? So you have a very, sort of, intuitive thing, you just, you know, a straight line is just something straight, and if you want to build a house, you put two perpendiculars, and of course, you put a roof on it, that makes parallels, and all kinds of, just sort of very intuitive like that. So that's a definite revolt against the tradition of . Time for one, or two questions? Any questions? Yesterday, there comes a plane, but in higher humidity, sometimes some questions will need to construct a solid or something, you know. I think to a lot of things that mentioned, to see and touch, and even in a more high dimension, we need pictures, moving pictures, to move the other dimension. Did you use this kind of painting or...? textbooks are printed on flat pieces of paper, which means that all the printed diagrams are flat. In algebraic chemistry, we deal with varieties in n-dimensions. So some of those pictures I showed you, the one up here in Arthur's example, it's a three-dimensional variety in some very large space, but we still draw a picture in flat because we don't have a three-dimensional model in the view. The only attempt I've really seen to do that is Tom Banshoff at Brown one time made some pictures of surfaces over the complex numbers in four dimensions a video screen and he had it moving so you see you see you see different three-dimensional first of all it looks three-dimensional on the screen but it's actually a four-dimensional object and it moves so that you get different three-dimensional angles looking at it but there's no very efficient way of representing three-dimensional things that we can do. We don't want to use it very much. Maybe we have time for two quick questions. Saul, and then... So, just to point out the dynamic character,

1:17:30 that we tend to think of diagrams as finished objects, but actually both in the Euclid thing about an isosceles triangle having equal angles we had it put in, But in the diagram chasing argument, not only do we have to think about various things being commutative, but we have to keep putting in new lines and new arrows to show where things go. And so we have really a succession of diagrams that we should be thinking of rather than a single one. I never thought of it as a succession of diagrams. I thought, the diagrams on the page, you tell the reader, at this point you have to do a diagram chase. And if the reader has some experience, he says, oh yeah, I'll take X, go over here, go over here, take that. Okay, but the reader is filling in by constructing a new diagram. Or working on the given diagram with imaginary elements moving around. I don't know, is that constructing a new diagram? Or is it building something on the existing diagram? Well, what you did was, you constructed new, new arrows, and now you have to think. Diagrams are helpful. Very helpful. Yeah. I could have said it without, without constraints. It would have been hard to follow. But it's going to be zero there, and then it's going to be zero there. I could have said it without scrolling, but that would be harder to follow. That would be, yeah. Diagrams of composition, as well. Yeah. And then code chalk would be wonderful. Use colors as well. Yeah. Mm-hmm. I have tried to make a kind of a list of things which appeared in your lecture. Let's take formulas, diagrams of the commutative kind, metrically inexact diagrams, and pictures. And I wanted to know if you consider that some of them must be identified and if not, where would you make a distinction of nature between those kinds of things? Or do you think that there is some continuity between one and...

1:20:00 Where would you make the gap? Certainly the diagram chasing and the community of diagrams to me is of a different nature than the pictures of varieties. So that's certainly different. Other than that, I'm not sure. I don't think I'd put on some side, I think I'd just say there are all these different things, and maybe that's your job in this kind of classifying, but I just see all these different things and I hesitate to put them in boxes. Okay, thank you very much. So we have a 15-minute break, or 14-minute break, at quarter after 12. A new geometrical reasoning is rightly claimed to be necessary for the deployment of the demonstration, because without them, not only our knowledge of spatial relations would be extremely poor, but many proofs would turn into non-sequitur. Things are not so straightforward, though, even if perceptual cues a primary role in nuclear-splained geometry, and texts of mathematical works in ancient Greek are almost always accompanied by figures, not only in the geometric and arithmetic books of the elements, exclusively visual proofs are to be found nowhere in the elements. On the contrary, geometric arguments are always composed by a verbal path, which we may call discursive text, and which interacts with the diagram via cross-reference, letters of groups of letters to indicate points, lines, and so on. Even when the claims to be proved are considered diagrammatically evident, it It is the case of the proposition of elements 120. In any triangle, the sum of two sides is greater than the third, which seems to have aroused, in antiquity, disputes around the necessity of the fully-fledged proof that we need in the elements.

