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Book Review

Piero della Francesca. A ’s Art Reviewed by Michele Emmer

earlierin La Voce (1916)? And could . A Mathematician’s Art Roberto Longhi [14] have discerned J. V. Field the “problem” of Piero without his Yale University Press, 2005 curiosity about the avant-garde (I 256 pages, US$55 am thinking of his great text on ISBN 0-300-10342-5 Boccioni’s sculptures and on Sev- erini’s pailettes)? And could the PierodellaFrancesca(c.1420–1492)wasthepainter academic historians have ever felt who “set forth the mathematical principles of per- it legitimate to re-engage the 15th spective in fairly complete form...Piero was the century (the pre-Raphaelists, as painter-mathematician and the scientific artist par they were called at the time) with- excellence, and his contemporaries so regarded out the ongoing debate fostered him. He was also the best geometer of his time.” by the painters’ discoveries and So writes Morris Kline in his monumental work the contributions of the letterati Mathematical Thought from Ancient to Modern (Bontempelli, for example)? The Times [11]. Clearly, the title of J. V. Field’s book, reason is that these artists of ours Piero della Francesca. A Mathematician’s Art, is were clearly neither nostalgic nor amply justified. interested by antiquity: simply, antiquity did not exist and they The Rediscovery of Piero della Francesca were perhaps constructing it them- Giotto and , Signorelli selves, searching the territory of and Piero della Francesca practical- a new and unpredictable avant- ly did not exist in the first decade garde. De Chirico did not look for of this century. They returned to museographic : he himself public attention due to the criti- underlined its value for individuals cal taste characteristic of Italian in the 1920s. No one would have , diametrically opposed been interested in , had to solid German philology. Would it not been for the research of an- Lionello Venturi’s book (1926) [18] other pentito futurist painter, Gino have come to life if the ex-futurist Severini. All in all, Art History itself Carlo Carrá had not written of would have been different without Giotto and Paolo Uccello ten years the contribution of these painters and technicians of image. Michele Emmer is professor in the Dipartimento di Matem- So wrote Maurizio Fagiolo dell’Arco in the cat- atica, “Castelnuovo”, Università di Roma “La Sapienza”, alogue of the exhibition Piero della Francesca e . His email address is [email protected]. il Novecento (Piero della Francesca and the 20th

