Instruments of Thought: Another Classical Tradition

Bernard Cache developed the concept of nonstandard architecture in his book Earth Moves, published by MIT Press in 1995. This concept was given the name “objectile” by Gilles Deleuze in his book Leibniz: The Fold. In 1996, Bernard Cache founded the company Objectile in order to conceive and manufacture nonstandard architecture components. He teaches nomadically at many universities and currently at Cornell University Department of Architecture.
The Antiquities of Athens, James Stuart and Nicholas Revett. Elevation with sundial lines and plan
The Antiquities of Athens, James Stuart and Nicholas Revett. Elevation with sundial lines and plan

Parametric design and mechanics are not new in architecture: they were actually there from the very beginning of the written sources of our discipline. To understand this, we must return to classics such as works by Euclid, Plato, and Vitruvius. A renewed reading of those classics has the potential to bring contemporary architecture back into a historical tradition. In order to demonstrate this, I will mainly rely on an intermediary character between our time and antiquity: Albrecht Dürer, the great artist of the German Renaissance, who lived at the turn of the 16th century.

Instead of discussing his paintings, however, I will mainly focus on his treatise of geometry, Underweysung der Messung or Instruction in Measurement. It is a rather unique case in the history of mathematics, since it is written by a non-mathematician for non-mathematicians. Rather than showing geometrical figures, Dürer often shows mechanical instruments that are intended to be used to draw curves. These instruments take on so much importance that sometimes the curves themselves are not even represented. Thus, in order to read this classical text, we can use cad/cam software, which enables us to establish parametric relations between elements in the figures, model the mechanical instruments, and simulate their movements in order to generate their trajectory.

Mechanics has been part of architecture since antiquity. Vitruvius’s second definition of architecture in De Architectura[1] consists of three parts: the construction of buildings (aedificatio), the construction of solar clocks (gnomonica), and the construction of machines (machinatio). The first building mentioned in this treatise is the Tower of the Winds.

While this looks like a building, it is actually a masterpiece of gnomonica and of machinatio, designed by a master clockmaker. There was a sundial on each wall of this octagonal tower, the lines of which were clearly indicated by Stuart and Revett in their book The Antiquities of Athens.2 Unfortunately, the tower is empty today, but there are traces of a mechanism on the ground. American archaeologists discovered that the cylindrical tower at the rear of the building was a water tank, from which water spilled into another tank. There, a float lifted a cable which, in turn, wound a rope around an axis to cause a large bronze disk to rotate.

The disk was a stereographic projection of the celestial sphere, being used as a clock in the Tower of the Winds. The problem was that the water was spilling at a constant rate, but in antiquity, hours did not have a constant duration. The day was divided into 12 hours, both in summer and in winter, so the hours of each day varied. Thus, to measure a varying time with a mechanism rotating at a constant speed, a network of conversion curves was built.

The heavy black line is the stereographic projection of the horizon. Above this horizon circle are the hours of the day, and underneath are the hours of the night. The curves divide the arcs into 12 equal and varying parts along the year. This was one of the first clocks to enable the time to be known during the night or when there was no sun.

Thus, this Tower of the Winds incarnates quite well the three parts of the Vitruvian definition of architecture: it is a building (aedificatio), it has at least eight planar sundials (gnomonica), and, above all, it is a machine (machinatio). It was a machine that produced information; what we today would call a computer, from the Latin verb computare, or to calculate.

Dürer inscribes himself within this antique tradition of variation, both parametric and mechanical, by proposing to us a variety of instruments. Since he is the first to write a treatise in German, he has no scientific vocabulary at his disposal. He invents names for his drawing instruments by taking names of animals: the snail, the spider, and the snake. Besides the drawing instruments, he also proposes calculating instruments that provide solutions to complex mathematical problems.

The snail line (schneckenlinie) comes directly from Vitruvius’s description of the ionical volute in De Architectura, book III. However, this particular chapter of Vitruvius’s text is very difficult to understand, if only because the ten figures at the end of the treatise have disappeared. As a result, Dürer turned toward Alberti’s interpretation, which is based on the same principle but is slightly different from Vitruvius’s.

The device that Vitruvius described could be generated by winding a thread around four nails positioned on the corners of a square inscribed within the oculus of the ionic volute.

A thread is rotated around four nails. While rotating, each time the thread is intercepted by one of the nails, the moving section is shortened by the distance between two of the nails. Hence, the free end of the thread traces a spiral with a diminishing radius. This is typically what mathematicians call a “thread construction” (the nail holes are actually visible on a number of ionical capitals such as those from the “teatro maritime” of the Villa Hadriana at Tivoli).

The solution proposed by Alberti was with two nails, vertically placed, rather than four. This is only the first of the many snails Dürer will propose. Having completed this one, he writes: “Now, I will draw another spiral by a different method which can be put to many uses and is very useful. Much can be learned from it, and it can be used instead of the one described before.”[3]

Actually, we can take this as a general principle in Dürer’s geometry: There is nothing asserted in a figure that cannot be submitted to variation in a subsequent figure.

