Professor of mathematics

Klingenstierna found that his return had been eagerly awaited. On October 15, 1730, he delivered his introductory address at the University. During the spring of 1731 he served as preses for six dissertations. One of them was written by Mårten Strömer, De Arte Conjectandi, treating the theory of probability that appears in Jacob Bernoulli's work by the same name.

In the catalog of lectures we can follow what was planned for Klingenstierna's teaching for each year. It is remarkable that infinitesimal calculus is only planned as part of instruction during the years (1729 and 1730) when Klingenstierna was not in Uppsala.

In Uppsala University's Catalog of Lectures, which tells what lectures were planned for each coming year, we can find Klingenstierna's planned teaching during his tenure as professor of mathematics 1729–1750. Instruction was divided into Public Lectures, that is, open, and Private Instruction (usually in the home). Private instruction usually provided a good supplemental income.

	Public:	Private:
1729	Not determined	Infinitesimal calculus	Not carried out
1730	Not determined	Infinitesimal calculus	Not carried out
1730	Klingenstierna in Uppsala
1731	Euclid's Elements	Natural philosophy (actually experimental physics)
1732	Plane trigonometry and construction of sinus and logarithm tables	Not determined
1733	Elements of conic sections	Not determined
1734	Elements of conic sections	Not determined
1735	Continued study of algebra and geometry	Not determined
1736	Applications of algebra and geometry in mechanics	Not determined
1737	Selected examples of applications of algebra and geometry	Not determined
1738	On conic sections and Euclid's Elements	Elementary analysis of both algebra and geometry
1741	Remaining parts of Euclid's Elements, selected parts of Archimedes' propositions, properties of conic sections	Experimental physics
1742	Klingenstierna was vice-chancellor of the University for half a year
1742	Geometric loci (curves with a certain property), both synthetic and analytic	Not determined
1743	Geometric loci, conic sections continued	Elementary algebra, mechanics and optics with experiments
1744	Euclid's Elements	Algebra, General physics, and plane trigonometry
1745	Explanation of Euclid continues, plane trigonometry, and about geometric loci	Algebra, physics
1746	Euclid's Elements, Analysis of geometry in both the ancient and algebraic manners.	Statics and mechanics together with the theory of geometric loci
1747	Plane trigonometry, elementary optics, catoptrics and dioptrics	Algebra, physics, and geometry
1748	Elementary algebra as presented in Introduction to Algebra by Palmquist	Euclidean geometry and the theory of infinite series
1749	Continued explanation of Palmquist's Algebra	Not determined
1750	Mechanics	Continuation of general physics, optics

As we can see, Klingenstierna had a great penchant for physical applications. It is not surprising that he became the first professor of experimental physics in 1750.

[...] eagerly awaited

Mårten Strömer says:

”Finally Uppsala had the pleasure of regaining its eagerly awaited Klingenstjerna.  Word of his imminent return preceded him, of the honour and respect he represented out among the greatest Mathematici then living. … I recall the change, not without emotion, when I think of how the Auditorium, previously attended by two, three, or four, now was filled with people.”