We are just north of Paris and a very long time ago. The year: 1140. For some time now, workers have been sledgehammering and digging and mortaring great stones into place at an already ancient abbey church.

The abbot is a remarkable man, Abbe Suger. For a time, he ran the entire kingdom, while the king was away on a Crusade. But Suger is, above all, devoted to his abbey, which is dedicated to the patron saint of France, St. Denis. Legend has it that Denis was burned and beaten and broken and none of it killed him. So he was beheaded. Even then, the martyr picked up his head and carried it a ways, to this spot.

The walls of the church are cracked and crumbling. Suger is rebuilding.

As the new entrance rises, with its pair of towers and three large doors, the trained eye can see something remarkable. The new arches climb like converging reeds to high and tapered points, and you have to crane your neck to see the upper bank of windows. These windows are filled with many-colored glass -- and, most striking of all, there is a huge, round window the cool fiery color of sapphire.

When the work is finished and the townspeople come inside, sunlight falls through the revolutionary rose window in an ethereal shimmer. From the smaller windows, light drops in shafts into the once-dark, now glowing interior. The beauty of it stuns them.

Suger is not surprised to see peasants and tradesmen and even his monks stand frozen inside this new building. He has created one of the world's aesthetic marvels, the first Gothic church, a soaring, stony space suffused with light, a window opening onto God. Of course there is awe. Awe is the proper relationship to the Almighty. Awe is not a byproduct of the structure; it is the whole point. As the abbot puts it: "The dull mind rises to truth through that which is material."

Why start a reflection on beauty at St. Denis? Why not in Japan, where the millennium arrived in the midst of a golden age of art, before the rise of the stern and self-denying shogun? In Japan in the 11th century, virtue was wrapped up sumptuously with art, music, poetry and love.

Why not start with the dizzying gilt and blood-red arches of the Alhambra, at the zenith of Islamic empire? With the lush seize-the-day poetry of the Persian Omar Khayyam? It was all happening about the same time.

We start in the church because the spirit of this millennium has been, in large measure, the spirit of science and technology -- of practical brainpower. And the beauty of St. Denis is a scientific, technological marvel. Gothic style, with its abundant space for those magical windows, is a function of engineering. To the science of the arch was added the technology of the flying buttress -- and the end result was beauty.

The German philosopher Immanuel Kant once warned that "to seek for a principle of taste which shall furnish, by means of definite concepts, a universal criterion of the beautiful is fruitless trouble." Beauty, you might say, is in the eye of the beholder.

For some, beauty is wrapped up with craftsmanship (whether the artist's or God's) -- a thinker, for example, sees beauty in a well-formed idea, a writer in a well-turned sentence, a mason in a straight, strong wall. Beauty has been found in color, and eros, and youth. In music and in literature there is euphony, rhythm, pattern. Beauty has to do with S-curves and diagonals; alliteration and tonic chords.

Passion? Yes. Pathos? Sometimes. Many critics have noticed that beauty has a startling quality -- arresting, immediate -- but somehow it bears a sense of the timeless, too. Beauty can soothe, but doesn't always; it can be agitating or ecstatic.

Beautiful things, critic Elaine Scarry has written, are "small tears in the wall that pull us through to some vaster place."

Whatever beauty is, exactly, its story in this millennium has a lot to do with science and technology. Sometimes, as at St. Denis, the loveliness and the engineering work in harmony. Sometimes they clash. There have been artists who speak to the head, and artists who speak to the heart. And, across 1,000 years, endless anguished hours have been spent striving to reach the whole person all at once.

A Matter of Perspective

A figure stands in the shadowy doorway of the magnificent Duomo -- the Cathedral of Santa Maria del Fiori -- in Florence. It is 1425. He is Filippo di Ser Brunelleschi, a short fellow nearing 50 years old. Everyone knows Brunelleschi around here. He is the architect in charge of this city's greatest construction project, the raising of an extraordinary dome over the cathedral altar.

But what is Brunelleschi doing? Painting. Hmmm. Brunelleschi is famous as a sculptor, an engineer, a builder -- not as a painter. Yet here he is at an easel, working on a little canvas. He's making a picture of the building just across the piazza.

Even more strange -- he's facing the wrong way. Next to his canvas is a mirror, in which he sees the building's image.

