Well, good evening, ladies and gentlemen. My name is Michelle DiMarzo. I'm the curator of education and academic engagement here at the Fairfax University Art Museum. And it's my pleasure to welcome you all to the final lecture being presented in conjunction with the exhibition, The Holy Name, Art of the J. Zoo, Bernini and His Age, which I'll take this opportunity to remind you is only on view until May 19. Immediately following tonight's lecture, we'll be presenting Ekrasis 6, a series in which Fairfax University MFA students perform poetry and prose works directly in response to works of art. In this case, they'll be performing them in the Bellarmine Hall galleries directly in front of works from the J. Zoo exhibition that inspired their work. So we invite you to join us up in the Bellarmine Hall galleries directly after the lecture, and the galleries will be open until 7 PM tonight. So you have a chance for a little evening sneak peak of the exhibition or a happy return, if you've seen it already. And I'll also mention that our final Art in Focus event will be taking place on May 17 at 11 AM. So please feel free to mark your calendars for that as well. And it's now my pleasure to introduce tonight's speaker, art historian and author, Charles Scribner, who received his PhD from Princeton University and taught Baroque art in their Department of Art and Artiology, and later served on the university's advisory council before joining his father in the family publishing house, Charles Scribner's sons, founded all the way back in 1846. As an art historian, Scribner has lectured widely on Baroque art at universities and museums across the United States, and has also branched into new media, including producing a program for PBS on the subject of Rubens Eucharist apostries, which subsequently became the topic of his first work. Other publications include Rubens Baroque artist, Renaissance Man, as well as genres of vermini impresario of the Baroque, copies of which are available for sale and which he has generously offered to sign for you following the lecture. You all please join me in welcoming Charles Scribner. Thank you. I want to emphasize. I don't get any of the proceeds from the sale of these books. It all goes to the museum. So I hope that encourages sales. And I just caught my eye on this first slide. I felt like I heard in my mind, Dorothy, we are no longer in Heidelberg. Fortunately. Their meaning embodies the Baroque. The consummate impresario, he dominates the 17th century. His audience comprised Europe's leading patrons, proletes, and princes. But Rome was a stage and stage enough. Today, vermini's presence around the eternal city remains as resonant as the ancient ruins. In marble, travertine, bronze, stucco, and gilt, he left his enduring imprint the stamp of genius. Within the walls, he created another realm, one of imagination incarnate that still shapes our experience of Rome and transfigures it. Born in Naples, 1598, Gian Lorenzo arrived in Rome as a six-year-old. His father Pietro was a successful Florentine sculptor, and he'd been called there to work on the full of his burial chapel in Santa Maria Maggiore. Well, like Mozart, vermini Bambino was a prodigy. From his father, he inherited a natural facility with a chisel and ability to shape stone as though it were pasta or dough. Well, perhaps it was from his mother, his Neapolitan mother, that he acquired his fiery personality and dramatic flair. By age 10, he'd been presented to the pope. Who announced this child will be the Michelangelo of his age? Following Michelangelo's example, the boy studied, sketched, and absorbed into his fingers the Greek and Roman masterpieces in the Vatican museums. And one of his earliest sculptures, here carved in about age 12, is this mythological goat, Amalthea, and the infant, Zeus, and Cedar. For centuries, it was cataloged as a Hellenistic piece over 2,000 years old. A deliberate forgery? No, but surely a most precocious debut. Now, his adolescent ambition was then shown stretched to the limit in this faun tease by Puti in the Metropolitan Museum. Its complex configuration suggests that Papa Pietro served as consulting designer. The matterist corkscrew of shapes beckons us, the viewer around it, in seeking an optimum view. But no one vantage point is wholly satisfying. This piece, true to its subject, is one grand sculptural tease. Now, separating the hands of father and son, is it easy? Well, we tend with reason to credit to young John Lorenzo those areas of convincing naturalism, where smooth flesh is supported by a solid structure of bone and sinew. Pietro, Papa, had been warned by a future pope, watch out, the boy will surpass his master, to which the enlightened father and teacher replied, well, I don't mind. In that case, the loser wins. The painter, Anibale Carracci, is said to have urged young Bernini to study Michelangelo's last judgment for two years in order to burn the proper representation of muscular term. Well, two of John Lorenzo's early religious pieces pay homage to the Renaissance master. Here we see St. Lawrence, who meets his fiery martyred harmonic gridiron, and his poses we see as a variation of Michelangelo's Pietà. But his expression is true to life. For Bernini, we read, placed his bare leg and thigh against a burning torch, then sketched his agonized reflection in a mirror. Carved a year or so later, here the St. Sebastian reveals a firmer grasp of Michelangelo as life still courses through the veins of this wounded young saint. And before turning 21, Bernini carved the first of a series of life-sized sculptures of Cardinal Borghese's villa. It shows Aeneas, the founder of Rome, carrying his old father and Pisces from the burning city of Troy, followed at his heel by the infant son, Ascanius. Bernini here translates Virgil's text into a marvel metamorphosis of the three ages of man, a spiral ascent from infancy to manhood to venerable old age. The mannerous torsion, this precarious twisting and piling of forms, still suggests the influence of Papa Pietro. But the brilliant contrasts of skin textures, the effects of age on flesh and bone, these proclaim the son's handiwork. Bernini then raised the human figure to a hybro pitch in his next action for Cardinal Borghese, the Pluto and Proserpina. It's a potted dirt and stone. As the god of the underworld stands at the threshold of Haiti's garden is received by that three-headed dog, Cerberus. And on the brink of consummating is abduction of fair Proserpina. Bernini has transposed, at the right, the manner of St. Malonia's rape of the Sabine women into a strongly frontal composition. This is a return to the Renaissance ideal of sculpture designed for a single primary viewpoint. Now, there are, to be sure, enduring peripheral views of this lusty configuration, but these are the sensual side shows. From Proserpina's tear-stained cheeks to that tactile indentation of Pluto's fingers, grasping her thigh, it's as though Bernini has translated a rubens into marble and color into chiaroscuro. In fact, we might compare it with Ruben's northern chillier rape of Oridia biborius, the North Wind, which, as you see, comes complete at the right with a snowball fight. Bernini's third group for the Villa Borghese describes a little literal metamorphosis from Hobbit. The moment the frightened nymph Daphne, presumed by the love-struck Apollo, is transformed into a laurel tree. About to be overtaken, Daphne cries out to her father, a river god, for help. Change and destroy this body which has given too much delight, she cries. Suddenly, Hobbit writes, her limbs grew numb and heavy. Her soft breasts were enclosed by delicate bark. Her hair turned into leaves, her arms, branches. Her speedy feet took root and her head became a treetop. Everything gone except her grace, her shining. Now before Bernini, only painters had tackled these verses. Adapting the ancient Apollo Belvedere, Bernini brought that classical statue to life as he froze Daphne's cry, her flesh, the marble itself, yielding to branches, bark, roots, and leaves. It is for us to decide which metamorphosis is the more miraculous, the gods or the sculptures. The final Borghese masterpiece is an injection of biblical heroism into the carpels Pagan Pantheon, Bernini's David. It emulates neither Donna Tello's boy triumphant nor Michelangelo's posturing adolescent. No, this David is full grown as we see and fully engaged both physically and psychologically. It's contra posto that wound up body about to spring loose derives both from the ancient discus thrower and from that early Baroque fresco on the right of muscular mythology, the Farnese ceiling, where we see the giant polyphemus preparing to hurl a stone at the youth acus. Well, Bernini has adapted Karachi's image with a brilliant reversal of roles. It is a giant, his youth will kill. For David takes aim at an implied Goliath looming directly behind us, the viewer. The pursed lips and furrowed brow reflect Bernini's own face, which he studied in the mirror, held by Cardinal Barberini. The future poem was present at the creation, but we are present and fully involved in the day anymore. For David is a split second away from hurling the stone toward us, the viewer. The statue penetrates our space and electrically charges it. It's this anticipation of violence that creates the dramatic tension. The David is a milestone in Bernini's art, the first physical and psychological engagement of the viewer. I'm showing from the exhibit up the hill the Cardinal Bellamy, which was done just about the same time as the David, and you can see on the right, Philippe de Montabello in front of the Cardinal, the director in Cardinal, Olfaz. I don't want to go into detail about this, because you can see this in the flesh, literally. But one thing I will point out for Bernini busts, it's unusual in that it's longer and it includes the hands, which you see forward in prayer. And as my mentor Irving Levin was the first to discover, this isn't an accident, because what is represented is that portion of Cardinal Bellamy's body that was presented or appeared without any decay when the coffin was opened, when he was reburied a year before the sculpture was begun in the Jezu. And that was considered a mark, a sign of sanctity. It was during this period that the Bellamy was being advanced for saintdom. And so Bernini is paying tribute to that fact of the uncorrupted body in what he has, if you will, brought to life again in this marvel. Well, by the time of its completion, the Cardinal, who held the mirror, had done the papal tiara as Pope Irving the eighth. The day he took office, he summoned Bernini then old, 24 years old, and announced, your luck is great to see mafia about Bernini pope, but ours is much greater to have Cavalliere Bernini alive in our pontificate. Cavalliere, because he'd already been knighted. Urban was now to give him commissions that would epitomize his triumphal restoration of Rome. This pope proved true to his chosen name. For the next 20 years, he just demonstrated the importance of being urban. He transformed the eternal city into Baroque's, into Europe's Baroque capital. Now, eager as he was to make Bernini his Michelangelo, Urban had him study painting, and this is on the right a painting of the pope by Bernini. Again, on the right, this early self-portrait of the artist reveals his natural facility in the medium. With striking economy of brushstrokes, he models his hands and features. There's an immediacy, a spontaneity, even an impressionistic quality that I think foreshadows the mature Velazquez. On the left is the painting in the Barberini of David, to which I think he's also given his own features, David