It would be necessary to review the paper line by line in order to give substance to this criticism. I think however that it can be briefly summed up in two points:
Let us look at some examples. What exactly is meant by the
statement, "during this time, reasoning shifted from the
isolated to the integrated" (p. 38)? Almost nothing; it
is a statement that could be applied to any scientific theory
that assembles and corrects knowledge that was previously scattered.
The phrase, "in Newton's The truly troubling point, central to the discourse as a whole,
comes when the paper deals with "transformational geometry".
In the history of mathematics, the theory of transformation is
a precise point of view that goes back to the second half of
the 1800s, and whose general formulation was first given by Felix
Klein in 1872. Citing Euclid's demonstration of the Pythagorean
theorem in this context shows that Clagett has a weak grasp of
the material. There is no doubt that he intends a reference to
the theory of groups of transformation, since he talks constantly
of rotations, translations, reflections, and other transformations
of the plane.
Let us take an example. Close to the beginning of the article is set forth a table which, according to Clagett, serves to place the Central European Baroque church in the context of the mathematical development of the seventeenth and eighteenth centuries:
What can be deduced from this table? Absolutely nothing. There
is no relation whatsoever between the If Clagett wished to demonstrate, for example, that a given architect of the 1600s was influenced by the Pantheon, no proof would be necessary beyond stylistic resemblances. The importance and notoriety of the model would be in itself sufficient proof. The situation changes when it is necessary to demonstrate that a given architect took as his point of departure the most abstract mathematics of his day: in this case it is necessary to find direct connections. Are there letters in which the architect in question declares that he has read a certain book? Do we know if he was actively engaged in mathematics, or if he was in contact with mathematicians? In the case of Thomas Jefferson, for example, it is known that he had in his library various texts of higher mathematics. If we were to find in Monticello specific geometric coincidences then we would be justified in deducing that these are not chance occurrences. These are the methods with which history is written. I don't see any particular signs of the influence of higher
mathematics in the architecture that is discussed by Clagett.
He speculates, for example, that the attempt to make squares
and circles coincide derives from the problem of the quadrature
of the circle (p. 41). In reality, there are no necessary connections
between these two forms: every high school student can draw circles
in squares and squares in circles without wanting the prove any
theorem by doing so. More generally, the use of circles, squares,
and ellipses arranged in various ways doesn't indicate a knowledge
of higher mathematics. It is obvious that particular symmetries
exist in Baroque architecture: a good portion of art has to do
with symmetry (or its absence). There are certainly examples
of figures that are rotated or deformed in Baroque decoration,
derived almost certainly from the artist's need to give movement
and variety to architecture elements used thousands of times
before. Is it possible to see how this phenomenon is connected
to Descartes's A last note on Clagett's weak thesis is shown on p. 49, on
which is discussed the influence of Desargues's projective geometry
on architecture. The works cited go from 1580 to 1766, and thus
show how Clagett confounds the study of perspective, which goes
back at least as far as Masaccio, with the mathematic discipline
known as projective geometry (and perhaps the descriptive geometry As the above arguments show, my critique
concentrates on two principle points, and this makes Clagett's
ideas appear to be better defined than they actually are. In
fact they are often so imprecise that it is difficult to analyze
them. This is in fact the main weakness of the paper: it is not
possible to say that Clagett has given even the slightest demonstration
of a possible connection between Baroque architecture and advanced
mathematics.
Spiritus Rector of the Nexus conferences,
had told me that John Clagett's paper on the "Transformational
Geometry and the Central European Baroque Church", published
in the first book, Nexus: Architecture and Mathematics
(1996), had been criticised by Sandro Caparrini. As I myself
had retained a good impression of this contribution, which I
had found stimulating at the Nexus 1996 conference in Fuccechio,
I expressed my surprise. Upon reading Caparrini's critique and
then twice re-reading Clagett's article, I had to admit that
Caparrini's critique, as far as it goes, is justified. But I
think that he does not sufficiently take into account that the
Baroque is a style particularly difficult to investigate and
to analyze, especially from the Nexus Architecture-Mathematics
point of view.And furthermore, I say, one has to take into account that
while, due to the lack of precise mathematical notions, Clagett's
views are not solidly enough grounded in the architecture side
of the Nexus, his remarks, nevertheless, seem to me stimulating
for further research, and to point in fruitful and promising
directions. Whereupon the One must consider, that most, if not all, concrete AM confrontations -- and the same holds even more so for the really established AM connections -- refer to either Greek, Roman, Romanesque, or Renaissance buildings, and sometimes also to modern ones. But rarely has there been a penetrating AM study of a Gothic or a Baroque building, especially of one that presents most of the characteristics of either of these two styles. The reason for this is obviously that the Gothic e and the Baroque styles present a more complex and difficult situation for AM research. Therefore AM investigations on either of these styles must always be considered as pioneer work; assertions in a direction that promise to be fruitful are meritorious, and in spite of their shortcomings and the criticism they deserve, must be welcomed by students of the Nexus-AM. And indeed, among the numerous points made by John Clagett, there are many that do point in a possibly fruitful direction for further research, and may even guide future studies to something more. In the following I shall discuss a few examples. - Baroque architects often tried to hide the geometric calculus, which underlies the plan of a building already before the details are worked out. This, I guess, was a reaction to the ideal of clarity and transparency that guided the great renaissance artists. And this is probably what Clagett has in mind, when he says "Yet Neumann's section.... ; as if Neumann once again shifted the layers of the chapels plan" (pp. 45-46). This deliberate hiding of the basic geometric idea is one reason that makes it so difficult to grasp the "geometric calculus" behind baroque buildings.