He looks silly, painting backward, but Brunelleschi is deeply serious. He is conducting a scientific experiment, perhaps the most important in the history of art. Around the world over the past four centuries there have been many very beautiful pictures made -- from the serene inks and watercolors of Sung Dynasty China to the oil-and-gold-leaf altarpieces of medieval Europe. They have one thing in common: They don't look entirely real. They are as flat as the boards and scrolls and fabrics they're painted on.

Brunelleschi, like many artists and intellectuals of his time, is fascinated by optics, the science of seeing. He has noticed certain facts about the way lines of sight seem to converge in a mirror. Mirrors fascinate late medieval scientists -- like canvases and frescoes, they show reality in a two-dimensional plane.

So here in the Duomo doorway, Brunelleschi is trying to translate these insights into rules for painting. By day's end he will have discovered the modern science known as perspective. He will show that, by the application of strict Euclidean geometry, it is possible to make flat pictures that appear three-dimensional. This will be, perhaps, the signature revolution of Renaissance art.

But in this doorway, we see something even more important about the efflorescence of Renaissance beauty than mere horizon lines and vanishing points. This artist is a gearhead. He has a beauty problem to solve -- how can one capture the order of Creation? -- and he tackles it like an engineer. Within months of Brunelleschi's experiment, Italian artists like Masaccio and Masolino will be skillfully applying his discovery to new pictures; Masaccio's "The Tribute Money" has, for Renaissance viewers, such dramatic impact that it is as if a window had been thrown open and just outside was Jesus Christ.

It's common now to think of the first four centuries of the millennium as dreary, ignorant, diseased and bloody -- a dark age of plagues, marauding Vikings, doomed crusades and Genghis Khan. But along with the Ottoman and Japanese glories of the early millennium, there was a creative and intellectual movement building in Europe. The 13th and 14th centuries were an explosion of universities and cathedrals, the age of Roger Bacon and Thomas Aquinas, of Giotto and Petrarch -- most of all, in the context of beauty, the age of Dante.

System and order were the obsessions of Dante's age. Artists believed deeply in the symbolic, mystical power of numbers. Dante's eternal poem, "The Divine Comedy," was carefully composed in three parts of 33 cantos each, with each verse containing three lines. Behold the Holy Trinity, and Christ's three days defeating death -- as with the Abbe Suger, the purpose of his structure was to direct dull minds toward the truth.

The artists who, over the next two centuries, embodied the Renaissance continued to hold this conviction -- that order and structure were partners with sensual beauty in conveying the truth of Creation. The rigorous geometry of Brunelleschi's perspective system was such an instant hit because everyone shared his belief in the power of visual realism.

The artist-technophile was more the rule than the exception. Everyone knows that Leonardo da Vinci, painter of the Mona Lisa, was also a scientist and inventor. Fewer people realize that Michelangelo -- painter of the Sistine Chapel, sculptor of David and the Pieta -- was an engineer. He built a bridge across the Arno River and, at St. Peter's in Rome, raised a dome even larger than Brunelleschi's. His reputation was so great that an Ottoman prince tried to lure him to Constantinople -- not to make statues, but to build public works.

And what about the increasingly complex counterpoint in Renaissance music? In the work of Palestrina, for example, multiple lines of mathematical complexity weave themselves into gorgeous tangles before resolving into perfect solutions. They are like exquisite geometric proofs.

Brunelleschi's breakthrough was so instantly popular because he gave artists a tool to show the world in all its natural order. This is the way they saw things: Humans could draw their moral lessons from the perfect geometric design of God's universe.

That perceived unity of the head and the heart, which peaked in the middle centuries of the millennium, was a precarious, endangered thing.

A Dirty Mutt

Roll forward 500 years. We are on the eve of the largest art exhibition thus far in American history. It's 1917. There is buzz and purpose in the air, a sense that Europe, mired in a ghastly war, is done as the center of Western art and the door is open for America. There is much to be done: Conventions require smashing, hierarchies need demolishing, hypocrisies must be laid bare.

Everyone is still talking about the 1913 Armory show here in New York, when modernism hit the United States like a bomb from an aeroplane. Imagine: pictures that no longer looked like pictures! This wasn't just the impressionists, whose pictures were blurry, but bright and beautiful. No -- this was perplexing and outrageous. Imagine people standing in line for the better part of an hour to spend a few seconds regarding a cubist painting -- "Nude Descending a Staircase," which looked nothing like a nude and not much like a staircase -- by a young Frenchman named Marcel Duchamp.

Now it's 1917, and the world is even more modern. This show, organized by the cutting-edge Society of Independent Artists, aims at an even bigger splash than the Armory.

