Studies at Uppsala
Klingenstierna started studying law at Adjunct Castovius' lectures. But it happened once that ”during the Lesson he and one of his acquaintances were noisy.” He was reprimanded by the lecturer. This ”hurt him so much that he vowed never to go there again.” Nevertheless, the study of law led him to the study of mathematics. It is said that he found it difficult to grasp the term “quantitates” in a work on natural and international law by Samuel Pufendorf. He was recommended to read Euclid's Elements to find a comprehensive definition of the concept of quantity. He is said to have studied Gestrinius' interpretation of Elements from 1637.
Klingenstierna had no difficulty understanding Elements, but he was amazed that Euclid demonstrated how to divide an angle in two parts down the middle but not how to divide the angle into three equal parts. It was proven in the 19th century that it is indeed impossible to divide any given angle into three equal parts.
When Klingenstierna posed this question, “there was no professor with whom so curious a youth could take refuge. For both Matheseos Professors, Elvius and Wallerius, had died, in 1718. On the other hand, there was a disciple of Professor Elvius living just outside Upsala, Master Anders Gabriel Duhre, who was a famed Mathematicus, at that time, and an assiduous champion of the discipline.”
Duhre recommended Klingenstierna to study the new mathematics, especially Analyse Demontrée by Charles Reyneau, which was the first textbook on both differential and integral calculus. This is a book that was also studied by Swedenborg.
It is said that he shut himself up in his chamber for six months in order to learn the new calculus. When he returned to his social life, his friends wondered whether he would now become Duhre's student. Klingenstierna is said to have replied that he “need not become Duhre's disciple, as he was already further advanced.”