**Johann Rahn** was a Swiss mathematician who was the first to use the symbol ÷ for division.

- Johann explains in his book that in his youth he enjoyed opportunities for study both in Switzerland and abroad.
- Johann Rahn followed in the tradition of having major roles in the administration of the city of Zürich.
- Now Rahn was certainly interested in mathematics, and he was carrying out small investigations, but it was only after he came in contact with John Pell that he developed a deeper interest and considerably more expertise.
- We do not know exactly when Rahn and Pell first came in contact but, given their respective roles, it is likely to have been soon after Pell arrived in Zürich.
- The first definite information is a letter which Rahn sent to Pell, dated 4 November 1654, in which he thanks Pell for sending him his publication, the two page A Refutation of Longomontanus's Pretended Quadrature of the Circle (1644).
- Rahn also enclosed a piece of his own mathematics.
- In early 1657 Rahn began to receive regular tutoring from Pell.
- Pell, in a letter to Thomas Brancker dated 5 March 1666, describes Rahn as his 'disciple'.
- The tutorials came to an end in early 1658 when Rahn was appointed as governor of the Kyburg district, about 20 km north east of Zürich.
- Once he was settled into his new role, Rahn wrote to Pell explaining that his duties as 'Landvogt' kept him very busy and this meant that he would not have time to devote to his mathematical studies.
- However, although Pell did not realise it, Rahn was finding more time to study mathematics than he expected from his earlier letter, for in 1659 he published his famous text Teutsche Algebra, oder algebraische Rechenkunst, zusamt ihrem Gebrauch Ⓣ(Examples in algebra and arithmetic, together with their application).
- Rahn was the first to use the symbol ÷ for division in Teutsche Algebra, a symbol which Pell had probably used when giving Rahn tutorials.
- The text is important for the innovation in algebraic symbolism that it contains but exactly how much credit is due to Pell and how much to Rahn is hard to determine.
- Clearly this refers to Pell, but the fact that Pell is not mentioned by name is almost certainly for the reason stated; because Rahn was aware that Pell would not wish to see his name in lights.
- The German-born Theodore Haak, a scholar and translator of wide scientific interests, gave a copy of Rahn's book to Pell in 1660.
- Two translations into English of Rahn's text were started, neither aware that the other was happening, and at the same time Rahn was translating his German text into Latin.
- Rahn left Kyburg in 1664 and returned to Zürich to take up the post of Secretary-Councillor there.
- His Latin translation had the title Algebra Speciosa seu Introductio in Geometriam Universalem Ⓣ(Special introduction to geometry or univesal algebra) and in the Preface Rahn explains that he chose not to publish the translation since he had, by this time, learnt that an English translation was about to be published.
- Pell modified a small part of Rahn's book but also greatly expended it to about twice its original size.
- Before it was published he tried to find out up-to-date details of Rahn.
- learn from Zürich, concerning Johann Heinrich Rahn, lately Landvogt der Graffschaft Kyburg, whether he be yet alive, whether he be now at Zürich, what titles or offices he hath now.
- Haak clearly did not manage to find out anything, for no further information about Rahn was inserted when An Introduction to Algebra was published.
- Rahn's name does not appear on the title page, but he is mentioned in the Preface.
- Although Rahn knew something of an English translation, Rahn did not resume contact with British mathematicians until 1671.
- In the above Rahn refers to three treatises he has written.
- The other two treatises on optics referred to by Rahn are Tractatus von der Dioptrica oder Durchstrahlung Ⓣ(Tract on dioptrics and radiation) and Catoptrica oder Gelbstrahlung in ebenen Flächen Ⓣ(Catoptrics or reflections in plane surfaces) but sadly neither has survived.
- On 17 April 1675, Rahn wrote to Pell.
- Finally let us mention that when Brancker translated Rahn's Teutsche Algebra he changed the notation (probably to make it easier to print) and so the division sign ÷ had been replaced.
- When Pell came into the project to make his additions and corrections, he insisted that the division sign be reinstated as Rahn had written it.

Born 10 March 1622, Zürich, Switzerland. Died 25 May 1676, Zürich, Switzerland.

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Algebra, Origin Switzerland, Number Theory

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