with the head of Goliath. According to his son, Bernini painted some 200 such pictures. Well, only a dozen or so have so far been identified, but enough to demonstrate that in painting too, he could have earned his place in the annals of art. Still, this painting was to remain a divertimental, a creative diversion, for the major papal commissions that now engaged him were three-dimensionally. At the center of Catholic Christendom, Bernini was commissioned in 1624 to erect over the high altar and tomb of St. Peter a Balanchino, a ceremonial canopy enlarged into a landmark. His chief challenge was to design a structure that would not be dwarfed by the colossal proportions of the Basilica or swallowed up in the void under Michelangelo's dome. His inspired solution was to fuse architecture and sculpture, and rather than reuse the original Salamander colors, so called because they were thought to have come from King Solomon's temple in Jerusalem, Bernini instead imitated and vastly enlarged them. Their rich dark bronze highlighted with gilt creates a colorful contrast with the surrounding walls and indeed the original colors, which you might be able to make out in the little niche above on the right. This Balanchino is crowned by a globe and cross and four huge Barbarini beads on the top of these scroll-like volumes. These exotic baroque flourishes evoke the sensation of soaring up into the heavenly dome, and the entire structure stands some 95 feet tall as high as the Palazzo Farnese. Where was enough bronze to be found? Some of it under the portico of the Pantheon. Local tongues wagged. Quod non fechurund Barbarini. Fechurund Barbarini. What the barbarians didn't do, the Barbarini did. In the center, playful pudding hovered with the papal tiara over the grave of Peter the First Pope. Urban was himself commemorated in all those Barbarini beads, sons and laurel branches that decorate the columns. Laurel, because the Pope was also a published poet. But not all were these contemporaries appointed his inventive hybrid. A canopy cast in bronze effortlessly suspended by angels, poised on freestanding columns. The Peter Ciampelli disapproved of what he called grave breach an architectural etiquette. And two centuries later, an American tourist dubbed it a huge uncouth structure that resembles nothing so much as a colossal four-post bedstead without the curtains. But the critical consensus at its completion, and it had taken nine years, was most favorable. Barbarini was richly rewarded for this miraculous suspension of gravity as a soaring symbol of papal power. He was unveiled in June 1633, just one week after Galileo had been silenced. In each of the four surrounding piers, Barbarini then designed a towering saint in a niche. One of these on the right long ginus, he carved himself. The others were carved by collaborators for already Barbarini was establishing himself in Rome as its artistic impresario. One sculptor said that to work in Rome is to work for Barbarini. The centurion long ginus converted to the foot of the cross is here projected two and a half times life size. This heroic image evolved through some two dozen boats empty, small play sketches which Barbarini modeled with amazing speed. Compared with the early Bonsetto on the left in the Fogg Museum, the final statue as we see more impassioned, more extroverted, more baroque in its appeal to our senses. We're looking up at the cross, long ginus throws out his arms in imitation of Christ like some Caravaggio saint at the moment of divine revelation.  According to Michelangelo, a sculpture shouldn't remain true to the original block from which it has been liberated. It should be able to roll downhill without damage. Well, nothing could be further from Barbarini's ideal. Long ginus is thrust into the surrounding space and then Barbarini ruffs up the marble by groove, strokes of the chisel in order to simulate varying textures of skin and fabric when seen from a distance. In effect, the blurred shimmering surfaces that a painter achieves with broken brushstrokes. The internal movements of the soul are expressed by drape-breed and engulfs this classical body in a most unclassical, indeed, supernatural fashion. The hallmark of Barbarini's hyper-rope style. Now, under Pope Urban, Barbarini had little time for private commissions or portraits. One notable exception. For his early patron, Shibioto Borghese, he carved a definitive speaking likeness. His method was to sketch the cardinal while they moved about and taught. This vivid drawing you see in the Morgan Library provides a glimpse of the man behind the marble. And this caricature reveals in a lighter vein Barbarini's genius in catching with a few depth strokes the essence of identity. He solves the problem, the inherent problem, in a portrait bust of amputation, no arms. By turning the shoulders, we see an active way that applies the rest of the body. Small drill holes in the pupils fix the cartel's gaze and engage the viewer. Well, Borghese was happily surprised by the result. In fact, doubly surprised. Before he had finished carving, Barbarini noticed a flaw across the marble. Can you see it straight across the forehead? But he completed it anyway and then ordered a new block and secretly, within two weeks, carved a copy. This he hid behind a cloth during the unveiling of the original. Well, Borghese tried bravely to conceal his disappointment that his stunning bust was flawed. And after prolonging the agony through small talk, presto, Barbarini unveiled the second bust to the cartel's explosive delight. Like his sculptural technique, Barbarini's dramatic timing was flawless. Three years later in 1635, he worked up a far more intimate portrait of his mistress, Costanza Buonorelli, that she was the wife of one of his assistants. And their affair was a stormy one. Barbarini's casual glimpse of this passionate woman with open blouse and hair gathered back in a bun as she looks to the side with lips parted, it's something of a false dawn. We must wait a century and a half for more such informal portrait busts by the great Hudo. Well, their affair ended badly with jealous accusations, violence, and legal threats. The pope himself had to bail out Barbarini and he advised him to settle down and marry. Barbarini stalled for a while, insisting that his statues would be his children and keep his memory alive for centuries. But in the end, he complied, and with typical results, his wife, Catarina and he, were to produce 11 children during the 37 years of marriage. Needless to say, Costanza's bust was given away. Barbarini was made for Rome and Rome was made for him. So proclaimed the urban pope. In this city of mountains, Barbarini left his signature writ in water. The first of these refreshing landmarks is the so-called Bancaccio or Old Boat in La Piazza di Spina. And it makes a virtue of necessity for the low pressure of the water brought in by aqueduct didn't permit spouts higher than ground level. So, Barbarini designed a ship, that ancient symbol for the church, half submerged in a basin. As if rising and falling with the sea, it's cannons spouting water. Urban himself composed verses to explain the symbolism. The papal war machines shoot forth not flames, but water sweet that quench the fires of war. They also quench thirst. Today, a more famous fountain, I think Rome's finest, was erected outside the Pope's founding palace in the Piazza Barbarini, Bernini's Triton. Free from all architectural trappings, this organic sculpture of marble and travertine emerges from water. The sea god Triton's son, Neptune, kneels on the shell, raised as we see by four dolphins. He blows his conch, as in Ovid's Metamorphoses, to signal the end of the great flood and the restoration of order. Wet, bearded Triton's son, lived to the great shell as Neptune ordered, sounding retreat and all the lands of waters heard and obeyed. Triton was also an emblem of literary immortality, an appropriate symbol for urban, the poet pope. The dolphins symbolize beneficence, the Barbarini bees, industriousness and prominence. But these footnotes in no way undermine the unity and clarity of the whole as Bernini's Triton sounds of visual fanfare for a pope, poet and princely patron, distilled in one soaring, sparkling case. It was perhaps the best label sculpture in Rome too. Bernini spent almost 20 years on urban's tomb inside St. Peter's and here again, he took traditional symbols that breathed new life into them. This bronze effigy of the pope is raised on a pedestal, a papal blessing frozen in eternity. Behind him, skeletal death, sits on his sarcophagus and inscribes urban's gilded name in the book of life. Then on either side, we see the gleaming figures of charity and justice. As they lean casually against the sarcophagus, these are the living virtues that survive the deceased pope. And charity on the left. Well, she's been called a Marble Rubens. She's also been recognized as Bernini's mistress, Costanza. The three bronze beads scattered haphazardly over the monument and a dynastic touch of trompe l'oeil, as if liberated from Barbarini's escutcheon. Or have they flown over from the Malbecchino? There, as we saw Bernini, made a sculpture of architecture. Here, he erects a pyramid of three robust statues framing ephemeral death. A visible echo of Dunn's death be not proud. One short sleep pass, we wake eternally, and death shall be no more.  Urban's death left his successor innocent to tent with a papacy on the brink of bankruptcy. Such was the price of Rome's and the Barbarini's artistic glorification over the past two decades. It was said of the Emperor Augustus that he found Rome brick and left it marble. Though urban had found Rome stone and left it baroque. As impresario of Barbarini extravagance Bernini fell from favor with the new administration, the fact that his bell tower at the Vatican at St. Peter's was causing cracks in the facade and alarm in the Vatican while this didn't help any. His rivals, including the jealous Borromini, seized their chance. The unfinished tower was pulled down and down with it went Bernini's reputation. While he sought private vindication and stone, he carved an allegorical statue of truth. Her airborne father, Time on the right, well, you see, he never got off the drawing board. But then in another revolution of fortune's will, truth was soon to reveal herself, not through time, but through art. Bernini now turned to private patrons. And for Cardinal Coronaro's family chapel in Santa Maria della Vittoria, he carved his most famous sculptural group, the ecstasy of St. Teresa. In this dim, shallow chapel, Bernini admits us to St. Teresa Davilismo's intimate vision, the moment in which a beautiful young angel has withdrawn his golden arrow from her breast. Filled with the love of God, Teresa swoones unconscious, elevated on a cloud, her lips parted, her lip pan dangling at her side. A mystical experience is described in overtly physical metaphors, a visual consummation of the sensual and the spiritual that has elicited mixed responses down the ages, especially as you can imagine from those Prudish Victorians. Well, even in Bernini's own day, one anonymous critic accused him of dragging that most pure virgin not only into the third heaven, but into the dirt, to make a Venus not only prostrate, but prostituted. Yet the vast majority of clergy applauded his achievement, wherein according to his son, he conquered art. Bernini himself considered it his best work. Those who were scandalized literally missed the point. Bernini has faithfully translated into three dimension the saint's own words. Teresa described an angel in bodily form, not tall but short and very beautiful. His face all aflame and his hands, I saw a great golden spear at his tip, a point of fire. This he plunged into my heart several times so that it penetrated to my entrails. When he pulled it out, I felt he took them with it and left me utterly consumed by the great love of God. The pain was so severe it made me utter several moments. This is not a physical but a spiritual pain, though the body has some share in it. Even a considerable share, she concluded. Teresa's tumultuous drapery reveals her state of soul. The fiery cherub's robe flutters like flames. Above frescoed clouds spill over the rim falls materialized below in golden bronze shafts of light suffused by daylight through a concealed tinted window. The columned tabernacle of the open portal of heaven it bows outward as if in response to some divine force within. And then on either side the carnaros gather at their prejuice, meditating on this apparition located deliberately beyond their sight lines. Blessed are they that have not seen yet to believe. And these are not theater boxes as so often described. In fact, Bernini's chapel is theatrical only in the sense that it realistically simulates Teresa's experience so that we, the viewers, may participate vicariously in her ecstasy. The biographer Baldenucci praised Bernini as the first to combine painting, architecture, and sculpture in such a way that forms a pleasing whole. That Bernini himself described this ideal as a bell composto, a beautiful blending of the arts.  