One hundred and fifty years ago the Swiss art lover and historian Jacob Burckhardt published his master work, The Civilization of the Renaissance in Italy. I believe this anniversary is as important as last year's of Darwin's On the Origin of Species. These two great 19th-century books are still at the living heart of their subjects. The study of the Renaissance can no more forget Burckhardt than biology can leave Darwin behind.

Both classics began in journeys. Darwin sailed to the Galápagos; Burckhardt merely went to Italy. His book drips with love of Italy and Italians. It is, among other things, one of the most passionate homages ever paid by a northern European to southern Europe, and yet herein lies its strangeness. Northerners, from Thomas Mann in Death in Venice to Martin Amis picturing the gilded English young on holiday in a southern castle in The Pregnant Widow, have tended to imagine Italy as a languorous, sleepy, timeless and archaic place – the slow, hot unconscious of the European continent, drooping out into the Mediterranean like a surrealist appendage. Burckhardt saw things very differently. The fascination of reading his book is its vision of Italy as the birthplace of modern individualism, political calculation, science and scepticism. In 1860 Burckhardt looked at Italy and saw the shock of the new, secreted in sleepy ruins.

The ruins, at that moment, were becoming less sleepy. Italian cities were discovering art history as a commodity. Burckhardt, who studied history in Berlin before returning to work as a journalist and university teacher in his native Basel, was very much part of the 19th-century discovery of Italy by the bourgeoisie. His book The Cicerone – a cicerone was an early tour guide – offered travellers a practical account of Italy's aesthetic riches. Where 18th-century aristocrats on their grand tours had seen themselves as lineal descendents of Roman senators and admired the classical tradition as their own, eternally connecting men of taste across the millennia, the women and men of the new middle classes of the industrial age were more alive to the otherness, the exotic sensuality, the mystery of the paintings and sculptures they travelled to Italy to see.

It is hard for us to comprehend the rapture these Victorians in their frock coats and high-collared dresses felt in front of the nudity of David. To get a sense of the obsession of 19th-century culture with Renaissance Italy, you only have to look up the name Savonarola in the British Library's digital catalogue. Today, this Ferrarese friar who exerted a charismatic grip on Florentine politics in the 1490s is studied by historians, but is no longer a household name. In the 19th century, by contrast, novels, plays and popular biographies of Savonarola streamed off the presses – books for the many, not the few. One that has endured is George Eliot's Romola (1862-63). To read this novel is to get some insight into the allure of the Renaissance for Victorians.

In 1860 there was not yet any agreed corpus of Renaissance art, so at the Uffizi you could gaze on Leonardo da Vinci's shocking painting of the Medusa – sadly now exiled from his oeuvre. There were none of today's legions of curators and scholars arguing over the attribution of works. The Civilisation of the Renaissance in Italy would ignite the spark of art history as an academic subject – but its greatness as a book lies in its imaginative intoxication. It is not a critique, but the supreme expression of the 19th-century fantasy of the Italian Renaissance.

"Don't be so gloomy. After all, it's not that awful," says Orson Welles as the black marketeer Harry Lime among the bombed wastes of Vienna in The Third Man. "Remember what the fellow said – in Italy, for 30 years under the Borgias, they had warfare, terror, murder, bloodshed, but they produced Michelangelo, Leonardo da Vinci and the Renaissance. In Switzerland, they had brotherly love. They had 500 years of democracy and peace, and what did that produce? The cuckoo clock."

Burckhardt felt the same way – and he was Swiss. Indeed, he is presumably the fellow Welles meant. From his vantage point of a Swiss citizen of conservative politics and modest habits, Burckhardt envied 16th-century Italians their wars and assassinations.

The Civilization of the Renaissance is a disturbing book. It is a vision of modernity – but a dark and haunted one. The first section is titled "The State as a Work of Art". Burckhardt sees the source of the Italian Renaissance in politics, for in the middle ages, while France and Britain centralised their monarchies, Italy resisted control by either the Holy Roman Empire or the Papacy and instead became a barbed collection of micro-states. "In them," Burckhardt argues, "for the first time we detect the modern political spirit of Europe, surrendered freely to its own instincts, often displaying the worst features of an unbridled egotism, outraging every right, and killing every germ of a healthier culture."

This is his theme – the birth in the Italian cities 600 years ago of an "egotism" that begins in politics and war and flows into art and culture and everyday life. In Europe in 1860 it was impossible not to wonder about the origins of the modern world. Life was changing at an unprecedented rate. Factories, railways and the triumph of capital, photography and iron-clad ships erased the immobility of the ages. In 1859 Darwin published his evidence that even nature is defined by ceaseless, unsettling change. In 1867 Karl Marx would publish the first volume of Capital, in which history is a forward movement driven by the engine of class conflict.

Burckhardt, like Darwin and Marx, wrote an epic of turbulence, change, transformation – he found in the Italian Renaissance the very birth of what he saw as the most striking aspects of the modern world. Italians never really knew feudalism, he argued. They had no time for the corporate character of medieval life. The second section of his book is called "The Development of the Individual" and portrays the typical Renaissance man as "the first-born among the sons of modern Europe."

Burckhardt's panorama of the ruthlessness of the Italian despots relies heavily on Machiavelli's writings. Indeed he sees the entire Renaissance through Machiavellian, meaning political, eyes. In contrast to Marx and today's historians of the consumerist "material culture" of the Renaissance, he starts with politics and holds that the development of the Machiavellian state liberated Italian energy. Another source he cites is Francesco Guicciardini, a friend of Machiavelli whose great History of Italy, written in the 1530s, compares with Tacitus for its disabused gloom and which flavours Burckhardt's own cynical melancholy.

The Civilization of the Renaissance in Italy is a classic of modernism. Its discomfort – the abrasive stress on violent change – is akin to the works of art that in Burckhardt's day were at once quoting and mocking the past in an effort to represent the new. In 1863, in Paris, Manet painted Olympia, a portrait of a naked young woman reclining on a bed. Contemporaries saw her as a prostitute and recognised, with shock, that she is imitating the pose of Titian's Venus of Urbino in the Uffizi. Manet's painting is identical in mood to Burckhardt's cultural history. Manet reaches back to the erotic art of the Italian Renaissance to create an ironic, shockingly unsentimental image of his own time. In just the same way and just as provocatively, Burckhardt finds in the schemes of Machiavelli a mirror of the new world of atomised individualism into which his own time was hurled. It is no coincidence that Sigmund Freud, whose unveiling of the unconscious was central to the collapse of Victorian self-confidence, reached back to Burckhardt in writing his own Renaissance study, Leonardo da Vinci and a Memory of His Childhood. For in the Swiss scholar's haunting and eerie masterpiece, there is a madness lurking.