Anyone who ponies up $6 is entitled to hang two pieces; the response is so great the show ends up hanging two miles of artwork. No judges, no prizes. It's the newest thing: Art is what an artist says it is. An artist is anyone who claims the title.

We're on the eve of the show. Entries are still arriving. Here is a box from an unknown artist in Philadelphia, a Mr. "Richard Mutt." There is no Richard Mutt -- his name is an amalgam of French slang for "moneybags" and the title of a comic strip. But the organizers do not know this.

Inside is the entry fee, and a piece of porcelain. The work is elegantly titled: "Fountain."

It is a urinal. From a plumbing supply store.

A urinal! Signed flamboyantly, R. Mutt. Some board members insist that it must be shown -- the fee was paid, and art is whatever an artist wants it to be. But after much debate, the board refuses to display the piece -- a decision that quickly becomes the most talked-about aspect of the exhibition.

And who is R. Mutt? None other than Marcel Duchamp, a board member himself. A trick? A masterstroke?

Here are the seeds of the artistic forest we live in at the end of this millennium: the purely conceptual work, the artistic navel-gazing, the manufactured scandal, the inside joke. The urinal is not about God, or Nature, or life, or even reality. It is all about art -- what defines it and who decides. Though many critics will write thousands of serious words about the lovely feminine curves of R. Mutt's "work," and though Alfred Stieglitz will lovingly photograph the pissoir veiled in shadows, there is no fooling the public.

After the urinal, art is no longer about beauty.

How did we get from the Sistine Chapel to R. Mutt? One simple way of answering might be that the head came unstuck from the heart.

Should we talk at length about the Protestant Reformation, the rise of the individual and the decline of orthodoxy? All these were powerful eroding influences on the common purpose, the shared assumptions, of artists at work.

Should we talk about the age of skepticism -- a very long age -- the age we still live in -- a hostile age for the simple truth of the Abbe Suger? Should we talk about the Copernican revolution, which upended the belief that God's universe orbited around the central eye of man? About Descartes? Revolution? Industrialism? Darwin?

We should talk about all of these forces of change, but there isn't time. Crashing waves of history and ideas swept us, over 350 years, from Michelangelo to Duchamp's urinal. The certain, functional universe of Suger, of Dante, became a world -- as F. Scott Fitzgerald would put it in 1920 -- with "all gods dead, . . . all faiths in man shaken."

The revolution was a long time coming. The beauty world rocked back and forth. There were periods of beauty for beauty's sake -- the saccharine cupidons of Fragonard's France, the frilly viscounts of Van Dyck's England and, later, the crushed purple velvet and languid lilies of Oscar Wildean aesthetes.

There were reactions against all that: The Romantic rebellion of the early 19th century, for example, called painters and poets away from their pillows, pipes and courtly essays and into the rocky redoubts of Nature. Looking back, you wonder if this was really such an enormous rebellion at all. For a time, large figures occupied small landscapes; then small figures occupied large landscapes. Certain artists hunted the quietest beauties in the most mundane things, until photography was invented in the 1830s, and the snapshot replaced the sketch.

This back-and-forth added up to more than stylistic change. By the early 1800s, the influential Romantic poet and critic Friedrich von Schiller worried passionately over a deep split in the human condition between the head and the heart. Schiller hoped that art could mend the breach, but in England, in the same years, poet Percy Shelley was less sure. "Spirit of BEAUTY," he wailed, "where art thou gone?"

What would the engineer-artist Brunelleschi have made of that? What would Suger have to say to John Ruskin? The abbot, in 1140, imagined that the beauty of his church would guide common folks to reverence. By 1850, Ruskin -- England's dominant Victorian critic -- was arguing almost the exact opposite: that the real beauty of Gothic churches was not their magnificence, but the flaws left by forgotten, common artisans. Depressed by the machined perfection of English factories, Ruskin wrote: "neither architecture nor any other noble work of man can be good unless it be imperfect."

What a struggle it was, century after century of increasing division and alienation. Then along cames Duchamp's urinal, which seemed to say: Why fight?

Why should art have to unify the human head and heart? If science could wrap itself into a closed fraternity, if historians and lawyers and philosophers could gather into little sects, art, too, could become a secret society, with its own lingo, its own inside jokes, its own obscure controversies, its own disdain for the uninitiated.

Leave the Renaissance Man to the long-lost Renaissance.