For the piazza La Bona, outside the family palace, Innocent had announced a competition to design a triumphant fountain to feature a recently excavated obelisk. Well, Bernini was pointedly excluded, but Ludivisi prevailed on him to create a model which the prince then secretly set up at the end of a hallway where the Pope would be sure to see it after dinner. The unsuspecting, the innocent Innocent admired him for a full half hour before he realized he'd been set up. This design cannot be by any other than Bernini, he announced, and this is a trick of Prince Ludivisi so that in spite of those who do not wish it, we will be forced to make use of Bernini because whoever would not have his designs executed must be sure not to see them. The Pope was converted. In seeming defiance of gravity, this massive pagan obelisk, the ancient symbol of light, here crowned, as you see at the right, by the Pamphili dove, the Pope's family symbol, it's raised on a hallowed mountain from which spring the four rivers of the world, each symbolizing a continent. Now these beefed up river gods were carved in marble by Bernini's assistants. The maestro himself tackled the travertine mountain and engineering marvel of dovetail blocks that support this massive obelisk, and he also carved the exotic wildlife. Oh, the horse, the lion, the palm tree, crocodile, and even this fanciful armadillo. At the Nile on the right, his head is shrouded since its source was then unknown. Romans like to jest that the Rio reacts in horror at Boromini's façade of San Danyese, but in fact, his gesture signifies America's conversion by the dazzling light of faith. And on a human level, he responds with fearful wonder to the obelisk, which seems about to topple when viewed sharply from below. And there were in fact complaints about its apparent instability. Irredited by these groundless fears, Bernini one day arrived at the piazza, studied his fountain with feigned apprehension, and then ordered his assistants to string ropes and attach the obelisk to the buildings at either side. That done, he studied it again, heaved a great sigh of relief, and then departed smiling. Yet years later, he used to pull down the blinds of his carriage when passing by, lamenting that he was ashamed of it. Well, a showy tour de force, a splashing tabloid reform, rendered permanent in stone. Well, yes, it appears admittedly in hindsight, like years from the profound spirituality of the artist's late style. Yet still, we continue to succumb to that magic of the glistening centerpiece that for the past three and a half centuries has celebrated piazza Navona and Rome as an immovable feast. Its rococo offspring bubbles up a few streets away, the trekkie fountain, which like almost every subsequent fountain fantasy finds its baroque source in Bernini. When, in 1655, cardinal Kiji became Pope Alexander VII, Bernini was finally given full reign as an architect. Here at San Andreas Quirinale, he fully exercised his powers as impresario, now combining and coordinating architecture with painting and sculpture. The English writer, John Evelyn, wrote the hearing heard in Rome about an opera for which Bernini painted the scenes, cut the statues, invented the engines, composed the music, ripped the comedy, and built the theater. Well, San Andreas is, architecturally, a sacred opera in two acts. In the sanctuary framed by these rose-colored columns, the crucified Saint Andrew looks up to his final moments from a painting to a fresco God the Father in the lantern from which natural light as we see materializes into a gloria of gilded shafts and stucco angels. And then Andrew's pure white soul, also a stucco, brings through the pediment and ascends unaccliled to the Holy Spirit in the lantern among the main dome above the congregation. This soaring bell composto fills and defines the oval church with the intrusion of the divine into our own space. It's Bernini's baroque revision of the pantheon, his favorite ancient temple.  Boromini's expressive violation of architectural grammar was anathema to Bernini. Better to be a poor Catholic than a brilliant heretic, Bernini said of that, quarter cutter, some quarters, some cuts. Boromini molded space. To him, architecture was sculpture. To Bernini, architecture provided a setting for his sculpture and it had to conform to a classical canon based on human proportions. And that is why, in his view, sculptors made the best architects. Nowhere is this architectural ideal more clearly articulated than in his Colonnais for St. Peter's, which took more than a decade to complete from 1656 to 1667. Crowned with 90 statues of saints, these 284 freestanding columns four deep enclose at Ovalpiazza. At the center is an obelisk flanked by two candlestick fountains. And he then pinched the wide facade of the basilica with these walls in order to create the illusion of height. This was his optical solution to the loss of his bell towers. Bernini himself described the Colonnais as the embracing arms of mother church. And his concept was as functional as it was symbolic. Evoking a huge open air church under the dome of heaven, the oval space permitted crowds to receive papal blessings dispensed from two distant windows, one above the main portal of the church and the other in the Pope's palace at the right. It still does. If you want to see what a man can do, you must give him a problem Bernini maintained. The highest merit he added goes to the architect who makes use of a defect in such a way that if it had not existed, one would have to invent it. But he faced an apparent insurmountable problem in redesigning the scholar aegia, the staircase to the papal apartments, a dark steep flight that narrowed toward the top. In yet another triumph of illusion, he camouflaged the narrowing by introducing colonnades of seemingly uniform width and then halfway up as we see he broke the tunnel effect by this brilliantly illuminated landing. Down below at the foot of the stairs rears the equestrian monument to the emperor Constantine who founded St. Peter's and it's Bernini's sculptural solution to that awkward transition, a sharp left turn along the ceremonial route from the Basilica to the pope's rooms upstairs. Constantine had been converted to Christianity by a luminous vision of the cross, here recreated by Bernini's visionary blending of marble, stucco, and light. Then inside the Basilica at the far end, he composed a brilliant finale. Four fathers of the ancient church effortlessly raise a huge bronze reliquary in case the cathedral painted the ancient throne of Peter. Now, in fact, St. Peter's is not a cathedral. No, the pope's official seat as bishop of Rome is across the river at the latter room. But Bernini's elevated relic here reinforces its status as the symbolic cathedral of all Christendom. He exploited the bronze columns of the Baldechino now as a frame for the cathedral. Tomb, altar, and throne are mystically combined in a single painterly composition illuminated by a miraculous burst of spiritual sunlight. Bernini brilliantly converted an awkward window, yet another architectural obstacle into the source of radiance. Passing through this oval alabaster lens, natural light is transmuted into golden shafts and a heavenly cloud of witnesses. His cathedral epitomizes what Wagner two centuries later would call a Gesamtkunstwerk, a total artwork, here of painting, sculpture, and architecture that converts dogma into tactile reality. Cast in tons of bronze, this throne hovers weightlessly on patriarchal fingertips as it suspends our disbelief. Work on the colonnaded cathedral was punctuated by Bernini's summons to France in 1665 by young King Louis XIV to design a new Louvre. Well, his Italianate version of the King's palace ultimately fell casualty to French architects. France was not to be ruled from a Roman palazzo. But on a more human scale, Bernini left behind a masterpiece that alone would justify this six-month intermezzo, his portrait bust of the king. It was prepared by dozens of life sketches at over 20 clay borsettis. He sketched Louis and council meetings even on the tennis court in order to attain a speaking likeness as he had done as we sold with Cardinal Borghese. At these, he then discarded when he finally chiseled the brittle marble in the king's presence. He explained that he didn't want to copy himself, but rather create an original. Miracle of miracles, he explained at the end of one setting, that a king so meritorious, youthful, and French should remain immobile for an hour. But the result is the baroque apotheosis of the portrait bust. And to the complaints of those carping French critics that the face didn't look enough like Louis, now there was a symbolic dose of Alexander the Great. The Bernini retorted, my king will last longer than yours. Bernini longed to return to Roman to his neglected children, as he called them, but Pollenet and Cathedra. He was 67 years old, but retired, was nowhere in sight. Once back, he designed an outdoor monument to the frail, peachy pope, not a fountain this time, but an exotic landmark, an obelisk raised on the back of an elephant. An erudite and witty emblem of Pope Alexander's wisdom, the obelisk, and intellectual strength, the elephant, erected outside the church of Santa Maria so propinae. The pope's soul, Bernini then later immortalized inside St. Peter's, and for this final papal tomb, he composed an allegorical frail for faith over death. Once again, he confronted a physical obstacle, a functional doorway which he transformed into the would-be door to Alexander's tomb. And through it, as it were, the skeleton death flies upward, brandishing an hourglass. And now I can testify from personal experience that as a 24-year-old graduate student, I had the sobering experience of escorting on one arm a wobbly, cardinal right from the mass at the Baldequino, on the other arm was his good-looking, young assistant, Father Whirl, who's now the Cardinal Archbishop of Washington, and we were escorting this 80-year-old wobbly cardinal out to his car to go back to his office. And all of a sudden, I looked up and there was Bernini's skeleton overhead, and we were walking through the tomb of Alexander VII to