the invention of infinity: mathematics and art in the renaissanceby j. v. field

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The Invention of Infinity: Mathematics and Art in the Renaissance by J. V. Field Review by: Lyle Massey The Art Bulletin, Vol. 83, No. 3 (Sep., 2001), pp. 564-566 Published by: College Art Association Stable URL: http://www.jstor.org/stable/3177245 . Accessed: 15/06/2014 13:18 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . College Art Association is collaborating with JSTOR to digitize, preserve and extend access to The Art Bulletin. http://www.jstor.org This content downloaded from 91.229.229.13 on Sun, 15 Jun 2014 13:18:37 PM All use subject to JSTOR Terms and Conditions

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The Invention of Infinity: Mathematics and Art in the Renaissance by J. V. FieldReview by: Lyle MasseyThe Art Bulletin, Vol. 83, No. 3 (Sep., 2001), pp. 564-566Published by: College Art AssociationStable URL: http://www.jstor.org/stable/3177245 .

Accessed: 15/06/2014 13:18

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

College Art Association is collaborating with JSTOR to digitize, preserve and extend access to The ArtBulletin.

http://www.jstor.org

This content downloaded from 91.229.229.13 on Sun, 15 Jun 2014 13:18:37 PMAll use subject to JSTOR Terms and Conditions

564 ART BULLETIN SEPTEMBER 2001 VOLUME LXXXIII NUMBER 3 564 ART BULLETIN SEPTEMBER 2001 VOLUME LXXXIII NUMBER 3

this book can be translated into Italian and English the better.

The book is beautifully illustrated and de- signed, with numerous of its color photo- graphs taken by the author himself. The im- ages include many works that have been since lost, as well as a fascinating selection of draw- ings, often little known. Early documents and printed books provide textual evidence to in- form the visual remains. The text is supported by a rich and up-to-date bibliography, which contains many useful titles in German that have not so far been incorporated into inter- national scholarship. Wolters has consulted a remarkable range of secondary literature, many of these titles, stretching back over more than a century, now nearly forgotten. The evidence for every point made in the text can be traced to its source through the pre- cise system of references. Although he is fas- tidiously cautious in his analyses of evi- dence-when he speculates he fully admits this-Wolters is not afraid to offer ideas on the interpretation of the subject matter of decorative elements.

The arrangement of the book is transpar- ent and generally logical. After a statement of purpose and a general review of the impor- tance of building materials in Venice, Wolters moves on to a survey of the current state of knowledge about the control of the architect over the design of details in Renaissance Venice. Workshop organization, the prepara- tion of models and drawings, and the extent of the architect's control over his craftsmen are carefully analyzed. This section is crucial in keeping the architect's role in the fore- front of the discussion.

The rest of the book is primarily organized according to decorative medium. Chapters fol- low on the cladding of walls, painted facade- decoration, ornamental stone carving, balu- strades and iron grilles, painted decoration in secular buildings, the design of doors and chimneypieces, wall painting inside churches, church furnishings, windows, floors, and cof- fered ceilings. The conclusion is merely a Nachwort of courteous acknowledgments. A final section to draw together the threads that define the relationship between theory and practice might have provided a more memo- rable climax.

Because frescoes in Venice deteriorate rap- idly, the sections on wall painting leave a particularly deep impression on the reader. Tragic losses include Giorgione's own house in Campo S. Silvestro containing his own fres- coes and the palace facade in Campo S. Ste- fano painted by Tintoretto with a Saint Vitale on horseback. Intriguingly, in 1556 an un- known parish priest bequeathed a house painted with Adam and Eve to designs by Raphael. The richness of the allover painted facades of the Palazzo Trevisan on the island of Murano and the Palazzo d'Anna, known only from drawings, can do no more than hint at the full extent of painted exteriors in

this book can be translated into Italian and English the better.

The book is beautifully illustrated and de- signed, with numerous of its color photo- graphs taken by the author himself. The im- ages include many works that have been since lost, as well as a fascinating selection of draw- ings, often little known. Early documents and printed books provide textual evidence to in- form the visual remains. The text is supported by a rich and up-to-date bibliography, which contains many useful titles in German that have not so far been incorporated into inter- national scholarship. Wolters has consulted a remarkable range of secondary literature, many of these titles, stretching back over more than a century, now nearly forgotten. The evidence for every point made in the text can be traced to its source through the pre- cise system of references. Although he is fas- tidiously cautious in his analyses of evi- dence-when he speculates he fully admits this-Wolters is not afraid to offer ideas on the interpretation of the subject matter of decorative elements.

The arrangement of the book is transpar- ent and generally logical. After a statement of purpose and a general review of the impor- tance of building materials in Venice, Wolters moves on to a survey of the current state of knowledge about the control of the architect over the design of details in Renaissance Venice. Workshop organization, the prepara- tion of models and drawings, and the extent of the architect's control over his craftsmen are carefully analyzed. This section is crucial in keeping the architect's role in the fore- front of the discussion.