1:22:30 Indeed, it is hard to see why Euclid troubled himself with such a proof, if in other logic of the elements he would also allow himself to draw conclusions from diagram and a constrained wave. I will propose in my talk a twofold explanation. At first, in Euclidean plane geometry, diagram-based inferences are subject to a set of constraints that relieve from the strictly controlled use of diagrams in the genetic practice. Because of these constraints, visual inspection of the diagram, despite its persuasiveness, cannot alone stand as a proof of the claim in 120. The second point is that proofs enter Euclidean practice to serve broader explanatory purposes than justificatory ones. Proof of 120, for instance, besides justifying the attribution of certain properties to given geometric entities, plays a decisive role within a strategy for enhancing control on diagram appearance. I will now give a brief description of the set of constraints in action within Euclidean Euclid's plane geometry. As recent works have shown, in Euclid's plane geometry, claims can be read directly in the diagram when they are based on those diagrammatic conditions insensitive to the effects of a range of continuous variations in diagram entries, the so-called coaxial diagrammatic conditions. Coaxial claims regard, for instance, part-hole relations of regions, segments-bounded regions, lower-dimensional counterparts, intersection of curves. On the contrary, claims are always assumed or established explicitly in the text when they are based on those diagrammatic conditions which fail upon the slightest variation in the appearance of a diagram. These last conditions are called exact diagrammatic conditions and exact claims include equality and inequalities between lines and angles, except the case of coincidence, parallelism, rightness of angles, and proportionality. I have basically shared Kenneth Mander's position, according to which these strictures on inferential standards are directly related to the physical and cognitive capacities participants deployed in attaining uniformity for controlling, production, and reading of diagrams in a shared way.

1:25:00 A little practice with diagramming can show how agreement on their appearance is easily attained in responding to several diagrammatic conditions. I suppose that any Euclidean geometer, as agent and or with normal cognitive capacity, has enough cognitive skills to agree on many coexist conditions indicated in a diagram and add enough physical skills to produce diagrams that meet many coexist conditions. On the contrary, the appearance of a diagram is extremely sensitive to exact conditions. It is humanly impossible to trace segments equal one another within the least degree of accuracy, perfectly straight or parallel lines, circles or right angles, so that disagreement on judging those features in a diagram is common and expected. Since written language can be employed successfully to express exact properties, Euclidean geometers can invoke textual resources to supply weakness in appearance control. In the elements, the construction of straight lines and circles is postulated. Lines are said to be parallel or perpendicular in a given geometric context, Equality and inequality between these ten elements in the diagram are usually obtained via fire entries in the text. However, in Euclid's Plane Geometry, control of diagram appearance does not limit itself to charge the text with the responsibility for that claim. Diagrams are compositional objects, and it is common that spurious subdiagrams just pop up at any step of a geometric construction. It is only to trace a line inside the triangle, and two new triangles appear without their being recorded in the text, or to cut two circles and their intersection point happens in the diagram. It has been rightly remarked that auxiliary constructions, being an essential part of geometric proof, ancient geometric proof, always introduce new individuals with respect to the beginning and the end of a geometric argument. However, to admit that something just happens when we draw a straight line of cross the circles, it is to admit a break of control along a broad font.

1:27:30 in fact as constructions pile up heuristic uncertainty about their outcome will probably grow leading to the possibility of multiple and dissimilar results under the development of a decent proof we look then for alternative measures that might have been available within Euclid's plane geometry in order to overcome what might have been felt as an impotence or failure in control. To provide critical scrutiny for variant diagrams, or, as it may be said, to probe them, seems to be one of the most promising means for challenging control within ancient geometric practice. In the following part of my talk, I will try to elaborate according to the guidelines in these papers which I think he has been published this year so 2008 but it's called the Mathematic Euclidean Diagram 1995 Ninety-five. This is the manuscript. Ninety-five is there. It came out in 2008? Okay. It means a local strategy for probing diagram alternatives in order to show how control on spurious subdiagons can be attained. I call this strategy local because it applies only to the emergence of spurious triangles under a geometric construction, but it is not of small importance. A little practice with diagram drawing shows that they emerge easily from a given consideration of line. Here, it is a list of all propositions on the first book, where Spolio's triangle appears. So I guess the challenge of the situation represents alone a great enhancement in control. An example of controlling diagram appearance is based on testing diagram alternatives. It is the effect of intuition that if we take a triangle and if we stretch its base, preserving

