So let's introduce the derivative. Hello, derivative! So our path to the derivative starts with this notion of the average rate of change, after which we define the instantaneous rate of change as the limit of this average rate of change. Since that instantaneous rate of change is always at some particular point, one important feature is that we want to make sure that we have an interval that contains that point, and the easiest way to do that is to either begin or end the interval at the point we want. So suppose our function is f of x and we begin our interval at x equal to a. Let the length of our interval be h. Then the interval runs between a and a plus h, and our average rate of change is going to be our function evaluated at the end minus our function evaluated at the beginning over the difference between the two endpoints, a plus h and a, which is just going to be h. Now our instantaneous rate of change is going to be the limit of the average rate of change as the length of an interval containing a goes to zero. So we're going to take the limit as h goes to zero of this quantity, which is called the difference quotient. And this particular limit is so important that we give it a very specific name. We call it the derivative at a point. And like every definition in mathematics, it is vitally important that you understand and remember this basic definition. So here it is, let f of x be a function. The derivative of f of x at x equals a, written f prime of a, is going to be defined as the limit as h goes to zero of the difference quotient. Except no substitutions. So for example, we might want to use the definition to find f prime of 3 for f of x equal to 5x plus 3. Now a little later in the course, we'll find fast ways of finding the derivative. So this naturally raises the question of how do we know when we're supposed to use the definition to find the derivative? That's a very complicated question to answer. And unfortunately the only way to tell that you have to use the definition instead of some other fast method is that the question says, use the definition. So we'll pull in our definition of the derivative. And so we need to find f of 3 plus h and f of 3. Now generally speaking, calculus is easy. It's the algebra that causes problems. So here's a quick reminder of how to find a function evaluated at a point. Now there's many ways of doing this, but here's one that I strongly recommend. Let's take our function definition. f of x equals 5x plus 3. And what we're going to do is we're going to take every occurrence of x and replace it with an empty set of parentheses. And the important thing to remember here is that whatever goes in any of these parentheses has to go in all of these parentheses. So I want to find f of 3 plus h. So I put a 3 plus h inside this set of parentheses. And I have to put a 3 plus h inside this set of parentheses. Likewise, I want to find f of 3, so I'll substitute that in. So I have my definition of the derivative. I'll substitute in the values I have for f of 3 plus h and f of 3. I'll do a little bit of algebra. I'll do a little bit of algebra. And finally, I get to a point where I have to do some calculus. I want to find the limit as h goes to 0 of 5. And since 5 is a constant, I know that the limit as x approaches anything of any constant is just the constant. And so my limit is going to be 5. Now it's worth pointing out that in all of this, we only use calculus at the very beginning where we set down the definition of the derivative and at the very end where we actually take a limit. All of the intervening steps are pure algebra. So this is a good time to brush up on your algebra skills.


So we're going to be talking about pretty much Rome under Pope Julius II, Julius de la Rovere, whom we mentioned briefly last time. And Julius is a really interesting figure. You could pretty much say the high Renaissance corresponds, more or less, with Julius's papacy. It was a period when some of the ambitions of the earlier Renaissance, let's say, this interest in proportion that we already saw with Brunelleschi, become more elaborated. So instead of simply having a proportional system that is based on the arithmetic adding together of like modules, there's an increasing interest in complex proportional systems. And this is just the golden rectangle, which shows you this system whereby if you take a diagonal from the midpoint of the square to the corner of the square and swing it down, you get a rectangle that begins to create this kind of endless spiral of nested similar proportions so that this A rectangle would yield another square and a golden rectangle and so forth. Over here is a Renaissance drawing of something called Plato's lambda. Lambda is the letter L in Greek. And Plato's a Greek, so he has very few choices when he picks letters. But what this basically shows you is something like the system of golden proportions that are used in the Renaissance and again and again. So not only are simple volumes, simple platonic solids, Euclidean geometries authorized by antique thought, but also these kinds of ratios, these complex ratios. As we look at the papacy of Julius, we're talking about the Cinquecento, bye-bye Quatrocento. And the Cinquecento, Cinque is Italian for five, and Cinquecento simply means the 1500s. And the 1500s associated with the high Renaissance are the first couple of decades of the Cinquecento because by 1527, there is a great political upheaval and Rome is sacked and the pope goes into hiding and pretty much the Renaissance falls apart at that moment. But during these glorious first couple of decades of the 1500s, you have this amazing flourishing of art, quite a lot of it sponsored by the patronage of Julius. Does anybody know what patronage means? Yes? Exactly, they're funding art, they're funding architecture, and you need that, right? Particularly you need that for architecture because you could have terrifically good ideas. One of you sitting in this room could be the best architect in the world. I suspect that many of you are. Let's say we have the top 10 best architects in the world right now. All that you're lacking is a commission and money to build it so nobody knows how good you are. So patronage becomes really interesting because it means there are money people who believe enough in this new art project or this new architectural project to fund it. We already saw that when we looked at Florence and the Quattrocento, that you had these new money people in the guilds and in the banking families who are willing to take a bet on this new kind of art. And in the Cinquecento, Julius becomes this kind of, I don't know, patron on steroids in terms of commissioning great projects, great projects for the glory of Rome and great projects to build his own legacy. Just to backtrack a little bit, we looked at centralized plans last time and we looked at the ideal centralized church and various attempts to build it. Anybody know what this one is? Yes? It's the Tempietto by Bramante, right? Often thought to be the ideal super perfect version of a centralized church and one reason that it's so capable of attaining ideality is that it doesn't have much work to do. All it does is mark a point. It doesn't need to deal with these things that for the most part create the failure of other attempts to make centralized churches. It's not a parochial church. There are no masses celebrated inside of it. You don't need to have a congregation. You don't need to have a priest. It's simply marking the spot where St. Peter, it takes this opportunity to really look at a Roman precedent. The precedent of the Tholos, which is a round temple and try to build it as carefully and accurately as possible. This section through the Tempietto I think is really interesting because it begins to show you what happens as you keep getting narrower and narrower from the perimeter of the freestanding columns to this carved wall. And you go from a kind of normative AAAA rhythm to a really syncopated rhythm on the inside in order to make the squeezing happen. We looked last time at this elevation where regulating lines, these diagonals, had been drawn on the volumes of the Tempietto. What are those parallel regulating lines the rectangles represent? This is a geometry quiz. Yes. What do you mean by proportions? Exactly, they're similar rectangles. Exactly right. And similar rectangles is a term from geometry class that means each of these rectangles has the same proportion. And I threw this little curly cue in here, which represents the golden rectangle spiral to show you that it's not just any old rectangle, but it is specifically the golden rectangle that Bramante is using here to understand the proportions of the base, the proportions of the drum, and also the proportions of this vertical that intersects with this horizontal. So when you look at these Renaissance buildings, there's really nothing intuitive and crazy going on. At least at this moment, at this moment of the high Renaissance, there is still this ambition to allow reason and logic and proportion and the lessons of the ancients to call the shots and to simply refine the application of these techniques. I mentioned last time that St. Peter was crucified upside down. And this is a painting by Caravaggio, which is he's an amazing Baroque painter. And he's showing the crucifixion of St. Peter as they're about to put the cross upside down. Worth looking at carefully, but we're not going to do that. Sadly. Maybe we will when we do Baroque. I want to show you one Cinquecento Centralized Church just as a point of comparison for the Quatrocento Centralized Churches we looked at. And that is Sandbaggio in Montipulcano. Montipulcano is a Tuscan town, magnificent wine. You could pretend to go for the architecture, urge you to stay for the wine. And by now you should be really familiar with what these long-distant views of ideal centralized churches look like, because the only ones really that get built are out in the middle of nowhere or out at the edge of town or out at the bottom of the hill. And that's the case in Sandbaggio. Montipulcano is a hill town. And at the base of the hill, you have this astonishingly beautiful church by Sangallo. Not the same Sangallo that did the Quatrocento Church that we saw last time. That was Giuliano Sangallo. This is Antonio Sangallo, so many Sangallo. This is the plan. You should notice a couple of things immediately. One thing, Greek cross, right? This is something we've seen. And this is something really suggested by Alberti that you want to take the circular plan. There are a couple of ways that you could amplify the circular plan. One is the Greek cross. And what's nice about this Greek cross is it begins to make possible some kind of axial relationship to the nave, and it begins to create a little bit more space. It also does a couple of other things. I mean, I think that Sangallo here has gone beyond what the Quatrocento architects have suggested, and he's begun to add things, right? He's thrown in this little sacristy in the back. And at the same time, round it off one lobe so that you can read that this is where the altar goes. He's marked it. He's made it visible. He's also done this thing that I think is incredibly sly, and that is he's added two towers. Not both of them got built, but he's added two towers to the front, and they don't touch the church. They don't break the cruciformness of the church, which you perceive in an unadulterated state on the interior. They're simply next to it so that they form a plane. When you look at the exterior of Sangallo, it's like a west work. It's like a big facade. But at the same time, it is this super perfect, ideal, centralized