阿尔布雷希特·丢勒Albrecht Dürer

阿尔布雷希特·丢勒是丢勒和芭芭拉·霍尔弗的第三个儿子。他是他们十八个孩子之一。丢勒家族来自匈牙利，老丢勒出生在那里，当时家族姓氏是Ajtos。Ajtos这个名字在匈牙利语中意为“门”，当老丢勒和他的兄弟们来到德国时，他们选择了Türer这个名字，听起来像德语“Tür”，意为门。名字改为丢勒，但老丢勒总是把自己签名为Türer而不是丢勒。

他父亲和母亲的肖像在THIS LINK。

老丢勒是一位珠宝商，曾在Hieronymus Holfer手下当学徒，后来娶了Holfer的女儿。小丢勒写到了他的父亲和他的成长经历（例如见[3]）：-

我的父亲一生饱受苦难、辛勤劳作，因为他除了自己行业的收入之外没有任何资源来养活自己和妻儿。他过着诚实、敬畏上帝的生活。他的性格温和而耐心。他对所有人都友好，对造物主充满感激。他很少在意社交，对世俗娱乐毫不在意。他寡言少语，极为虔诚，非常重视子女的宗教教育。他最殷切的希望是，他灌输到他们心中的高尚原则能使他们更加配得上神的保护和人类的同情。他每天都告诉我们，我们必须爱上帝，在与邻人的交往中要正直。

丢勒小时候在圣爱德华·诺顿·劳仑次的拉丁学校接受教育，他也在父亲的作坊里工作，学习金匠和珠宝商的行业。到13岁时，他已经是一位熟练的画家，从当时他画的一幅自画像中可以看出。

你可以在THIS LINK看到这幅自画像。

这是丢勒绘制的众多自画像中的第一幅，它们提供了绝佳的记录。我们收藏的此类自画像位于THIS LINK。

1486年，丢勒成为迈克尔·沃尔格穆特的学徒画家和木刻设计师，后者是当时领先的祭坛画制作人。经过四年的学徒期，丢勒已从沃尔格穆特那里学到了所有能学的东西，其艺术水平已超过了他著名的老师。沃尔格穆特建议丢勒外出旅行以拓宽经验并结识其他艺术家。遵循沃尔格穆特的建议，丢勒推迟了访问意大利（沃尔格穆特本人从未去过那里），那里有着非常不同的艺术风格，直到他充分发展了自己的风格并从其他德国艺术家那里学到了更多技巧。

沃尔格穆特的一幅肖像画在THIS LINK。

丢勒首先前往Nördlingen，在那里他遇到了施瓦本画派的艺术家。施瓦本风格受到荷兰艺术设计的影响，这是丢勒以前没有接触过的。他的下一站是乌尔姆，在那里他遇到了更多施瓦本画派的艺术家。丢勒[3]：-

……他兴致勃勃地参加同龄艺术家们在低矮天花板的酒馆里、在冒着泡沫的大杯啤酒旁的讨论。这些年轻的热衷者，与历史上所有民族的同类人一样，一心要复兴世界的艺术。他们喜爱丢勒的画作、他最初的版画以及他已经画成的小幅画，这些作品独立于Wolgemut的指导或意见。

丢勒前往康斯坦茨，那里的仙境般外观令他着迷。巴塞尔是丢勒访问的下一个城镇，他发现它与家乡纽伦堡颇为相似。最后丢勒返回家乡，途中访问了科尔马和斯特拉斯堡。

对丢勒来说，这是一次历时近四年、极为重要的长途旅行，但他在1494年回到纽伦堡后，对自己没有访问意大利感到失望。他还确信[1]：-

……新艺术必须建立在科学之上——尤其是数学，因为数学是科学中最精确、最合乎逻辑、最具图形构造性的。

意大利不仅是一个能在艺术上为丢勒提供新观念的国家，而且当时在数学复兴方面也引领世界。然而，在出发前往意大利之前，丢勒娶了Agnes Frey，她是一位博学之士Hans Frey的女儿，Hans Frey通过制作珠宝、乐器和机械装置赚了不少钱。

Agnes的肖像在THIS LINK。

这桩婚姻似乎更多是阿格尼斯和丢勒的父母的意愿，两人于1494年7月7日结婚。这桩婚姻帮助提高了丢勒在纽伦堡的地位，并为他提供了资金，帮助他建立了自己的工作室。

1494年底之前，丢勒再次踏上旅程，把Agnes留在纽伦堡。他首先访问了奥格斯堡，在那里第一次接触到强烈的意大利艺术影响。他穿越蒂罗尔，到达特伦托，第一次看到意大利。

