What do you give the American who has everything? In 1919, the French chess player and soon to be retired artist Marcel Duchamp, preparing for one of his many transatlantic crossings, wanted to take his patron a present. The patron, Walter Arensberg, "had everything money could buy", recalled Duchamp, so he took him something money couldn't buy. "I brought him an ampoule of Paris air." At a Parisian pharmacy, Duchamp obtained an alchemical-looking bottle that contained "a physiological serum", and asked the pharmacist to empty and reseal it. Arensberg treasured the gift as a Duchamp readymade, and in 1950, with the rest of the Louise and Walter Arensberg Collection, an authentic re-creation entered the Philadelphia Museum of Art.
At that moment, when Duchamp's pharmacist was sealing the little bottle, the air of Paris was, for anyone interested in the arts, the most precious substance in the world. It was magical, and redemptive. Paris could make you a genius. It made Gertrude Stein a genius. It did F Scott Fitzgerald and Ernest Hemingway no end of good. Not just writers and artists, but all kinds of besotted enthusiasts of avant gardism made their way across the Atlantic to kiss the boulevards. In 1919, Sylvia Beach, the daughter of a New Jersey Presbyterian minister, opened the bookshop Shakespeare and Co at 8 rue du Dupuytren. Three years later, she published James Joyce's Ulysses.
If you couldn't survive the heady Paris air, you might at least help someone else to breathe it. Harriet Shaw Weaver, feminist, communist and editor of the Egoist, who described herself as "hopelessly English", sent thousands of pounds from dreary London to enable Joyce to live in Paris. Meanwhile, rich Americans bought into the avant-garde big time.
The wealthy and the curious who flocked to Paris after the first world war were, however, merely joining a migration that had already begun in 1900, when artists from all over the world sought footholds in grotty studios, congregating in rat-infested warrens with names like the Bateau Lavoir ("the laundry boat") and La Ruche ("the hive"). At the beginning of the century, Pablo Picasso came to Paris from Barcelona, followed by Amadeo Modigliani from Italy, Marc Chagall from Russia, Jacques Lipchitz from Lithuania, Diego Rivera from Mexico. Stein remembered that her Saturday night parties round about 1907 "were frequented by many hungarians, quite a number of germans, quite a few mixed nationalities, a very thin sprinkling of americans and very few english".
Yet of all the great migrations to Paris in the early 20th century, none was more extraordinary than that undertaken by Constantin Brancusi. This carver of Endless Columns and caster of magic birds was to become one of the most authentic of modern artists. But first he had to get to Paris.
Brancusi was born a long way from the arcades and boulevards, in the village of Hobitza in Romania, in 1876; his father was estate manager of lands belonging to the local monastery, a job that probably had not changed much since the middle ages. In May 1904, after mastering the rules of 19th-century sculpture at the Bucharest School of Fine Arts, Brancusi set out to walk from Romania to Paris. He hiked through Austria and Germany, reaching Paris on July 14 - Bastille Day - as a medieval pilgrim might have arrived with bleeding feet at the gates of Rome. Anyway, that's how the story goes.
Like all tales of journeys and exchange and impossible distance, Brancusi's expedition raises the question of what Paris was, that lost Paris of the modern movement - what kind of place could harbour, like some monstrous, endless Bateau Lavoir, Romanian sculptors and Irish polyglot novelists, Mexican cubists who would one day be suspected of murdering Trotsky, and matadors Spanish and American?
The capital city of modernism was not even topographically identical with the Paris of today. Montparnasse, which in the 1910s started to replace Montmartre as the suburb of artists, was still essentially a village. Modernist Paris was not so much a fixed, physical place as an imaginary, shifting map of haunts, hideouts, dens, clubs and brothels. The centre of Paris was Stein's home, at 27 rue de Fleurus. Or it was Picasso's studio in the Bateau Lavoir. A woodcut carved by Pierre-Antoine Gallien in 1922 maps Montmartre as an archipelago of cafes - Napolitan, Parnasse, Vavin, Rotond, Caméléon - between which you could roll in the sea of night. And if the places were transitory, the events - the history - were as insubstantial as a recollection the morning after.
Avant-garde Paris was already a myth when Stein amplified it in her great modernist memoir of 1933, The Autobiography Of Alice B Toklas. Perhaps it was the opium, or the absinthe, but no one seemed to agree about anything that happened. Alice tells the tale of a banquet that Picasso hosted in his studio in honour of Henri Rousseau, a customs official who painted delirious visions of jungles and deserts and flat, bold portraits. Was the feast for the douanier an honest tribute or a cruel joke? The latter, said Toklas/Stein: "Guillaume Apollinaire solemnly approached myself and my friend and asked us to sing some of the native songs of the red indians. We did not either of us feel up to that to the great regret of Guillaume and all the company. Rousseau blissful and gentle played the violin... "
Others denied that the occasion was, as she made it sound, a burlesque. On the contrary, said André Salmon, "we sincerely admired Rousseau". Anyhow, the dinner party took on the colossal proportions of a Historical Moment. Which may be why a recent biography places Brancusi among the guests, although none of the sources I have read remembers his being there.