1:30:00 the length of the other sides, at some point the triangle disappears or splits up. If you imagine that we have a triangle, and just if this is made of rubber, and you can maintaining the length of the other side the same, at a certain point, the triangle would break. This may suggest that the appearance of a triangle in a configuration of lines is tied with metric condition concerning the length of its size. Do we have to clarify how these metric conditions govern the production of such a bounded...? I don't know, okay, bounded. In the region. Okay. For example, I may start by moving to segments around the endpoints in B to recall circumstances in which a triangle appears and subsequently try to get them to fail. segment AB and the moving segment AM and DN, and we imagine that they are made of rubber too, so they can dispatch free. We notice that if we decide to circle the moving segments, we will soon arrive at three situations distinct from a coexus point of view. In one case, the two certain segments intersect the given initial segment in one recognizable point. Otherwise, they intersect it in two points, M and N, such that either point M is between A and N, or point N is between A and N. I think another time. So, you can interpret it like this.

1:32:30 Or... like this. So if we consider the trace circle instead of the circling segment, we notice that three pages can occur. Circle with ray AM either touches externally, circle with ABN, it's tangent, or circle with ray AM is second, a circle with ray BN is crossed with a circle with a BN. Or finally, circle AM does not intersect circle BN. Now, when the circles cross each other, a triangle, or exactly a couple of them, can be constructed by joining the intersection points with the and B. In the other two cases, no possibility to connect the end point is given in order to follow the end. We have thus come up with the following result. If circle A, M, and have an intersection point, so that point N is between A and M, and the centers A and B are joined to it by straight lines, a triangle is produced. The clear fact that this geometric context does not exhaust all the imaginable contexts in which a triangle can pop up, shows that it is a sufficient, but not a necessary condition for presenting a triangle. Secondly, condition concerning the point-wide circle perception is indeed a coaxial diagrammatic property, and the second one is a diagram entry not controlled propositionally. There no need to be controlled propositionally. That is the reason of my own space line. Quotation mark. Yes. In other terms, lines may not be drawn for this space or equal to one another to

1:35:00 produce a clearly-readable diagram. We have enough elements, I think, to try and ask the question about the metric relationship that hold between the three sides of the triangle that's produced. Inspection of the diagram shows that when circles do cross each other in exactly one point, not AB, and if we take just the region of the plane outside AB, then they cross the given segment AB on two points, M and N, such that N is between A and M. In Euclidean plane geometry, adjustments of equality and inequality among segments can be made directly on the diagram, when a segment is a proper part of another or two segments as coincidence. There are both cohesion conditions. Both coaxial situations are obtained in our diagram, so that we can conclude that a, m, n, m, and m, and b, taken together, are equal to a, b. From this, we can infer via simple reasoning that a, m, and b, n, taken together, exceed a, b. Thus, we can get the following results, which is coaxial. If circles, we weigh a, m, and b, n, do cross each other in exactly one point, and thus a triangle can be produced, a, m, and bn taken together at greater than the segment a, b. On the contrary, the following statement, which can be considered as the converse of the preceding one, even a segment a, b, and two circles with a, a, m, and bn, if the sum of a, m, and bn exceeds a, b, then the circles with a, m, and bn and two intersects in exactly one point, is no more based on quite a diagrammatic condition. In fact, we can get it to face, imagining that circle BN has been drawn, has been drawn so as to cross circle AM in more than one point. The circle, that circle, Bn, is in itself coaxed at the equivalent to another circle,