church. So this is a view coming down the hill and looking at what this thing looks like. It's very gutsy volumetrically. You read this cube with this cylinder, with this half sphere, and a couple of temples clipped on the four axes. This is how the tower begins to work with a facade to create this sense of plane, but at the same time to break the plane. And notice the plasticity of the wall. And also, I think this is something you'll notice often with the Sangallos, there's also a more canonical use of the architectural language. When we were looking at some earlier churches like Santa Milet de la Consolazione in Toti, we saw that the column capitals could become little cow heads, winged cows, or almost anything. And here, there's more of an attempt to get some kind of Doric going on with some kind of typical Doric freeze. The lessons of antiquity are being played straight rather than being played in an inventive way. But I think what is so strong about this church is the attitude toward the wall. The planarity, the flatness of the wall that we saw in the Quattrocento stuff is really becoming liberated in two directions. Here, we're pulling out with these engaged pilasters and columns to give us an outer layer and then we're pushing in and carving into the wall. So this is a really kind of deep surface of the wall with all of these different planes making relationships to each other. This is the little church of Sambiago, and when I say little, it's pretty big, at the base of the hill town of Montipolcano. And I'm just showing you the painting of good government that we looked at before to show you that the landscape of Montipolcano is pretty much the landscape that the Lorenzetti were painting and not so different even today. Just compare Sambiago and the Sambidella Consolazione. Here, it's really all about the flatness and it's really all about honesty too. You're doing your Greek cross and you're not adding things and here there's a lot more license and a lot more plasticity. Both of these things are out in the middle of nowhere or at the edge of town. Showing you the flatness of one, pilasters barely relieved from the surface and the incredible carving, thickness and depth, plasticity of the other. Ditto, interior of Sambidella Consolazione where there's a kind of interest in articulating the wall but not exploring the depth of the wall and this hyper plasticity of Sambiago. One of Bramante's first projects, when he got to Rome, when he left Milan and went down to Rome, was a courtyard for a convent called Sambidella Pace. And this is pretty much what it looks like. This is the plan of the courtyard of Sambidella Pace. Can anybody guess what the geometrical big idea here was? Any takers on that? Pretty much a circle in a square, right? It's not difficult to see that. Bramante is building an ideal form. In many ways, this is strongly related to his project for the tempietto, the one that didn't get built with all the concentric rings where not only an individual object but an entire space gets organized by this ideal geometry. Here you see it, cranking off the church of Sambidella Pace. And the church that you see here is how it looks now and all of this part's baroque, so Bramante would not have seen that. But the axis of the church would have been like this and Bramante rotates it, of course there's a street here and makes this perfect square. What is considered to be important about this courtyard by Bramante is its geometry but also that Bramante, the serious smart guy, Cinquecento guy now, is tackling the question of what do you do in the corner? How do you solve the question of turning a corner on a courtyard? Bramante does it in a way that's kind of similar to the way that we saw it in Orbino but even more sophisticated. And that is he gets this kind of complex double rhythm going. If you notice the lower story has this superposition of the traviated and the arcuated so that there are little columns slipping down and having the ground plane as their base and pilasters pushing up and being situated on plinths. A plinth is like a platform or an artificial ground plane. You could say a stylobate in a Greek temple, the platform that a Greek temple on is like a plinth. But you could also say that these pedestals that the pilasters rest on are plinths. So he's got this syncopation going of two scales, the big scale of the plaster, the little scale of the column. When he gets to the corner, he's already gone and placed this kind of doubling mechanism that we saw introduced in a somewhat artificial way in Orbino. So he has already built into his system a little column to come to this end and a little column to come to this end. Fabulous. And I think another nice thing that he does in the courtyard is that when he gets to the upper story, he doesn't simply take the same dimension and have it repeat. But he recognizes that this is a lower height. Bramante is all about proportion, desperately interested to lock as many proportions in as possible. So he splits apart the two kinds of elements that are conjoined on the lower order and has this kind of rhythm of peer column, peer column. Beautiful, such a smart map. By the way, this is where it's located. I put a little circle here. I don't know if you could read my red writing. I thought that would be extra clear, but I guess there's so much junk on this map that you can't really tell what it is. But it's a great map. This is the Noli map of Rome engraved by a guy called Giambattista Noli in 1478. And what's so nice about this is that it really gives you a sense of what the urban space is doing. We see the river down here. And we see lots of black stuff and lots of white stuff. Any idea what the black stuff is and what the white stuff is? Yes, good. The black stuff is like homes. It's private space. It could be even like an ice cream