372 Notices of the AMS Volume 54, 3 Century) [4]. The 500th anniversary of the artist’s long been one of my favorite masters. I have also death (1991–92) was the occasion of several im- been so lucky as to own a house in , in the portant conferences and exhibitions. Four volumes Marche region in the center of Italy, a few hundred worth mentioning were published: Piero and the meters away from the church where Piero’s famous 20th Century, the catalogue of the 1991 exhibition di Senigallia was housed until the begin- in San Sepolcro (Piero’s birthplace) [12]; Piero and ning of the Second World War, when it was moved , Piero and the Courts, the to the Palazzo Ducale in Urbino. It has never been catalogue of the 1992 Urbino exhibition [3]; Milton returned. But Urbino is quite close to Senigallia, and Glaser’s catalogue Piero della Francesca, for the there, in addition to the Madonna, one can see The exhibition held at in 1991 [16]; and final- Flagellation of Christ. I have therefore seen these ly Through Piero’s Eyes: Clothing and Jewelry in works by Piero dozens and dozens of times. In my Piero della Francesca’s Works[1].Theseconstituted own work, I have cited Longhi on severaloccasions. an important celebration of a great Italian Renais- As Dell’Arco says, that is where, for the most part, sance artist who influenced twentieth-century art, everything started. as witnessed by De Chirico’s metaphysical period Longhi was born in 1890. In 1911 he graduated (1910–20), with his and Modena cathedrals, from the University of Turin with a thesis on Car- histrains,thoseemptypiazzasthatconveyamyste- avaggio. His interest was not only in ancient art but rious and perturbing motionlessness, an enigmatic also in the avant-garde, specifically futurism and stillness. and so many others are Boccioni in particular. In 1914 he published a long also clear examples. articleon“PierodeiFranceschi[i.e.,dellaFrancesca] Dell’Arco added, in his chapter entitled “Styles and the Development of Venetian Painting” in the of the return to order: and , light, journal2 published by Adolfo Venturi, who had ”: been his professor in Scuola di perfezionamento Right after the [First World] war, (school for advanced studies). In 1927 he revised researchresumes in exactly the op- and republished this article in the journal Valori posite way from the past. Painting Plastici under the title “Piero della Francesca” [14]. returns to being precise, mentality Longhi proposed a look with fresh eyes at the ex- to being ancient, structures med- periments in painting carried out by part of the itated and color orderly. Giotto artistic culture of at the beginning of the and Paolo Uccello were evoked by fifteenth century, “when some artists, too much a master of futurism, Carlo Carrá. in love with space to come back to a superficial Frescoes became the subject of spazialitá (sense of space), too much in love with meditation, cassoni and predelle color to rely on a chiaroscuro intended to create an were reread, and Carrá and De illusion of space, understood that there was only Chirico point a clear finger to Mi- one way to express in a picture shape and color at lano’s neoclassicism, Mastrini to the same time: perspective.” Longhi also wrote: “By Etruria (ancient Tuscany) and Oppi constructing, throughaneffortatsynthesis, shapes to Giovanni Bellini. All of these are according to their countable and measurable sur- clear sources. The most secret, al- faces, [perspective] succeeded in presenting them beit the most fertile, will continue all asa projection upon a plane, readyto be clothed to be the meditation on Piero della in calm, broad of color” [14, p. 11]. Giaco- Francesca, the name which Roberto mo Agosti added, in his essay3 “From Piero dei Longhi brought to the fore starting Franceschi to Piero della Francesca”: “This inter- in 1914.1 Many artists welcomed pretation of perspective as the art of synthesizing Longhi’s book on Piero with memo- ‘shape and color at the same time’ allowed him rable texts. In fact, when the topics (Longhi)tobeginbyidentifyinginPaoloUccelloand of proportion, number, color-light, perspective, regular bodies are re- the Florentine models closest ferred to, Piero della Francesca is to Piero della Francesca; and this interpretation always implicit. [4] brought him to recognize the most evident man- ifestations of the visual influence of Piero in the I was very happy to have received J. V. Field’s worksofAntonellodaMessina,‘Themost disperato book on Piero della Francesca for review, because prospettico (enthusiastic expert in perspective) of I too have been influenced by reading Longhi’s the human figure’ and those of Giovanni Bellini, essays, by de Chirico’s metaphysical works, and by the of Morandi and Carrá. Piero has 2L’Arte; see footnote 1. 3 1R. Longhi, Piero dei Franceschi e lo sviluppo della pittura G. Agosti, Da Piero dei Franceschi a Piero della Francesca veneziana, L’Arte, XVII (1914), pp. 198–221 and 241–256. (qualche avvertenza per la lettura di due saggi longhiani), Reprinted in [14]. in [12], p. 199.