And the months and years immediately after the urinal -- when the full horror of the First World War began to register on Western intellectuals -- were even worse for beauty. The high-born, refined, art-loving elite of Europe pompously sent millions of young men to die in the gassy, barbed-wired trenches of a pointless war. Fat lot of good beauty did for them.

Ponder a few lines from the most influential poem of the 1920s, T.S. Eliot's "The Waste Land": "A rat crept softly through the vegetation/ dragging its slimy belly on the bank/ While I was fishing in the dull canal/ On a winter evening round behind the gashouse . . ."

You could trace the same pattern in any art. Music went from Palestrina's polyphony to Bach's fugues to the sonata form of the classical symphonists -- and wound up in the music-about-music realm of atonalism and endless repetition of the same note. Literature had its heroic couplets and its Shakespearean sonnets -- but the hot thing in the 20th century came to be writing-about-writing.

One critic compared the chasm between ancient and modern that came around the turn of the 20th century to having your neck broken. Philosopher George Santayana certainly felt it. He had written, in 1898, an important essay called "On Beauty." In those not-so-distant days, he reflected later, beauty was "a living presence or aching absence night and day."

By 1928, Santayana would observe sadly that his old friend beauty had become a pariah, something "which we must not speak of today."

The Pile, and Dung

Arts patron Walter Arensberg and painter George Bellows were there the day Duchamp's urinal arrived. They argued over it. Bellows could not believe that anyone could take a bathroom receptacle seriously as art.

"It's for the artist to decide what is art," Arensberg argued.

Bellows erupted. "You mean to say, if a man sent in horse manure glued to a canvas that we would have to accept it!"

"I'm afraid we would," Arensberg answered.

It took about 80 years. But eventually all that Bellows foresaw came to pass. The manure came from an elephant, not a horse. It was part of the most sensational art exhibition in London and New York in these waning years of the millennium.

And yet, here's the strange thing: For ordinary people, there is more beauty around now than ever before. Tally it up: There are literally millions of works of art collected in public museums around the world. Access to fine art was extremely limited, even in the 19th century; today, there are museums and galleries in most major cities.

Before 1960, even the most important books on beauty had few, if any, color illustrations. Even educated people experienced Titian, say, in black-and-white. Today, you can find nearly any significant picture, building, statue or piece of glass reproduced in a high-quality book. Art books can fill entire libraries.

A person living in a small town before electricity might never hear any music lovelier than Aunt Mabel's parlor crooning. Now, a set of 10 compact discs will give you an A-to-Z of classical music, from Bach to Wagner, for $42.

In the 17th century Shakespeare's sonnets could be purchased by a very few people at considerable cost. The Internet revolution is now making them available to billions for free.

Washington Post art critic Paul Richard calls it "The Pile" -- the constant presence at millennium's end of all the pictures and all the music and all the literature of the world, easily accessible to the ever-expanding developed world. People are bombarded by masterpieces. We are saturated with stunners.

For artists, The Pile can be a resource, but also an oppression. What is there to make or say that feels new? How can one possibly out-paint Rubens or out-compose Mozart? What's most interesting these days for many creative people is critiquing, deconstructing, experimenting with or poking fun at The Pile itself. In this, too, Duchamp was a pioneer: Among his best-known works is a copy of the Mona Lisa with a mustache.

For consumers of beauty, surely, The Pile is almost entirely a good thing. Almost entirely -- not quite. The ubiquity of beautiful creations has a way of draining each one of its original power. As anyone who has ever walked with heavy legs and a wandering mind through the 25th room of Old Masters at a major museum knows, there can be too much of a good thing.

In our lives, archetypes of the beautiful are constantly put to work selling everything from movie stars to bedsheets, designer bathrooms to underwear. Is that Martha Stewart centerpiece evoking a Dutch Master still life? Is the Calvin Klein underwear boy posed like a Renaissance Saint Sebastian? It is inevitable that some of the passion -- some of the shock -- that beauty once bore begins to leak away.

So take a moment to get back in touch. Consider the picture at the top of this essay. She is Venus, the Roman goddess of beauty and love, daughter of Jupiter, mother of Eros and all things erotic, lover of Adonis, love-object of Mars, author of Helen's destiny, which was to launch the thousand ships of the Trojan War.

We see her here as rendered by Sandro Botticelli at the dead smack middle of our millennium. We see woman. We see youth. The flowing hair. The contemplative mood. We see color, and many curves, and exquisite craftsmanship.

We see the result of layering colored paste onto stretched fabric with hair tied to a stick. We see the animating juice of genius.

After all these years, you still feel it, don't you?