Mexico City, by contrast, looked to its indigenous past for national icons. A statue of Cuauhtémoc, the last Aztec emperor, stands on a pedestal overseeing the new city that had arisen on the ancient Tenochtitlán. Redrawn by the Spanish, it was now a source of republican pride.

We discover many of these statues in hazy prints of silver and sepia, and indeed this is as much an exhibition of early photography as it is of architecture and urban planning. By 1900, shutterbugs in these Latin American capitals were snapping plush urban parks in Buenos Aires and Mexico City, spaces of leisure for the new bourgeoisie. Both of these cities received radical makeovers at this time, with ostentatious, oversized thoroughfares built on the Parisian model. An unnamed photographer’s family pictures capture Buenos Aires’s lordly Avenida de Mayo, a boulevard built in commemoration of Argentine independence and in palpable imitation of Baron Haussmann’s extended axes.

And as for the vintage photographs of Rio, it is hard to look without desperate jealousy. The most beautiful city on the planet appears, by some measure, as the most sophisticated of the capitals examined in this exhibition, though the photographers and mapmakers here had far more interest in Rio’s boulevards and gardens than its favelas, the first of which arose in the 1890s.

In two exquisite photographic panoramas by the Brazilian photographer Marc Ferrez, dating to around 1895, Sugarloaf Mountain looms gently over Botafogo and Flamengo — the boulevards and alleys of the capital in seductive harmony with the beach and the hillside. Ferrez also shot Rio’s ritzy cinemas and lush botanical gardens, as well as new infrastructure. A tramway cuts through the underbrush to Corcovado, the verdant peak that would soon be topped by the statue of Christ the Redeemer.

Eventually a new generation of Latin American architects and urban designers, less hung up on European models, would transform these capitals once again. There is a hint of what’s to come in a 1927 photograph of Rio’s Praça Marechal Floriano, whose undulating mosaics of white and black would soon be replicated in Roberto Burle Marx’s wavy boardwalk on Copacabana Beach. Burle Marx would go on to landscape a new capital, Brasília, that would take national ambition to new extremes. Buenos Aires would invite Le Corbusier, the Swiss apostle of towerblocks and bulldozers, to reimagine the Argentine capital from scratch. Cuba would take a different approach to national reinvention, one that, ironically, has left much of Havana’s earlier architecture deteriorating but in place.

Latin American cities would become spaces of utopian fantasies, frequently followed by political nightmares. But before the new towers arrived — and before, more recently, Rio’s white-elephant Olympic stadiums and Mexico City’s big-money museums — they were already cities of dreams.