store, but it's not something that has a great public value. And the white stuff is public space. And the way public space is understood here I think is interesting. That public space is the piazza, the square. And this is piazza Navona, one of the great squares in Italy, or even the world for that matter. Here you see the pantheon. The pantheon at this point is a church. And the interior space of the pantheon is considered to be an extension of the public space. Courtyards that relate to different buildings. Like this is the little square courtyard, the cloister of Sanctum de la Pace by Bramante. Comes in here. So you get this elaborate knitting together of figure ground conditions. Figure and ground. You look at this thing and you have this immediate sense that there's some kind of pattern going on. And it begins to show you relative densities. Places that are expansive, places that are contracted. And it's a very different kind of figure ground condition than we would have, say, in Columbus, Ohio. In fact, this is the countryside over here. And you see a couple of pavilions in this open field. And this is more or less what we have. This is more or less what Ohio looks like. Freestanding objects in open terrain. We live in a world of figural objects and not figural spaces. But when you move across from the countryside into the city, there's a figure ground reversal. And that which is figural, which can be named as square or rectangle or circle, tends to be a space, not a perimeter. All the perimeters are subsumed into this fabric. And that's kind of nice. It really, I think, clarifies what the city looks like. It makes you get a good sense of understanding of what's going on. So here is this perfect little square of Santa Maria della Pace. And even in the world of perfect little squares that you have in Italy, this one is exceptionally perfect, I have to say. These are the proportions. Brumante, once again, running with a module of the square and beginning to put these more complex, superimposed organizations of other geometries on it. The red things here are golden rectangles. And the black things here are squares. Even the space of the upper corridors and lower corridors that wrap the pollinate have as their section the golden rectangle. Thoreau going in. I think if I had been slightly, I don't know, more ambitious in throwing these things in here, we would begin to find that lots of secondary things, lots of things that seem fairly minor are also organized by the same network of proportions. It's not like he breaches it for a moment. He finds a way that the system can be expanded to organize everything. Fabulous little Brumante in corner. Now let's get to St. Peter's, which really is Julius' big project. Julius is a megalomaniac, and Julius wants a fabulous legacy. Julius' first project is one that we'll talk about a little bit later on, and that is to build himself the best tomb ever. And he gets this sculptor from Florence to start quarrying marble for him. And this sculptor goes out to the quarries for like about four years of his career, finding marble to build Julius' tomb. And at his own expense, has it shipped to Rome where he can begin to execute something like 32 statues for the Julius tomb. This sculptor, by the way, is Michelangelo. And Julius got distracted with the new project, which is rebuilding the Vatican. Brumante comes in to rebuild St. Peter's. And all of Michelangelo's marble is just like piled up in the square in front of St. Peter's where all this work is going on. And it's kind of sad because Michelangelo claimed, and who knows, I don't think he's a liar, that he could see already in an uncut block of marble the figure that would emerge from it. When he goes to places like Carrara, the great marble quarries in Tuscany, to find himself a chunk of marble, he picks that piece of marble because he knows what the figure is. And if you spend four years of Michelangelo time quarrying marble, that's like 12 lifetimes of the rest of us because he's pretty good. He's pretty fast. Julius didn't quite get built. Julius, meanwhile, is fussing around with the rebuilding of St. Peter's and he brings Brumante in to do it. Julius is from the north. Julius is from the town of Genoa. And he's a little bit suspicious of the Florentines. You know, they're squabbling going on like crazy. So Julius always thinks like the Medici are at his throat and they're gonna take over. And in fact, after Julius dies, it's a Medici that takes over the papacy. So Julius was right, the Medici were in fact at his throes. And Michelangelo had been under the patronage of the Medici. So there's a kind of suspicion there. Brumante's from the north. Julius feels much more comfortable working with this northerner. We looked at the ambitions of these two projects that Brumante did. One, the extension of a garden, a kind of walled garden to connect the Belvedere villa with the Vatican apartments. And two, the rebuilding of St. Peter's itself, which is an amazing project if you think about it. The most important church in Christendom torn down and reconstructed. Of course, some of the things have to be conserved, right? Because there is a crossing. At the crossing there's a crypt and that's where the tomb of St. Peter's is. So you have to rebuild the new St. Peter's over the old St. Peter's. So certain dimensions are conserved in the new St. Peter's. This is a drawing by a Dutch artist called Martin van Heemskerk showing you St. Peter's under construction. St. Peter's as designed by Brumante under construction. And it's sort of spectacular because you see these big barrel vaults coming up and these sort of nicely articulated walls with old St. Peter's still intact, hunkering down underneath these soaring new vaults. As we mentioned last time, that's about as far as it got because Brumante really couldn't build