Brancusi inherited the innocent charisma of Rousseau, who died in 1910. Rousseau's art is strong and alive in its freedom from any convention of what a painting "should" look like: this is why it mattered so much to artists such as Picasso, who were breaking free of what they were taught in 19th-century academies. Brancusi, too, had an academic training, but he was a Paris peasant, a Romanian with a heritage of folklore and folk art as wild and unbourgeois as Rousseau's fantasies. Like Rousseau, he played the violin. You can see Brancusi's violin in his reconstructed atelier in Paris. You can picture the bearded sculptor playing Romanian folk music in the evenings accompanied by his friend, the anarchic composer and cabaret pianist Erik Satie. The music and personality of Satie in his little round spectacles shaped the playful spirit of French dada, the only art movement to which Brancusi - loosely - belonged.
On his death in 1957, Brancusi left his studio and its contents to the French state, including key examples of every work he ever made. Like Duchamp, when Brancusi coined an idea, he replicated it many times. Their art is utterly unalike except for this taste for repetition. Duchamp selected "readymades", pre-existing humanly produced objects, which he nominated as works of art; most notoriously, the porcelain urinal he submitted as Fountain, by R Mutt, to the American Society of Independents' Exhibition in New York in 1917.
America again. The oddest quality of that bygone modernist Paris that makes its history so elusive as you wander today through its rues, its places, as you browse at Shakespeare and Co - complete, the last time I was there, with an expatriate North American writer trying to sell secondhand books to purchase, what, a square meal? a typewriter ribbon? - is that it existed, in part, only as a projection in the American mind. Paris looked exotic and modern and revolutionary from whichever corner of the world you longed for it in 1900, or 1922; but it blazed most brightly in American eyes. For an artist such as Brancusi, his life in Paris was made possible solely because of the spectacle it presented to an American audience.
When people talked about modern art in the early 20th century, they meant art from Paris. And it exploded in American culture, with burnings in effigy, critical denunciations and the birth of a luxury market at the Armory Show in New York in 1913. Held in an old civil war armoury on Lexington Avenue, the show was the sensational launchpad of modernism in America. Brancusi was included after Duchamp sent the organisers to his Paris studio. The Romanian rural misfit who had walked to Paris now exhibited in Manhattan. When the show reached Chicago, Brancusi was burned in effigy by conservative art students. His works, including Mademoiselle Pogany, The Muse and Sleeping Muse, drew some of the poison that circulated around the show's most controversial work, Duchamp's Nude Descending A Staircase.
Even today, it's not hard to see why Duchamp's painting appalled Americans brought up on beaux arts and Tiffany art nouveau. It was one thing to distort the human body, as artists had done one way or another since Michelangelo; it was another to say the body was a machine. Duchamp's painting, more or less his last painting, depicts the fluttering motions of a woman walking down steps as a vibrating mechanical dance, a robotic symphony: coolly elegant, cybernetically sexual and disturbingly inhuman.
What is perhaps less recognised, because his art tends to be thought of as organic and carved and abstract, is that Brancusi's works disconcerted in exactly the same, science-fiction way. His polished bronze Sleeping Muse (1910), today in the Musée National d'Art Moderne in the Centre Georges Pompidou, is startling because it is a head without a body, lying on its side. It is like an egg, an alien egg, and also like a beautiful abstract woman. A long, slender, sharp nose, closed, delicate eyes, moon lips, decorate a shining ovoid with short hair swept back. She's so 20th century, Brancusi's dreamer, and so erotic.
Brancusi was a legendary lover. His bearded intensity and violin playing attracted, apparently, artists, collectors, princesses. He is often spoken of as a spiritual artist, but the shock of his art lay in its combination of metal, stone and sexuality. Just like Duchamp, he seemed to be saying that the new century was engendering a new humanity, with smooth, metallic desires. The work that really scandalised Americans in 1913 was Mademoiselle Pogany, an abstract portrait of one of his lovers, the Hungarian-Romanian painter Margit Pogany. Today, surviving versions of this face, with its vast almond eyes, make you think of an alien - a bug-eyed, sexy alien.
It's a kind of profanity to say that the new creatures painted and sculpted in Paris 80 and 90 years ago look like sci-fi monsters. But absolute otherness, reinvention of the human, was part of their scandal and meaning. Picasso's Demoiselles d'Avignon (1907) stare out of the painting like a new species. Mademoiselle Pogany, with her even huger eyes, inhabits a place that you, the viewer, has yet to reach.
Perhaps that new planet was Paris. If some visitors to the Armory Show reached for denunciatory slogans, others, rich ones, set sail with cash in hand.
Brancusi's work was collected primarily by one American, John Quinn, who also bought the manuscript of Ulysses from Joyce. He was a crude thinker, claimed Joyce: "With a person like Quinn you should never hint at any imperfection in your work; he wouldn't understand it." When Quinn died in 1924, a potential crisis for Brancusi was averted by the ever-loyal Duchamp, who, together with the writer and art collector Henri-Pierre Roché, bought Quinn's 33 Brancusi works and managed their sale, at increasing prices, to collectors and museums.