1:37:30 B first, B prime, N prime, which respects the point-wise circle-circle-plagative condition. but in the overall diagrammatic context it's been right to actually different geometric situation exactly one point you mean two points are symmetric not there okay but in the good answer two intersections, two symmetric ones. But before I decide to proceed, I should have... No, no, I just want to be clear, yeah. But in the overall diagrammatics context, it gives right to other and different geometric situations. Given the same disposition of the points in AB, in order to trace circles that only intersect in one point only, taking the upper part of the plane, it is required that rays am and bn hold the same length when circling around. And the last condition is an exit one. According to Euclidean standards, we need a verbal argument in order to prove this plane. The necessary condition. And this is done in the first book of the elements. Where with right variation in the terminology, Proposition 120 subsumes the preceding claim B. The letter, I mean claim B, is indeed explicitly proved in the Euclidean text as a necessary condition for the regulated construction of a triangle, given its three sides. And the proposition is 122, the construction of a triangle. Now, we could expect that if probing had ever been an historical practice, Euclid's proof-of-claim-B should, in principle, recover our preceding steps. However, the peculiar procedure adopted by Euclid is utterly different. But I think this thesis may be motivated by methodological reasons, reasons of deductive orderings, for instance, could have forced the author to place a proposition as a topical of Poinject in the structure of the element, for book, limiting the available resources to those claims already proved in the text.

1:40:00 What's more, as an historical practice, probing can only conjecturally be described at Euclid's plane geometry, where we can find, in Euclid's plane geometry, at most, responses to it. For instance, the diurism of 122, which is the statement proved in 120, the necessary condition for the construction of the triangle, can also be seen with Brockels as having an objection-refuting role, thus excluding all the variance diagrams examined before. So I will return briefly to my initial question. How can you explain that you will give proof even when you see clearly what's going on in the diagram? A proceeding discussion shows that even if we admitted that the non-propositional acquaintance with the diagram generates a reliable belief state about the assertion stated in 120, it does not automatically license the proof of the claim. As it was shown, if we want to prove the claim in 120, we must invoke a verbal argument, because the claim rests on excellent diagrammatic properties. And these constraints on diagrammatic properties are, as far as we know, enforced not only within Proposition 120, but they represent a standard of practice within all the geometric groups of the elements. To return to the second part of the question, once our claim is demonstrated, plane geometry, by accepting a number of discursive results containing from position 1 to 1.19, is endowed with adequate resources to predict whether a given configuration of lines will realize the necessary access conditions in order to produce a triangle. Since the practice of testing diagram alternatives gives sufficient access conditions too, It is reasonable to suppose that in the impossibility of stating all construction postulates needed for Euclidean plane geometry, a significant enhancement of controlling the emergence of a certain class of spurious diagons has been nevertheless acquired. Thank you. Thank you. Question.

1:42:30 You mentioned Procris, and you said to me that in Procris we have such arguments related to how this diagram varies. So do you restrict your own analysis to the kind of variation from which you can find evidence? Or how do you connect the kind of variation from which you can find evidence in the photos with the kind of analysis you want to develop from? I think, as a starting point, we should consider not only historical practice, but even imagined practice. So how could we possibly interact with the text of the elements? For instance, you were going to say that in this case, you have something in the process, right? Looking somehow, producing the same kind of variation, but in a different way. So how do you articulate this? I started by making a conjecture about how proposition 120 could, how we could have been arrived to proposition of 20, a conjecture based on Euclidean's side, in what were the resources at this position. And the fact that Brotruth presents something which may some points of connection with the results of this conjecture is a way of showing that this may be a reasonable, a reasonable way of attacking the proposition of the element. At the same time, Procure does not exactly what I did, because there is enough, what concerns the text of the element, there is a fact that objections are hardly mentioned. They are incorporated into the elements.

