”our honest Klingenstierna”

Samuel Klingenstierna (1698–1765) was the most brilliant Swedish mathematician of the 18th century. He was professor of geometry at Uppsala 1728–1750. When a professorial chair in experimental physics was established in 1750, Klingenstierna became its first holder. A few days after his death, he was briefly described by his friend and colleague Carl Linnaeus.

Carl von Linné (1707–1778)

”So our honest Klingenstierna is dead; he was a meek and even man; nulli gravis [trouble to no one]. Had he worked to his capacity, he could have been 1,000 times greater.  He had an uncannily steady head.”


Samuel Klingenstierna (1698–1765)

Many of Klingenstierna's students also became professors at Uppsala. One of them, Mårten Strömer (1707–1770), delivered the commemorative address about Klingenstierna. Much of what we know about Klingenstierna's life comes from this speech.

Störmer's commemorative address about

Childhood and youth

Strömer says in his commemorative speech:

”Our departed KLINGENSTJERNA was born on August 18, 1698, on Tolefors Estate, in Kjärna Parish, just outside Lindköping. His father, Zacharias Klingenstjerna, Son of the Bishop of Götheborg, Zacharias Klingius, formerly Queen Christina's Court Chaplain, and graced by her with Letters of Nobility for his children, was not at home long enough to see his Son's disposition, or take part in his upbringing. His profession required him to accompany the Swedish Army, wherefore he shared in the glory that our forces earned under the leadership of the Heroic King Charles XII. He never returned; for he left his life [in 1708] on his sickbed, in Saxony, as a Major, having defended it in manly fashion in many epic battles, in the service of his Fatherland.”

His mother, Helena Maria Gyllenadler, died two years later, but Samuel ”was already placed in such good hands that his future fame could be predicted, as his inborn genius developed.” It was the bishop of Linköping, later archbishop in Uppsala, Haqvin Spegel, who took over the education of the future mathematical genius. Early on Samuel learned by heart classical Latin poems by Ovid, Horace, and Virgil. After graduating from upper-secondary school in Linköping he headed for Uppsala and registered at the University on January 11, 1717.

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.

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.

How to dissect an angle is shown in
Proposition 9 in the first book of Elements.

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.”

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
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.

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.”

Klingenstierna's writings

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.

The experimental physicist

We know that from the time he returned from his educational tour Klingenstierna often introduced physics, optics, and mechanics into his lectures. His remarks about the translated book Inledning til Naturkunnigheten (Introduction to Understanding Nature) (1747) revealed his fascination with experimentation, for instance when he describes various experiments with people holding charged glass bottles and giving off sparks. The book concludes with some advice to the reader:

”One must take pains to carry out these trials with the greatest assiduousness and make note of any further remarkable phenomena that might occur.”

When he became our first professor of experimental physics in 1750 he himself had new opportunities to find those further remarkable phenomena.

Klingenstierna had seen to it that the University purchased the most modern equipment for his experiments. In 1738 he spent 6,000 daler in copper coins on a considerable collection of instruments from England. The collection included e.g. a vacuum pump, an electricity machine, a Laterna Magica (a projector of sorts), and a microscope.

The interplay between physics and mathematics that we see in Klingenstierna also bore fruit in another connection. In both his Principia (1687) and his Opticks (1704) Newton had expressed doubt as to whether it was possible to construct a combination of lenses that would eliminate the diffusion of color that occurs in a lens owing to the diffraction of light. Klingenstierna demonstrated mathematically in 1754 how an achromatic optical instrument could be constructed. For this discovery he was awarded a prestigious prize from the Academy of Science in St. Petersburg in 1762.

Royal tutor

After merely two years Klingenstierna was granted a leave of absence from his professorship in physics. He was commissioned to carry out experiments to develop the artillery. One reason for this change in his activities was no doubt the fact that a great misfortune had befallen him at home, and he was in poor health. One of his sons, Zacharias, drowned in the Fyris River in June 1752, and three days later his wife Ulrika died. It is said that, feeling frail and devastated, he visited his father-in-law, but fell ill there and after that felt pain at the slightest touch and was experiencing ”shortness of breath.” In spite of this, when he was summoned to serve at the court in 1756 he answered the call.  Duty called!

There was an attempted coup d’état in Sweden in 1756. They tried to introduce an autocracy with the king as ruler. The coup failed. The Hat Party (the other party being the Cap Party) came into power. The royal advisers were replaced, against the wishes of King Adolf Fredrik and Queen Lovisa Ulrika. Prince Gustaf's tutor, Olof von Dalin, was replaced by Samuel Klingenstierna, a man ”of whom the country can boast regarding the respect of other countries.” At first he was received grudgingly, since he had been forced upon the royal court and the prince.

