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ARCHITECTURE AND GEOMETRY IN THE AGE OF THE BAROQUE
By GEORGE L. HERSEY The University of Chicago Press Copyright © 2000 The University of Chicago
All right reserved. ISBN: 978-0-226-32784-6
Chapter One INTRODUCTION
This book deals, in what I hope is a refreshing and useful way, with a well-worn subject: European architecture in the seventeenth and eighteenth centuries. I will not be attempting a geographical survey, nor will I be interested in style, iconography, patronage, or urbanism; nor in such juicier matters as deconstruction, commodification, gender studies, or body imagery.
My interests will be centered on something equally fundamental to architectural design, if not more so: geometry. I will look at the ways in which designs were laid out on paper and at building sites, and at the geometric figures that stand behind or within those designs. I will be interested in what the likes of Bernini, Borromini, Blondel, Guarino Guarini, and Wren learned from, or had in common with, the likes of Descartes, Galileo, Kepler, Desargues, and Newton.
The book consists of separate episodes of forgotten lore. This means that the book is not a summation of what is known about Baroque architecture, plus a summation of what is known about geometry in the period, with the two then somehow interwoven. Nor is it even limited to the specific ways in which architecture and geometry interacted. Given the advanced state of research now at the turn of the twenty-first century, these approaches would have produced a book that was almost more than massive. Instead, my "episodes" are exactly that-brief independent units that make up a longer narrative. Each episode, whether it has to do with epicycles or with "pregnant" Platonic solids, is sharply focused on a specific problem in architectural design: in these two cases, first, the geometric matrix of architectural spirals, and second the way one basic type of solid relates to another. The episodes are grouped into chapters based on their themes-for example, the idea that architecture is musical, or that it involves principles derived from optical instruments. I will readily admit that another author might write with equal justification about different episodes.
I also need to explain my idea of geometry itself. Partly this will be a matter of the science as we study it today; but it also comes out of a wider, less "scientific" geometry that was known and practiced in the seventeenth and eighteenth centuries. Not only geometry proper but geometrics (number and shape games) and geomancy (number and shape magic, prophecy) held sway during most of the period I write about. It is almost a truism to complain that in the 1600s the leading lights of the "century of genius" wrote as much about astrology, numerology, and alchemy as about what we would call science. Rather than decrying such beliefs, however, I will be showing how they could condition and guide both "science" and architecture. These wider kinds of geometry make up much of the book's "forgotten lore."
An important aspect of this geometry, as I hope to show right here in this introduction, is that some shapes and numbers were considered better than others. Johannes Kepler used the word "effable"-capable of speaking, of being expressed-to describe this aristocracy. Using a more common word for the same thing, the Modenese architect Guarino Guarini called these nobler shapes and numbers "rational." Such effable or rational shapes and numbers dominated Baroque architecture. I will also look at other forgotten Baroque terms for geometric and architectural forms. These too were hierarchical. I conclude this chapter by looking at another phenomenon deeply connected with these ideas, and that helped justify them: the invisible architecture of the heavens. This preeminently embodied effability and rationality. It was often advocated as a model for buildings on planet Earth.
BAROQUE ARCHITECTURE AND BAROQUE GEOMETRY
Though few modern scholars make use of the fact, or even seem to realize it, Baroque architecture was above all mathematical. It comes from an age when architects and patrons could think of buildings as "studies in practical mathematics," to use Virgilio Spada's phrase about a proposed Pamphilj villa in Rome. Any number of the major architects of the period were as much mathematicians as architects, and this when all of architecture itself could well have been called a subset of geometry. In this same spirit the preface to Guarino Guarini's posthumous Architettura Civile says that the book shows how "excellent a geometer Father Guarini was, how versed and profound in all parts of mathematics, and especially the part that constitutes civil architecture."
Once again the mathematics of architecture, after a lapse of several centuries, is beginning to fascinate people. In writing the following chapters I have been particularly aware of my debts to three recent magnificent studies. The first, by the late Robin Evans, is The Projective Cast: Architecture and Its Three Geometries (Cambridge: MIT Press, 1995). As its title indicates, this is about projective geometry in architectural design. The second book is Architectural Representation and the Perspective Hinge, by Alberto Pérez-Gómez and Louise Pelletier (MIT Press, 1997). This deals with more or less the same things as Evans's book (linear perspective as a form of projective geometry), but with instructive differences. The third book is Lionel March's Architectonics of Humanism: Essays on Number in Architecture (London: Academy Editions, 1998). This, more than any previous publication that I know of, makes the architectural geometry of the Italian Renaissance crystal clear to non-mathematicians. And it does so in a completely visual way. It is my hope, in the following pages, to do the same for Baroque architecture.
