Budapest is a city that happens to sit on top of hot water — and everything else follows from that.

01 A capital with a boiler underneath

You notice it first as a smell. Somewhere along the Buda embankment, in a doorway on a January morning, there is a faint mineral warmth in the air — sulphur, wet stone, something older than the traffic. Then you start seeing the evidence everywhere: a nineteenth-century pump room, a drinking fountain of thermal water with a plastic cup chained to it, an Ottoman dome pushed up against a six-lane road. Budapest does not treat this as exotic. It treats it as plumbing.

And the plumbing is the story. Almost every capital sits on something — a harbour, a ford, a defensible rock — and the thing underneath usually stops explaining the city after the first few centuries. In Budapest it never stops. The water explains why the Romans settled here, why the Ottoman period left domes rather than palaces, why the great building boom of the 1890s produced bathing halls the size of railway stations, and why on a Tuesday morning in February two retired men are playing chess up to their chests in steaming water while the snow lands on the board.

02 The water arrived first

Look at the two halves of the city and you are looking at a geological seam. Pest, on the east bank, is flat: the beginning of a plain that runs for hundreds of kilometres. Buda, on the west, rises immediately into limestone hills. The Danube runs along the join, and along that join the ground is broken. Rain that falls on the Buda hills sinks into the limestone, travels down, is heated at depth, and comes back up through the fracture — which is why the springs cluster on the Buda side, at the foot of Gellért Hill and along the river.

That circulation has been running for far longer than anyone has been drinking coffee here, and it hollowed out the hills on its way. Beneath the residential streets of Buda lies one of the world's more improbable urban features: a system of caves dissolved out of the limestone by the ancestors of the same hot springs that still feed the baths. The Pál-völgyi system alone runs for close to thirty-two kilometres, the longest in Hungary; Szemlő-hegy, smaller, is famous for mineral formations that grew out of rising thermal water. The city has a warm, wet basement it did not build.

People have been using the surface end of that system for about two thousand years. The Romans built their provincial town of Aquincum on this bank and bathed in it. In the sixteenth century, under Ottoman rule, the practice was rebuilt in stone: the Rudas baths were raised between 1550 and 1556 around an octagonal pool set beneath a dome ten metres across, carried on eight pillars and pierced with star-shaped panes of coloured glass that drop shafts of light onto the water. It is still working. You can pay at a turnstile and get into a room designed before anyone in Europe had seen a potato.

03 The day hot water crossed the river

For centuries everyone accepted an obvious fact: the hot water belonged to Buda. The springs came out on that side, the bath houses stood on that side, and flat Pest — the commercial half, the half that was about to explode — had nothing underneath it but farmland. Then an engineer called Vilmos Zsigmondy spent ten years drilling into the Pest plain and, in 1878, struck thermal water at a depth of 970 metres. The hills had no monopoly. The whole city was sitting on the same warm system; it was only a question of how far down you were willing to go.

The timing could not have been better. Buda, Óbuda and Pest had merged into a single city only in 1873, and that city was about to become the fastest-growing in Europe: from around 296,000 people at unification to some 733,000 by 1900, eighth largest on the continent. A capital in that state builds enormous things quickly, and the millennium celebrations of 1896 gave it an excuse — the boulevards, the underground railway beneath Andrássy Avenue, the whole eclectic skyline that visitors now photograph. The hot water arrived precisely as Budapest was learning to build at scale.

So it was built at scale. The Széchenyi, opened in 1913 in the City Park and drawing on that Pest-side water, is a neo-baroque palace the colour of egg yolk with pools indoors and out — the outdoor ones fed by springs at seventy-four and seventy-seven degrees, cooled before they reach a bather. The Gellért followed in 1918, at the foot of its hill, all glass roofs, majolica and columns. These are not spa hotels dressed up for tourists. They were built as public infrastructure, in the same civic mood and the same decades as the market halls and the railway stations, and they were meant for everybody.

04 The bath is not a spa. It is a room in the neighbourhood

This is where the misunderstanding usually happens. A spa, in most of Europe, is a place you go to once, expensively, to be quiet. A Budapest bath is a place people go to on a Wednesday, on a season ticket, to see who is there. The register is closer to a municipal swimming pool than to a wellness retreat: turnstiles, cabins, a smell of chlorine competing with the smell of minerals, a caretaker who has run the same corridor for twenty years, and prices set so that a pensioner can come regularly.

Watch the outdoor pool of the Széchenyi in winter and you can read the whole social contract in ten minutes. Men in their seventies stand at waist height around floating chessboards, playing games that appear to have been going on for decades. Women do slow lengths in the swimming pool next door with their hair pinned dry. Somebody's shoulder is being treated on the recommendation of a doctor. Somebody else is simply out of the flat and talking. The steam makes everybody roughly anonymous, and the water makes everybody roughly the same height.

That flattening is the point, and it has consequences above ground too. A city with a habit of undressing in public together is a city with a particular tolerance for the unglamorous. It shows up in the plain neighbourhood baths nobody photographs — the Lukács with its wall of marble tablets left by grateful bathers, the small district houses where the ticket costs less than two coffees. It shows up in the way Budapest, for all its imperial-era grandeur, is fundamentally unbothered about being seen in a dressing gown.

05 What the water costs

None of this is free, and Budapest is currently negotiating the bill. The great baths have become international attractions, which pays for their upkeep and, at the same time, changes them: at the Széchenyi and the Gellért there are afternoons when the water is full of visitors and the chess players have gone home. Locals answer the way locals always do, by retreating to the less famous houses and the early hours. Ask anyone here when to go and the reply is a time, not a place: before nine, or on a weekday, or in the cold.

There is a physical bill as well. Thermal water is heated over a very long time and pumped up in a hurry; the flow is generous but not infinite, the historic buildings around it are enormous and expensive to keep, and heating, treating and replacing that much water is an industrial operation hidden behind the mosaics. The palaces of 1913 and 1918 were built by a city convinced it was about to become one of the capitals of Europe. Keeping them open is a promise the present has to renew every year.

And there is a subtler cost, the one every city with a signature pays. The baths photograph so well that they risk becoming a costume — a yellow facade, a steaming pool, a picture. The thing worth protecting is not the picture. It is the fact that this is ordinary here: that the water is a public service, that people arrive with a plastic bag and a towel from home, and that nobody involved considers any of it remarkable.

06 What stays

You come to Budapest for a skyline that looks like a nineteenth-century idea of a capital, and you leave with something much smaller: the memory of a February morning when you walked out of a heated pool into freezing air and the steam came off your own shoulders, and around you, in a city where you did not speak a word of the language, several dozen strangers were doing exactly the same thing without looking up.

That is what a capital built on hot water produces in the end. Not luxury — the tiles are chipped and the changing rooms are functional — but a habit of sharing warmth in public, inherited from the Romans, rebuilt by the Ottomans, monumentalised by the boom of the 1890s, and still running on a Tuesday. Other cities have their water in fountains, where you can only look at it. Budapest put its water where you can get in.