the dome. The domes that he built in Milan were largely aided by the offices of the masons up there. And his biggest dome, Santa Maria delle Grazie, the one that has the last supper in it, was really only about 88 feet in diameter, which is good, don't get me wrong. But something like the Florence Cathedral was about 144 feet in diameter. So a big project, but not as ambitious as that. This was to be even more ambitious. So too bad. The project gets stalled. This is Brumante's drawing for it and it just could not be executed. But you see that there's a kind of drum here that's not so different from the drum of Santa Maria delle Grazie. And by drum, I mean vast cylinder on top of which a dome is positioned. And the idea of a drum is kind of great. You get more verticality that way. And you also get something that plays a stronger role urbanistically because when it's that tall up, it's almost like a tower. And you can see it from all over the city. Of course, you couldn't see this one because it didn't get built. We admired the Brumante plan for its incredible laziness and delicacy. And this idea of the centralized plan becoming elaborated with multiple organizations, a square, a rotated square, a Greek cross, lots of little Greek crosses so that you have simultaneously the possibility of experiencing this thing as a centralized organization or as a longitudinal organization. Tragedy never got built. Everybody came in to do projects. This is a project by Sangallo where he tries to rectify the problems of St. Peter. St. Peter's. And you can see that Sangallo is adding structure like crazy. So much so that the central circle almost seems to be of the same visual weight as these perimeter circles here. These are projects by, here's Brumante. I believe this is Peruzzi. This is Michelangelo. This is Maderno. Everybody's trying their hand at it. And Michelangelo one eventually got built and we'll look at that one a little bit later. But notice the amount of structure that Michelangelo put in his plan compared to what Brumante had anticipated in his plan. It's like tripled. The Maderno plan represents something that was to happen in the later part of the Cinquecento and that is mistrust of the centralized plan. Everything that we've been saying about the centralized plan as a perfect idea means that it's a perfect humanist idea. It plays well with classical philosophy. It plays well with Platonic theory. It plays well with the ideas of Pythagoras. But it doesn't play so well with Christian liturgy. And it doesn't also play so well with Christian liturgy to have people rationally engage the building that they're in. Gothic architecture worked very well in a Christian sense because it was able to persuade people in an emotional and a kind of magical sense that something great was going on. When you start thinking too much about your architecture or if you start thinking too much at all you're on the road toward Protestantism. And so ultimately a nave was added to St. Peter's. I mentioned that Julius was a crazy patron and he's doing everything. He's working on the tomb kind of. He's working on St. Peter's kind of. He's redecorating the apartments, the papal apartments. And if you want to redecorate your apartments well get Raphael. That would be a good way to do it. And at this point Raphael is not the greatest artist yet. Raphael is maybe the greatest copycat. Raphael is incredibly young. He dies quite early, he dies in his mid 30s. And so much of the work that he did that you admire and you think is fabulous, he did when he was younger than you. So live with that. Luckily Michelangelo died as an old man so you have time to catch up with him. But this is Raphael's most famous painting and this is School of Athens. And this is one of the paintings that he did in the Vatican apartments. And the thing about this that's pretty interesting is the architecture. And your first take would be to say, well, this is the classicizing architecture that we always see. This is the architecture of antiquity. This is the architecture of Barrow vaults and articulated walls and carved niches. But it's more than that. It's also the architecture of the St. Peter's that's under construction. He's sort of building as much of St. Peter's as Bramante managed to build. And he's putting this cast of classical characters into that space. And you might say, well, who are these fabulous classical characters that are in that space? And it's called School of Athens. So that suggests, well, maybe these are the great Greek philosophers. These are the heroes of antiquity. These are the people that thought the thoughts and conceived the ideas that the entire Renaissance is aimed at recovering. And in fact, in the center, you have Aristotle and Socrates walking around and having a fabulous discussion. Over here you have Archimedes figuring out how to do geometry. And here you have Heraclitus, who was just grumpy. He was a skeptic. There's not simply a matching of the architecture of contemporary Rome, the architecture of Bramante St. Peter's with the architecture of classical Greece, but also portraits are encoded in the School of Athens of contemporary figures. So you look at Socrates over here. And the Socrates over here is pretty much Leonardo da Vinci. It's a portrait of Leonardo da Vinci, whom Raphael would have known quite well. If you look down here, the grumpy, skeptical guy, Heraclitus, is a portrait of Michelangelo. And Raphael includes himself. This is Raphael, self-portrait. There is a real boldness, you might say, in the conception of saying, this is the School of Athens. These are the great thinkers of antiquity, and we are them. We are them again. I'm just showing you a couple of the other paintings of the Vatican stanza. And the reason I'm showing you this is to show you how clever Raphael is and what a quick learner he is. This is a kind of early one. And you might say, compositionally, it's not so different from the School of Athens. We have this technique of lining everybody's head up and making a plane, lining everybody's head up and making a plane, locking a center line. This is School of Athens is composed, as a Renaissance painting should be composed. And the disputa, the disputation of the sacrament, big argument about, what is this sacrament? Is it the blood of Christ? Is it the body of Christ? Is happening here?  