The rest of the book is primarily organized according to decorative medium. Chapters fol- low on the cladding of walls, painted facade- decoration, ornamental stone carving, balu- strades and iron grilles, painted decoration in secular buildings, the design of doors and chimneypieces, wall painting inside churches, church furnishings, windows, floors, and cof- fered ceilings. The conclusion is merely a Nachwort of courteous acknowledgments. A final section to draw together the threads that define the relationship between theory and practice might have provided a more memo- rable climax.

Because frescoes in Venice deteriorate rap- idly, the sections on wall painting leave a particularly deep impression on the reader. Tragic losses include Giorgione's own house in Campo S. Silvestro containing his own fres- coes and the palace facade in Campo S. Ste- fano painted by Tintoretto with a Saint Vitale on horseback. Intriguingly, in 1556 an un- known parish priest bequeathed a house painted with Adam and Eve to designs by Raphael. The richness of the allover painted facades of the Palazzo Trevisan on the island of Murano and the Palazzo d'Anna, known only from drawings, can do no more than hint at the full extent of painted exteriors in Renaissance Venice. All that remains of Por- denone's frescoed facade of 1531-32 for the palace of the Flemish merchant Martino d'Anna on the Grand Canal, much lauded by

Renaissance Venice. All that remains of Por- denone's frescoed facade of 1531-32 for the palace of the Flemish merchant Martino d'Anna on the Grand Canal, much lauded by

Vasari, is a drawing in the Victoria and Albert Museum.3

The reader is treated to many wonderful visual surprises, especially from the interiors of buildings not normally open to the public, such as the ceiling joists covered with painted paper still visible in a house in Campo S. Stefano and the interiors of Palazzo Trevisan (a building probably designed by Daniele Barbaro) and the Casino Mocenigo, both on Murano. The recently uncovered fresco cycle in the Scuola dei Calegheri at S. Toma is now more easily accessible, thanks to its use as a public library. The illustration and discussion of many elements from the Palazzo Grimani at S. Maria Formosa, currently undergoing restoration, prove most welcome. Wolters publishes a drawing of 1598 by the German artist Heinrich Schickhardt, now in Stuttgart, of the remarkable illusionistic ceiling decora- tion by Cristoforo and Stefano Rosa formerly on the ceiling of the Madonna dell'Orto, de- stroyed in the 19th century. Because of the classification by medium, individual buildings receive piecemeal attention, but careful cross- referencing holds the structure together.

Wolters draws attention to other eye-open- ing details found in familiar buildings. How many of us have passed these without seeing them? Among such revelations he considers the ornate majolica tiles on the floor of the Cappella Lando in S. Sebastiano, the iron- work in the first-floor loggia of the Doge's Palace, and the red and white clay floor tiles in S. Giobbe. Now we shall give more atten- tion to the stunning marble pulpit in S. Gia- como dell'Orio and seek out the delightful frescoes of birds in the branches of trees in the sacristy of S. Salvatore. The sections on wall plastering, the application of marmorino stucco, and the discussion of when white- washed church interiors came to acquire pos- itive aesthetic associations all make invaluable contributions to our present understanding of the field. Even the section on capitals makes clear the previous neglect of this sub- ject; despite the prominence of the orders in most of the earlier literature, capitals that do not conform to canonical principles have been sadly marginalized.

Perceptive allusions to the treatise litera- ture pervade the whole book. Why, for exam- ple, did Serlio's recommendations for chim- neypiece design have so little impact? Following the disapproval of spatial illusion- ism voiced by Vitruvius and Alberti, Serlio criticized painted decoration that dissolved the wall surface, except on ceilings, where he considered views of the sky acceptable. Por- denone's dramatic rearing horse plunging outward over the Grand Canal from the Palazzo d'Anna thus perfectly exemplifies the sort of fresco decoration that Serlio was soon to censor. Even Carlo Borromeo's Instruc- tiones of 1577 furnished advice on details such as the design of doors.

No one involved in the field can fail to be

Vasari, is a drawing in the Victoria and Albert Museum.3

The reader is treated to many wonderful visual surprises, especially from the interiors of buildings not normally open to the public, such as the ceiling joists covered with painted paper still visible in a house in Campo S. Stefano and the interiors of Palazzo Trevisan (a building probably designed by Daniele Barbaro) and the Casino Mocenigo, both on Murano. The recently uncovered fresco cycle in the Scuola dei Calegheri at S. Toma is now more easily accessible, thanks to its use as a public library. The illustration and discussion of many elements from the Palazzo Grimani at S. Maria Formosa, currently undergoing restoration, prove most welcome. Wolters publishes a drawing of 1598 by the German artist Heinrich Schickhardt, now in Stuttgart, of the remarkable illusionistic ceiling decora- tion by Cristoforo and Stefano Rosa formerly on the ceiling of the Madonna dell'Orto, de- stroyed in the 19th century. Because of the classification by medium, individual buildings receive piecemeal attention, but careful cross- referencing holds the structure together.