the Cardinal's fiat outside. Of course, the Cardinal's dried fiat's, what else? But so it is a sobering experience to pass under that skeleton. But, and indeed, Alexander himself had a penchant for these memento mori. He slept in the casket, designed, of course, by Bernini, and the 80-year-old plates, decorated with skulls, again, designed by Bernini. Yet this skeleton here is not ultimately terrifying, but really uplifting. For doubling his father's time with wings and an hourglass, he begins to restore Alexander's reputation as he pulls the Jasper shroud to uncover naked truth. Well, she was eventually given a few more clothes and stuck up by a later Prudish poem. Serene and elevated, year to eternity, surrounded by these virtues, the poem humbling deals in prayer bareheaded below a halo of Keejee stars. We have prowled a long way from urban's tomb across the sanctuary on the right. Now it is faith, not fortitude, that permeates Bernini's twilight masterpieces. For the next, quote, Clement the Ninth, Bernini renovated the Ponte San Angelo. Before Mussolini bulldozed his avenue through the borgo, this bridge was the main approach to the Vatican. Ten exquisite angels bearing instruments of the passion line Bernini's Via do Rosa, his penitential prelude for pilgrims to St. Peter's. Originally, two of the angels were carved by Bernini himself, but the pope ordered them kept indoors to be spared exposure to the elements. Bernini then secretly carved one of their replacements for the bridge. He would have the last word. But today, as we see here in Sant'Andrea de la Frate, which was the Bernini family church, the original pair stand in counterpoint, the closest any sculptor has ever come to capturing pure spirit in stone. Consumed by garments of flame there, the attenuated bodies and emotional intensity seem almost neo-gothic. Yet their source was from classical antiquity, the so-called, there's the copy on the bridge by Bernini. Here it is, the so-called Antinous, to which in his youth, Bernini had often turned for inspiration, he said, as to the oracle. The critic, Jakob Burkhardt, described, not as a compliment, their fantastic, unclassical draperies carved with a spoon in almond jelly. But our modern dogma of truth to material is not a valid criterion for judging such creations. For otherwise, as the sculptor Henry Moore observed, a snowman made by a child would have to be praised at the expense of a rodent or a Bernini. Another huge pair of angels this time in bronze frames Bernini's final masterwork at St. Peter's, the altar of the Blessed Sacrament. Completed for the holy year 1675, it combines painting, sculpture, and architecture in one sublime summation of the Eucharist. This bronze tabernacle, the symbolic tomb of Christ, is an elaborate and eloquent reduction of Romante's tempietto outside on the genicolo. There you see it on the left. In this vertically staged drama, Christ is shown crucified at the altar, and then above his body the sacramental bread is buried as it were in the tabernacle tomb on top of which he is resurrected in a bronze statuette before taking his place in heaven in Peter and Aquatola's background painting. Twice life-size, the two angels direct the viewer to the mystery they embrace. One turns in rapt adoration toward the tabernacle, while the other partakes of divinity. Delicately counterpoised and psychologically contrasted, the pair evolve through a virtuoso series of Bozzetti. Here is one of them, and they rank among the glories of Harvard's Fog Museum. Now in his first altar work at St. Peter, perhaps actually earlier as we saw, Bernini had to cast architecture as colossal sculpture of bronze tabernacle erected above an altar itself above a tomb. Now he retranslated sculpture into miniature architecture, a tabernacle in the form of a tomb raised above an altar. And then he confronted the subject of death, face-on in his last full-bodied marvel, Blessed Ludovica Albertoni. The holy woman lies on her deathbed and at the threshold of eternity. The walls of the small memorial chapel converge as we see like wings of a giant triptych open to reveal Bernini's most luminous and painterly tableau.  Physical agony and metaphysical turbulence resonate through the folds of her dress. Cherubs float like snowflakes down streams of light from concealed side windows. While the background painting provides the window as it were into Ludovica's ecstatic vision of paradise, the mystical dying that preceded her physical death. A comparison with the Saint Teresa underscores the deepening of Bernini's vision over the quarter century. A tactile apparition has modulated into an internal transposition from body into soul. Diagonists are suppressed into sustained horizontals, architectural isolation, I mean Teresa's tabernacle, has yielded dramatic imminence as Ludovica's pole cascades toward us, the viewer, our perspective channeled by this foreshortened arch. No family effigies are here admitted as eternal witnesses. No, the blessed Ludovica is contemplated by us alone. Requiem internal donae domine, this is Bernini's Requiem. And he approached his own death with no less artistry or imagination. For according to his son Domenico, he undertook a final sculpture an over life-sized bust of the savior. In it, he summarized and condensed his art, Domenico writes, adding that the weakness of his wrist prevented it from embodying the boldness of his idea. This parting work was not intended for his own tomb of simple marble slab in San Antonio Maggiore, but as a gift to Rome's pre-eminent convert, Queen Christina of Sweden, who had exchanged her throne for a mass. Lost for two and a half centuries, the bust was finally identified in 1972 at the Chrysler Museum in Norfolk, Virginia by Irving Leyvin. And now the story gets a little complicated because 72, 30 years later, the original bust almost literally came out of the closet, came out of a back sacristy in a little church of San Sebastiano on the Appian Way in Rome, and there it is. That's the brilliant Bernini original, and I had to write a new edition of my book as I ended with the Rome one. Back in 1991, I had to update it. And in this beautiful bust in Rome, which conforms exactly to the dimensions in the inventory and the same pedestal of Sicilian Jasper, it's absolutely brilliant and wonderful and well worth the trip on the Appian Way, and only a handful of perverse outliers, scholars who cling to the more bizarre bust on the left. I guess they're vision shaped by the age of Picasso. Prefer the Norfolk vision, the version, but the Norfolk Museum, to its great credit, labels it copy after Bernini. Well, it was originally intended to be raised on a pedestal by kneeling angels. Here's a similar drawing by Bernini. Elevated as if one huge white host, the Eucharistic body of Christ, the bread of angels. I want to just add one comment about the newly discovered bust. The hand is so much more prominent and beautifully articulated. And this hand has caused lots of ink to be spilled in scholarly journals, trying to interpret this gesture as some say, warding off the chalice at the Garden of Gethsemane or damning the souls at the Last Judgment. It's just one theory after another. But Bernini's son, Domenico, after saying that it was his father's favorite sculpture, his Veniamino, his darling, his last and favorite child, if you will, Benjamin, he describes, and you can read it in Franco Ramondo's brilliant translation of Domenico's biography, as in Anto di Bernini, in the act of imparting a blessing. And I always think with Bernini, there are depths of meaning, but the surface always makes sense. There's no mystery to the surface of a meaning of a Bernini. And how, in a sculpture, a frozen block of marble, do you convey the act of blessing? Well, anybody who's been to a papal blessing or even a mass knows how the gesture goes, down to the side and across. And what Bernini has done here to convey that sense of movement, just like the David wound up like a Roger Federer backhand about to hurl the stone, he's got the arm off to the left as it's about to complete the sweep across his body and complete the blessing. It's arrested movement. That is how you convey movement in a sculpture. You have to arrest it. And this is why Bernini arrests it off-center so that we may fill in the remaining movement mentally. And so we see the striking realism of the child prodigy here transfigured in the end by this vision of a genius for whom life and art were as inseparable as fact and faith. To the French king and court Bernini had insisted, let nowhere once speak to me of little things. And now, in accounting for his life, he said he would have to deal with the Lord who infinite and superlative in his attributes would not be concerned to count in Venice. At the foot of his deathbed, Bernini set out the painting of the Sangue di Cristo, The Blood of the Redeemer. The deadly fever lasted 15 days. His wit remained undimmed. When paralysis struck his right arm, he declared, it is only right that even before death that arm rest a little that worked so much in life. He had rearranged with his confessor a language of expression and gesture of which he, of course, was the master. In case, as indeed it turned out, he finally lost the power of speech. Moving only his eyes and left hand, Bernini gave his final blessing to his nine surviving children. He died on November 28th, 1680, just short of his 82nd birthday. To his friend, Queen Christina, who had refused to accept so precious a gift while he lived, Bernini bequeathed the bust of the Savior. Christina, as Pope Innocent XI, who was the eighth pope Bernini had served, the value of the artist's estate. 400,000 Scoody, the Holy Father replied, a fortune. Her response? I would be ashamed if he had served me and left so little. But, Bernini left to roam into the world, or be at or be a priceless legacy. And we are all ultimately his heirs and the custodians of the treasures he called his children. Thank you. That was a good question as well. Okay, so if you have any questions, you'll find all the answers in the books. Hi. In the back of the room. But also our wonderful speaker has agreed to answer a few questions. What would you possibly not know after that lecture? So, I'm very happy that you featured the scholar Regia. I've tried and failed for 30 years to visit it. I wonder if you have, and what your impression of it is in real life? And whether you compare it contrasted to Borromean's perspective gallery and the Spada gallery? Yeah, well the Borromean, of course, you have a much Borromean. You know the Borromean, it's an optical illusion. You think you're looking at this long tunnel, right? Column A, but it actually is like five feet long. The scholar Regia, I've been down, but never have. And you can sneak down, or if you go to the Sistine Chapel and kind of wander out the wrong way when they're not looking. Okay. You probably shouldn't have done that. You probably shouldn't have done that. But you know, I was kind of a loose cannon when I was in the Vatican. I was much younger then. Okay, just one more, nobody, no other questions? Does everybody believe that Borromean is better than Borromean? Okay, but most of those, how do our sculptures, I imagine, are subject to decay? So what is the prognosis for their future? They get cleaned up from time to time, just like the buildings. I mean, I remember you probably remember too, the Triton, once I went there, and it was just covered with barnacles and it was brown and it was horrible. And the next year I came back, it was gleaming white and beautiful. The Romans really know how to clean things up. But they have to. Okay, well thank you very much.


