After studying at Bern, then in 1729 at Lausanne, König went to Basel in 1730 to study under Johann Bernoulli. After three years of study with Johann Bernoulli he also began to be taught by Daniel Bernoulli. Along with Clairaut and Maupertuis he studied Newton's publications, in particular he did thorough and detailed work on the Principia. In 1731 Hermann left the chair of mathematics at St Petersburg to return to Basel where he was appointed to the chair of ethics and natural law. König studied under Hermann from 1731, in particular he learn of Leibniz's philosophy. In 1735 König went to Marburg to continue his study of Leibniz's philosophy under Christian von Wolff's supervision.
In 1737 König returned to Bern, having begun to publish mathematical articles two years earlier. In Bern he wrote on dynamics, and two articles on this topic appeared in 1738, but he practiced law (which earned him more money than mathematics). Near the end of 1738 König went to Paris where he met Maupertuis. Through Maupertuis, König was introduced to Voltaire and to the Marquise du Châtelet. He began to give the Marquise du Châtelet lessons in mathematics and on the philosophy of Leibniz.
Together with Voltaire and du Châtelet, König visited René-Antoine Ferchault de Réaumur, the leading entomologist of the day who was conducting research in widely varied fields. Through his discussions with Réaumur, König was led to publish a work on the structure of honeycombs. This work was so highly thought of that König was elected to the Paris Academy of Sciences.
Soon after this König fell out with the Marquess du Châtelet and, although the reason is not certain, it has been suggested that they argued over payments for the lessons that König was giving to the Marquess. Certainly König did not leave Paris after his disagreement but continued to live there for about 18 months. After returning to Bern, he continued to earn his living from his law practice but also continued to study mathematics, in particular studying the works of Clairaut and Maupertuis. König wrote a work on the shape of the Earth, which was published in 1747, based on what he had learnt from studying Clairaut's Théorie de la figure de la Terre Ⓣ and other works on this topic.
In 1744 König was exiled from Bern for 10 years because he signed a liberal petition. König obtained a chair of philosophy and mathematics at Franeker in the Netherlands. Then, in 1749, he went to The Hague where he was librarian to Prince William IV of Orange.
Maupertuis proposed König for the Berlin Academy and he was honoured by election to the Academy in 1749. However, he was to cause a split in the Academy through an essay, written before 1749 but published in 1751, on the principle of least action. This work attacked Maupertuis accusing him of having plagiarized Leibniz's work on this principle. König had touched on a sore point with the Berlin Academy for a number of reasons. Firstly the Academy had just been involved in a dispute on the prize question for 1746, which Euler had made it clear required the entrants to criticise the philosophy of Leibniz and Wolff. Euler was a strong opponent of the Leibniz and Wolff philosophy. Also for König, as a newly elected member of the Academy, to be seen to be attacking its President was not going to be well received.
König had studied the philosophy of Leibniz under Wolff so he was a strong follower of their philosophy. He claimed in his 1751 article that Leibniz had stated the Principle of Least Action in a letter of 1707 to Jakob Hermann. A heated dispute followed the publication of König's article and he was asked to produce the letter. However König failed to do so and, after an investigation, Euler gave a report on the affair accusing König of fraud.
As a result of the dispute Maupertuis was greatly upset and soon left the Berlin Academy. König's article was to have other repercussions as Voltaire, d'Alembert and Euler all ended at the centre of the dispute. König lived for six years after his article causing :-
... the ugliest of all scientific disputes.These last six years of his life were completely dominated by the dispute.
An important kinetic law of energy, published in a work of 1751 (the same year as his controversal article appeared), is named after König and this law is a major contribution which he made to mathematics. König's character and role in science is summed up in  as follows:-
A candid and amiable man, he was distinguished by erudition of unususl breadth even for his time. ... The opinion is occasonally voiced that were it not for the controversy over the principle of least action, König would be completely forgotten in the history of science. His formulation of the law (named after him) of the kinetic energy of the motion of a mass point system relative to its centre of gravity is sufficient in itself to refute this view.
Article by: J J O'Connor and E F Robertson