The next spiral is that of Archimedes. At the time, there was no device with which to draw this spiral.[4]

So, in order to draw the Archimedes spiral, Dürer took the radius of a circle and divided it into 24 equal parts. Then, he divided the circle into 12 equal angles, and he associated an angle with a length on the radius. This gave points around the circle and enabled him to draw interpolation curves between the points in order to draw this spiral.[5]

These interpolation curves are the ancestors of nurbs and B-spline used in cad software. Dürer introduces the third snail as follows: “Now I shall alter this previously made spiral once more. … But if you want to increase the spaces between the outer lines of the spiral even more and make those closer to the center narrower, then incline the vertical line ab at its upper end toward point c.”[6]

Dürer starts by drawing a varying diagram. It is an arc of a circle that is divided into equal parts that project from the vertex g onto the vertical line ab into varying segments, instead of the equal segments we had before with the Archimedes spiral. Then, not only is the aperture angle a variable parameter, but the line ab can also be slanted in order to get a different projection. We understand where this variable diagram comes from when we switch to the drawing of a tower with a written inscription in book III of Underweysung der Messung.

In this inscription,[7] the letters have a varying height in order to appear at an equal angle from point c. This brings us back to the optics of antiquity. Plato’s Sophist[8] explains that artists are perfectly entitled to deform the proportions of a statue that is placed at the top of a column in order that when seen from below, the statue appears to have the correct proportions. Dürer was among one of the first theoreticians of perspective but, interestingly enough, he resisted this new technique, even though he had invented the first mechanical imaging device that did not require a human eye.

In all previous perspectival devices, the eye of the observer played a fundamental role. In this particular one,[9] the “eye” is actually a nail in the wall from which you pull a string toward an arbitrary point in the object. The person on the right then marks the coordinate of the intersection of the string through the perspectival window. The person on the left turns back the sheet of paper and places a point onto the drawing. If these people were blind, it would work all the same. This is the first computer image in history.

It is important to understand that Dürer also uses this antique optical diagram for curves that he develops in 3d.[10] This leads directly to architectural applications. The diagram with equal arcs that project into varying segments reappears[11] where it is used to generate the spiraling fluting on the vertical shaft of the column. Three figures later, Dürer puts into variation everything proposed above, and makes the column itself spiral.[12]

Now we have reached the limit of the snail, and we come to the spider. We understand this image better[13] if we remember that Dürer was interested in simulating the movements of the human body. In the Codex Huygens,[14] we find drawings by Carlo Urbino di Crema, where the body movements are based on the a diagram of epicyclic circles similar to those generated by Dürer’s instrument.[15]

(L–R)<br />
Dürer’s configuration = one single retrogradation loop<br />
Inversion of the rotations = eccentric circle<br />
Generation of an ellipse as an epicyclic curve<br />
Barocco curve generated by Dürer’s spider instrument
Dürer’s configuration = one single retrogradation loop
Inversion of the rotations = eccentric circle
Generation of an ellipse as an epicyclic curve
Barocco curve generated by Dürer’s spider instrument

The parameters of the instrument vary, making the joints of the arm rotate at different speeds to get many kinds of curves:

Copernicus’s ideas circulated in Germany and Poland much before the astronomer published De Revolutionibus Coperinicus: De Revolutionibus.[16] Dürer was, thus, perfectly aware of the many uses of the curves that could generate this instrument in astronomy. For instance, the eccentric circle could be used to explain the varying duration of the seasons. Looped curves could be used to simulate the apparent retrogradation of the planets. Thus, in the Renaissance, the same principles were applied to simulate the movements of the human body as for the celestial bodies. Does this same principle of motion necessarily lead to a conception of harmony between the microcosm and the macrocosm? I am doubtful.

Let us look at the very famous “Vitruvian Man” that lies behind the magnificent drawing of Carlo Urbino di Crema. If we look closely, we realize that there is nothing that coincides in this drawing.

There is no particular relationship that would stipulate that there should be, at the hands, a small corner going outside the circle, and, at the feet, a large corner. The center of the square and the circle do not coincide. The latter is on the navel, while the previous is on the genitalia. Leonardo da Vinci knew how to draw, so if he had wanted to do it another way, he would have. Instead, he made a montage between two consecutive paragraphs of Vitruvius’s De Architectura, where Vitruvius describes first the human body lying on the ground, inscribed in the circle, the center of which is the navel. In the other paragraph, he speaks of the body standing up, which is inscribed in a square. He describes two different figures, one horizontal and one vertical. Leonardo combines the two to make us realize that nothing coincides in the body. There is a varying center: a discordance within the human body just as in the sky. The heliocentric conceptions of the Pythagoreans and then of Aristarchus of Samos were never totally eclipsed by the prevailing geocentric view,[17] and even without considering the persistence of the heliocentric system, we can find several instances of a discordant conception of both the celestial and human bodies.