Hello and welcome to the session. In this session we will discuss about the pair of linear equations in two variables and graphical method of solving these equations. We know that two linear equations in the same two variables are called a pair of linear equations in two variables. Now the most general form of a pair of linear equations is given as a1 x plus b1 y plus c1 equal to 0 and a2 x plus b2 y plus c2 equal to 0. Where we have a1, a2, b1, b2, c1, c2 are real numbers. Also a1 square plus b1 square is not equal to 0 and a2 square plus b2 square is also not equal to 0. We have considered a pair of linear equations in two variables. Linear equations in two variables can be solved by the graphical method which we shall discuss now. We know that the graph of a pair of linear equations in two variables is represented by two lines. If these two lines intersect then we say that the system has unique solution and moreover the system is said to be consistent. As you can see these two lines of a pair of linear equations in two variables intersect at this point which would be the unique solution of the pair of linear equations. That is we have x equal to 4 and y equal to 2. So these pair of linear equations have unique solution and hence they are consistent. And if these lines coincide then we say that the system is said to have infinitely many solutions and the system is said to be dependent or consistent. And in this case each point on the line would be the solution of the pair of linear equations. This is the graph of a pair of linear equations in two variables. As you can see that the lines of both these equations are coinciding. So this pair of linear equations will have infinitely many solutions and each point on this line would be the solution for this pair of linear equations. And thus the system is dependent or you can say consistent. Next we have if these lines are parallel then we say that the system will have no solution and the system is said to be inconsistent. This is the graph of a pair of linear equations in two variables and as you can see that these two lines are parallel to each other. So this pair of equations will have no solution and thus the system is said to be inconsistent. This completes the session. Hope you have understood the concept of pair of linear equations in two variables and the graphical method of solving these equations.


Being here, I notice we have a little bit less attendance on Fridays for some reason, I don't know why that is. There you get three day weekend anyways, I guess. Y'all are trying to do a four day. Any questions? Before we get started, we're going to get started on the syntax analysis. Do it. Okay. So syntax analysis. So, what did we just talk about the last week? Good expressions. And yes, regular expressions for what? Purpose. Lexical analysis. Yes, what does lexical analysis do? Because tokens out of bytes, perfect. Input, bytes, output, tokens. And how does that work if you zoom into the box and rip it apart? How does that actually work on the inside? Regular expressions.  I always want to say, I think it's the longest matching prefix rule, is that right? Yes. Because I want to say the longest matching prefix rule. For some reason that sounds more natural to me, but I guess my instincts are bad there. Perfect. So, our goal now is we're now at the next stage of the pipeline. So we've got a sequence of tokens coming from the lexer. And so now we need to transform this into something that's useful, right? So just a list of tokens, a list of id.id, if, curly brace, id, dumb, right? It doesn't really mean anything. We want it to kind of turn it into, in the most vague term possible, something useful, right? But, so let's think about it this way. Is every sequence of tokens valid? No? No. So let's think about, I'm not going to define the token. Let's say we have tokens for num, we have tokens for dot, we have tokens for id, we have a token for plus, and let's just leave it at that, right? So these are our tokens, how would we describe it to these tokens? Using English? Regular expressions. Yes, we would define regular expressions here to define what each of these things mean, right? And so the lexer, right, gets in a series of bytes, I'm going to stop drawing these edges, right? And then outputs a sequence of tokens, okay? So the question is in, let's say a programming language, right? So what would, let's think about what this kind of means when you have, let's say like id.id. Is this something that happens in your programs anyway? Yes. How, when, in one language? Java. Java? When doing what? Calling methods. Calling methods or accessing fields, right? So this would be something like foo.bar, right? So in Java this would be like, on object foo I'm accessing field bar. So is this a valid sequence of tokens? Yes. All the time? Right? Well, so when would it not be valid in Java? If the methods don't exist, so it would, well, okay, let's hold on to that for a second. Yes, because that's definitely a problem. The method could not exist, right? And that's going to affect our compilation. But what I'm talking about here is kind of trying to get a feel for what does it mean to be a sequence of tokens to be valid or not valid? So let's think about this, right? What happens at the very start of our Java programs? Yeah, but like at the very, so we're talking about bytes, right? At the very top of your program, what does it look like? Import. Import. We have an import, whatever, whatever, whatever, whatever, right? And then maybe other imports. And these are actually dotted. This would be an id.id.id, right? So if you think about this in symbol, in tokens, there would probably be an import token, an id token, followed by a dot, followed by an id, followed by maybe a semicolon. This is going to be faster. But that's fine. Right? So this would be that one. So here, an id.id is valid. But on the next line, right after an import, can I have foo.r? Why not? Not in a main method. That's not inside of a method body. Even if it was inside of a method, let's say I'm inside a class, right? So in Java, you'd have to, everything has to have a class with some name, right? I have a curly brace all the way to the class. If I just have the next thing after this curly brace is foo.r. Is that valid in Java? So what do I have here? So let's go back to this original question. So is id.id a valid sequence of tokens? Yes. Sometimes. But what times? Yeah, so it's depending on, in some sense, the context of where it's used. Right? We can't just arbitrarily throw in any id.id and expect it to be valid Java code. There's only certain places where we can actually use that. Cool.  Like dot plus adopt. Right? I think so. Tell me a thing of a reason why this would be a valid token. You can think of a language maybe where this does happen, right? What was that? Would this be maybe a valid? Yes. What would this mean semantically? What does this actually mean? It's a basic addition. Yeah, basically an addition of what? Of one variable up to another. Yeah, one variable to another, right? This would be different maybe than a num. That a num wasn't a num, right? But they're all valid syntax. And going back to the, can you do it, right? So id plus id, we already said depending on context may be valid. But what if you've never declared that id as a variable? Okay, same job. Then it wouldn't compile, right? So it would be semantically invalid. So those are additional checks that we're going to look at later about, okay. So right now we're worried about is this sequence of tokens that the lexer is giving us is this a valid sequence of tokens? And we already saw we can't answer this question in isolation, right? Because this id.id problem. We can't just say, yes, any id.id is definitely okay, right? Okay, so this is our whole goal. This is the goal of, part of the goal of syntax analysis is checking is this sequence of tokens a valid sequence of tokens? So we already kind of looked at some of these things, some of these things. Like decimal.num, does that make sense? No, right? It's like a, it's syntactically invalid. That doesn't mean anything in our language. Cool. Dot dot dot num id dot id, right? You can have any combination and the lexer will correctly tell you, hey, this is a dot, this is a dot, this is a dot, this is a num, right? But now we need to decide is that right? And if not, we need to throw in here. Okay, so we already learned about regular expressions, right? Yes, please say yes, you learned. We can also agree we are learning, you are currently also learning and refining your regular expression skills. We studied and looked at regular expressions. So if we already said, hey, we want to tell if a sequence of tokens are valid, we kind of already have a way to describe a sequence of characters, right? In terms of a regular expression, we can use a regular expression to define a set of strings. Why don't we use regular expressions here to define what are valid operators here and what are valid sequences of characters. Sorry, sequences of tokens. So let's use regular expressions. So let's say, okay, so we have a program. So let's, let's go like this. Let's do a regular, I like that. So we have a program, let's try to program regular expressions. What's a program composed of? Let's say like a very basic language. So it has maybe a main method, does it have to? No, not necessarily. You can write a C, a dot C file that does not have a main method which you're going to use and import. You can compile that to an object file and then link to it later, which is exactly the dot O files that we gave you in project one, right? So let's think about each line, most lines in a C program, what do they end with? Semi-colon. Semi-colon, right? So let's call that a statement, right? A statement ends in a semi-colon. So program is composed of statements and can we have just one of them? How many statements can we possibly have? Infinitely many, how do you do that with major expressions? Star, statement, star, right? So this would say a program is zero or more statements. So what kind of things can a statement be? What makes it a statement? Tokens, yes. So here we're only talking about tokens. If you think of our basic characters, they're going to be tokens. Maybe a keyword ID. So what would a keyword ID be? Oh, like the ID. Ah, so we can declare a variable. Okay, what are some other things that statements can be? Function calls, what else? Operators, like plus. Operators, like plus. What else? Something that can end the program. Something that can maybe end the program. Like a break or a halt or something like that. A control statement. So if, while, unless in some languages, loops, forms. Right? Let's go back to the operator. So for statements, so let's think about plus. So we have plus here. Check, cheat real quick. Okay, cool. Okay, so we have plus. On the left-hand side we have like an operator. Sorry, plus is an operator. So on the left-hand side, what should we have? Why is it not going away? I don't know what. I didn't want to use the, maybe if I do, literally. Nope. You're like burned into the page. That's what it is. You know, we're all seeing this, right? I'm not like excited. So statement is going to be, ignore the ghost. So what can be on like the, so we have a plus sign. Right? So what can be on the left? It could be an ID. What else? A num. A num. Decimal. Decimal. Right? It could be a lot of different things. Okay, cool. Let's think about it like this. So we have here, so let's just go with nums. So we have num plus num. Right? Okay. So let's think. I have a statement. So I have five plus five. Right? So five is going to be a token. What? Num. Num. And plus is going to be a token. Num. And num is going to be a token. Num. So are these tokens in the language described by the regular expression proger? Yeah. Yeah. Right? I have num plus num. That's in there. That's described by the regular expression. Awesome. So what about like this? Is this a valid expression? Yes. No?  Sorry. Okay. Well, do you want it to be? Yeah. Right? We want five plus five plus five. Yes. Right? And what does this mean? Well, so we have like the plus operator. We have five. Right? And then we have another plus operator and that's going to be five plus five. Right? So the left-hand side of this plus operator is really a statement. Right? Yeah. Not just a number. Well, okay. We already said we can't have recursive regular expressions. Right? So is there... So how do we change this so that we can now match this plus? Is that it? Oh, sorry. The plus num. Plus num? Yeah. Into the bracket and star. Into the bracket and star. Interesting. Okay. Now I'm plus num so we can do plus num plus num. So one problem this gets us into, right, is just num by itself. Is that a valid statement? Maybe. Maybe not. So we can do plus num plus num. Okay. That gets us out of this problem. Now we want to do... Can we have as a statement... We have parentheses. Just the open parentheses. Well, sorry, sorry. Can we do a statement like five plus five? And then could I add any number of parentheses here? What's the key thing that I need though? Equal number of opening and closing parentheses. Right? They should be what? Balance. Balance. Yes. This was the scale. Yeah. They should be balanced. Right? And I want that... I want to know that this sequence of tokens is valid or not valid and I want to know if these parentheses are balanced or not. So how do we write a regular expression of balanced parentheses? So do we need a opening for a left and a right? Call it LT and RP. Okay. What goes in the middle? Why can't we do this? It goes recursive. It goes recursive. Can I do like maybe... What would this describe? Any number of left and right? What specific form is this in? So let's think. So I can do left parentheses and then any number of left, right. So this would be like this. And then a right parentheses. But does this match? Yeah. This one does. Does this one? Where do you put it? Like the stop. It should be at the left. Because LT is... It could be at the left. This matches? No. Why not? It has to be left, right. All of my left, right. All of my left, right. So what are we trying to say here? Like for this one here? We just said bounds. What does bounds mean? Same number of left and right parentheses. So let's try to boil this down. So this is when you're trying to think about, okay, what are the bounds of what I can do? Let's just call these A and B. So I want to describe... I want a regular expression to describe the set of all strings where they're a number of A's followed by an equal number of B's. How do I do that? So this is the strings that I want. So I want some regular expression R. I want it to be... We can include sigma or not. It doesn't matter. I want it to be A, B, A, A, B, B. A, A, A, A, B, B, B, and so on. So what's R like? A star dot B star. What string is this produced that is not in here? A, A, B, A, A, A. A, A, B, right? What else? Something like R is equal to capital A, capital B, whole star, which then calls A, alphabet A and B. So something like a token which calls specifically A and another token that calls specifically B. Like this? No, not a star. Just like that? Yeah. And then we can call R as equal to A, B. No, A, B, whole star. Like this? Cool. So what does this do? Yeah, this is A, B, which is in there, but it's A, B, A, B, and then A, B, A, B, A, B. You'll notice this is exactly the same thing we tried up here with this. So what's the problem? What are your brains hitting when you try to think about a regular expression to match this? We can't forget a single multiple. What is it about that number though? We're trying to make them repeat the same number of times, but if you separate the stars, that's... So what do we get? Think about the operators, the regular expression operators that you have. What does star give you? Zero or more. Do you have any way of specifying how many times do you want that repeated? No. Do you have any way of capturing how many times did that star capture and repeat? No. Not in any of our regular expression operators. So fundamentally, we cannot write a regular expression. This R does not exist. We cannot write a regular expression to capture this. And this is a very basic part of the program. If we broke the rule that we couldn't have recursion, we kind of already saw we could easily deal with this. We can say, yeah, we can have a statement here. And I think it'll prove that in 355, but I don't know. Right now you can take my word for it. And so that's kind of what we have here. So this would be a little bit more expanded an example because I wasn't writing everything down. We have program, composed of statement star. Statements are expressions or if statements or while statements. Operators are plus, minus, multiply to five. Expression is either a number, an ID or a decimal, followed by an operator, followed by a number ID decimal. And so we can see that an expression is num operator num that matches foo minus far ID minus ID. But now one plus two plus three is we saw we can't do this. But we can maybe do opt num ID decimal star like we did. And that kind of gets us into a good place. So the expressions and the ideas of statements that we want to describe, regular expressions fundamentally cannot capture. They are not expressive enough. You cannot use regular expressions to define that set that we want of balanced parentheses. So we need something that's a little bit more, has more expressiveness. But when we gain an expressiveness, we lose a little bit in performance. So there's trade-offs there. So this is why I said at the beginning, regular expressions are very fast and very performant because they're not as expressive. Does that make sense? Can I say what I mean by expressiveness? I mean, literally you cannot describe this R here. In English, we can describe this set R, right? Or this set L, this language described by R. I can describe this set very easily. The set containing all strings that have equal numbers of A's and B's, right? That set exists. But we cannot define that set using a regular expression. I'll show you other things where we tend to find, using context-free grammars, which is what we're going to learn about, you tend to find that set in a formal way. Okay. Get that. Parentheses. See, look at all this. Where do I do this? Boom. Okay. So context-free grammars. We need a new way of describing this. You can kind of think in some way that this is relaxing that restriction on recursion essentially. And saying we're going to define things that look kind of like regular expressions, but are definitely not regular expressions. And we'll see that they can be recursive. Okay. And the way we're going to do this is with rules. So we have context-free grammars. So let's try to think, let's try to think if we can define maybe something that could do this. So let's say we have a starting symbol that we're going to start with. We have some starting symbol. And so I want balanced lower-case a's and b's. Right? So s, whenever I see a capital S, I'm going to replace it with something. So what would I want to replace it with to capture balanced a's and b's? A, s, b. Or a, b. Or s. Alright, we'll do it. Or s. So what am I kind of saying here? What I'm saying is start from s. Apply it. Well, let's get rid of the ors for a second. So I have three rules. S can either be a, another s, or a lower-case b. S can either be... A, b, or s can be... We'll go to the example that we were... Or epsilon. So, I start from s. I apply one of these rules randomly. Let's say I choose rule three. What string is that going to generate? Empty string, epsilon. Is that in my set? I start with s again. Let's say I try to apply rule two. It's going to produce what? A, b. So I have the two can kind of get together. The string A, b. Is that in the language? Okay. I have s. Let's apply the first rule. What am I going to get? Can I stop here? Could this be about string? Why not? Yeah, because I still have some way to get rid of s. So now which rule do you want to apply? Let's go with rule two. Yeah, good call. So what string does this represent? A, a, b, b. So, what would happen if I randomly generated and applied every single one of these rules to generate all possible strings that could come starting from s? Would the string, would epsilon be in there? Yes. Would a, b be in there? Yes. Would a, a, b be in there? Yes. These are all definitely yes, because we just showed that, right? But would this string be in there? Sure. Yeah, it should be there. It would be this s produced again by a, b, s, and then that s produced by another a, b, and so on and so forth, right? If I keep applying rule one, you can generate strings of matching a's and b's. So we talked about this very intuitively and informally. So now we need to actually define things. Yes, please. What about a, b, a, b? A, b, a, b is not what we're trying to describe right here, because here we're just trying to describe matching a's and matching b's. And I can replace the a with the left parentheses, the b with the right parentheses, and now we have balance for this type of balance, where we have the equal number of a's followed by the equal number of b's. Could we get rid of the middle row? Yes. Is this the same? No. Why? Does someone want to raise your hand? Yeah. A epsilon b is b, b. Yes. Right, so we go a, b. a, b, this s would go to now epsilon. What's a concatenated with epsilon concatenated with b? a, b. And if instead of this I chose a, s, b, and then that s went to epsilon, that would be a concatenated with a concatenated with epsilon concatenated with b concatenated with b, which is a, a, b, b. Yeah, so actually surprisingly that rule is