All three books, in short, attack the same mountains of unstudied material that I do, and do so with finesse and learning. My approach, however, will be different from those of Evans, Pérez-Gómez, and Pelletier. They take in wide swaths of architectural history. I limit my undertaking to the European Baroque, much as March limits his to the Italian High Renaissance. In addition I will be making new inroads into these mountains, into rocky and forbidding terrain. I go inside the geometrical principles, I anatomize the theorems, I work out (at least some of) the problems. And I think I say things-in the field of architectural analysis-that have not been said before, and do things not done before, at least not in the last 250 years or so.
More specifically, the following chapter, "Frozen Music," is predicated on the ways in which heaven-derived musical sounds, and music itself, can be translated into visual form. The chapter concentrates on the geometry of the heavens and on the idea that this geometry can be read musically and, indeed, that it actually embodies the unheard melodies that Keats wrote about. As a corollary, we will see how the geometries of architectural structures, such as Bernini's baldacchino in St. Peter's, may be musically understood. I go so far as to write out sample chords and tunes.
Chapter 3, "The Light of Unseen Worlds," fastens on optics, the science of light. Here again, heaven, the main source of earthly light, is the key. Light-projection, conic sections, and projective geometry were studied as heavenly emanations. These studies determined the shapes of the lenses and reflectors used in optics. We will see how, in architecture, domed structures also used these forms-how in a way they were telescopes for imaging their painted vaults-and how the instruments designed to exploit light (the camera obscura, the magic lantern, the microscope, and the telescope) had architectural as well as scientific influence.
The fourth chapter, "Cubices Rationes," deals with the Baroque mystique of the cube. The cube was considered the parent-literally-of all other forms. More curiously still, theorists like Kepler and Athanasius Kircher interpreted the reproduction of geometric forms, notably the Platonic solids, sexually. Like lusting gods and goddesses these shapes (which were particularly effable) kiss, have intercourse, and above all give birth-though some of them are gay. These geometrical joinings and unions lead onward to Kepler's ideas about tiling. Tiling is the interlocking and interpenetration of exactly fitting plane and solid shapes. It is, we will see, an important Baroque architectural phenomenon.
We also find intensive development of what at the time was turning into the modern science of symmetry. Architectural examples of reflective, translatory, and glide symmetries will be investigated, by themselves and in architectural settings. Spiral symmetry, especially, was of key importance in the design of buildings. And it spoke out with special strength in that characteristic piece of Baroque decor, the Solomonic or twisted column. All this constitutes the fifth chapter, "Symmetries."
Chapter 6, "Stretched Circles and Squeezed Spheres," investigates the beauties of distortion. For all its reverence for effable shapes, Baroque architecture loved to pull, push, squeeze, and stretch those shapes. We look at some of the results in the work of Borromini, Bernini, and Blondel. And then, returning to the subject of tiling, we look at the ways in which these "distorted" shapes, usually unstable in themselves, could be packed or tiled into stable lattices to form buildings or parts of them.
Chapter 7, "Projection," deals with one of the great mathematical achievements of the French seventeenth century: projective geometry. This was discovered or invented by Girard Desargues but has a lot to do with earlier concepts such as Descartes's picture of physical reality as a geometrical and algebraic lattice; and, indeed, with Renaissance perspective, which, as I will note, is a form of projective geometry. I will deal with this latter subject, emphasizing the so-called costruzione legittima; with Desargues's treatises on perspective and geometry; and in a wider context with Desargues's architecture, showing how spiral staircases are projective.
In chapter 8, "Epicycles," we turn to a type of planetary movement that was ultimately rejected by Baroque astronomers but that continued to have plenty of fallout in earthly geometry. We look at simple epicycles, epicycloids, eccentric epicycles, elliptical epicycles, and related forms of broken symmetry. These serve to analyze and explain several further aspects of Baroque architecture.
The last chapter, "Unforgotten Lore," considers how all this plays into the earlier twentieth-century architectural "modernism." This happens, first, when the Baroque ideas survive, unsuspectedly, into later periods. I instance two giants, Frank Lloyd Wright and Le Corbusier. These men were geometers -neo-Baroque geometers, I will claim-in an architectural world that was decidedly antigeometric.