Wolters draws attention to other eye-open- ing details found in familiar buildings. How many of us have passed these without seeing them? Among such revelations he considers the ornate majolica tiles on the floor of the Cappella Lando in S. Sebastiano, the iron- work in the first-floor loggia of the Doge's Palace, and the red and white clay floor tiles in S. Giobbe. Now we shall give more atten- tion to the stunning marble pulpit in S. Gia- como dell'Orio and seek out the delightful frescoes of birds in the branches of trees in the sacristy of S. Salvatore. The sections on wall plastering, the application of marmorino stucco, and the discussion of when white- washed church interiors came to acquire pos- itive aesthetic associations all make invaluable contributions to our present understanding of the field. Even the section on capitals makes clear the previous neglect of this sub- ject; despite the prominence of the orders in most of the earlier literature, capitals that do not conform to canonical principles have been sadly marginalized.

Perceptive allusions to the treatise litera- ture pervade the whole book. Why, for exam- ple, did Serlio's recommendations for chim- neypiece design have so little impact? Following the disapproval of spatial illusion- ism voiced by Vitruvius and Alberti, Serlio criticized painted decoration that dissolved the wall surface, except on ceilings, where he considered views of the sky acceptable. Por- denone's dramatic rearing horse plunging outward over the Grand Canal from the Palazzo d'Anna thus perfectly exemplifies the sort of fresco decoration that Serlio was soon to censor. Even Carlo Borromeo's Instruc- tiones of 1577 furnished advice on details such as the design of doors.

No one involved in the field can fail to be grateful for the clarity, visual sensitivity, and meticulous scholarship that characterize this fundamental work. Wolters's new book will be of immeasurable value for many generations.

grateful for the clarity, visual sensitivity, and meticulous scholarship that characterize this fundamental work. Wolters's new book will be of immeasurable value for many generations.

It will also enable all of us to look at Venetian Renaissance buildings with new enjoyment.

DEBORAH HOWARD

St. John's College Cambridge CB2 1TP

United Kingdom

Notes

1. Sebastiano Serlio on Architecture, vol. 1, trans. and ed. Vaughan Hart and Peter Hicks (New Haven: Yale University Press, 1996), 378.

2. Antonio Sagredo, Sulle consorterie delle arti edifi- cative in Venezia (Venice, 1856); Andre Wirobisz, "L'attivita edilizia a Venezia nel XIV e XV secolo," Studi veneziani (1965): 307-43; Susan M. Connell, The Employment of Sculptors and Stonemasons in Venice in the Fifteenth Century (New York: Garland Press, 1988); Egle Renata Trincanato, Venezia minore (Mi- lan: Edizione del Milione, 1948); Richard J. Goy, Venetian Vernacular Architecture: Traditional Housing in the Venetian Lagoon (Cambridge: Cambridge Uni- versity Press, 1989). Neither of the first two titles is listed in Wolters's bibliography.

3. Also associated with the d'Anna family is the complex of houses at S. Maria Maggiore, recently attributed to Sansovino by Manuela Morresi, for which a drawing in the Uffizi (U 203A) illustrates an elaborate scheme of painted decoration. See Mor- resi, Jacopo Sansovino (Milan: Electa, 2000), 327-32.

J. V. FIELD The Invention of Infinity: Mathematics and Art in the Renaissance Oxford: Oxford University Press, 1997. 264 pp.; 100 b/w ills., 32 color, 200 line drawings. $35

In Two New Sciences, a dialogue published to- ward the end of his life in 1638, Galileo ad- dressed the philosophical problem of mathe- matical infinity, essentially taking the Aristotelian position. Simplicio says to Salviati:

"From this [the problems posed by the mathematics of indivisibles] immediately arises a doubt that seems to me unresolv- able. It is that we certainly do find lines of which one may say that one is greater than another; whence, if both contained infi- nitely many points, there would have to be admitted to be found in the same category a thing greater than an infinite, since the infinitude of points of the greater line will exceed the infinitude of points of the lesser. Now the occurrence of an infinite greater than the infinite seems to me a concept not to be understood in any sense."

To this Salviati replies, "These are some of those difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and bounded things."1 Like Simplicio, most of us feel intuitively that the very exer- cise of quantifying infinity is paradoxical,

It will also enable all of us to look at Venetian Renaissance buildings with new enjoyment.

DEBORAH HOWARD

St. John's College Cambridge CB2 1TP

United Kingdom

Notes

1. Sebastiano Serlio on Architecture, vol. 1, trans. and ed. Vaughan Hart and Peter Hicks (New Haven: Yale University Press, 1996), 378.

