Exercise your mind in these cities
Sometimes a city can change the way we think. Even in hindsight, it's hard to define how it happens, but mix together the right ingredients - education, or money, or even, sometimes, religion - and occasionally, something miraculous happens.
A previously undistinguished backwater suddenly becomes a distillery of the mind, fermenting ideas that spread around the globe.
Over the centuries, a number of cities have held the role of intellectual beacon; in many of them, the legacy continues, albeit in a lower key.
So if your brain needs a bit of a workout, skip the walking holiday; head for one of the cities below, and enjoy a thinking holiday instead.
The Scots don't like to brag, which may explain why so few of us realise that for a time, Edinburgh was the intellectual capital of the world. During the 18th and early 19th centuries, local thinkers generated new ideas in economics, geology, chemistry and philosophy that kick-started the industrial revolution, and changed the way we look at the world.
Some, including economist Adam Smith and philosopher David Hume, have kept their place in the history books. Others have fared less well. Does anyone, outside university geology departments, remember James Hutton, the first modern geologist?
Doctors may remember that William and John Hunter turned Edinburgh into the global centre for medical training. In the late 18th century, 85 per cent of British doctors were educated at Scottish universities.
The influx of students wanting to be part of this heady intellectual ambience also changed the nature of Edinburgh, weaving intellectualism and internationalism into its DNA.
That side of the city is still in evidence, from its thriving universities to its massive Festival every August.
Visitors to Edinburgh can say hello to Adam Smith and David Hume, close friends in real life, whose statues gaze toward each other on the Royal Mile which they once walked. (Locals rub Hume's big toe for luck).
The real monument to the Enlightenment's intellectual boom, however, is Edinburgh's elegant New Town.
The grid plan and the Georgian buildings are a celebration of rationalism, modernity and a city that, for a while, was the centre of the world.
Some emperors are celebrated for their military conquests, or their able administration. Rudolf II was known for holding court.
Holy Roman Emperor in the second half of the 16th century, Rudolf had plenty of flaws - his psychological problems ultimately saw him deposed by his brother - but he turned Prague into the most exciting city in Europe, thanks to the artists and scientists he gathered around himself.
Among the careers that Rudolf sponsored were two of the founders of modern astronomy, Tycho Brahe and Johannes Kepler.
Rudolf also supported mathematicians such as Joost Burgi, who invented logarithms, as well as indulging a collection fetish that included coins and ivories, scientific instruments and natural objects.
Rudolf II's greatest love, however, was art. He was the patron of a wide range of painters, including Giuseppe Arcimboldo, whose portraits made of fruit and vegetables remain surreal even today. Rudolf's art collection included thousands of paintings by the likes of Veronese, da Vinci and Brueghel the Elder.
The collection is long dispersed, but the beautiful architecture Rudolf commissioned, including the Archbishop's Palace and the new wing of the imperial palace - built to display his art collection - remains. So too does the astronomical clock on the Old Town Hall, which predates Rudolf, but is a reminder of his passions.
If he were alive today, however, what would really delight Rudolf is Prague's thriving art scene.
At museums such as the Czech Museum of Cubism and the Futura Centre of Contemporary Art, and galleries such as the Manes Exhibition Hall and the Leica Gallery, Prague's current artistic blossoming is on full display.
Washington has the politicians, LA has the film stars, but Boston has always had the thinkers. Pick almost any period of US history, and you'll find the brains trust was probably based in Boston.
The American Revolution started here, with the Boston Tea Party and Paul Revere's midnight ride. In the lead-up to the American Civil War, Boston was the capital of the Abolitionist movement. Later, the first African-American regiment, the Massachusetts 54th, was recruited here.
Then there's the cultural scene. The city had the first public park in America (the Boston Common, still worth a visit today) and the first public library (established in 1636, although today's Boston Public Library was founded in 1848.)
In Victorian times, it was also America's literary capital, home to writers such as Henry James, Nathaniel Hawthorne, Louisa May Alcott, Ralph Waldo Emerson and Henry David Thoreau.
The city's grand past remains remarkably accessible. Start with the Boston Tea Party Ships and Museum, and the Old South Meeting House, where the Tea Party was planned.
Then move on to the Black Heritage Trail, which features historic sites such as the first public school for black children, and the African Meeting House.
The oldest surviving black church in the US, this is where Frederick Douglass spoke and where the first anti-slavery society was founded in 1832.
And if that's not enough intellectual stimulation, you can always head across the Charles River to Cambridge, where Harvard University and MIT continue to keep brains buzzing.
Florence may have been the artistic centre of the Renaissance but, when it came to science, Padua was where it was happening.
The city's esteemed university, established almost 1000 years ago, enjoyed a golden age in the 15th and 16th centuries. Among other things, it revolutionised the teaching of anatomy.
While other institutions taught from ancient books without pictures, anatomists at Padua did their own dissections and vivisections, discovering in the process such basic bodily functions as pulmonary blood circulation.
Equally celebrated was the chair of mathematics which, from 1592 to 1610, was held by Galileo Galilei.
When he wasn't teaching classes, Galileo kept busy with his own work: during his sojourn in the city, he discovered the parabolic path of projectiles, invented a horse-driven pump and a primitive thermometer.
He also built a telescope after hearing a description of one, and used it to discover three of the moons of Jupiter.
These days, Padua's university students have become adept at weaving their way around tourists: the university runs tours that take in the anatomy theatre and the hall where Galileo lectured.
From there, lose yourself in the beautifully preserved old town, with its array of bridges and network of arcaded streets, and imagine yourself studying here four centuries ago, on your way to learn from a lecturer who was rewriting the rules of the game.
When you've made a pile of money, there are plenty of ways to spend it.
In 18th century Manchester, however, the city's cashed-up industrialists seemed locked in a competitive spending spree, each striving to prove that they could spend the most money on other people.
At the time, Manchester was the city of the future. Unlike London, here it didn't matter what you were born; what counted was what you made of yourself. And the city's self-made men were positively eager to help others make the most of themselves.
They set up and funded scholarships at the vocational colleges such as the Mechanics' Institute and the Manchester School of Design; they endowed institutions such as the Portico Library, the Literary and Philosophical Society, and the Manchester Guardian newspaper.
When the world's first blockbuster exhibition, Art Treasures, took place in the city in 1857, many factory owners organised trips for their workers, underwriting the rather steep entrance fees.
Much of their legacy is evident in today's Manchester. You can still attend lectures at the Literary and Philosophical Society, and pick up a copy of the Guardian newspaper. The most remarkable relics, however, are the city's libraries, such as the beautiful Chetham Library.
Inside its mediaeval interiors, two German migrants, Friedrich Engels and Karl Marx, researched theories that would later shake the world.
Then there's the John Rylands Library, an exquisite neo-Gothic concoction with an extraordinary collection that ranges from an ancient fragment of Homer's Odyssey to a stunning Japanese woodblock-printed book of poetry, with illustrations done in powdered mica, gold dust and mother of pearl.