One reason Klingenstierna is nearly forgotten as a mathematician is his unwillingness to see his writings into print. He took part in the publishing of two books, a Latin version of Euclid's Elements in 1741 and six years later an annotated translation of Peter van Musschenbroek's Elementa Physicawith the Swedish title of Inledning til Naturkunnigheten (Introduction to Understanding Nature).
In Klingenstierna's foreword to Inledning till Naturkunnigheten we can read:
”I am certain that Mister Musschenbroek would not begrudge that in some of my comments on this lovely work I am of another opinion than he, in as much as I have regarded his Physica as the Principal Book of its kind, the Swedish Translation of which could in the highest degree benefit and edify not only the Youth of the Fatherland but also more mature persons, of all ranks and circumstances.”
Klingenstierna wrote 20 articles in scientific journals. Roughly half of them were written in Swedish for the proceedings of the Royal Swedish Academy of Sciences (abbreviated KVA in Swedish).
In Mathematiskt Spörsmål, om en kroklinie, som återförer en ljusstråle, efter tvänne reflexioner til, des ursprung (Mathematical matters, on a curved line that returns a ray of light, after two reflections, to its origin) (KVA's proceedings 1749), Klingenstierna solves a problem that Euler had presented in 1745 in Acta Eruditorum. There were other mathematicians who had solved the problem, but they had used, as Klingenstierna says, ”Calculatory methods.” Instead, he demonstrates a solution ”that merely by looking at the figure, without calculation, goes straight to the purpose.”
He then explains why he prefers geometric solutions to algebraic ones.
Under Klingenstierna's presidency just over 70 dissertations were printed and ventilated during the years 1731–1752, many of which he had written himself. These dissertations were primarily in the realms of mathematics and physics, but several of them address philosophical matters.
Why Klingenstierna preferred geometric solutions to algebraic ones
”To be sure, Algebraic calculations are a secure Ariadne's string, but, as I see it, they should only be used when one find's oneself in a Labyrinth from which one sees no other exit. The Geometric path, when one can take him, is always light; one sees the destination one wishes to reach throughout the entire journey, and how each step contributes to its attainment. But with the Algebraic path one often has the displeasure of so losing sight of one's subject that one hardly sees more than a shadow of it represented in symbols and formulas. On the other hand, one must grant this latter method the advantage of being more wide-ranging than the Geometric, which must often stand aside as it moves by. Each has its advantages, and it behooves the master to make use of each in its place.”