So both of these paintings are essentially the same in terms of composition. The subject matter has been switched out from the great thinkers of philosophy in antiquity to the great fathers of the church and this pantheon of heavenly figures. But then you get to something like this, and this is later, this is 1514, Raphael firing the borgo. He's switching things up a little bit in a number of ways. One, it's not so static anymore, crazy stuff is happening. You have somebody carrying someone, you have these incredibly muscular nudes and these torsioned twisted positions that seem unthinkable based on the kinds of figures that we saw in the earlier Raphael. And one answer for that is that right next door, literally right next door, if Raphael were in this room and he opened up that door right over there, he would find Michelangelo painting the Sistine Chapel. And so Raphael is sneaking in and looking at Michelangelo and stealing. And by stealing, I mean realizing that there is this way to conceive of the human figure that's much more than these very decorous, very static, very hierarchically posed figures that Raphael had been using up until that point. For example, this is a famous Raphael painting, the Madonna del Prato.  Is that true? Who's got a grandma with this picture in her house?  It's a great painting.  Fire in the Borgo still has some of that stuff, but it's more about disequilibrium than equilibrium. And it's more about the disunity of action than the unity of action. A new sensibility is being expressed here and a new technique in representing the human figure. Let's look at some architecture by Mr. Raphael, the Medici, who wants to have a country villa. And he wants to have something out in the hills. Just means three miles away on a hill called Montemagno. And so he gets Raphael to build him something, this fabulous villa, not completely finished. This lower image shows you how much of it actually got built and Raphael stops work and then ultimately Raphael dies and the building is only half finished. The original conception probably would have been more like this, which is to say, this large circular courtyard with garden going off in one direction and an entry courtyard going off in another direction. The thing you see here is not the entry, it's a porch. And the reason we know that it's just a kind of, let's say balcony, is that there's a big sectional change here. So we're way up high here and we're way down low here. But it's a kind of great looking thing in the way that all of these classical typologies are slammed together and reconfigured. You've got a theater, you've got a rotunda, you've got a Belvedere skewered across the axes and this kind of argument between a cultural condition, the paved entry court and a natural condition, the garden. So it's an ambitious project and it picks up on a lot of the themes that are going on in the Belvedere. It also has an interesting decorative program. This is what the Lodja, these big arches right over here are painted by Raphael. And you might say, well, where did he ever get those ideas about how to paint a Lodja? And the answer is that he and another great architect, artist, were doing what they always did, walking around the forum, sketching, taking notes, having a conversation about Roman antiquity and the lessons they could learn. The other architect was Giulio Romano, who also worked on the Vatican stanzae, the Vatican paintings with Raphael. So they're walking along and suddenly they fall through the ground. The ground of the forum opens up and they fall into Nero's Golden House, a Roman interior that had been covered up for hundreds of years. And suddenly they see wall paintings, Roman wall paintings, something that nobody in the Renaissance had seen up until that point because sculpture lasts a long time, architecture lasts a long time, but painting really doesn't. If it's exposed to the weather, it flakes off. Here Raphael is really trying to imitate not only the space of a Roman vault, but also the kind of wall painting. The unfinished that would have been is actually more in a Roman house. So that at least Vila Madama would have been if it had been finished because it begins to suggest some kind of tension, some kind of axis that doesn't complete itself, some kind of engagement with distant objects or pulling apart a villa that is not unified like this one around the circle, but just splitting apart at the seams. And I think this kind of tension became an interesting thing for the architects in the later part of the Cinquecento as we shall see anon. The material of the Vila Madama is also pretty interesting. This is what it looks like, but the color is sort of creamish pink and it's a kind of stone called tufa. And you look at it and you would have to say, well, that is a bizarre choice of material for a classical building because Mr. Raphael, I see you have ionic column capitals. And Raphael would answer you that I don't know what he would say, but he might say, I'm following decorum and that if I were building a country house, I would engage in this more kind of decorative lighthearted sense of materiality. If I were building a urban house, I would be more serious minded about what kind of ornament and what kind of material I use, material I use. So he is translating a classical language into a more vernacular material by choosing this heavily rusticated polychromatic tufa stone instead of using the more refined stone that you would see on a Raphael building in the city. Let me just show you what is often considered to be the super duper greatest Renaissance palace. And that is a Farnese palace in Rome. Fabulous.  