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

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

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

... 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.

... 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.

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.

I wish you could ... 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. 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.

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.

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.

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.