redundant. Cool. Any other questions on this, in this example? All right, let's define some things. So, these are called a context-free grammar. We're going to define a grammar. And so I know so far we've been talking a lot about very concrete things about bytes turning into tokens. Here, we're going to talk about context-free grammars a little bit abstractly because they're a little, I mean, I think they're a little bit more complicated than regular expressions and a little harder to build up an intuition of what they are and how they work. So we're going to kind of first looking at, just like we did here with symbols of s's and a's and b's. And then towards the end when we finally mastered this, I'm going to show you how real programs use context-free grammars to do parsing of really cool stuff, using tokens. So for context-free grammar, we need an s. What did we use for, why do we use this s here? What did the s mean? All right, you got me. Okay, let's phrase it a different way. Where did we start whenever we were building one of these trees? Or when we were starting and we wanted to say, okay, let's find a string that this language, this, sorry. We want to find a string that this grammar defines. Where did we start by? S, yes. Because it was a statement. So we need some kind of starting symbol. So how come I never expanded this a or this b? What was that mean? Why? Like why did I not expand an a or b? What was that? There's no rule for that? Yes, there's no rule. I don't have anything about how do I translate an a to something else, right? I know I can replace here. I can replace an s with an a followed by an s followed by a b. Or I can replace an s with an epsilon. But how do I replace an a? I can't. There's no rules for that. So I just leave it as is. I'm going to get back to that in a second. We have an a, so we're going to call this a, oh man, I just almost messed it up again. Terminal, right? So if you think about when we're breaking out all these rules, when we get to something that's a terminal, we cannot go any farther, right? That's as far as we can go. Go to the opposite of a terminal b. Yeah, something that's non-terminal. I hope I'm smelling this right. Cool. And so s is the start, right? That's where we're going to start. So it's going to be our starting non-terminal. Now we're going to do a little bit less on the formally defining exactly what a context free grammar is and what it must have to be a context free grammar. But these are the information we need. What is the starting non-terminal? Most cases, by convention, we will use s. s will always, so if you see a context free grammar, if it has a capital s in it, that is the starting non-terminal. The other thing, in all of our more abstract examples, all uppercase characters will be non-terminals. Lowercase characters will be terminals. And then we also have our epsilon. What is epsilon? There's a question you don't know the answer to. Well, it's definitely not a non-terminal, right? We can't replace epsilon with anything. But it doesn't, unless it's by itself, it doesn't end up in the output strain. So terminal, I guess we can call it a terminal. But it's something to be aware of that epsilon will need to be treated a little bit specially, especially when we start going over algorithms of how to calculate things about a context free grammar, which is what you're doing project three. So for any context free grammar, we should look. So even if it's not 100% specified, like here, why do we know A and B are terminals? Yeah, there's no rules, right? We didn't specify any rules to tell them where to go and what to do, right? If we had something like A goes to S, something like that, then it would be a non-terminal in this example, right? But I will not try to trick you like that explicitly. I mean, it could be a mistake, but that's how you should think about it. Okay. So we have terminals, non-terminals, starting non-terminals. So the syntax basically of a context free grammar is we have a left-hand side, followed by an arrow, followed by a right-hand side. So you actually see when we get to project three, it'll be really cool. You're going to be reading it, you're going to write a program that reads in a context free grammar, and the description of how to read that in is itself a context free grammar. So it's pretty cool. So the way to figure out this, there's some left-hand side. But can you have terminals on the left-hand side? No, it doesn't make sense, right? That's part of the reason why it's a terminal. Can you have multiple non-terminals? Can I have, let's say I can replace two S's with a B? So this is a type of grammar, but in context free grammars, no. So in context free grammars, you only have a single non-terminal on the left-hand side, and you have an arrow character followed by any number of terminals and non-terminals or an epsilon on the right-hand side.  Cool, all right. The rules, we can call rules productions.  You can also think of it as a rule. We did all this, started with an uppercase, good. Oh, so we already did this example for matching parentheses. And we can use the bar symbol to do for just like in regular expressions. So we'll say this is red S can go to left parentheses, S right parentheses, or epsilon. So it's exactly the same as this. It's just a little bit easier to write. Okay. So we can do, from a context-free grammar, when we did these trees, what were we doing? How did we do this? And why? What was the first step? What was the first thing I wrote down when I wanted to do this? S, Y, S. It starts somewhere. Yes, you do have to start somewhere. And it is our starting non-terminal. So it's very easy to remember that, right? You've got to start somewhere, start at the starting non-terminal. Start with the thing, and then start at the same. Right, we wrote down S and then what did we choose? What did we do? I'm going to go like this. We chose one of the rules, and then we basically replaced S with that rule. Right, so here we're building, in some sense, we'll get back to it, but it's almost like we're building a tree. Right, by expanding each non-terminal and replacing it with one of the valid rules. Another way to think about that is a derivation. So in a derivation, so let's say I have S so left parentheses, S, right parentheses, or epsilon. So a derivation would be, start from S and apply one of the rules. Exactly. So we use the double arrow. So I start with S, I apply one of these rules, which rule? Sure. Epsilon. Then do I apply another rule? No, no, no. No, so my derivation is done. So let's apply the other rule. Left parentheses, S, right parentheses. Am I done? Then apply one of the rules. Let's go to epsilon, the second rule. Left parentheses, epsilon, right parentheses, which is the same thing as this. Right? So each of these, the double arrows, is a derivation. So this is one step in our derivation. Basically, we're saying, how do I get a string that is represented by this context-free grammar, meaning this context-free grammar? Well, I started by starting on terminal and I keep applying rules until I can't apply any more rules and that string is definitely in my context-free grammar. So we can do this, we can do all kinds of things, we can do it, this would be two, three, four steps. We can just keep applying this, right? Every step is a derivation. Cool. So now if I get a little more complicated, so here I have, so here I'm changing a little bit, back to tokens and lexers. So here the token would just be num. So let's say you have the token num and the token multiply and plus, but we are leaving the characters here. So I'm saying expression produces an expression plus another expression, or an expression is an expression, produces an expression, multiplies simple another expression, or an expression is a number. Right? So can I generate all possible, so can I derive an expression? So let's start with expression. Right? So I can apply one of these rules, let's say the second one. It's going to be expression times expression. But now we have a question. Which of these two expressions do I replace first? Does it matter? No? Yes? Maybe? Why? So let's say I change expression and I change expression to three. That's some number.  I exchange that to the two and then I change that to present one. So I have one plus two times three. So assuming these are both num should really be num plus num times num. Is that in this context for grammar? Yes, and I know because in every step of my derivation I correctly apply one of these rules. So as we saw, there are different ways of how to derive this string. Right? At this step, we had a choice. Which expression do we reduce, do we derive first? And so this will help us answer questions about context-free grammars that are very interesting. Or interesting in the terms of syntax analysis and eventually parsing. So left-most derivation sounds like a lot. What would you assume this is just on the title? Which thing do we choose to derive? The left-most for you at that time. Always expand the left-most non-terminal. This is incredibly easy to remember. So if we say do derivations show derivation that is left-most. Just always do the left-most one. So same example. So you could ask a question is this a left-most derivation? Why not? So is this, okay? Is this the left-most? Yes. Is it also the right-most? It's only one. Okay, so now in this one we're deriving the right one, not the left-most one. This is not a left-most derivation. Cool. Is this one on every step? Yes. Double technique. Which would lead us to a right-most derivation, which is what? Yes, always derive the right-most non-terminal. It's incredibly easy. Always expand the right-most non-terminal. Everybody know their right from their left? The left hand makes an L. Okay. I don't know, you read through that in physics, right? Like the right hand rule or whatever and like trying to figure out the forces and you're trying to take a test if you actually do it through your left hand and you're hearing the wrong answers. Okay, so is this the right-most derivation? Yes. Okay, cool. Okay. So, the cool thing basically here we're kind of drawing a tree, right? To do derivations and showing deriving things in the trees, right? I mean, I drew arrows which kind of messes up the tree in five but if you just think of them as lines like here so if I think about like S and then I have S goes to the left Paren, S right Paren which this goes to epsilon right? So this is a tree every node a tree is what's a tree? People know how you had to it has nodes it has nodes you guys are describing actual trees it's a computer science tree some of those terms do apply so I'm mostly messing with you a root, what's a root? the start of a tree which is weird that we draw them on top we're going to start at the bottom so this would be S in this case would we have a root for this tree? right? so what are some of the things? leaves what does the leaf mean? node with no children, what's a children? we have a child, we didn't talk about that there's no nodes how do you know? like an oddly philosophical over here what are these things? edges they establish the parent-child relationship without edges we'd just be drawing circles or not even circles here right? so I have these nodes I have S left Paren Epsilon this is the root it's the top of the tree S has three children left parentheses, S right parentheses does the order of these matter? of the siblings? sometimes in what we want to do in this time yeah why? convince me that it is important what was that? we're concatenating the role we're concatenating the role what are we concatenating though? the leaves we're concatenating all the leaves together to get the resulting strain and so if I have this tree is this tree the same as do these produce the same strain? no so the order of siblings is important and where does it come from? from our production rule we're concatenating all the leaves together and so from our production rules exactly from this rule cool so a little preview drawing this tree is the same thing as doing this derivation so we'll do that on Monday thanks for watching


