Linné on line Mathematics in Linnaeus’ time ”our honest Klingenstierna” A journey of learning is started On the infinitesimal in Paris

## On the infinitesimal in Paris

Klingenstierna arrived in Paris on about April 1, 1729. His first contact seems to have been with the Swiss mathematician Gabriel Cramer (1704–1752), who was also on an educational tour of Europe. Cramer had spent five months with Johann Bernoulli and several years in London after that. He had now become professor of mathematics at Geneva.

In a later letter to Cramer, Klingenstierna refers to earlier discussions they had about geometry and series. He also shows that the sum of the inverted values of whole-integer squares can be written as an integral in the following way:

if *x* = 1.

He says he is unable to solve the integral.

Euler solved the problem later, but did not do so using elementary methods.

The sum is .

In the letter Klingenstierna remarks that mathematics is boring nowadays. He probably did not have much contact with French mathematicians. According to Strömer in his commemorative speech, there was a meeting with Bernhard Fontenelle (1657–1757), secretary of the French Academy of Science. Fontenelle claims in one of his books that the infinitesimal was something definite and predetermined, which can be obtain by division.

Klingenstierna argues against this. He imagines a rhomb, on which he connected the midpoints of each side. A rectangle is then formed. By connecting the midpoints of each side of the rectangle, another rhomb is created. The process continues. The question is: Is the infinitesimal a rhomb or a rectangle? This can hardly be predetermined, argued Klingenstierna. Fontenelle is said to have agreed.

Klingenstierna left Paris around July 1, 1729, going on to London.