One of my interests is the development of scientific publishing following the invention of moving type printing in the middle of the 15^{th} century. The first real maths book, that is excluding reckoning books which are textbooks for commercial arithmetic, to be printed and published in German was written by the artist Albrecht Dürer who was born 540 years ago on the 21^{st} May 1471.

Born in Nürnberg the son of a Hungarian goldsmith he originally started to learn this trade with his father but then apprenticed to the leading Nürnberger artist Martin Wolgemut. As a journey man painter and printmaker he made his first journey to Italy from 1494 to 1495 where he learnt much about the new Renaissance style of Italian art and in particular the then still relatively new method of linear perspective. Dürer became convinced that this mathematically based style of composition was the true core of art and returned to Italy from 1505 to 1507 specifically to learn more about this technique.

Dürer devoted his artistic life to the study of linear perspective and proportion and to developing the mathematical knowledge necessary to carry out this type of drawing. He was one of the people most responsible for introducing this technique into Northern Europe. Convinced that the secret of true beauty lies in the theory of mathematical proportions he devoted twenty years to writing his major thesis on the subject his *Hierinn sind begriffen vier bücher von menschlicher Proportion* (Four Books on Human Proportion) printed and published posthumously in Nürnberg in 1528. When this book was almost complete Dürer realised that the book was much too mathematically advanced for apprentice artists, its targeted readership, and so he compiled an elementary geometry textbook, his *Underweysung der Messung mit dem Zirkel und Richtscheyt* (Instruction in Measurement with Compass and Straightedge), which was printed and published in Nürnberg in advance of his magnum opus, to which it was supposed to provide an introduction, in 1525.

As already mentioned this was the first maths book to be printed and published in German, all previous mathematical works having been printed in Latin. Although aimed at apprentices of art the book found a much wider audience and went on to become a scientific bestseller not only in Germany. It was translated into Latin and several major European languages and became a standard textbook throughout Europe for most of the next century. It was still actual at the beginning of the 17^{th} century as Kepler was writing his geometrical works and he criticises Dürer’s construction of the heptagon as only an approximation and not mathematically precise. One of Dürer’s illustrations from the book on how to construct linear perspective is thought to have inspired Kepler to his experimental solution of the pinhole problem in optics.

Dürer’s textbook also plays an important role in the history of typography. It is printed in a typeface especially created for the book by the Nürnberger writing master Johannes Neudörfer (1497 – 1563) brother in law of the printer publisher Johannes Petreius. The typeface is a fraktur script and although this was not the first fraktur typeface, all fraktur typefaces developed and used since then are derived from Neudörfer’s creation.

Several years ago the City of Nürnberg decided to relaunch itself on the world market as “Dürer City”. The local newspaper conducted a survey and asked the people on the street if they knew who Albrecht Dürer was. As usual with these things something like 70% of the good citizens of Nürnberg didn’t have the faintest idea of the identity of the city’s most famous son. I was not asked but if I had been I would have answered, “an important German Renaissance mathematician”, which would almost certainly have convinced the questioner that I was one of the clueless!

Illustrations taken from, Albrecht Dürer, *Underweysung der Messung mit dem Zirkel und Richtscheyt: *Faksimile-Neudruck der Ausgabe Nürnberg 1525 mit einer Einführung von Matthias Mende, 3. Aufl., Verlag Dr. Alfons Uhl, Nördlingen, 2000

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I knew of Dürer as a wonderful artist but not of his mathematical work so my thanks for informing me of something I did not know before (which for me is the

propermeaning of ‘information’).While mindful of all the drawbacks to life in those days, Dürer was lucky enough to live during a period of European history when it was possible for a talented man (or woman, if she could circumvent the prejudices of her society) to become a master in both what we now call the sciences and the humanities. Today, in the sciences, it seems to be as much as one person can do to keep abreast of the latest developments in his or her own narrow field, let alone what is happening in neighboring fields. They seem to have little time or energy left for anything else other than as a hobby. As for those in the humanities, they seem to be almost positively discouraged from involvement in science. Which makes me wonder when the gulf between the two cultures began to emerge. Was it the early nineteenth century with the emergence of a wealthy and independently-minded middle class?

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Are there any pictures of the book, so we can see the typeface? That’s pretty neat.

Thanks for the post. The constructive geometry Dürer advanced puts me in mind of another seminal figure in that department Jakob Steiner, whose method was a closely guarded military secret once upon a time.

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Looks like Dürer is giving a construction for a regular septagon (among other polygons) in the second illustration. Presumably approximate, or maybe he’s employing means other than ruler and compass? I’ll have to spend a slow afternoon sometime deciphering the text. (I got as far as “gleych eckert figure”, equal angled figures, nicht wahr?)

The McTutor entry on Dürer is pretty good, and they do give a reference for his polygon constructions, but alas in Russian:

E A Fribus, The construction of regular polygons in A Dürer’s geometric treatise (Russian), Moskov. Oblast. Ped. Inst. Ucen. Zap. 185 (1967), 201-211.

On further searching, I find that this page gives Dürer’s approximate construction for the regular septagon. And it seems Dürer even gave two constructions for the pentagon, one exact, and one only approximate but more practical for artists and stonemasons.

Dürer’s heptagon construction is indeed an approximation, which Kepler famously criticised. Before you ask I’m not sure where but probably Harmonices Mundi. There’s a nice arXiv paper on his polygons

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