**Johann Lambert** was the first to provide a rigorous proof that $\pi$ is irrational.

- Heinrich attended school in Mulhouse, receiving a reasonably good education up to the age of twelve, studying French and Latin in addition to elementary subjects.
- Most young boys would have ended their education at that point, but not young Heinrich who continued to study in his own spare time.
- It was a natural occupation for Heinrich since he had acquired great skill in calligraphy and he was given a job at the ironworks at Seppois, which was south of Mulhouse and almost due west of Basel.
- When he was seventeen years old Lambert left his position at the ironworks to take up a post as secretary to Johann Rudolf Iselin who was the editor of the Basler Zeitung, a conservative daily paper.
- This position was ideal for Lambert who could now concentrate even more deeply on his own study of mathematics, astronomy, and philosophy.
- In 1748, when he was twenty years old, Lambert took up a new position, this time as a tutor in the home of Count Peter von Salis in Chur.
- Lambert could now use the excellent library in the Count's home and was in an even stronger position to continue his studies of mathematics, astronomy, and philosophy.
- It was in Chur that Lambert first came to be noticed by the scientific community.
- In 1756 Lambert left Chur with the two older boys whom he had tutored during the previous eight years; they were now 19 years old.
- There Lambert met Kästner and Tobias Mayer, and was elected to the Learned Society of Göttingen.
- The French and Austrians began to get the upper hand in 1757 and occupied Göttingen while Lambert was studying there.
- Lambert's first book, which was on the passage of light through various media, was published in The Hague in 1758.
- Before returning to Chur, Lambert took his pupils to Paris, where he met d'Alembert, and to Marseilles, Nice, Turin, and Milan.
- The exponential decrease of the light in a beam passing through an absorbing medium of uniform transparency is often called 'Lambert's law of absorption', although Bouguer discovered it earlier.
- 'Lambert's cosine law' states that the brightness of a diffusely radiating plane surface is proportional to the cosine of the angle formed by the line of sight and the normal to the surface.
- Lambert was asked to organise a Bavarian Academy of Sciences in Munich along the lines of the Berlin Academy, but he fell out with other members of the project and left the new Academy in 1762.
- Lambert's book is also remarkable for the modernity of its methodological stand: his systematic survey of the differences among facts, theories, predictions and possible verifications was not emulated in cosmological literature until the 20th century.
- After returning from Munich, Lambert took part in a survey of the border between Milan and Chur, and also visited Leipzig where he was able to find a publisher for a work on philosophy Neues Organon (published in 1764).
- Lambert was therefore delighted to go to Berlin in 1764 at the invitation of Euler.
- However, although Lambert joined the Huguenot Church, of which Euler was a staunch member, differences between the two men soon arose, mainly concerning the income of the Academy, which depended on its privilege to sell calendars.
- In 1766 Lambert wrote Theorie der Parallellinien Ⓣ(The theory of parallel lines) which was a study of the parallel postulate.
- Above all, Lambert carefully considered the logical consequences of these axiomatically secure principles.
- Lambert's physical erudition indicates yet another clear way in which it would be possible to eliminate the traditional myth of three-dimensional geometry through the parallels with the physical dependence of functions.
- A number of questions that were formulated by Lambert in his metatheory in the second half of the 18th century have not ceased to remain of interest today.
- Lambert is best known, however, for his work on π.
- However Lambert was the first to give a rigorous proof that π is irrational.
- Wallisser shows that Lambert's proof is not only complete but is an outstanding mathematical achievement for its time.
- It was Pringsheim in 1898 who first noted that Lambert's proof was absolutely correct and exceptional for its time, since the expansion of the tangent function was not only written down formally, but also proved to be a convergent continued fraction.
- Also, remarkably, Lambert conjectured in this paper that eee and π are transcendental.
- Lambert also made the first systematic development of hyperbolic functions.
- Lambert is also important for his study of the trigonometry of triangles on surfaces, his work on perspective and cartography, as well as his contributions to the theory of probability.
- Lambert's vast, multifaceted activity covered optics, cosmology and geodesy and put him in contact with the major scientists and philosophers of his day (Kant).
- In his own fundamental philosophical opus, the "Neues Organon", Lambert developed a noteworthy theory of logical probability that, to our knowledge, has thus far escaped the attention of eminent scholars in the field such as Keynes.

Born 26 August 1728, Mülhausen, Alsace (now Mulhouse, France). Died 25 September 1777, Berlin, Prussia (now Germany).

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**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive