Albrecht Durer was a painter and print maker in Germany, 1471-1528. He also wrote books, and traveled widely in western Europe. He
was a superb draftsman. I have enjoyed the compositions, the details, and the lines in his engravings and
woodcuts for 50 years.
His woodcuts were for the people, most of whom were illiterate. Their livelihoods did not require writing. They could read the images: here the rough stable, the well dressed men coming to see a baby, the angels and the star.
The engravings were not so easy to make. They were for books, for people who could read.
These plates come from the Dover Publications reprints of Durer's wood cuts and engravings.* This is print #183.
He drew what he saw around him. His plates are full of the life he knew, including construction details.
This detail from #183 shows a truss which includes a collar tie with angled tenon joints and pegs. The purlins and rafters are structurally correct.
The detail from Plate #190, shows the grain of the wood brace running in the right direction. The angled cuts for the joints could serve as templates for repair.
Here in Plate #185, the brace is tied. a peg serves to tighten the tie as needed. The thatch for the roof is properly applied.
When I began to write this post I was thinking about Durer's knowledge of construction and his use of geometry in his compositions. (More about that in another post.)
I was side tracked as I began to read Durer biographies. The scholars who wrote them rarely knew about construction. To them the structures are allegories or useful for his compositions.
I saw practical and abstract geometry.
This is my exploration, as a Geometer, of one of Durer's most important engravings.
Durer put many construction tools around his melancholy angel. She holds a compass. Tucked beside her hand is a gauge. The putto sits on a mill wheel next to a ladder. Set beside the rhombus is a pot for hot liquid metal, possibly lead, on a brazier. Under the angel's skirts is a pair of pliers.
In the lower left corner is a profile gauge.
Then: a plane, a saw, and a straight edge that can be used to draw arcs. It may also be a level.
Lastly: nails and a nail punch.
The theory was that Melancholy, influenced by the planet Saturn, was part of the inherent character of artistic and philosophical people. There were 3 levels. The lowest was artists and artisans. The next was scholars, natural scientists, and statesmen. The highest level was theologians and those who studied the secrets of the divine.
Here the putto is taking a nap after doing some numbers (perhaps) on a slate, the lowest level. The angel, on the other hand, is intently studying the rhombus, an abstract shape. She is a scientist. The dog - which I learned represents 'faithfulness' - is waiting. The sphere is an abstraction, perfect as the tools and mill wheel are not. They are all 'things',
scales, the hour glass, the bell are said to refer to the knowledge of artisans:
they understand weight, measure (of time), music (as an expression
I have very little understanding of the Magic Square. I do know that all the lines add up to 34 and no number is used twice.
I do wonder if in a largely illiterate society, the fairly recent acceptance of Hindu-Arabic numbers, as opposed to Roman Numerals, might be part of its purpose here: to illustrate the scholarship, the mathematics that numbers made possible.
Roman Numerals in the grid of the Square.
Can you quickly add up the amounts? How would you do it as an arithmetic problem on paper?
16 +5+9+4 = XVI+V+IX+IV .
I will write about the rhombus in the next post.
*images from The Complete Woodcuts of Albretch Durer, edited by Dr. Willi Kurth, 1963 and The Complete Engravings, Etchings & Drypoints of Albrecht Durer, edited by Walter L Strauss, 1972. Both republished by Dover Publications: www.dovderpublications. com.
Walter L. Strauss' analysis is the best I've found. I wish he had known more about geometry.