A journey of learning is started

In 1720 Klingenstierna was employed at the Royal 'Kammarkollegium' [now the Swedish Legal, Financial, and Administrative Services Agency] in Stockholm as a clerk. But this did not mean he abandoned mathematics. In his spare time he studied Newton, Leibniz, Huygens, the Bernoulli brothers, l’Hospital, and many others.

_{Guillaume de l’Hospital's Analyse des Infiniment Petits (published in 1696) is the first textbook on differential calculus.}

He was also commissioned to review texts in the newly started scientific journal Acta Literaria Suecia, which was published by Bokwettsgillet (Society of Book Knowledge) in Uppsala. This shows that he already enjoyed a good reputation in scientific circles in Sweden.

Back in Uppsala in 1725 he probably taught the new calculus at Duhre's school of theory and practice.

In 1727 he was awarded the Helmfelt travel scholarship, which gave him an opportunity to undertake an educational tour of the scientific centers of Europe. He first went to Marburg and the tutelage of the philosopher-mathematician Christian Wolff in December 1727. Wolff had written several textbooks in mathematics, logic, and mechanics, and many students from all over Europe traveled to Marburg to learn from him.

_{Wolff wrote some of the first textbooks on differential calculus. Elementa Matheseos first appeared in 1713.}

During his time in Marburg Klingenstierna wrote a dissertation on third-degree curves, elaborating some of Newton's theories from 1704. On the strength of this and a recommendation from Christian Wolff, Klingenstierna was appointed professor of mathematics at Uppsala after Elof Steuch in August 1728. He was allowed to continue his educational tour and now also had a professor's salary.

 
_{Stations on Klingenstierna's journey 1727–1730. From the collections of Uppsala University Library.}

As Leibniz, Newton, and Jacob Bernoulli were dead, Johann Bernoulli was the man people turned to if they wanted to understand the mysteries of calculus. It was common, therefore, for young mathematicians traveling around Europe for their education to go to Basel to be taught by Johann Bernoulli. Klingenstierna arrived there in September 1728, just a few weeks after being appointed professor.

Johann Bernoulli wrote in a letter to Johann Jakob Scheuchzer in late October 1728:

_{”At present the professor of mathematics at Uppsala M. v.
Klingenstierna is here studying with me. He has travelled
so far merely to benefit from my feeble light, although,
truth to tell, he already has an excellent understanding
of the most sublime geometry, so I know not whether rumour
has lied about me, to bring him all the way from his
northern country.”}

_{But Klingenstierna t}ruly did benefit from his time with under the care of Johann Bernoulli. A great many of his manuscripts derive from his time in Basel. Many of them are copies of Bernoulli's writings. Bernoulli also gave him problems to solve. As his teacher, Bernoulli then corrected Klingenstierna's manuscripts.

_{This manuscript describes the motion of a body in liquid under special conditions. We can see how Johann Bernoulli corrected Klingenstierna's text, especially on the second and third pages.}

Many of the problems Klingenstierna grappled with have to do with the motion of bodies, for example in a liquid resisting this motion. This often leads to problems in the so-called calculus of variations. A common problem in differential calculus is to find the maximum and minimum values for a given function (or curve). In calculus of variations, you look for a function (or curve) that has given maximum and minimum points.

The so-called brachystochrone problem is a motion problem that had previously been solved for motion in a vacuum. It attracted some attention that Klingenstierna solved the problem in Basel, but now under the condition that the body was falling in a liquid or the like, which would offer resistance to the motion.

A few years after Klingenstierna, Euler also solved the expanded brachystochrone problem.

Also important to Klingenstierna during his sojourn in Basel was his meeting other mathematicians, especially Johann's nephew Nikolaus Bernoulli. Klingenstierna's more anglophile contacts would later bring him to London. After six months in Basel, however, he first went to Paris.

The brachystochrone problem

The brachystochrone problem was presented as a challenge to mathematicians throughout Europe by Johann Bernoulli in 1696. The problem is to find the fastest path for a body to fall from one point to another point at a lower level. In the original problem, gravity was the only force affecting the body. Among others, Jacob Bernoulli, l’Hospital, and Leibniz submitted solutions. Newton submitted anonymously. Johann Bernoulli is said to have remarked about this solution: “You can tell the lion from his claws!” The curve in the solution is a so-called cycloid curve.

The new condition in Klingenstierna's problem was that the resistant force of liquid also affected the motion of the body. The solution could thus be given as a differential equation whose appearance was dependent on whether the resistant force was set as being proportional to the velocity, the velocity squared, or a higher power of the velocity.

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.

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.

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