1:45:00 Procluf discusses the case and objections, which, I think, is the only historical evidence of the practice of probing. It is not exactly this term. My question is very naive, so I'm sorry to put such a question, but you're talking about the possibility of tracing a segment equal to another one, and now of course we'll do So what exactly do you mean by equality of two segments, say, in a diagram, or in a larger ecosystem, which is an explanation? I think that equality in a diagram can be read looking at the diagram when two segments are coincident. So when you arrive for some reason... They are not coincident? They are not coincident. Equality cannot be seen directly in the diagram. So you understand the future? Yeah, I am the other thing. What is the meaning of, oh, a bugbear by the grammatical conditions if you have to pass in a sentence to explain what it would be? Because equality is not correct in the grammatical conditions, unless segments are coincident. I don't know if... My question was really bad. No, it's... I have another many questions. Me too. So if you're doing the Manders thing, you're saying, well, we have triangles, and we want that we have a particular triangle, we want to see if the result is general, so we imagine kind of arbitrary continuous variations of the angle from the sides, something like that.

1:47:30 I would think that when we're doing that, we would allow all of the sides to stretch, so that it stays with the triangle as you do this. So how did we get to the broken triangle where we allow one side to stretch and we preserve the lengths of the other two sides? In other words, I would think we'd want to kind of stretch it and move it and keep it being a triangle. It's a way of placing only one and moving the other. Right. And what motivates to do that if we're probing exactly 1.20 and we want to know what are the conditions under which a triangle is possible, you might stretch one side and keep the other two sides the same length. But if you were just kind of probing the generality of a statement, say the angle sum theorem instead, you want to say, well, I can do it for this triangle, can I do it for that triangle, can I do it for that triangle? What theorem? The angle sum theorem, what is it 132 132 so i would i would keep it being a triangle but i would let it be sometimes you know scaling sometimes obtuse sometimes bustleys and i would keep it being a triangle and i would let the i would let the everything stretched so that the stay the triangle and the line stays straight but they can stretch and the angles can rotate i wouldn't i wouldn't consider I don't know if they have an answer, but maybe it depends on the, on the, on the, on the geometric context here on the right. There's something, you have to imagine a sort of interactive situation when doing, discussing a geometric proof. And for example, there are participants that make objections and propose cases, and based for example on intuition. So on this case we may start to see what we have to modify in order to let the triangle disappear into fate. In the case of the sum of the angles, you may start from another point and see what happens if you...

1:50:00 You want to vary the angle. Because if you stretch the distance, the angle disappears too. It may be depends on the context. So here you're just imagining that it's 1.20 that is being approached. Yes, I mentioned one of the tools, he was suggesting the end of the page, without developing it. I've got from what you're saying, you can't infer an exact future from life. I think so. Not only. When an angle is a part of another, you can. So that's when an angle is part of another. I thought you said when you have one region, one segment, one kind of partisan, a proper one. Okay. Let's go back. It is so, yes. Okay. Yes, I agree. So, are you, do you think that it's always okay to infer, how is that? It is not always okay. It is not always okay because, well I can make an example, I think maybe a right example. you have a triangle you draw that particular and you say that you can be inside the triangle feature of the diagram then you take a triangle which is quite easily equivalent

1:52:30 So in this case, you cannot infer directly that this line goes inside the triangle, because there is a possible objection. And so you have to reformulate your statement doing what Mandel's called branching, tape branching. uh they looking at the consequences from this geometric situation from this one at the same if they are the same uh the figures are equivalent now can i continue yeah okay right okay so so now that so you can't always if uh a co-exam So, now the question is like, when can we involve people in our lives? Thank you for that. Because I think, where are the cases that I think this is the point that we don't know before. Before examining, doing this sort of reason, we cannot know how, where, what are the legitimate corrective features. Maybe in new physiological practice, for every figure, this kind of reasoning was made, testing and checking alternatives. Maybe sometimes things are obvious. They're obvious that you can change the corrective features in your figure that you like, but result is the same at other times uh this process of testing alternatives may lead to different results it is the case the only so i think this is the the fact that we cannot know before what are the so the way that we cannot control them is a break in control in the geometric practice Can I just follow that up a little bit? I mean, Mander says something like, we can infer it if it's true for all continuous variations

