Born: 10 March 1622 in Zürich, Switzerland
Died: 25 May 1676 in Zürich, Switzerland

Johann Rahn's father was Hans Heinrich Rahn (born 7 July 1593, died 21 September 1669) and his mother was Ursula Escher vom Glas (born 1591, died 28 April 1663). Hans Rahn was Bailiff to Töss at the time his son Johann was born and later (1655-1669) he was Burgermeister of Zürich. At this time Töss was a town in its own right but today it is incorporated into the city of Winterthur; it is situated north east of Zürich. In fact when Johann was born his grandfather Hans Rudolf Rahn was Burgermeister of Zürich. Hans Heinrich and Ursula had four children, two boys and two girls: Hans Conrad (born 1616) was Johann's elder brother, with Regula (born 1628) and Ursula (born 1635) being his two younger sisters. Johann explains in his book that in his youth he enjoyed opportunities for study both in Switzerland and abroad. It is thought that Johann Rahn's interest in mathematics came through his uncle Johann Georg Werdmüller who was an engineer. In 1642, with Europe in the middle of the Thirty Years War, the city of Zürich decided to construct fortifications. Various proposals were evaluated and Werdmüller's proposal was accepted. This was also the time when Johann Rahn married Elisabeth Holzhalb (born in 1626); they had two children Hans Heinrich (born 9 March 1646) and Hans Rudolf (born 18 April 1669).

Johann Rahn followed in the tradition of having major roles in the administration of the city of Zürich. He [5]:-

... was appointed a 'Schützenmeister', which meant that he supervised shooting practice, and a 'Zeugherr', responsible for military supplies and artillery.

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. In May 1654 Pell arrived in Zürich, sent there by Oliver Cromwell on a diplomatic mission. Cromwell wanted to split the Protestant cantons of Switzerland off so they might join a Protestant League with England at its head. 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.

Pell's negotiations in Zürich were long drawn out and he remained there for a number of years. In early 1657 Rahn began to receive regular tutoring from Pell. John Aubrey explained in Brief lives that Pell had told him that [3]:-

Rahn was Dr Pell's scholar at Zürich, and came to him every Friday night.

Pell, in a letter to Thomas Brancker dated 5 March 1666, describes Rahn as his 'disciple'. He writes that he kept:-

... copies of the most considerable papers that he wrought in my presence or that I gave him to transcribe.

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. In June 1658 Pell returned to England to give his report to Cromwell only shortly before Cromwell's death. 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. He writes in the Preface that [4]:-

... in the preceding summer he met at the watering place, Taynach, a nobleman, Leonard Weiss, with whom he discussed algebra and to whom he promised to prepare a work in the German language which should contain the advances due to Vieta, Descartes and others, which were at that time accessible only in the Latin and French languages. He says that he took his problems partly from Vieta, Descartes, van Schooten, Diophantus and Clavius.

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 book, written in German, contains an example of Pell's equation. 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. Both were modest men so we must not necessarily take their comments too literally, but nevertheless the credit for much of the mathematical innovation must be due to Pell. Rahn writes in the Preface that:-

... in the solutions, and in the arithmetic too, I make use of a completely new method, which has not been used by any writer on algebra in a published work, that I first learned from an eminent and very learned person to whom I should very gladly acknowledge indebtedness and humble respect, had he permitted.

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. We see plenty of evidence of Pell's humble attitude below when discussing the English translation of the famous text.

The German-born Theodore Haak, a scholar and translator of wide scientific interests, gave a copy of Rahn's book to Pell in 1660. Pell was very impressed by the work and thought that an English translation should be made to provide a more modern algebra book than William Oughtred's Clavis Mathematicae(1631) which was still the standard English algebra text. 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. He then completed his translation and deposited the manuscript in the library in Zürich in 1667. His Latin translation had the title Algebra Speciosa seu Introductio in Geometriam Universalem 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.

In fact Thomas Brancker had completed his English translation by May 1665 and it was in the hands of the publisher when John Collins asked him to delay publishing until he had discussed the matter with Pell. After meeting with Brancker and his publisher, Pell said (reported in a letter by Brancker):-

He hoped to be at leisure, to review some of Rahn's Problems, and to work them anew; and that he would send them to me, with leave to publish them or to keep them by me.

