Born: 21 May 1471 in Imperial Free City of
Nürnberg (now in Germany)

Died: 6 April 1528 in Imperial Free City of
Nürnberg (now in Germany)

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.

Here are
portraits of his father and mother.

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):-

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. This was the first of many
self-portraits which Dürer painted and they provide a wonderful record.
Here is our collection of such
self-portraits.

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.

Here is a
portrait of Wolgemut.

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:-

... 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:-

... 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.

Here are
portraits of Agnes.

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.

Here is one of
his paintings of Trento.

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:-

... 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 (On Architecture) 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:-

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. He was
deeply attached to Nürnberg, painting
these views of the city in 1497.

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

Список
литературы

Для подготовки
данной работы были использованы материалы с сайта http://www-history.mcs.st-andrews.ac.uk/