2. Antonio Sagredo, Sulle consorterie delle arti edifi- cative in Venezia (Venice, 1856); Andre Wirobisz, "L'attivita edilizia a Venezia nel XIV e XV secolo," Studi veneziani (1965): 307-43; Susan M. Connell, The Employment of Sculptors and Stonemasons in Venice in the Fifteenth Century (New York: Garland Press, 1988); Egle Renata Trincanato, Venezia minore (Mi- lan: Edizione del Milione, 1948); Richard J. Goy, Venetian Vernacular Architecture: Traditional Housing in the Venetian Lagoon (Cambridge: Cambridge Uni- versity Press, 1989). Neither of the first two titles is listed in Wolters's bibliography.

3. Also associated with the d'Anna family is the complex of houses at S. Maria Maggiore, recently attributed to Sansovino by Manuela Morresi, for which a drawing in the Uffizi (U 203A) illustrates an elaborate scheme of painted decoration. See Mor- resi, Jacopo Sansovino (Milan: Electa, 2000), 327-32.

J. V. FIELD The Invention of Infinity: Mathematics and Art in the Renaissance Oxford: Oxford University Press, 1997. 264 pp.; 100 b/w ills., 32 color, 200 line drawings. $35

In Two New Sciences, a dialogue published to- ward the end of his life in 1638, Galileo ad- dressed the philosophical problem of mathe- matical infinity, essentially taking the Aristotelian position. Simplicio says to Salviati:

"From this [the problems posed by the mathematics of indivisibles] immediately arises a doubt that seems to me unresolv- able. It is that we certainly do find lines of which one may say that one is greater than another; whence, if both contained infi- nitely many points, there would have to be admitted to be found in the same category a thing greater than an infinite, since the infinitude of points of the greater line will exceed the infinitude of points of the lesser. Now the occurrence of an infinite greater than the infinite seems to me a concept not to be understood in any sense."

To this Salviati replies, "These are some of those difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and bounded things."1 Like Simplicio, most of us feel intuitively that the very exer- cise of quantifying infinity is paradoxical, since quantities seemingly must be tied to actual, limited objects rather than theoretical, unlimited objects. And yet it is precisely in the leap from an empirical insistence on the ge-

since quantities seemingly must be tied to actual, limited objects rather than theoretical, unlimited objects. And yet it is precisely in the leap from an empirical insistence on the ge-

This content downloaded from 91.229.229.13 on Sun, 15 Jun 2014 13:18:37 PMAll use subject to JSTOR Terms and Conditions

BOOK REVIEWS 565

ometry of objects in the world to the analytic insistence that mathematics need not be tied to visualizable objects that allowed for a math- ematics of the infinite. In 1641, Evangelista Torricelli presented the sizable community of European mathematicians with a proof show- ing that a solid could have infinite length but finite volume, his "acute hyperbolic solid." Imagining that "infinity could be measured using a solid of infinite length but finite vol- ume"2 seemed then, and still seems now, quite impossible (how could it be possible to measure the volume of something that ex- tends to infinity?), and yet this turn toward the imagined hypothetical in which the finite becomes measured by the infinite proved to be a crucial moment in the history of analytic geometry.

For art historians, however, to whom the arcane details of early modem mathematics may seem inscrutable at best and uninterest- ing at worst, the notion that infinity can or cannot be represented may be better associ- ated with the invention and exploration of linear perspective from the 15th century on- ward. It was Erwin Panofsky who initially sug- gested that perspective made possible the idea of representing the ostensibly unrepre- sentable infinite extension of space: "For it is not only the effect of perspectival construc- tion, but indeed its intended purpose, to re- alize in the representation of space precisely that homogeneity and boundlessness foreign to the direct experience of space. In a sense, perspective transforms psychophysiological space into mathematical space."3 Panofsky's famous essay on perspective was one of the first and most important attempts to show the epistemological links between art, science, and philosophy in the Renaissance and as such represents one of the best examples of interdisciplinary work that benefits each of the disciplines it draws from. It is a testament to the strength of interdisciplinary endeavors that in the fields of Renaissance and early modern studies, historians of science, com- parative literature specialists, and art histori- ans often seem to have a great deal to say to one another about the epistemological status of the image. Authors such as Barbara Staf- ford, Paula Findlen, and Eileen Reeves cross the boundaries of these disciplines on a reg- ular basis and to great advantage, opening up the discussion of image making and its rela- tion to science.

J. V. Field's book forges another inroad into the cross-disciplinary hybrid, this time from the direction of mathematics and its history. Field is a historian of science and senior visiting lecturer in art history at Birk- beck College in London. Her background in mathematics (she has a B.A. in mathematics as well as a Ph.D. in the history of science) gives her a purchase on early modern geom- etry that most of us studying perspective do not have. She has published books on the geometry of Johannes Kepler and Girard De- sargues as well as important articles on per- spective and mathematics, Piero della Francesca, and Masaccio's Trinity. Thus, she may be better placed than any art historian to

assess the relationship between Renaissance art and mathematics. In The Invention of Infin- ity, she attempts to do just that.