他的一幅特伦托画作在THIS LINK。

他继续旅行到维罗纳，然后到达他的主要目的地威尼斯。在威尼斯，丢勒像在整个旅程中一样，写生风景，参观画廊和教堂，并会见当地艺术家。他在威尼斯遇到的艺术家之一Giovanni Bellini对丢勒产生了重要影响，因为[3]：-

……威尼斯能教给他的一切，都能在乔瓦尼的画作中找到。因此，他怀着既热烈又恭敬的虔诚，与这位艺术家交往，终生全心全意地对这位向他揭示了一个如此奇妙世界的人怀有无限的感激之情。

丢勒于1495年回到纽伦堡，尽管他在旅途中似乎没有遇到任何重要的意大利数学家，但他确实遇到了雅各波·德·巴尔巴里，后者向他讲述了卢卡·帕西奥利的数学工作及其对美与艺术理论的重要性。丢勒在意大利期间也没有遇到列奥纳多·达·芬奇，但他了解到这位艺术家对数学的重视。回到纽伦堡后，丢勒开始认真研究数学。他阅读了欧几里得的Elements以及维特鲁威（公元前1世纪）的重要论著De architecturaⓉ（建筑），后者是著名的罗马建筑师和工程师。他还熟悉了莱昂·巴蒂斯塔·阿尔伯蒂和卢卡·帕西奥利关于数学与艺术的作品，特别是关于比例的作品。

当丢勒在纽伦堡开始其艺术生涯时，影响他的不仅是这种对艺术的科学态度，他还受益于看到了不同的艺术风格和他在[3]所见过的不同风景：-

丢勒在旅行过程中所经过的地区的多样性，以及他对其中最迷人或不熟悉的地区所作的素描和水彩画所付出的细心，为他提供了来自最多样化来源的大量绘画主题。

1495年，丢勒在最高层的圈子里还不是一位知名的艺术家，但他技艺的消息传到了萨克森选帝侯智者腓特烈那里，丢勒受委托为他绘制肖像。腓特烈喜欢丢勒于1496年4月腓特烈访问纽伦堡时为他绘制的肖像。尽管腓特烈试图说服丢勒搬到魏玛并成为宫廷画家，但这位艺术家不愿离开纽伦堡。

你可以在THIS LINK看到这幅肖像。

他对纽伦堡有着深厚的感情，1497年绘制了该城的风景画，你可以在THIS LINK看到。

大约从1500年起，丢勒的艺术显示出比例数学理论的影响，他继续花费大量时间研究这一理论。据称他1500年制作的self-portrait in a wig具有按比例构造的头部尺寸。见THIS LINK。

对于1504年制作的版画Adam and Eve，丢勒描述了他为构造人物形象而进行的复杂尺规构造。见THIS LINK。

在这一时期影响丢勒艺术的不仅是比例数学理论，还有他通过学习几何学而掌握的透视法。这在他1502年至1505年间制作的木刻Life of the Virgin中最为明显。见THIS LINK。

在1496年之后的十年里，丢勒从一位相对不知名的艺术家变成了作为艺术家和数学家都享有广泛声誉的人。他的个人境况发生了巨大变化。他的父亲于1502年去世，丢勒被留下来照顾他体弱多病、几乎失明的母亲。他建立了自己的印刷所，同时他，或者常常是他的妻子，在当地集市上向买家出售他的作品。生活艰难，丢勒的健康开始受到影响。事实上，在他余生中，他再也没有完全恢复健康。

从1505年到1507年，丢勒第二次访问意大利，再次在威尼斯度过了大量时间。这次访问与他的第一次截然不同，丢勒现在更关心自己的国际声誉，而不是学习艺术。他如此意识到自己的名声，以及他可能对当地艺术家构成的威胁，以至于[3]：-

……他拒绝赴宴，以防有人试图毒害他。

丢勒现在希望从意大利人那里学习的不是艺术，而是数学。他前往博洛尼亚会见卢卡·帕西奥利，他认为卢卡·帕西奥利掌握着艺术中的数学秘密。他还拜访了Jacopo de Barbari，而丢勒为会见de Barbari所作的巨大努力表明，丢勒越来越重视数学知识。丢勒从这第二次意大利之行回到纽伦堡时感到，他必须更深入地钻研数学。

大约在1508年，丢勒开始为一部关于数学及其在艺术中应用的重要著作收集材料。这部著作最终未能完成，但丢勒确实在后来的出版作品中使用了部分材料。他继续创作质量卓越的艺术品，并于1514年创作了他最著名的版画之一Melancholia。