- But then Clagett also points to another important characteristic of the baroque style, e.g. (p. 45) "the intersection of geometric curves, surfaces, and figures, such as circles, squares, octagones, cupolas, etc. In the Romanesque and Renaissance styles, these geometric figures are presented as beautiful in themselves, each one contributing individually to the desired beauty and harmony of the building. But Baroque architecture is not satisfied by this "individual presentation," favouring M-constructions of "interpenetrating" curves, surfaces and 3-dimensional bodies, all chosen from a great variety. It is through the well-thought-out interplay of these M-elements that the architect expresses his ideas. This is an important point, even if Clagett does not sufficiently analyse the mathematics used in each case. However, for a student it is a useful starting point for penetrating into the puzzles presented by the Baroque style.
- The same must be said about his " five intersecting
quasi-ellipsoidal domes" (p. 45), also something deeply
characteristic for the Baroque style, where A makes quite important
use of M. But then, what exactly is "quasi-ellipsoidal"
? After his conference I had myself a very interesting conversation
with Clagett, during which he told me that the presumed ellipses
observed in architecture are often in fact "ovals".
Ovals, contrary to ellipses, are not curves defined by one law
only, but are put together from several arcs of circles in a
"smooth" way such that the tangent to the curve never
changes abruptly, but always continuously: an oval has no corners.
But when much later I showed Sylvie Duvernoy's very interesting paper on arenas from the Nexus 2002 conference in Obidos, "Architecture and Mathematics in Roman Amphitheaters" to my friend, mathematician B. Marzetta, he discovered and proved that any ellipse can be approximated as closely as one wishes by an oval and vice versa. This, of course, puts many often-made statements, whether on the Roman or the Baroque style, into question! Whether for 3-dimensional ellipsoids an analogue statement holds, i.e., whether there are "ovaloids", I do not know. A three-dimensional ovaloid could be generated through the rotation of an ordinary two-dimensional oval around its major axis of symmetry. Cutting this ovaloid in two by means of a horizontal plane of symmetry, the upper part may serve as the cover of a Baroque church. A possible candidate for this, among many others, is Borromini's S. Carlo alle Quattro Fontane in Rome. This situation presents a series of interesting questions and problems for Nexus research, which range from measurement techniques, through stylistic investigations of historic buildings, up to aesthetic theories on Baroque art. - I must admit that the meaning of the term
*Zweischaligkeit*, literally "bi-shelledness" (p. 48), borrowed from the German, did not become clear to me; neither did I understand to what exactly in the building it refers, nor what in general its aim and function in Baroque architecture are. - Clagett uses explicitly the term
*Gesamtkunstwerk*( p. 37), which was coined by R. Wagner who introduced it into his theory of the*Musikdrama*. Here Clagett makes a very fortunate point. That the Baroque style artistically combined architecture, sculpture, painting, and especially the art of decoration to a degree not ever seen in Europe before or since, was, of course often noted. But I wonder, whether the full importance of the ambition of the Baroque*Gesamtkunstwerk*was always grasped sufficiently and seriously enough. For here we probably find the highest ambition of the Baroque art, and this too is an open field for all kinds of Nexus research. - It is curious that Clagett does not mention at all the "artificial architectures" created in so many Baroque churches and palaces, together with the sister-art painting, and, of course, also with the sister-art M, namely, the understanding of the rules of perspective. This heralds a first, of course only intuitive, understanding of projective geometry. The first example of such an artificial architecture, extending and covering the built one, may be Michelangelo's ceiling of the Sistine Chapel, and the most virtuoso performance is probably Pozzo's ceiling in San Ignazio in Rome. Later we find, of course, many such painted extensions in Austrian and South German churches and palaces as well. Incidentally, when one manages to see and appreciate the built and the painted architecture as one single building, one realizes that one criticism often made of Baroque art is unjust: Baroque decorations, which are often felt to be overloaded and even bombastic, will be appreciated according to their just value if one realizes that they belong to one building only, which is, however, about twice as high as the purely architectural structure!
These examples shows, I think, where the merits of Clagett's stimulating conference as well as its weaknesses lie. He points to quite a few, important characteristics of Baroque architecture and style, and they are a good starting point for Nexus reflections as well as for concrete AM- research, but indeed, much remains to be done! For, unfortunately, and this holds especially for the end of the article, the respective roles of A and especially M as well as their meeting points, i.e. the Nexus are not precisely enough analyzed and worked out. While he leads our attention to many an interesting question, his somewhat cavalier attitude to mathematical vocabulary and theorems, suggest to the reader often another answer to it.
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