Field's goal in the book is to chart a course from 15th-century practical applications of mathematics in ordinary, everyday matters as well as artistic practice to the mathematiciza- tion of infinity and the projective geometry of Desargues in the 17th century. The book's implicit question is whether we can trace an epistemological connection between the art- ist's employment of geometry to achieve cer- tain pictorial ends and the development of higher mathematics. Field's interest in per- spective follows from the premise that this two-dimensional, essentially visual technique seems to provide a base from which a three- dimensional, purely analytic geometry is de- rived, a fact that annexes her project to Pa- nofsky's in at least one respect. Like Panofsky she is engaged in a retroactive archaeology, trying to mine the early history of perspective for what it can tell us about modem thought and science.

As a history of science, the book is an out- standing introduction to the complexities of early modem geometry. From this art histori- an's perspective, understanding Desargues is tantamount to deciphering hieroglyphics, and yet as she carefully takes us from Piero's ele- mentary proofs to Guidobaldo del Monte's seemingly impenetrable codes, and thence to Desargues's radical reformulation of geometry in three dimensions, it is possible to follow and to appreciate both the beauty and precision of mathematical inventions in this period.

In the first five chapters of the book, Field presents a fairly standard and somewhat cur- sory account of the development of pictorial perspective, starting with the medieval sci- ence of optics, or perspectiva, and the Renais- sance interest in Euclid and Vitruvius, moving on to Filippo Brunelleschi's experiments and Masaccio's Trinity, then to Leon Battista Al- berti's De pictura and other treatises on per- spective written and published in the 15th and 16th centuries. She does not have much that is new to add to this narrative, except perhaps the sense of how practical mathemat- ics was applied by artisans, craftsmen, mer- chants, and bankers alike. Her concern with artistic invention and the mathematics of per- spective centers for the most part on Piero della Francesca. It is obvious that she favors Piero precisely because he is the only painter whose mathematics is quite sophisticated, and thus he provides a bridge for her between artistic perspective and geometry. This serves her own purposes but also skews the history of perspective somewhat, with the result that many intervening figures, such as Leonardo, get little or no treatment.

Her work on Piero is detailed and illumi- nating, as she has spent a good deal of time laboring over his proofs and his three extant texts, the mathematics textbook Trattato d'abaco, the geometry treatise Libellus de quinque corporibus regularibus, and, of course, the treatise on painting and perspective, De prospectiva pingendi. While the abacus was pro- duced for purely instructional purposes, the

Libellus and De prospectiva pingendi were writ- ten as companion texts, demonstrating the link that Piero himself saw between geometry and the painter's perspective. Field notes that the Libellus is the first geometry treatise writ- ten since antiquity, and thus it demonstrates that "painters, using their skills at visualising structures in three dimensions, made a signif- icant contribution to the revival of a piece of geometry [the Archimedian solids] that was recognized as having its place in the learned tradition" (p. 78). This is not in and of itself sufficient evidence that the pursuit of per- spective led to a pursuit of geometry. Accord- ingly, Field shows that Piero was the first to actually provide a theorem, based on but transcending Euclid, for the distance point construction, although Piero in fact does not use either a centric or distance point in his proofs (pp. 39-40). In this Piero surpasses Alberti, whose mathematics, I would agree with Field, are shaky at best. Piero actually provides the mathematical justification for perspective that Alberti maddeningly leaves out of his elegant but largely uninstructive treatise.

However, in the end, Field does not suc- ceed in fully disclosing the geometric signifi- cance of perspective itself. Pictorial perspec- tive may or may not have been based on precise mathematical theory, and Field's book by no means clarifies this particularly murky historical problem. Most likely it was a combined application of Euclid's theory of proportional triangles and subtended angles of vision with a series of artisanal shorthand practices such as distance or bifocal construc- tion that led to the production of a theory of linear perspective. This was first articulated by Alberti in general terms and then made bet- ter sense of by subsequent authors. But even after it became a codified technique with its own history and literary and mathematical justifications, it seems clear that it was used by artists as a flexible tool for adjusting visual images rather than for making them conform to some quantifiable, mathematical template. Field demonstrates this herself in her metic- ulous attempt to decode the perspective of Masaccio's Trinity, only to find that this paint- ing, which "appears" to obey the strictest laws of perspective, cannot be mathematically jus- tified in any way. As she points out, a painting is not a theorem, nor is it generally intended to be. This is true even in the case of Piero della Francesca, whose mathematics seemed to have slightly overshadowed his artworks in his own lifetime.

Field's discussion gets more interesting when she finds herself on her own turf, so to speak. This comes when she begins examin- ing what she calls the "professional" en- croachment on perspective by mathemati- cians such as Egnatio Danti, Federico Commandino, Giovanni Battista Benedetti, and Guidobaldo del Monte in chapter 7. Here we get a sense of how perspective is simply one aspect of geometry, although an aspect that can lead in suggestive directions for mathematicians. For instance, Guido- baldo del Monte was interested in the point

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566 ART BULLETIN SEPTEMBER 2001 VOLUME LXXXIII NUMBER 3 566 ART BULLETIN SEPTEMBER 2001 VOLUME LXXXIII NUMBER 3

of "concurrence," that is, the point at which orthogonals converge (the so-called vanish- ing point), leading him to the important geo- metric observation that all sets of parallel lines will appear in the picture plane as a series of lines meeting at a point. This then leads in turn to the purely mathematical problem of finding the "positions of the points of concurrence for various sets of par- allels" (p. 173).