We know it's a palace because it is a big flat facade, flat because it's interested in making edge. It's not interested in being object. The thing about the palace that's figural is its courtyard. And so here it's making edge and it articulates itself emphatically as base, middle, and top. Yay. By the way, this thing is the French embassy right now. And so good luck trying to get in. There's a really important Baroque fresco cycle right up here by Annibale Carrache. And I've been in, but they let you in like two days a year. So you probably will never get in. But I am old enough to have lived before terrorism. So I get in. This is what it looks like. The Farnese palace. It's really a square donut on steroids. It's just this giant thing. The Farnese family also was a family that produced a couple of popes. So giant thing with a courtyard in the middle. Here's a square, an urban square in front of it. And there's a garden behind it. So there is this kind of negotiation between one facade that is urbane and addresses the city and another facade that is more open addresses the garden and views across the river. This is the river over here. Of interest in the history of this facade is you can really see a transition going on because the Farnese palace was begun by Sangallo. Antonio Sangallo, the younger. If you really want to be careful about which Sangallo you're talking about and completed by Michelangelo. So there is a shift in the language that they use. You probably can't tell, I would say, but maybe you can. Maybe you don't have enough detail. All this stuff is the Sangallo. And when you get to the top, it's the Michelangelo. So I would say you can't see well enough to know what's going on there. Let's hope we get lucky. Yes, we get lucky here. These are details of the courtyard. What the beautiful interior of the square donut looks like. So Sangallo is a very good architect, but we already noticed when we looked at San Biaggio, which we admired quite a lot, Sangallo is pretty kanonic. Kanonic means he follows the Canon, C-A-N-O-N. And the Canon is a set of rules that can really be locked into a tradition. It's not so much invention as it is mastery of a tradition. So what Sangallo does, I think, is beautiful. He gives you this double structure of the traviated structure of the levels and the arcuated structure of the walls. And into that, he has this very decorous positioning of little pediments and little windows that have their own little infreement. It's really very nice. But when you look at what Michelangelo is doing, there's a kind of disruption going on. For example, on the Sangallo windows, the relationship of the pediment to the little columnar surround of the window is coherence. The pediment rests on an entablature which rests on these little columns. So you get little miniature temples that express their tectonic forces with clarity and honesty. Tectonic means constructional. So the constructional logic of these buildings is being revealed to you. If you look at what Michelangelo is doing, it's like he's pulling the thing apart. The round-headed pediment seems to be held up on little brackets which float above the windows down here and there's a kind of stretching. So if the Sangallo level is all about balance, it seems as if the Michelangelo level is all about some kind of tension, some kind of pulling apart, some kind of disequilibrium. Maybe you can see it better here where the pediment floats away from the window and the surrounds of the columns that you would expect in the Sangallo to rest very carefully on a string course that's placed there, slip down. So there's a shifty, shifty, shifty complexity going on and how all of these elements come together. This is the back facade of the Farnese Palace and in the back facade, there is an opening up of a loggia. There's a shift between how the building presents itself to the city, which is decorous and facade-like and normal and how it presents itself to the garden, which is much more playful and much more animated. Let me show you one more palace and right now we're at the edge of slipping between, let's say, high renaissance and mannerism. Manorism means a stressed and exaggerated application of the rules of classicism, not toward expressing clarity but toward expressing tension and toward not toward expressing equilibrium but toward expressing disequilibrium. And of all the mannerists that we will encounter, Michelangelo is the number one champion mannerist. But Peruzzi is also one of these transitional figures who floats between mannerism and classicism. This is one of his buildings in Rome. It's the Palazzo Massimo. For your convenience here on the Noly map of Rome, we see the Palazzo Massimo, tucked down in here below the Piazza Navona. And it's in a funny piece of the street. The street is curving and you have little street cutting in here. And so the response of the Palazzo Massimo was to bend with the curve. And so how do you make an entry if you're bending with the curve? How can you set that up? And Peruzzi wants to line up with a little street down here. So he carves in a little hot dog of a portico. The hot dog portico is called Portico in Antis, A-N-T-I-S, which just means subtractive portico. Carved into the bulk of the building. It's also a funny building because it's a double palace. These are for two brothers, Pietro and Angelo Massimo. And so you see two courtyards organizing these two houses, but then this incredible kind of push and pull, renegotiation for space, renegotiation for ideality. Because if we go back here to this, you'll notice that they have a square at the back of their building that they relate to and then they have the street here. So the ambition is to somehow lock into this street and to somehow create something like a facade when you get to the back. Here, notice how this one brother, I think this is Pietro, let's say, his facade has this little dog leg on it because he wants to lock the street. And so his brother's palace has this incredibly irregular sprawl of little triangular rooms. However, the space of the courtyard is so clear and so kind of figural that it holds together the irregular perimeter. Here's Angelo's palazzo.  