An interesting example is that of Varro, an important writer and political figure, who lived slightly before Vitruvius. In the section VII of his book De Lingua Latina,[18] Varro states that the situation is similar for celestial and human bodies. The apparent center of the Earth would be Delphi (a religious center at the time), whereas the effective center is deep in the center of the sphere. In the human body, the apparent center would be the navel, whereas the effective center is “the organ that makes the difference between man and woman.”[19] Hence, the correspondence of human and celestial bodies lies on the fact that neither of those bodies is organized around one unique center.

Returning to Dürer, it is important to remember that his main purpose is to write a treatise on painting. His ambition is that his treatise would be a new De Pictura (Alberti), but, finally he splits his project into several books. Besides Underweysung (1525), he publishes a treatise on fortifications (1527), and one on human proportions (1528).

In the treatise on human proportions, Dürer rejects the idea of one single perfect system of proportions. There are differences between men and women, between adults and children, and also between different temperaments. For Dürer, there is only a system of variation. Besides static topological deformations, there are deformations of the human body with motion. In order to show this, he needs an instrument that generates lines in 3d space. This is our third animal—the snake or the serpentine line.

This is the only picture that Dürer provides that is related to the serpentine.[20] Strangely enough, he does not provide any drawing of the line itself. We have to use contemporary software in order to visualize it. Below are only two examples of the infinite variety of curves that can be generated with such an instrument.

This is described by Dürer in a parametric way: “The rods shall be arranged in a manner that they can be advanced by degrees and can be shortened or extended. The instrument should also be made with few or many dials or rods, according to the intended applications. The rods can be pulled apart or pushed together, also by degree, so that they become shorter or longer.”[21] This is perfectly compatible with what we can do with mechanical software today.

It is also interesting that Dürer rejects all systems of perfect harmony, but he still uses proportion as a tool to control the variety of objects that he can generate. At that time, proportions were the horizon of mathematical knowledge. People were using proportions to study phenomena that were not proportional, and this only changed much later with Newton and Leibniz. The problem is that when those new mathematical tools were invented, artists were not able to understand them. They took Newton as an excuse to reject proportions. To be clear: I am not advocating a discourse in favor of proportions; I am advocating being able to use the mathematical tools at our disposal in order to control what we are doing in design.

As we have seen, the classical is not necessarily classic: it is not necessarily invariable, symmetrical, ideal, abstract, or harmonic. More precisely, there is a plurality of traditions in antiquity. Andronikos’s Tower of the Winds was already an information machine. Additionally, Vitruvius’s catapult was a war machine, the main parameter of which depended on a complex mathematical formula with a cubic root which, in its turn, needed another machine in order to be calculated. Dürer precisely focuses on the instrument enabling the calculation of cubic roots, one of which was falsely attributed to none other than Plato. Hence, parametric design is not new, and mechanics were already an essential part of architecture in antiquity. It is precisely on these grounds—parametric design and numerical instruments—that we can think of inscribing contemporary architecture within this other classical tradition.


1. Vitruvius, De Architectura I, 3, 1.

2. James Stuart and Nicholas Revett, The Antiquities of Athens, Vol. I, London, 1762.

3. Albrecht Dürer, Underweysung der Messung, (1525), bk. I, fig. 7.

4. Huygens had invented such a device in 1630 and had discussed it with Descartes while Descartes was writing his own geometry treatise, but Dürer was not aware of this.

5. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 6b.

6. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 7.

7. Albrecht Dürer, Underweysung der Messung, bk. III, after 27.

8. Plato, Sophist, 360 B.C., 235e–236a

9. Albrecht Dürer, Underweysung der Messung, bk. IV, last image of the 1525 edition.

10. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 15.

11. Albrecht Dürer, Underweysung der Messung, bk. III, fig. 7.

12. Albrecht Dürer, Underweysung der Messung, bk. III, fig. 10.

13. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 40.

14. To be found in Erwin Panofsky, The Codex Huygens and Leonardo da Vinci’s Art Theory. The Pierpont Morgan Library Codex M.A. 1139. London, The Warburg Institute, 1940.

15. Carlo Urbino di Crema, Codex Huygens (c. 1570) Folio 6 (right) and 12 (left).

16. Nicolaus Copernicus, De Revolutionibus, Nuremberg 1543. One of the people he sent his manuscript to was Johannes Werner, who was Dürer’s mathematics teacher.

17. As we can see in Theon of Smyrna’s Mathematical Knowledge Useful for Reading Plato (2nd Century A.D.) (120).

18. This is mentioned in De Architectura.

19. Marcus Varro, De Lingua Latina, (1474), VII, 7.

20. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 42.

21. Albrecht Dürer, Underweysung der Messung, bk. I, fig. 41.

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