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. Pell wrote to Haak on 13 June 1666:-

I wish you could ... learn from Zurich, 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. You gave me his book in November 1660. I suppose you know that it is turned into English, and that 15 sheets of it are printed. Some mention should be made of him, if we can get sufficient information concerning him.

Haak clearly did not manage to find out anything, for no further information about Rahn was inserted when An Introduction to Algebra was published. Pell was in two minds as to whether his name should appear as an author. He wrote (but never sent) a letter which reads [2]:-

I know not how my mind may alter but for the present I think it best not to name me at all in the title or preface: and yet you may be more ingenuous than Rahn was and not vent all for your own devices. You may say that the alterations and additions were made by the advice of one of good reputation in those studies.

In fact when the book appeared, the front cover stated "Translated out of the High-Dutch into English by Thomas Brancker M.A. much altered and augmented by D.P." The "D.P." stands for "Doctor Pell". 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. On 17 May of that year Rahn wrote to Haak (see for example [5]):-

Quite some time ago, relying on the training I received from Mr Pell (to whom alone I am indebted for whatever I know in algebra), I reduced and arranged the problems of Diophantus of Alexandria in algebraic form. Recently I have also finished a treatise on optics, catoptrics, and dioptrics, together with an appendix on practical dioptrics, showing the method (and the necessary instruments) for grinding and polishing glass lenses. These things, indeed, became clear to me not on the basis of mere theory, but on the basis of actual practice and use, when I was repeatedly preoccupied by the desire not only to know how to construct large and small telescopes, and microscopes, but also - above all - when I found a method (to do with the internal part or middle circuit of the wheel) of making hyperbolic sections, thanks to which objects are rendered very clear, and the field of vision is extraordinarily enlarged.

In the above Rahn refers to three treatises he has written. The first on the problems of Diophantus was entitled Solutio Problematum Diophanti Alexandrini. He completed it in 1667 and intended it as a companion volume to the Latin translation he had made of his Teutsche Algebra. Quite independently, both the English translation and Latin translation had doubled in length. The other two treatises on optics referred to by Rahn are Tractatus von der Dioptrica oder Durchstrahlung and Catoptrica oder Gelbstrahlung in ebenen Flächen but sadly neither has survived.

On 17 April 1675, Rahn wrote to Pell. By this time he was a Senator of Zurich, a Judge and President of the Criminal Court [5]:-

I have received the Algebra - no longer 'my' Algebra, but rather, on many accounts, yours - which, thanks to your generosity, was sent to me by a certain scholar; from it, I gather that my memory is still flourishing with you, after the passing of so many years. When you were worthily fulfilling the role of English Resident, you conferred so many favours on me, so often, that whenever they come to mind, it causes me no slight pain to think that I have no ways of repaying them that would suitable to my obligations - all the more so, since they are so augmented by this favour of a new kind, that I shall never be capable of deserving them.

The letter goes on to explain about the optical research he undertook before his public duties became so onerous that he had no time left for research.

Books:

1. N Malcolm, C Cavendish, J A Stedall and J Pell, John Pell (1611-1685) and His Correspondence with Sir Charles Cavendish: The Mental World of an Early Modern Mathematician (Oxford University Press, Oxford, 2005).
2. J A Stedall, A Discourse Concerning Algebra : English Algebra to 1685 (Oxford University Press, Oxford, 2002).

Articles:

1. J Aubrey, John Pell, Brief lives II (Oxford, 1898), 121-131.
2. F Cajori, Rahn's Algebraic Symbols, Amer. Math. Monthly 31 (2) (1924), 65-71.
3. N Malcolm, An Unpublished Letter from Henry Oldenburg to Johann Heinrich Rahn, Notes Rec. R. Soc. Lond. 58 (3) (2004), 249-266.
4. C J Scriba, John Pell's English Edition of J H Rahn's Teutsche Algebra, in R S Cohen et al (eds.), For Dirk Struik (Reidel, Dordrecht, 1974), 261-274.