In 1525 Albrecht Dürer, by then the most famous artist in northern Europe, published a book about compasses and straightedges. It was the first mathematics book printed in German rather than Latin, and it was written for craftsmen: painters, goldsmiths, stonemasons, and typefounders who worked by rule of thumb and had no access to the geometry that would have explained what they were doing.
Underweysung der Messung mit dem Zirckel und Richtscheyt, instruction in measurement with compass and straightedge, is one of the more unusual objects in the history of science publishing. It is a technical manual, an artist’s treatise, and a piece of deliberate popularisation, and it invented a good deal of the German mathematical vocabulary in the process because the words did not exist.
Why an Artist Wrote It
Dürer was born in Nuremberg in 1471, the son of a goldsmith, and trained as a painter and printmaker. His woodcuts and engravings circulated across Europe and made him wealthy and widely imitated.
Two journeys to Italy changed what he thought art required. In Venice and Bologna he encountered a tradition in which painters studied proportion, perspective and geometry as a matter of course, and in which the construction of a convincing space on a flat surface was understood as a mathematical operation rather than a knack.
He wrote in a letter from Bologna in 1506 that he was going there to learn the secret art of perspective, which somebody was willing to teach him. Whoever it was, the trip appears to have convinced him that German artists were working blind. In his own words, they had great skill in practice and no grounding in theory, and he set out to supply it.
What Is in the Four Books
Book I deals with lines and curves. It covers spirals, including logarithmic and conchoid forms, epicycloids generated by rolling circles, and the conic sections. His method for the conics is characteristically practical: he constructs the ellipse, parabola and hyperbola by taking a cone, imagining it cut, and projecting the section point by point onto a flat plane, essentially drafting the curve the way an architect would.
Book II covers plane figures and polygon construction. Dürer gives methods for regular polygons and is unusually honest about which are exact and which are not. The heptagon and the enneagon, which cannot be constructed exactly with compass and straightedge, he supplies as approximations and says so plainly. That distinction, between a construction that is provably correct and one that is good enough for a workshop, is a mathematician’s instinct in a craftsman’s book.
Book III applies the geometry to real objects: columns, towers, monuments and sundials. It contains his celebrated treatment of letterforms, constructing the Roman capital alphabet within a square using ruler and compass, with each letter’s proportions specified, alongside a geometric analysis of the Fraktur letters used in German printing. Typography as a geometric discipline begins here.
Book IV turns to three dimensions: the five Platonic solids, several Archimedean ones, and then perspective.
The Net
Book IV contains an invention so useful that it has become invisible.
To describe a polyhedron, Dürer drew it unfolded: the faces laid out flat, connected along their shared edges, so that the figure can be cut out and folded back into a solid. These are what we now call nets, and they are how every schoolchild is first shown that a cube has six faces.
Nobody had printed them before. The device solves a genuine problem, since a drawing of a solid in perspective hides half of it, while a net shows every face at true size and preserves the relationships between them. It is simultaneously a piece of mathematical notation and a set of workshop instructions.
The solids themselves were familiar from the classical tradition, and the five Platonic solids are the subject of the final book of Euclid’s Elements. Dürer owned a copy of Euclid, bought in Venice in 1507, and the structure of his own book, building from lines to plane figures to solids, follows the same progression.
The Drawing Machines
The best known images in the book are the woodcuts illustrating perspective devices, added in the second edition.
One shows a draughtsman looking through a fixed eyepiece at a lute, with a gridded frame between him and the object, transferring what he sees square by square onto gridded paper. Another shows the same arrangement with a reclining figure. In another, a taut string runs from a point on the object through a frame to a fixed point on the wall, and the position where the string crosses the frame is recorded, point by point, until the projection is complete.
These pictures are usually reproduced as curiosities. They are better understood as diagrams of a theory. Each one is a physical demonstration that perspective is a projection: rays from the object converge on the eye, and the picture is the section of that pyramid of rays cut by the picture plane. The machine makes the geometry mechanical, and in doing so proves that it is geometry.
Melencolia I
Dürer’s mathematical interests are not confined to the treatise. His engraving Melencolia I of 1514 is among the most analysed images in Western art, and much of what puzzles people in it is mathematical.
A large stone solid dominates the middle ground: an irregular truncated form, now known as Dürer’s solid, whose exact geometry has been the subject of a substantial technical literature. On the wall hangs a four by four magic square in which every row, column and diagonal sums to thirty four, and in which the two central cells of the bottom row read 15 and 14, giving the date of the engraving. Scattered around are a compass, a sphere, a balance, a plane and an hourglass.
Whatever the allegory means, and the interpretations are numerous, the props are the equipment of measurement.
The Proportion Book
Dürer’s other major treatise, Vier Bücher von menschlicher Proportion, appeared in 1528, shortly after his death.
It began as a search for the single ideal set of human proportions, in the manner of Vitruvius. It did not end there. Dürer measured a great many people and concluded that no single canon exists, so the book instead catalogues a range of body types, from thin to heavy, with proportional schemes for each.
The final section goes further and describes geometric transformations that can be applied to a figure or a face, stretching and skewing the underlying grid to generate variation systematically. The technique anticipates by four centuries the transformation grids that D’Arcy Thompson used in On Growth and Form to compare the shapes of related species.
Where It Sits
The Renaissance treatment of art as a mathematical discipline runs from Alberti through Piero della Francesca, Pacioli and Leonardo to Dürer, and he is the point at which it crosses the Alps and gets printed in a vernacular language for people who cannot read Latin.
The idea that geometrical structure can be taught visually, and that a diagram is an argument rather than a decoration, has a long afterlife. It reappears in the nineteenth century in Oliver Byrne’s coloured Euclid, which replaces letters in the proofs with coloured figures, and it is why the propositions of the Elements work as well on a wall as a set of fourteen posters as they do on a page.
Dürer’s own medium was the printed image, and the relationship between how science and mathematics are pictured and how they are understood is the subject of Portraying Science, which collects the visual conventions through which knowledge has been presented.
What He Was Doing
The Underweysung is not a research contribution. Very little in it was new to mathematicians, and Dürer knew that. He was moving knowledge across a boundary: out of Latin and into German, out of the universities and into the workshops, out of proof and into construction.
He also chose to write it at the height of his fame, when he had nothing to gain professionally from producing a technical manual for other craftsmen. His stated reason was that the German artists who came after him should not have to travel to Italy and beg for the secret, as he had.
No products in the cart. 