The sum of the infinite number sequence 1, 1/2, 1/4, 1/8, … tends toward the value 2, something that can easily be guessed by adding the first few entries of the series. But this should not lead you to believe that every infinite number sequence whose entries decrease in value has a finite sum. The so-called harmonic series, for example, which starts as 1 + 1/2 + 1/3 + 1/4 + 1/5 + … tends toward infinity. It does so very slowly: No less than 178 million entries must be added to arrive at the sum of 20. In mathematical jargon one says that the harmonic series diverges. Infinite number sequences whose entries add up to a finite sum are called convergent.

During the Enlightenment, number sequences and their sums were considered important areas of research. In 1644 the 19-year-old student Pietro Mengoli from Bologna, who later became a priest and professor of mathematics, asked whether the sum of the sequence consisting of reciprocal squared numbers (1, 1/4, 1/9, 1/16 …) converges and, if so, toward which value.

Over the years Mengoli amassed vast experience working with infinite number sequences. It was he, for example, who proved that the harmonic series diverged but that the alternating harmonic series—where the entries are added and subtracted on an alternating basis—converges toward 0.6931. But he had no answer for the series of reciprocal squared numbers. He had the suspicion that the sum approaches the approximate value 1.64, but even that he was not entirely sure about.

Some years later the mathematician Jakob Bernoulli, from the Swiss town of Basle, caught wind of this mysterious number sequence. The scientist, famous throughout Europe for his mathematical abilities, could not find

a solution either. Frustrated, he penned a notice in 1689 in which he wrote: “If anybody finds out anything and is good enough to inform us, we would be very grateful.”

At the turn of the 18th century, European intellectuals were absolutely fascinated by this particular problem. The number sequence and the mystery surrounding it became chosen topics of conversation in the salons of the social elite. Soon it was considered as significant as the Fermat problem, which by then was already 50 years old. Several mathematicians, among them the Scotsman James Stirling, the Frenchman Abraham de Moivre, and the German Gottfried Wilhelm Leibniz, cut their teeth on it. In 1726 the problem returned to Switzerland, to its hometown Basle.

Jakob Bernoulli’s brother Johann, who was a well-known mathematician in his own right, had an exceptionally gifted student named Leonhard Euler, also from Basle. Euler was regarded as a rising star in the world of mathematics. To encourage him, Johann asked him to work on the problem. Due to its connection with mathematicians from Basle, the problem of reciprocal squared numbers became known as the Basle problem.

Euler spent many years working on the problem, sometimes putting it aside for months only to pick it up again. Finally, in the autumn of 1735 he believed he had found the solution. Nearly half a century after Pietro Mengoli had first thought about this number sequence, Euler stated that the value of the sum, calculated to the sixth digit, was 1.644934.

What led him to this solution? Surely he did not just add up the entries of the sequence. To calculate the sum to only five digits, Euler would have had to consider over 65,000 numbers. Obviously, the Swiss mathematician had guessed the exact value of the sum—which turns out to be π squared, divided by 6—even before he was able to prove it.¹ For a while Euler refused to announce the solution to the public, since even he was surprised by the

result. What on earth does π, the proportion of the circle’s circumference to its diameter, have in common with the sum?

With the publication of De Summis Serierum Reciprocarum a few weeks later, Euler provided the proof to his assertion. In it he wrote that he had “quite unexpectedly found an elegant formula for 1 + 1/4 + 1/9 + 1/16 which is dependent on the squaring of the circle!” Johann Bernoulli was at once startled and relieved. “Finally my brother’s burning desire has been fulfilled,” he said. “He had remarked that investigating the sum was much more complex than anyone would have thought. And he had openly admitted that all his efforts had been in vain.”

The solution was unexpected because Euler had chanced on it while studying trigonometric functions. The so-called series expansion of the sine function is closely related to the reciprocal squared number series. And since trigonometric functions are related to circles, the number π occurs as part of the solution.

Euler’s proof established a relationship between number series and integral calculus, which at the time was still a young branch of mathematics. Today, it is well known that the Basle number series represents a special case of a more general function (the zeta function), which plays a significant part in modern mathematics.