Klingenstierna was not supposed to teach mathematics and physics primarily. According to the directives, he was chiefly to teach history, ethics, natural and international law, subjects that he himself had encountered during his early studies in law. He now tried to get the crown prince to understand and be interested in Pufendorf's theories, in which the word ”quantitates” had led Klingenstierna to mathematics.  However, the prince was mostly interested in drama and theater.

With time, Prince Gustaf came to be more affectionate toward his tutor. Queen Lovisa Ulrika, in particular, demonstrated her immense gratitude by later erecting a stately gravestone in honor of the crown prince's tutors, Klingenstierna and Dalin.

A further token of the great respect Klingenstierna enjoyed toward the end of his life is the fact that he was awarded the Order of the North Star in 1762 and was appointed State Secretary in 1762

Klingenstierna och Linné

Det berättas, i Th. M. Fries Linné Lefnadsteckning, senare delen (Stockholm 1903), att Klingenstierna och Linné på ålderns höst var mycket goda vänner. Under ett av Klingenstiernas besök på Linnés Hammarby sägs att de båda gick omkring ”endast uti lintyget och byxorna”.

Linné och Klingenstierna i gemytligt samtal på Linnés Hammarby. Konstnär: Fibben Hald (2007)

Death and grieving

Owing to his poor health, Klingenstierna had to give up his position at court in 1764. Even though he was bedridden for long periods of that year, he did not abandon mathematics, especially not geometry. There are many manuscripts preserved in which he attempts to restore theorems that were assumed to have been included in one of Euclid's missing, but talked about books, Porism. A ”definitive” reconstruction of the book was carried out in 1860 by Michel Chasles.

Klingenstierna died on October 26, 1765.

Strömer writes:

”In the evening when he was about to sit down at the table with his beloved daughters, he cried out, invoked the name of Jesus, his Saviour, and fell into the arms of his elder daughter, who had run to his aid, whereupon this precious and rare life was already ended.”


Porism is a book of geometry that according to ancient Greek sources comprised 171 theorems and 38 corollaries. A corollary is an auxiliary theorem that is needed to prove another ”higher” theorem.

Strömer's commemorative address

Every member of the Royal Swedish Academy of Sciences was to have a commemorative address delivered at a meeting of the Academy after his death. Normally, as in the case of Klingenstierna, it was a close friend that delivered the speech. The address was then published in print by the Academy. To facilitate the composition of these speeches, each member was asked to write his autobiography. We do not know whether Klingenstierna had written one.

The grave of a giant

Queen Lovisa Ulrika showed her gratitude to her son Gustaf's tutors, Olof von Dalin and Samuel Klingenstierna, by erecting an enormous stone column over the two men's graves in 1769. It stands at Lovö Church close to Drottningholm Palace.

An attempt at an Opera Omnia

In the manuscripts Klingenstierna left behind, only mathematical problems are treated, some with applications in physics. They comprise several thousand pages, the overwhelming majority of them written in Latin. These were cataloged by his student Fredric Mallet, who planned to publish them in an Opera Omnia, that is, as Klingenstierna's collected works. As a result of war with Russia, however, there was no money for the printing. In Mallet's catalog, according to a plan from 1788, they are divided into ten main groups, including geometry, algebra, integral computation, differential equations, statistics, mechanics, and optics. These, in turn, are divided into subgroups (a total of 292), each of which contains one or sometimes several propositions in mathematics or physics. The great majority of Klingenstierna's remaining manuscripts are preserved today in the manuscript collection at Uppsala University Library.

Klingenstierna's manuscripts

Klingenstierna's manuscripts divided into 10 main groups with a total of 292 subgroups with Mallet's categories:

  • Geometrica (geometry) G. 1 – 67
  • Algebraica (algebra) Alg. 1 – 22
  • De Serierum Summatione (on series sums) S.1 – 27
  • De Fluentium inventione ex data relatione (on integral computation) F. 1 – 21
  • De Æquationibus Fluxionum (on differential equations) N. 1 – 33
  • De Mensura Sortis (on statistics and probability) P. 1 – 8
  • Mechanica et Physica (mechanics and physics) M. 1 – 41
  • De Resistensia Fluidorum (on the resistance of liquids) R. 1 – 8
  • Optica et Dioptrica (optics and dioptrics) D. 1 – 58
  • Miscellanea (miscellaneous problems)  M. 1 – 6
Last modified: 2023-01-18