There's a little passageway here so that Angelo and Pietro can share the stables and all of the support buildings in the back. Look at all the recentering that goes on. We're coming down the street, we lock the center of the hot dog portico and we're coming along here and suddenly we're not in the center anymore, we're on the perimeter. We shift over, we're on the center, we're on another little hot dog of space and then we shift over and we're on the perimeter. This is what the front facade of the palazzo massimo looks like. Really planar, all about the edge, all about bar relief except for the portico which is really plastic, but plastic through subtraction. Freestanding columns, columns in the round but only understood as subtractions from the volume of the building, not added things. And this is the courtyard and the courtyard has all kinds of interesting archeological stuff going on because it's actually not an easy problem. The space is pretty tight. These are not giant Farnese Palace style courtyards, they're pretty little. So how do you get enough light in here? And Peruzzi is taking a tip from Hadrian's Villa where Hadrian has something called the cryptoporticus, this underground portico with these windows carved into the barrel vault. This is really just a good example of how the building negotiates all of these differences and that ends up creating a more or less resolved condition here at the far side where it reaches this other little palace condition over here. I'm gonna show you one more Peruzzi building before we stop and this is the Farnesina. And the Farnesina is another one of these little country houses and by country house, I mean, it's just across the river from the Farnese Palace. It's a seven minute walk from the Farnese Palace but because it's across the river, it's a country house. Who would ever cross the river unless you were looking for a country experience? And it's called the Farnesina but the actual client was not Farnese, it was Kiji but it's typologically a little bit different than the buildings we've been looking at because it is a country house, it's a villa rather than a palazzo. And so what's the difference between the villa and palazzo? And the answer is it's the difference between figure and ground. If you're talking about a palazzo, the figural thing is the courtyard in the middle of it. If you're talking about a villa, the figural thing is the object in an open landscape. So this thing is a little villa, little object in the open landscape. And so Peruzzi gets all of his friends to help him with it. And Peruzzi's friends are interesting as they are Raphael, for example, and they are Giulio Romano and they're even Michelangelo. People that he met because Peruzzi was also working on the Vatican Apartments. He was also painting the lunette frescoes like the School of Athens and the Fire in the Boregal. So if you look at it, it seems fairly decorous. There's a big loggia, luckily painted by Raphael. So you have some pretty good looking things. As is the case in many palazzi, the ground floor space is not the premium space. The better space might be up above where you get more view, where you get more breezes. So you walk upstairs to a large room called the Salla delle Prospective. And Salla delle Prospective means hall of perspectives. Let me just go back to a real palace to give you this vocabulary word. Here is the Farnese Palace. And I mentioned that there was a really great fresco cycle right up here. The condition of life on the ground floor in the city was really kind of chaotic.   So one floor up tends to be the premium real estate in a palazzo. In Palazzo Farnese, that is where the Caracci frescoes are located. And this is called the Piano Nobile. Piano like, ming ming ming ming, the piano that you play. And that just means level or story. And nobile means noble. So one story up is the Piano Nobile. And that's not so much the case in a villa because the villa is not full of the kind of clamor that city life would be full of. But even so, there's this spectacular room, one story up in the farnazina called the Salla delle Prospective. This is not it. These are some of the rooms that were painted by Raphael that we have down here with this elaborate astrological chart. But when you go upstairs to this room, there are these fake perspectives that have been painted by Peruzzi very accurately to extend the space of the room. So you're standing in the room and you think you're looking out the window. But in fact, you're not looking out the window. It's a painting of a window. And if you get off axis, you suddenly find the whole Trompe-Loi perspective fall apart on you. So these are similar devices to the devices that were being used by Bramante, another painter who also applied himself to architecture. And the Salla delle Prospective is the most thoroughgoing exploration of those kinds of techniques where you have this whole kind of, every surface conspiring to create the illusion which lasts just for a split second before it falls apart. And this notion that there is a instability about the way the visual world constellates itself is already planting the seed of mannerism which we shall explore next time.