1:55:00 some such remark, right? A co-exact. So here we imagine, you know, a continuous variation is changing that acute angle into an obtuse angle, and then it comes up. So the idea that this notion of all continuous variations is not antecedently precise. It's not clear. is the problematic idea within one group because what is the the the the limit the limit point when you have a line you imagine to deform it when you cannot no more accept it i mean all these are questions but at a certain point this is no more as a line where even if you you You can say that it is a straight line. You cannot say that it is a straight line, even if you are twisted. But there is no, there is no, it is a vague difference. John, do you want to say something about that? Making this notion of all continuous variations per time? Right, it was hard because, just like you said, is that you can make a line into, like, a curly circle, and it's still topologically close. But it seems like with 10 codes actually kind of topologically close, but you can still have certain roughs. So the second one down, you can still say maybe it's close actually close, but the last one, yeah, you didn't show that. The idea is that I think that with the drawing straight lines and drawing circles, you can be expected to get roughly the shape of the circle, so if you can be convected, let's draw convected. Or, um, these lines. What's the shape of roughly the shape of the circle? Right. Well, I don't... I think that you've asked the rest of it. No, yeah, it's roughly pleasant. It's not, it's not worked out. I guess what I did with the diagram, part of the idea was that there's a guy who has

1:57:30 two formalizations, one guy who finds everything, who's a purely topological entity. So you can get diagrams where a straight line looks like a thing in the bottom. To try to avoid that, what I did was have these underlying array entrants so they could be as refining as one. The idea was to kind of build in in a way that warms the previous one. So that's one way to make it so that it's interesting. But I don't have any sense about how it works. Mark? you see how hard it is to have a student so you spend a lot of time to follow them and then they agreed with Ken Mandos. There is one thing about Ken's account that I never understood while and you repeated it so perhaps you will explain me better than Ken it is this idea of diagram control you made the example here of circles that met in more than two points or more than one point on the one side Ken makes the idea of the proof makes the example of the proof that all triangles are is also with a diagram that is not a good one But it seemed to me that, at least in a great majority of cases, perhaps not in all cases, I don't know, but at least in a lot of cases, these proofs are based on diagrams that you can prove not to be correct based on precedent propositions. for example if you take the Robin one seven one seven the one seven you prove that a circle two circles can only meet in one or two points sorry one point three no but you can already you can already deduce it by the fact that which were

2:00:00 only on the on one segment you can only want triangles because if you add two points of intersection of the circle you have you would have two triangles well okay four if you want okay one on the one side okay yeah yeah yeah yeah yeah so you don't you want to say that this case is not excluded by one seven that there is a circularity here but is enough to exclude the case that is considered in 120, because 120 you are working exactly with this case. So it's enough to exclude this possibility. So it seems to me that, yeah, it seems to me that the fact that certain diagrams are not constructible is not only a question of controlling the diagram, but it's a infer from other population. For example, the fact that the proof that all triangular are is also less is not a good proof does not depend on the fact that you have a good control of the diagram. It depends on the fact that you can prove that the diagram with the two points on the other side, two external, one internal, or two internal, is not possible. You can prove that. So it me that it's a question of the geometry, not a question of controlling diagrams. So I don't see in which sense the controlling of a diagram is something the quality of segment is in Euclidean elements is clearly given by the definition of circle So, what means the two segments are equal? It means that they are radios of the same circles, or they can prove to be equal, based on the property of equality that are given in the common actions, to be equal to two segments of the same circle.

2:02:30 From the definition. It's given from the definition. You have not an independent definition of equality of segments. Two segments equal means that either they are two radius of the same circles, or they can be proved on pure and illogical deduction to be equal to two segments of the same circle. It means nothing else. It's not diagrammatic. It's not diagrammatic. It's not diagrammatic. This is my point. It's not diagrammatic. This is my thought, yes. But because of the definition, to prove equality, you do constructions of circles, which is diagrammatic in some sense. So I think that the problem is control. Thank you.