Linné on line 
Mathematics in Linnaeus’ time "our honest Klingenstierna" Professor of mathematics
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.