**Dürer** was a German artist who is known for his work as an engraver. The foundations of descriptive geometry are laid in Dürer's treatise on human proportions published in Nuremberg after his death in 1528.

- The name changed to Dürer but Albrecht Dürer senior always signed himself Türer rather than Dürer.
- In 1486 Dürer became an apprentice painter and woodcut designer to Michael Wolgemut, the leading producer of altarpieces.
- After an apprenticeship of four years, Dürer had learnt all he could from Wolgemut and had reached a level of artistic quality exceeding that of his famous teacher.
- Wolgemut advised Dürer to travel to widen his experience and meet other artists.
- Following Wolgemut's advice, Dürer delayed visiting Italy (which Wolgemut himself never visited), where there were very different artistic styles, until he had fully developed his own style and learnt more techniques from other German artists.
- Dürer travelled first to Nördlingen, where he met artists of the Swabian school.
- The Swabian style had been influenced by Dutch artistic design which Dürer had not met before.
- They were delighted with Dürer's drawings, with his first engravings and the small pictures he had already painted, independently of Wolgemut's directions or opinions.
- Leaving Ulm, Dürer made his way to Constance which charmed him with its fairyland appearance.
- Basel was the next town which Dürer visited, and he found it quite similar to his home town of Nürnberg.
- Finally Dürer returned home, making visits to Colmar and Strasbourg on the way.
- It had been a long journey of great importance to Dürer which had taken nearly four years, but after he returned to Nürnberg in 1494 he felt disappointed that he had not visited Italy.
- Italy was not only a country with new ideas to offer Dürer in art, but it was also leading the world at this time in the revival of mathematics.
- It was a marriage which helped raise Dürer's status in Nürnberg, as well as provide him with money which helped him set up his own studio.
- Before the end of 1494, Dürer was on his travels again, leaving Agnes behind in Nürnberg.
- In Venice, Dürer, as he had done throughout his journeys, sketched scenes, visited galleries and churches, and met with the local artists.
- Dürer returned to Nürnberg in 1495, and although he does not seem to have met with any of the major Italian mathematicians on his journeys, he did meet Jacopo de Barbari who told him of the mathematical work of Pacioli and its importance to the theory of beauty and art.
- Nor did Dürer meet with Leonardo da Vinci while in Italy, but he learnt of the importance which that artist placed in mathematics.
- Back in Nürnberg, Dürer began a serious study of mathematics.
- In 1495 Dürer was still not well known as an artist in the highest circles but news of his skill reached Frederick the Wise, Elector of Saxony, and Dürer was commissioned to paint his portrait.
- Frederick liked his portrait which Dürer painted in April 1496 when Frederick had visited Nürnberg.
- Despite Frederick's attempts to persuade Dürer to move to Weimar and become Court painter, the artist did not wish to leave Nürnberg.
- From about 1500 Dürer's art showed the influence of the mathematical theory of proportion which he continued to spend so much time studying.
- 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.
- 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.
- It was a difficult life and one in which Dürer's health began to suffer.
- 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.
- It was not about art that Dürer now wished to learn from the Italians, but rather about mathematics.
- 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.
- 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.
- Dürer met Maximilian personally for the first time in 1518 and, probably from one sitting in Augsburg, painted Maximilian's portrait.
- 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.
- Passing through Aachen, Dürer sketched the cathedral at Aachen.
- She had already given the book by Jacopo de Barbari to another artist so Dürer's quest was in vain.
- After returning to Nürnberg, Dürer's health became still worse.
- 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.
- This treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit Ⓣ(Training of measurement with the compass and straightedge), 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.
- 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!).
- Dürer also gave approximate methods to square the circle using ruler and compass constructions in this book.
- 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.
- 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.
- Dürer's remarkable achievement was through applying mathematics to art, he developed such fundamentally new and important ideas within mathematics itself.

Born 21 May 1471, Imperial Free City of Nürnberg (now in Germany). Died 6 April 1528, Imperial Free City of Nürnberg (now in Germany).

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