In its first issue of 2010, the scientific journal Nature looked forward to a dazzling decade of progress. By 2020, experimental devices connected to the internet would deduce our search queries by directly monitoring our brain signals. Crops would exist that doubled their biomass in three hours. Humanity would be well on the way to ending its dependency on fossil fuels.

A few weeks later, a letter in the same journal cast a shadow over this bright future. It warned that all these advances could be derailed by mounting political instability, which was due to peak in the US and western Europe around 2020. Human societies go through predictable periods of growth, the letter explained, during which the population increases and prosperity rises. Then come equally predictable periods of decline. These “secular cycles” last two or three centuries and culminate in widespread unrest – from worker uprisings to revolution.

In recent decades, the letter went on, a number of worrying social indicators – such as wealth inequality and public debt – had started to climb in western nations, indicating that these societies were approaching a period of upheaval. The letter-writer would go on to predict that the turmoil in the US in 2020 would be less severe than the American civil war, but worse than the violence of the late 1960s and early 70s, when the murder rate spiked, civil rights and anti-Vietnam war protests intensified and domestic terrorists carried out thousands of bombings across the country.

The author of this stark warning was not a historian, but a biologist. For the first few decades of his career, Peter Turchin had used sophisticated maths to show how the interactions of predators and prey produce oscillations in animal populations in the wild. He had published in the journals Nature and Science and become respected in his field, but by the late 1990s he had answered all the ecological questions that interested him. He found himself drawn to history instead: could the rise and fall of human societies also be captured by a handful of variables and some differential equations?