In chapter 8, Field gives a very compelling description of Desargues's work, which is one of the strongest and most important sections of the book. In it she illuminates Desargues's inversion of Alberti's "cone of vision," the basis of the foreshortened grid and its centric and distance points. Desargues imagines the grid as a "cone of which the curves are sec- tions" (conical sections or slices through cones that make circles or ellipses were a preoccupation with 17th-century mathemati- cians). For Desargues, the cone extends infi- nitely in two opposing directions, sort of like an hourglass, with each end functioning as a mirror image of the other. This double image extends infinitely in either direction and de- pends on "the proviso that the eye looks in both directions at once, so as to give us the true cone, to either side of its vertex, as in the work of Apollonius" (p. 205). Thus, De- sargues's theory of conic sections produces a definitive mathematical treatment of the grid as a projection into infinity, one that tran- scends all attempts at visualization. The eye, the presumed focal point of perspective, is now subsumed by the infinite extension of Desargues's "involution." Desargues, taking the perspectivist's preoccupation with fore- shortening objects in the grid, instead treats the grid as its own object, one that maintains the property of constancy under certain trans- formative operations. Or as Field puts it, De- sargues

... is not looking for what is changed by perspective, as artists were, but is instead looking for what is not changed. Perspec- tive is not being seen as a procedure that "degrades" but merely as one that trans- forms, leaving certain relationships the same. The relations that are left the same are what we now call "projective proper- ties" of the figure concerned. (p. 202)

In the long run, one has to ask what this book offers that is different from other at-

tempts to provide comprehensive surveys, such as John White's study of perspective in the Renaissance, Martin Kemp's history of science in art from the Renaissance through the modern era, or David Lindberg's history of vision and optics from the great medieval Arab opticians to Kepler.4 One thing that stands out about Field's book is that by spe- cifically foregrounding geometry instead of

optics or painter's perspective it attempts to show the effect that visual perspective had on

of "concurrence," that is, the point at which orthogonals converge (the so-called vanish- ing point), leading him to the important geo- metric observation that all sets of parallel lines will appear in the picture plane as a series of lines meeting at a point. This then leads in turn to the purely mathematical problem of finding the "positions of the points of concurrence for various sets of par- allels" (p. 173).

In chapter 8, Field gives a very compelling description of Desargues's work, which is one of the strongest and most important sections of the book. In it she illuminates Desargues's inversion of Alberti's "cone of vision," the basis of the foreshortened grid and its centric and distance points. Desargues imagines the grid as a "cone of which the curves are sec- tions" (conical sections or slices through cones that make circles or ellipses were a preoccupation with 17th-century mathemati- cians). For Desargues, the cone extends infi- nitely in two opposing directions, sort of like an hourglass, with each end functioning as a mirror image of the other. This double image extends infinitely in either direction and de- pends on "the proviso that the eye looks in both directions at once, so as to give us the true cone, to either side of its vertex, as in the work of Apollonius" (p. 205). Thus, De- sargues's theory of conic sections produces a definitive mathematical treatment of the grid as a projection into infinity, one that tran- scends all attempts at visualization. The eye, the presumed focal point of perspective, is now subsumed by the infinite extension of Desargues's "involution." Desargues, taking the perspectivist's preoccupation with fore- shortening objects in the grid, instead treats the grid as its own object, one that maintains the property of constancy under certain trans- formative operations. Or as Field puts it, De- sargues

... is not looking for what is changed by perspective, as artists were, but is instead looking for what is not changed. Perspec- tive is not being seen as a procedure that "degrades" but merely as one that trans- forms, leaving certain relationships the same. The relations that are left the same are what we now call "projective proper- ties" of the figure concerned. (p. 202)

In the long run, one has to ask what this book offers that is different from other at-

tempts to provide comprehensive surveys, such as John White's study of perspective in the Renaissance, Martin Kemp's history of science in art from the Renaissance through the modern era, or David Lindberg's history of vision and optics from the great medieval Arab opticians to Kepler.4 One thing that stands out about Field's book is that by spe- cifically foregrounding geometry instead of

optics or painter's perspective it attempts to show the effect that visual perspective had on modern mathematics. Thus, she is searching for specific connections between manual, vi- sual practices and purely theoretical ideas. For this reason, I believe that the book's

modern mathematics. Thus, she is searching for specific connections between manual, vi- sual practices and purely theoretical ideas. For this reason, I believe that the book's

project is unprecedented. I am not sure that it provides the clear links between these two things that Field herself seems to think it does, but the project itself is of great interest in a broader sense. That is, it actively chal- lenges the reader to account for how theories of image making transform scientific knowl- edge, rather than assuming that this chain of causality always operates in reverse. Her dis- cussions therefore have the virtue of being highly suggestive, beyond the scope of her own interpretations and analyses.