你可以在THIS LINK看到这幅版画

它包含欧洲首次出现的幻方，巧妙地将1514这个年份作为两个条目放在最下面一行的中间。Melancholia中同样具有数学趣味的还有画中的多面体。这个多面体的面似乎由两个等边三角形和六个略不规则的五边形组成。文献[19]给出了该多面体的一个有趣重构，更多细节另见[18]。

大约从1512年起，丢勒为神圣罗马帝国皇帝马克西米利安一世工作。然而，马克西米利安几乎没有财富来支付丢勒的工作，于是他请求纽伦堡的市议员免除丢勒的税款作为补偿。随后他又请求市议员代他向丢勒支付一笔年金，这当然令他们不快。大约从1515年起，市议员们试图逃避支付这笔年金。丢勒于1518年第一次亲自见到马克西米利安，并且很可能是在奥格斯堡的一次坐姿画像中，为马克西米利安绘制了肖像。见THIS LINK。第二年马克西米利安去世，这成了市议员们拒绝继续支付任何款项的最终借口，他们称新皇帝查理必须同意这笔年金。

此时丢勒已相当富裕，年金对他并非必需，恢复年金更多是关乎声望。1520年7月15日，他带着妻子和女仆动身前往安特卫普，去拜访皇帝查理五世。途经亚琛时，丢勒画了亚琛大教堂的速写。见THIS LINK

丢勒这次访问荷兰还有第二个原因，因为他相信马克西米利安的女儿有一本雅各波·德·巴尔巴里关于数学在艺术中应用的著作，而丢勒长期以来一直在寻求他相信这部著作所包含的真理。见到马克西米利安的女儿时，他把自己画的她父亲的肖像送给她，但令他苦恼的是，她并不想要这幅肖像。她已经把雅各波·德·巴尔巴里的书给了另一位艺术家，所以丢勒的寻求落空了。不过，他确实说服查理五世恢复了他的年金，这一点于1520年11月12日正式商定。

回到纽伦堡后，丢勒的健康状况变得更糟。他并没有放松数学或绘画方面的工作，但他大部分精力都投入到了他的著作《比例论》中。尽管这部著作于1523年完成，丢勒意识到它所需要的数学知识远远超出了任何读者可能具备的水平，因此他决定写一部更初级的文本。他于1525年出版了这部更初级的论著，共四卷，并通过自己的出版公司出版。

这部论著，Unterweisung der Messung mit dem Zirkel und Richtscheit Ⓣ（用圆规和直尺进行测量的训练），是第一部用德语出版的数学书（如果忽略更早的一部商业算术书的话），并使丢勒成为文艺复兴时期最重要的数学家之一。丢勒这部著作的资料来源在[21]中进行了讨论，其中提出了三个主要来源：(i)工匠的实用配方，(ii)来自印刷作品和手稿的古典数学，以及(iii)意大利艺术家的手册。文章[16]给出了该论著中所包含数学的许多细节。

四卷中的第一卷描述了大量曲线的构造，包括阿基米德、Equiangular or Logarithmic Spiral、Conchoid、Dürer's Shell Curves、Epicycloid、Epitrochoid、Hypocycloid、Hypotrochoid和Limaçon of Pascal（当然，丢勒并没有使用那个名称！）。关于丢勒对这些曲线描述的细节，特别是他称为“muschellini”的一条曲线，见[9]。

在第二卷中，他给出了构造正多边形的精确方法和近似方法。丢勒对5、7、9、11和13边正多边形的构造在[12]中进行了讨论。丢勒在这卷中还给出了使用尺规构造来化圆为方的近似方法。还给出了一种通过欧几里得构造获得trisector of an angle良好近似的方法。

第三卷讨论棱锥、圆柱和其他立体。这卷的第二部分研究日晷和其他天文仪器。最后一卷研究五种柏拉图立体以及半正阿基米德立体。这卷中还有丢勒的阴影理论以及透视理论导论。

1527年，丢勒出版了另一部著作，这次是关于筑城术的。他此时撰写筑城术著作有充分的理由，因为德国人民正担心土耳其人的入侵。包括纽伦堡在内的许多城市，都采用丢勒在本书中提出的方法来改进他们的防御工事。丢勒的最后一部杰作是他的Treatise on proportion，在他去世时正处于校对阶段。

画法几何起源于丢勒在这部著作中的工作，尽管它直到加斯帕尔·蒙日后来的工作中才被置于坚实的数学基础之上。克服投影问题并描述物体在空间中运动的方法之一就是画法几何。丢勒的非凡成就在于将数学应用于艺术，他在数学本身内部发展出了如此根本性的新思想和重要观念。

Albrecht Dürer was the third son of Albrecht Dürer and Barbara Holfer. He was one of their eighteen children. The Dürer family came from Hungary, Albrecht Dürer senior being born there, and at this time the family name was Ajtos. The name Ajtos means "door" in Hungarian and when Dürer senior and his brothers came to Germany they chose the name Türer which sounds like the German "Tür" meaning door. The name changed to Dürer but Albrecht Dürer senior always signed himself Türer rather than Dürer.