All of this transatlantic speculation and exchange sustained, and was inspired by, slow, painstaking labour in a back street in Montparnasse. Brancusi did not select readymades. He carved. He lived for his physical relationship with stone, wood, plaster, bronze. He organised his life in his studio, so that there was no gap between life and work - keeping his saws and chisels next to his bed. When he was working there, he said, there were days when "I would not have given up 15 minutes of my time for anything under heaven".
Where was Paris? If the fleeting locus of modernism was anywhere, it was in the ateliers, the studios that functioned as home and workspace, where artists even chose to exhibit instead of in the cold arena of the salons. If you wanted to understand Picasso's cubist innovations, you had to visit his studio home in the Bateau Lavoir.
The inventor of the studio as secret utopia had been Vincent van Gogh, who set up his Studio of the South, the Yellow House, in Arles in 1888. Van Gogh painted his blazing Sunflowers to decorate what he conceived as an artists' colony where he, Gauguin and the others who flocked to them would paint and live as brothers. The ateliers, the legendary dream spaces that artists inhabited in early 20th-century Paris, have gone now. In homage to Picasso, the government declared the Bateau Lavoir a historical monument in 1969. It almost immediately burned down. The magic world of the studios survives only in paintings such as Matisse's The Red Studio, now in MoMA, New York.
There is one exception. Brancusi's studio (in fact, a complex of studios) survives as a pristine, timeless, reconstructed exhibit. He gave his whole personal collection - sculptures, the photographs he took to document them, drawings, love letters, the lot - to the French state, on condition that his studio be preserved in its entirety. And so, in a low, rectangular building whose appearance couldn't be more of a contrast with the exoskeletal Gargantua of the Centre Georges Pompidou to which it is an appendage, Brancusi's entire life is preserved behind glass. His Endless Columns, giant birds and monumental gates now guard the delicate realm of memory. Peering through the windows into the white, top-lit space with its rustic wooden furniture carved by the artist himself, with its homely wooden archway leading into his sleeping quarters, where his violin and guitar hang in the shadows, is to contemplate a world bottled - like Duchamp's Paris air.
Brancusi's studio was his home - and what a home. Gradually expanding from 1916 onwards into a labyrinth of spaces in the Impasse Ronsin in Montparnasse, he created a secret, enchanted world that turned away from the life of the streets. Perhaps it was because he was an outsider, an immigrant, that he needed to create a refuge for himself and his art. The art he made here was of a unique, untranslatable reality, private to him, that existed somewhere between Paris, Romania and America. In the end, what drew people to Paris, what made it possible for a Brancusi to exist, was that in the ateliers, working at night, you could free yourself from the rules, deceits, expectations that operated in daytime society. His studio was his lair. He invited friends into his world, to eat Romanian food and to listen to folk music and jazz from speakers inside sculptures.
One of his visitors was Joyce, who sat for a portrait in 1929 and, according to Joyce, concurred in deploring "modern feminine fashions, the speed of modern trains, etc, etc". Brancusi did some touchingly characterful drawings of Joyce, but the image that was published was his Symbol Of Joyce - consisting of three straight lines and a spiral. "The boy seems to have changed a great deal," as Joyce's father in Dublin commented. The Symbol Of Joyce was commissioned by the (American, naturally) founders of Paris's Black Sun Press as a frontispiece to Joyce's Tales Of Shem And Shaun - an early publication of sections of Finnegans Wake, the "night piece" that Joyce began in Paris in the early 1920s.
There's a strong affinity between Joyce and Brancusi. Brancusi's Sleeping Muse is as troubling and glorious an evocation of the things we know in the night and cannot name in the morning as Joyce's dream novel: in a language of puns and garbled myths, experienced as abstract music, Joyce goes outside rational descriptions of the world into the realm of night. And that is where he meets Brancusi. Perhaps because André Breton and the surrealist movement colonised the idea of "revolution by night", it slips our notice sometimes that all the modern art and writing in Paris was, one way or another, an art of dreams; it just didn't put this in Breton's rational political speech. Van Gogh's Starry Night and Gauguin's Tahiti, Picasso's Demoiselles d'Avignon and Matisse's Dance: all these masterpieces are profoundly nocturnal. In the end, the best guidebook to modernist Paris is the slumberous text of Finnegans Wake, whose last words are "PARIS, 1922-1939".
Brancusi's art, in its quotation of Romanian folklore, is sometimes reduced to terms such as "primitivism". His most incomparable creation, the Endless Column, refers to a folk legend of the pillars that support heaven. But to constrict this sculpture to folklore would be like reducing Finnegans Wake to the drinking song from which it takes its title. The Endless Column streams up through the silent air of the studio. It speaks of indestructible creativity, a tower of beauty that will never be brought down, power without arrogance, strength without violence, body without weight. It finds beauty in repetition and seems to touch on something lost and impossible, a magic geometry, the key to the universe. Something you knew once, if you could only put it in words.
"All that was such a long time ago," said Stein when Picasso's wife was horrified by stories of the old days at the Bateau Lavoir. But it was once worth walking across the world for
· Constantin Brancusi: The Essence Of Things is at Tate Modern, London, from January 29. For details, call 020-7887 8888.