For instance, pictorial perspective seems to have been of minimal interest to those she iden- tifies as "professional" mathematicians, who in- stead concentrated on certain properties of ge- ometry from which perspective itself derives. One of the book's crucial problems is that we still do not understand how an essentially visual and representational practice, the art of per- spective, could serve as a springboard to ana- lytic geometry. While Field's description of De- sargues's theorems gives us an intimate sense of how he developed the projection to infinity of conical sections, it still is not clear how he did or did not use perspective as a springboard. One of the characteristics of perspective theory in the 15th and 16th centuries is that it is generally preoccupied with positioning the eye and de- scribing objects within a two-dimensional visual field. Treatises are typically concerned with de- scribing how to produce enclosed spaces, such as scene designs or ideal cities, or projections onto curved or domed surfaces, but even these are still imagined as visually circumscribed ob- jects, tied to the experience of seeing and to spatial orientation. The change from viewing perspective as a rule for foreshortening objects to a theorem for extending conical sections to infinity requires something like a paradigm shift, to use an overworked phrase borrowed from Thomas Kuhn, but one that seems highly appropriate in this instance. Perspective was in- vented to limit or define the parameters of vision in order to designate the viewer's rela-

tionship to objects in space. However, perspec- tive came to provide the theoretical foundation of a limitless and boundless space that is not "visualizable" but is instead purely mathemati- cal. The move from the first to the second case seems to me to require a radical epistemologi- cal shift. I think that Field's grounding in math- ematics adds a great deal to our understanding of how this shift might have taken place, but the book itself shows that there is a story that re- mains hidden from view, apparent only in the

gaps between the examples she explores. On the other hand, it is precisely this kind of in-

quiry that currently drives the fruitful intersec- tion of art history and the history of science.

Developing an adequate epistemology of the

image requires engaging in case studies similar to the one pursued here by Field. I am not

entirely sure that this is what Field intended to

produce with the book, but it is certainly one of its consequences as written, and a good reason, among many others, to read it as well.

project is unprecedented. I am not sure that it provides the clear links between these two things that Field herself seems to think it does, but the project itself is of great interest in a broader sense. That is, it actively chal- lenges the reader to account for how theories of image making transform scientific knowl- edge, rather than assuming that this chain of causality always operates in reverse. Her dis- cussions therefore have the virtue of being highly suggestive, beyond the scope of her own interpretations and analyses.

For instance, pictorial perspective seems to have been of minimal interest to those she iden- tifies as "professional" mathematicians, who in- stead concentrated on certain properties of ge- ometry from which perspective itself derives. One of the book's crucial problems is that we still do not understand how an essentially visual and representational practice, the art of per- spective, could serve as a springboard to ana- lytic geometry. While Field's description of De- sargues's theorems gives us an intimate sense of how he developed the projection to infinity of conical sections, it still is not clear how he did or did not use perspective as a springboard. One of the characteristics of perspective theory in the 15th and 16th centuries is that it is generally preoccupied with positioning the eye and de- scribing objects within a two-dimensional visual field. Treatises are typically concerned with de- scribing how to produce enclosed spaces, such as scene designs or ideal cities, or projections onto curved or domed surfaces, but even these are still imagined as visually circumscribed ob- jects, tied to the experience of seeing and to spatial orientation. The change from viewing perspective as a rule for foreshortening objects to a theorem for extending conical sections to infinity requires something like a paradigm shift, to use an overworked phrase borrowed from Thomas Kuhn, but one that seems highly appropriate in this instance. Perspective was in- vented to limit or define the parameters of vision in order to designate the viewer's rela-

tionship to objects in space. However, perspec- tive came to provide the theoretical foundation of a limitless and boundless space that is not "visualizable" but is instead purely mathemati- cal. The move from the first to the second case seems to me to require a radical epistemologi- cal shift. I think that Field's grounding in math- ematics adds a great deal to our understanding of how this shift might have taken place, but the book itself shows that there is a story that re- mains hidden from view, apparent only in the

gaps between the examples she explores. On the other hand, it is precisely this kind of in-

quiry that currently drives the fruitful intersec- tion of art history and the history of science.

Developing an adequate epistemology of the

image requires engaging in case studies similar to the one pursued here by Field. I am not

entirely sure that this is what Field intended to

produce with the book, but it is certainly one of its consequences as written, and a good reason, among many others, to read it as well.