Portraits of his father and mother are at THIS LINK.

Albrecht Dürer senior was a jeweller who had served his apprenticeship with Hieronymus Holfer, and then married Holfer's daughter. Albrecht Dürer junior wrote about his father and his upbringing (see for example [3]):-

My father suffered much and toiled painfully all his life, for he had no resources other than the proceeds of his trade from which to support himself and his wife and family. He led an honest, God-fearing life. His character was gentle and patient. He was friendly towards all and full of gratitude to his Maker. He cared little for society and nothing for worldly amusements. A man of very few words and deeply pious, he paid great attention to the religious education of his children. His most earnest hope was that the high principles he instilled into their minds would render them ever more worthy of divine protection and the sympathy of mankind. He told us every day that we must love God and be honourable in our dealings with our neighbours.

As a young boy Dürer was educated at the Lateinschule in St Lorenz and he also worked in his father's workshop learning the trade of a goldsmith and jeweller. By the age of 13 he was already a skilled painter as seen from a self portrait which he painted at that time.

You can see this self-portrait at THIS LINK

This was the first of many self-portraits which Dürer painted and they provide a wonderful record. Our collection of such self-portraits is at THIS LINK.

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.

A portrait of Wolgemut is at THIS LINK.

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. His next visit was to Ulm where he met more artists of the Swabian school. Dürer [3]:-

... participated with keen enjoyment in the discussions among artists of his own age, in the low-ceilinged taverns, over foaming mugs of beer. These youthful enthusiasts, in common with those of all nations throughout history, were bent on rejuvenation of the art of the world. 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. He had also become convinced that [1]:-

... the new art must be based upon science - in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences.

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. Before setting out for Italy, however, Dürer married Agnes Frey, the daughter of a learned man Hans Frey who had made quite a lot of money through making jewellery, musical instruments, and mechanical devices.

Portraits of Agnes are at THIS LINK.

The marriage seems to have been more the idea of the parents of Agnes and Albrecht, and the pair were married on 7 July 1494. 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. First he visited Augsburg where he met strong Italian artistic influences for the first time. Travelling through the Tyrol, he reached Trento and his first view of Italy.

One of his paintings of Trento is at THIS LINK.

He travelled on to Verona before reaching Venice which was his main objective. In Venice, Dürer, as he had done throughout his journeys, sketched scenes, visited galleries and churches, and met with the local artists. One of the artists that he met in Venice, Giovanni Bellini, had an important influence on Dürer for [3]:-

... everything that [Venice] could teach him was to be found in Giovanni's paintings. He cultivated the artist's society, therefore, with a devotion both impassioned and deferential, retaining throughout his life, with his whole heart and soul, unbounded feelings of gratitude to the man whose pictures had unveiled so wonderful a world to him.

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. He read Euclid's Elements and the important treatise De architectura Ⓣ by Vitruvius (1st century BC), the famous Roman architect and engineer. He also became familiar with the work of Alberti and Pacioli on mathematics and art, in particular work on proportion.

It was not only this scientific approach to art that influenced Dürer as he began his artistic career in Nürnberg, but he also benefited from seeing different artistic styles and the different scenery which he had viewed [3]:-

The variety of regions through which Dürer had passed in the course of his travels and the care he had taken with the drawings and water-colours he had made of the most attractive or unfamiliar of them had provided him with a great range of pictorial motives emanating from the most diverse sources.

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.

You can see this portrait at THIS LINK.

He was deeply attached to Nürnberg, painting views of the city in 1497 that you can see at THIS LINK.

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. See THIS LINK.

For the engraving Adam and Eve made in 1504, Dürer described the intricate ruler and compass constructions which he made to construct the figures. See THIS LINK.

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. See THIS LINK.

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 [3]:-

... 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.

You can see this engraving at THIS LINK

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 [19], see also [18] 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. See THIS LINK. 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. See THIS LINK

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 [9].

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.