Albrecht Durer

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From about 1500 Dürer's art showed the influence of the mathematical theory of proportion which he continued to spend so much time studying. It is claimed that his  self-portrait in a wig made in 1500 has the dimensions of the head constructed proportionally. For the engraving  Adam and Eve made in 1504, Dürer described the intricate  ruler and compass constructions which he made to construct the figures. It was not only the mathematical theory of proportion which influenced Dürer's art at this period, but also his mastery of perspective through his study of geometry. This is most clearly seen in his woodcuts  Life of the Virgin made between 1502 and 1505.

During the ten years after 1496 Dürer went from a relatively unknown artist to someone with a wide reputation as both an artist and a mathematician. His personal circumstances had changed greatly. His father had died in 1502 and Dürer was left to care for his invalid, and nearly blind, mother. He had set up his own printing press while he, or often his wife, sold his works to buyers at local fairs. It was a difficult life and one in which Dürer's health began to suffer. In fact he would never regain full health during the rest of his life.

From 1505 to 1507 Dürer made a second visit to Italy, spending much time again in Venice. It was a very different visit from his first, with Dürer now more interested in his international fame than in learning about art. He was so conscious of his fame, and the threat he perceived that he might hold to the local artists, that:-

... he refused invitations to dinner in case someone should try to poison him.

It was not about art that Dürer now wished to learn from the Italians, but rather about mathematics. He visited Bologna to meet with  Pacioli whom he considered held the mathematical secrets of art. He also visited Jacopo de Barbari and the great efforts which Dürer made to meet de Barbari shows the importance which Dürer more and more attached to mathematical knowledge. Dürer returned to Nürnberg from this second visit to Italy feeling that he must delve yet more deeply into the study of mathematics.

In about 1508 Dürer began to collect material for a major work on mathematics and its applications to the arts. This work would never be finished but Dürer did use parts of the material in later published work. He continued to produce art of outstanding quality, and he produced one of his most famous engravings  Melancholia in 1514. It contains the first  magic square to be seen in Europe, cleverly including the date 1514 as two entries in the middle of the bottom row. Also of mathematical interest in Melancholia is the polyhedron in the picture. The faces of the polyhedron appear to consist of two equilateral triangles and six somewhat irregular pentagons. An interesting reconstruction of the polyhedron is given in , see also  for further details.

Dürer worked for Maximilian I, the Holy Roman emperor, from about 1512. Maximilian, however, had little in the way of wealth to pay for Dürer's work and he asked the councillors of Nürnberg to exempt Dürer from taxes as compensation. He then asked the councillors to pay Dürer a pension on his behalf, which certainly did not please them. From about 1515 the councillors tried to avoid paying this pension. Dürer met Maximilian personally for the first time in 1518 and, probably from one sitting in Augsburg, painted  Maximilian's portrait. The following year Maximilian died and this was the final excuse for the councillors to refuse to make any further payment, saying that the new emperor Charles would have to agree to the pension.

Although Dürer was fairly well off by this time and the pension was not necessary for him, it was more a matter of prestige to have his pension restored. He set off for Antwerp on 15 July 1520 with his wife and their maid to visit the Emperor Charles V. Passing through Aachen, Dürer sketched  the cathedral at Aachen.

Dürer had a second reason for this visit to the Netherlands, for he believed that Maximilian's daughter had a book by Jacopo de Barbari on applications of mathematics to art, and Dürer had long sought the truths which he believed this work contained. On meeting Maximilian's daughter he offered her the portrait of her father which he had painted, but was distressed to find that she did not want the portrait. She had already given the book by Jacopo de Barbari to another artist so Dürer's quest was in vain. He did persuade Charles V to restore his pension, however, which was formally agreed on 12 November 1520.

After returning to Nürnberg, Dürer's health became still worse. He did not slacken his work on either mathematics or painting but most of his effort went into his work Treatise on proportion. Although it was completed in 1523, Dürer realised that it required mathematical knowledge which went well beyond what any reader could be expected to have, so he decided to write a more elementary text. He published this more elementary treatise, in four books, in 1525 publishing the work through his own publishing company.

This treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit, is the first mathematics book published in German (if one discounts an earlier commercial arithmetic book) and places Dürer as one of the most important of the Renaissance mathematicians. Dürer's sources for this work are discussed in [21] where three main sources are suggested (i) the practical recipes of craftsmen, (ii) classical mathematics from printed works and manuscripts, and (iii) the manuals of Italian artists. The article [16] gives many details of the mathematics contained in the treatise.

The first of the four books describes the construction of a large number of curves, including the  Spiral of Archimedes, the  Equiangular or Logarithmic Spiral, the  Conchoid,  Dürer's Shell Curves, the  Epicycloid, the  Epitrochoid, the  Hypocycloid, the  Hypotrochoid, and the  Limaçon of Pascal (although of course Dürer did not use that name!). Details about Dürer's descriptions of the curves, in particular one he calls a "muschellini", is given in.

In the second book he gave exact and approximate methods to construct regular polygons. Dürer's constructions of regular polygons with 5, 7, 9, 11 and 13 sides is discussed in [12]. Dürer also gave approximate methods to  square the circle using ruler and compass constructions in this book. A method to obtain a good approximation to the  trisector of an angle by Euclidean construction is also given.

Book three considers pyramids, cylinders and other solid bodies. The second part of this book studies sundials and other astronomical instruments. The final book studies the five Platonic solids as well as the semi-regular Archimedean solids. Also in this book is Dürer's theory of shadows and an introduction to the theory of perspective.

In 1527 Dürer published another work, this time on fortifications. There were strong reasons why he produced a work on fortifications at this time, for the people of Germany were in fear of an invasion by the Turks. Many cities, including Nürnberg, would improve their fortifications using the methods set out by Dürer in this book. Dürer's final masterpiece was his Treatise on proportion which was at the proof stage at the time of his death.

Descriptive geometry originated with Dürer in this work although it was only put on a sound mathematical basis in later work of  Monge. One of the methods of overcoming the problems of projection, and describing the movement of bodies in space, is descriptive geometry. Dürer's remarkable achievement was through applying mathematics to art, he developed such fundamentally new and important ideas within mathematics itself.

J J O'Connor and E F Robertson