While generally written at a very high level, Field does not assume that her readers are ter-

ribly familiar with mathematics in this period, and thus she writes as clearly as possible, obvi-

While generally written at a very high level, Field does not assume that her readers are ter-

ribly familiar with mathematics in this period, and thus she writes as clearly as possible, obvi-

ously trying to address a more general reader- ship beyond other historians of the same sub- ject. While this is admirable, one unfortunate consequence is that it is presented as a series of lectures (in fact, it is based on a series of lec- tures Field gave at Imperial College, University of London), and as such the book has not a single footnote and only one and a quarter pages of bibliography. Given the scholarly na- ture of her arguments, and also the necessary references to long-term debates (such as that over the nature of distance point construction versus Albertian "costruzione legittima," or the controversy over Abraham Bosse's defense of Desargues), this lack of footnotes has the dis- concerting and certainly unintended effect of making it appear as though Field believes she is writing in a void. The lacuna of citations can be especially aggravating. For instance, she refers to but does not name an art historian who mistakenly has said that Piero's second proof for perspective is incorrect (p. 89). By failing to name this author, she also fails to provide a forum for him or her to defend his or her interpretation, while at the same time she pre- vents her readers from comparing her argu- ments with those whose work she criticizes. It seems to me that the nature of this book is to provoke reasoned debate and constructive con- versations regarding perspective and its effects on Western thought and science. By omitting references, Field closes off opportunities for this kind of dialogue to take place. While I seriously doubt that this was intentional, it is an unfortunate result of the book's current pub- lished state.

LYLE MASSEY

Department of Art History Northwestern University

Evanston, Ill. 60208-2208

ously trying to address a more general reader- ship beyond other historians of the same sub- ject. While this is admirable, one unfortunate consequence is that it is presented as a series of lectures (in fact, it is based on a series of lec- tures Field gave at Imperial College, University of London), and as such the book has not a single footnote and only one and a quarter pages of bibliography. Given the scholarly na- ture of her arguments, and also the necessary references to long-term debates (such as that over the nature of distance point construction versus Albertian "costruzione legittima," or the controversy over Abraham Bosse's defense of Desargues), this lack of footnotes has the dis- concerting and certainly unintended effect of making it appear as though Field believes she is writing in a void. The lacuna of citations can be especially aggravating. For instance, she refers to but does not name an art historian who mistakenly has said that Piero's second proof for perspective is incorrect (p. 89). By failing to name this author, she also fails to provide a forum for him or her to defend his or her interpretation, while at the same time she pre- vents her readers from comparing her argu- ments with those whose work she criticizes. It seems to me that the nature of this book is to provoke reasoned debate and constructive con- versations regarding perspective and its effects on Western thought and science. By omitting references, Field closes off opportunities for this kind of dialogue to take place. While I seriously doubt that this was intentional, it is an unfortunate result of the book's current pub- lished state.

LYLE MASSEY

Department of Art History Northwestern University

Evanston, Ill. 60208-2208

Notes

1. Galileo Galilei, Two New Sciences, trans. and ed. Stillman Drake (Madison: University of Wisconsin Press, 1974), 39-40.

2. Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Ox- ford: Oxford University Press, 1996), 129. See chap. 5 for an expanded and remarkably lucid discussion of Torricelli's proof and its reception.

3. Erwin Panofsky, Perspective as Symbolic Form, trans. Christopher Wood (New York: Zone Books, 1991), 30-31.

4. John White, The Birth and Rebirth of Pictorial Space (1957; reprint, Cambridge, Mass.: Harvard University Press, 1987); Martin Kemp, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (New Haven: Yale University Press, 1990); and David Lind- berg, Theories of Vision from Al-Kindi to Kepler (Chicago: University of Chicago Press, 1976).

HARRY BERGER JR.

Fictions of the Pose: Rembrandt against the Italian Renaissance Stanford, Calif.: Stanford University Press, 2000. 624 pp; 32 color ills., 51 b/w. $85; $39.95 paper

The late, great Rembrandt, I am happy to

report, still lives. A veritable industry has

Notes

1. Galileo Galilei, Two New Sciences, trans. and ed. Stillman Drake (Madison: University of Wisconsin Press, 1974), 39-40.

2. Paolo Mancosu, Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Ox- ford: Oxford University Press, 1996), 129. See chap. 5 for an expanded and remarkably lucid discussion of Torricelli's proof and its reception.

3. Erwin Panofsky, Perspective as Symbolic Form, trans. Christopher Wood (New York: Zone Books, 1991), 30-31.

4. John White, The Birth and Rebirth of Pictorial Space (1957; reprint, Cambridge, Mass.: Harvard University Press, 1987); Martin Kemp, The Science of Art: Optical Themes in Western Art from Brunelleschi to Seurat (New Haven: Yale University Press, 1990); and David Lind- berg, Theories of Vision from Al-Kindi to Kepler (Chicago: University of Chicago Press, 1976).

HARRY BERGER JR.

Fictions of the Pose: Rembrandt against the Italian Renaissance Stanford, Calif.: Stanford University Press, 2000. 624 pp; 32 color ills., 51 b/w. $85; $39.95 paper

The late, great Rembrandt, I am happy to

report, still lives. A veritable industry has

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