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Series in London

Two books on infinite series were awaiting publication in London in 1730. The authors were James Stirling (1692 - 1770) and Abraham de Moivre (1667 - 1754).

James Stirling was a Scottish mathematician. His book was titled Methodus Differentialis: sive Tractatus de Summatione et interpolatione Serierum Infinitarum. In it we find, among other things, the so-called Stirling's formula for calculating the faculty n!.

Abraham De Moivre was born in France. He had to flee his country in the 1680s after the Edict of Nantes, which had granted religious freedom to the Huguenots, was revoked. He was involved in many areas of mathematics. Among other things, he developed the theory of probability in his book Doctrine of Chance from 1718. We recognize the name De Moivre's formula from the equality . His book on series was titled Miscellanea Analytica. Klingenstierna's name is found on a list of the few foreign buyers of the first edition of this book.

Klingenstierna was also involved in series in London, but this was a branch of mathematics that he had already encountered with Bernoulli in Basel. A well-known result that bears his name is Klingenstierna's arctangent series (or π-series). This is found in a manuscript dated “Londini d. 7. Aprilis 1730.” The series is (written in modern form):

Actually, it was Robert Simson (1687 - 1768), a professor at Glasgow, who first discovered this series. So it would be more realistic to give Klingenstierna's name to another π-series, which long remained undiscovered in one of his undated manuscripts.

Klingenstierna's arctangent series (or π-series).

Klingenstierna became a member of the Royal Society in 1730, and a year later he published a paper in its proceedings, Philosophical Transactions. The paper deals with a general solution to integrals of certain rational functions, where the nominator cannot be factored.

[Click to open in a new window. PDF file 403 KB]

Klingenstierna left London probably in the late summer of 1730. He was back in Uppsala in October that year.