EFFABLE SHAPES
I have said that in the age of the Baroque people believed in hierarchies of number and form. Indeed it was claimed that only these number-aristocrats, these rational or effable numbers, were "real" in the mathematical sense. Or as Guarino Guarini put it: "Some proportions are effable, and can manifest themselves by [rational] numbers, for example the proportion of an inch to a foot, 1:12. But other proportions are ineffable and cannot be expressed in [rational] numbers, but are called irrationals, for example the side of a square with its diagonal, as proved by Euclid Book 12, Proposition 4." In other words, the ratio 2:3 is effable but the ratios 1:[pi] and 1:[square root of 2] are ineffable. Of course, given the complexities of putting up a building, ineffable numbers and shapes could hardly be avoided entirely. They were, however, avoided as much as possible, and certainly so when the architect was laying out a building's principal proportions. People saw effable or rational shapes as part of a divine order involving all the regular or geometric solids-not only cubes but spheres, certain parallelepipeds, the Platonic and Archimedean solids, and other select two- and three-dimensional shapes. Add to this the fact that it was much easier to measure the area or volume of a "perfect" solid, such as a sphere, than those of an irregular shape. A sphere's volume can be neatly stated:
v = 4/3[pi][r.sup.3].
The volume of an irregular sphere-like form, on the other hand, depends entirely on its particular irregularities and distortions. To paraphrase Tolstoy on happy and unhappy families: all regular spheres are the same; each irregular one is different in its own way. This sense of the perfect sphere's greater measurability, and therefore superiority, applies to all the other effable forms.
The perceived superiority of easy-to-measure shapes still lurks in the language we use today. For example, we now know, thanks to Newton, that Earth is not a sphere but an oblate spheroid-a squashed sphere (see chapter 6). And the very word "spheroid" implies something that has degenerated from an earlier perfection. A spheroid is like a humanoid or a factoid-not the real thing, not effable.
Presumably we would feel a similar weakness, a similar decay from some original simple beauty, if, for example, a Renaissance prototype for any number of Baroque churches, the Gesù, instead of being built up of double squares and perfect circles, had been composed of square-oids and circle-oids like a twentieth-century building. Nevertheless, I repeat, just such squeezed and distorted shapes will play their role in the following pages; but often they do it to set off, to express more forcefully, some effable beauty, present or implied. We are told that in the eighteenth century a beautiful lady, as the ultimate touch in her toilette, would carry a tiny pug dog. This enhanced her own beauty by setting it off with a contrasting piece of ugliness. In the same way square-oids and circle-oids, in the lesser or marginal spaces of a Baroque building, enhance by contrast the perfect shapes in its main areas.
Then let us look at the Gesù-at its plan by Giacomo Vignola (fig. 1.1). If you measure it with a pair of compasses you will find that it possesses, packed tightly together edge to edge, the effable, rational shapes listed in table 1.1. In other words, the plan of the Gesù is what I will call a tiling of squares and double squares. The circles of its dome and apse correspond to the squares of the nave and transepts. The diameters of these circles equal the squares' sides. Two mutually interlocked geometries, one of circles and one of squares, subdivided, clustered, and overlapped, create just about everything that exists in this plan. Not every one of the Gesù's innumerable offspring have just these shapes and just these clusterings, but those things are the genetic pool of that progeny.
SQUARE-BASED PROPORTION
Effable shapes had other special names. In the age of the Baroque, proportions like 3:2 and 5:4 were not written out that way-at least not before Newton at the end of the seventeenth century. Instead, a cumbersome but revealing Latin terminology was used. Each author had his own variations, but the basics are the same. Though hard to grasp, the system opens the door to a really different way of thinking about numbers and forms-different from our way, yes, but it was Bernini's, Blondel's, and Wren's way.
We begin by looking at a Latin word for which there is no English equivalent: sesqui. It means "more by a half." We still say sesquicentennial to mean a 150th anniversary. A sesquicentennial is a centennial plus "more by a half." But sesqui can be used in other ways. Sesquitertial means the ratio of 4:3. In this case the "sesqui" is modified by the "tertial" to mean: "more by 1/3." That is, 3 plus 1/3 of 3. (That number, of course, would more commonly be known as 4.) Or you can say that a rectangle is 3/3 x 4/3. (But, in this period, you do not write 3:4. You can see why recent generations have considered this system problematic.) To continue: sesquiquarta means "more by 1/4," sesquioctava "more by 1/8," and so on.
In short, the classical and early modern tradition thinks of rectangles that are some unit along one side, and then, on the other side, that same unit plus a specific fraction of itself. The terminology can be applied to many other shapes besides rectangles.
(Continues...)
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