You have any update on judge and was that concerning it to you at all in terms of how he was feeling out there. I took him out of the game. So yeah, it was concerning. He's I just I just spoke with him. He's doing pretty good. So hopefully we'll be in a good spot for tomorrow. Just dealing with some dizziness. I got a stable level connection. Can you hear me okay? Yep. Can't see you though. Ever. Yep. Okay. I want to see him. I'll be back on right now. Aaron, do you have a problem with what Lindor did circling the bases because they were talking about it on TV quite a bit. Started with his second home run as he rounded the bases. Right. I mean, I'm not even going to go go down that road of this, that boys will be boys and you know, I'm just going to leave it at that. Okay. The next one is the Adler. And is the dizziness something that judge was experiencing before the game? Yeah, when he, I think right after like the first pitch of his first at bat, he just got a little bit dizzy and, you know, lasted for a little bit. Next to Randy Miller. Well, we'll judge to get some tests done. I know Frazier had the dizziness and hasn't played since I guess it's probably different situation but what kind of test can you guys do? Yeah, I mean, I don't know.  So I don't know what all that's already been done or what will be done. I know every loss this time of year stings, but when it's Lindor that hits a third home run to kind of do you and does that make it hurt even more? No, I mean, it's, you know, he put out a great performance and, you know, we lost a tough ball game that we fought our way back in. I thought the, you know, I thought I thought we competed really, really well. It was, you know, it was kind of a back and forth game a little bit and, you know, great player had finished off a great night and that ultimately was the difference, but I don't think it makes it any worse. These games are huge, so stings when you lose them and it's pretty joyful when you win. And last thing, I think this is not a job of a playoff spot for the first time in quite a while. There's a couple weeks to go, but what's your take on where you guys are right now in the standings? If we play well, we'll be where we want to be. I mean, so that's, that's about, I don't know, what are we, 20 games, 20-ish games left now? I don't even know. Bottom line is we play really good baseball. We'll get to where we need to go. Thank you. Yep. Go back to Ken Davidoff.   He, you know, he's not going to get suspended. He didn't throw any punches or anything like that. Did you like to see that? We're playing for a lot right now, Kenny. Again, I feel like you're trying to pull out of me and get the, make it a little controversial of, you know, I feel like we're fighting, we're competing really well. And, you know, I know our guys are in the right headspace and we're going through a tough stretch right now and a grueling stretch, but I like how we're competing right now. Thank you. Yeah. Go back to Tom Aram. Aaron, do you feel overall this was a more intense, more emotional series than most?   You know, those, and when you add in 9 11 and the two New York teams playing each other over that weekend, I think that does add a little something to it. But, you know, we're playing for a lot right now, regardless.


In a previous video, we wrote a class to simulate a toaster. Here's the UML diagram. Everything that the toaster needs is built into one class, as if a manufacturer built all the parts a toaster needs. In reality, no manufacturer builds the entire toaster. Instead, they buy the power supply and the dials from some other manufacturer and then put those into the toaster frame, which they themselves build. In an object-oriented world, we use what the book calls composition or aggregation, where instances of one class contain references to instances of another class. In essence, we have classes that are composed or made up of other classes. We'll create a power supply class and a dial class.  The dial class has a minimum and maximum value to make it more generally useful. For example, some companies might buy a dial that goes only from 1 to 4 instead of 1 to 10. Now that we have the power supply and dial classes, we can put those parts into our toaster. Composition is a has a relationship. A toaster has a power supply and a toaster has a dial. In the code, here's the power supply class and the dial class. There's nothing particularly new in them. But in the toaster, we now have a power supply attribute called power and a dial attribute called darkness. That means that in the constructor, we have to create a new power supply object for the power attribute and a new dial object for the darkness attribute. When inserting bread, we have to ask the power supply if it's turned on. That's because the is turned on method belongs to the power supply, not to the toaster. Similarly, in the to-string method, we need to get the voltage from the power supply attribute and the dial value from the dial object. Now let's look at what has to change in the main method. We can't say toslot.turnon because the turn-on method belongs to the power supply. We could change it to an expression like this. Read from right to left, we're telling the toslot toaster to get the power supply and tell it to turn on. That's called a chain of method calls. But that's fairly ugly, and it's sort of like asking the customer to access the power supply in order to turn on the toaster. Instead, we'd like all the dial and power functionality to appear to be part of the toaster. We'll add some methods to the toaster class to make it look like it's all one unit, and those methods will call the methods in the subparts of the toaster. Here's what the code looks like for the turn-on and turn-off methods. The turn-on method in the toaster will call the power supplies turn-on method, and similarly, the turn-off method for the toaster will call the power supplies turn-off method. To make the dial appear to be part of the toaster, we'll have a public int getDialValue, which will return the darkness controls getDialValue, and we'll have a public void setDialValue, which is given a dial value by the user, and we'll pass that on to the darkness controls setDialValue method. Once we do that, we can keep everything as it is, and we don't need to do a chain of method calls anymore. We can compile and run it successfully. Toaster will learn another way of adding functionality to objects, inheritance. While many Java books and tutorials strongly emphasize inheritance, current best practice says to prefer composition over inheritance.


What is the closest you've ever come to shedding a tear during the tournament? I think in 2014, I was so firm all season long that Shabazz Napier was going to be the best guard in college basketball. And when he put Connecticut on his back and led them to wins over Iowa State and Michigan State to go to a final four and that coincided with the day that Aaron Harrison made a three point shot in the final seconds for Kentucky to beat Michigan in roughly the same spot where he made a shot on the left wing two nights prior to beat Louisville. I just said I am so grateful that I get to take part in this. I keep going to the next thing to try and keep striving to be better on my next appearance.


Welcome folks, we have the Dow Industries trading down $5.57, we get the Nasdaq down $1.89, S&Ps are off $5.57, that's a hit of 2.3% in the Nasdaq, 2.1% in the S&Ps, 2% in the Dow Industries. Gold contract, gold contract up $2.40, trading at $12.86 an ounce. We have silver down a penny, $14.91 an ounce, light sweet crude off $0.91, $61.31 a barrel, notes and bonds. You get the 10-year note up $12.6, $124 even, 30-year bond up almost a full point at $148.24. We get open, we take a look at this 10-year, what you're going to see, both the 10-year in the 30 folks, they're breaking topside once again, they have volume behind the move. They have 5 million contracts in the 10, going after the $124.04 level, bottom line is that you are over the first tie, which was January 1st, that's what we had broken out on the 22nd of March, this baby wants to run up to this $124.31, wouldn't likely take that baby out. We take a look at the yield on the 10-year, right now you're at $2.446. King dollar, what do you have a king dollar, king dollar right now trading up at a price point of $97.400, that's not enough to take it up and over the highs of this consolidation, it's $97.705, that's the number right now, you're at $97.400. We go take a look at the S&P, what you're going to have out here folks, S&P, bottom line you're at down $6, you're at $286.80 and you can expect lower prices right into the close. The differential out here between yesterday and today, you can see it right in the tick index. That out here yesterday, bottom line, that tick index came in way too high. You had a plus 1,400 and now on the downside we only get a minus 12.63, a minus 12.63 is indicating that this baby can keep going. In order to turn around on that tick index, you got to get that thing up to about 1,400, minus 1,400 and then get a second one of about minus 12, minus 1,300 and you're not even close to that. So what you don't have out here, even though the market's down pretty dramatically, you don't have dramatic selling simultaneously. That's within minutes. You just don't have that out here. This has been pressure all day long inside the marketplace. I expect what you're going to see pressure coming right into the close out here. Dow industry is down $555, Nasdaq off $192, S&P is down $52. Stay right there folks, come right back.