March 2007 Notices of the AMS 373 ‘who, without rejecting the problem of form, con- the purpose of exposing their alleged underlying siders it as taking to higher levels the problem of geometrical structure.” More frequently than not, color.”’ Longhi wrote: “Here is what constitutes the these geometric constructions are carried out us- unity-distinction of Antonello and Giovanni: their ing reproductions that are evidently quite different synthesis a priori is Piero” [14, p. 39]. And regard- from the original, at least in size. Field comments, ing The Flagellation, he added “the perfect union and one cannot disagree, that “trying to find per- between architecture and painting that emerges spective schemes seemsto me like trying to extract should be understood as a mysterious combina- the sunbeams from cucumbers.” The book is full tion of and painting”. Piero is a great of imaginative comments, mostly in the notes. I painter;Piero is a greatmathematician. will give a few examples. At the risk of seeming unscholarly, Field forgoes correcting errors in the Piero as a Painter, Piero as a “apparently relevant literature rather than stir my Mathematician own and my readers’ unhappy memories of maths In the introduction to the book under review, Field homework that came back covered with a teacher’s writes (p. 1): “Piero della Francesca is now best commentsin red”(p.5). remembered as a painter. He has a secure position In order to have a deeper understanding of in the history of Italian . However Piero’s mathematics, the author refers to Luca Pa- inhisownlifetime,andforsometimethereafter,he cioli (1445–1517), the mathematician and author was also known as a mathematician. In his detailed of the famous book De (On the scholarly monograph on Piero’s painting, Eugenio Divine Proportion) [15], with drawings by Leonardo Battisti went so far as to say that there should be da Vinci, where of course divina proportione refers a further volume to consider Piero’s activity as to the so-called Golden Section. The most recent a mathematician.” Field concludes that since the research on his life seems to show that Pacioli was mathematician and the painter were one person, it not a pupil of Piero, although both were born in seems clear that the mathematics and the painting Borgo San Sepolcro. In any case, what is sure is that shouldbe takentogether,to seewhat one mayhave Pacioli knew the mathematical works of Piero, at to tell us about the other. “As the title of the present least after Piero’s death in 1492. They were very study indicates, I am inclined to think Piero’s math- likely acquainted: Field argues that in the Madonna ematics does have a recognizable relationship with dell’Ovo,nowattheBrerainMilan,thereisaportrait his painting. Of course there is no avoiding the of Paciolidoneby Piero. recognition that Piero is likely to remain a more In fact, in Piero’s mathematics there is no evi- significant figure in the history of art than in the dence of an interest in divina proportione. So one history of ” (p. 2). It is important to under- should first look elsewhere in seeking to identify score, and Field does so right away, that there is an traces of mathematics in Piero’s pictures: “The inevitable shift in style between the historical study present study will accordingly look much more of art and that of mathematics. In mathematics carefully at Piero’s mathematics than previous the course of history can reasonably be considered studies have done. By doing so it will be possible to in terms of progress. In art this is clearly not the offer an explanation of that famous ‘stillness’. It is case. Piero’s art was forgotten for many centuries not possible to give mathematical proof, but from becauseofa change in aesthetictaste. my failure to find serious deviations from math- Field begins by asserting that “since Piero was ematically correct perspective I strongly suspect known to have written on the mathematics of per- that the reason why Piero’s pictures look mathe- spective, art historians have generally [following maticallycorrectisbecausetheyareindeedcorrect. Longhi,Imayadd—M.E.] beenreadyto describehis Thus everything shownis representedas seenfrom pictures as in some sense ‘mathematical’, though the single ideal viewpoint, though, thanks to the thequalitymostoftenremarkeduponisa‘stillness’ tolerance of the eye, it will of course look convinc- that is rather hard to define in any precise way. The ing even when seen from points at a considerable assertion that Piero’s pictures are ‘mathematical’ distance from the ideal viewpoint” (p. 7). The main is usually so vague that it is understandable that thesis of Field is that Piero’s very considerable some arthistorians have preferredlargelyto ignore mathematicaltalentdoes seem to have contributed it”(pp.3–4). Evenifthe greaterpartofPiero’smath- to hispersonality asa painter. ematicsisratherelementary,Iamnotshockedatall as a mathematician by the loose usage of the word Field’s Book “mathematics”. We are in the field of art history, Thebookstartswiththebackground:thetrainingof where opinions and preferences(suchas, for exam- an artist at the time of Piero, including the study of ple, the rediscovery of Piero) are important. Field the abacus. Probably Piero was sent outside of Bor- adds that “there has also been a most unfortunate go San Sepolcro to study in an abacus school. “Since fashion for drawing lines over Piero’s pictures with Piero never actually taught mathematics, his own

374 Notices of the AMS Volume 54, Number 3 interest in the subject has something of the same Field recalls one of Piero’s most famous master- purely intellectual motivation in it. Pieromust have pieces, The Baptism of Christ (, enjoyed mathematics.” Piero himself wrote a Trat- London). She carefully examines the proportions tato d’abaco (atreatise on the abacus). of the painting, measuring from photographs by The second chapter is a history of perspec- means of an ordinary ruler, without any claim to tive, starting naturally with precision and taking no notice of the scale of the (1377–1446) and Alberti’s treatise actual picture. This is not the only occurrence of (1413–1472). Field reconstructs through draw- this practice in the book. But Field reminds readers ings the perspective constructions of Alberti, later that the Baptism is often and entirely reasonably called costruzione legittima (correct construction). cited as an example of Piero’s beautiful rendering Investigating the work of (1386–1466) of light. Piero’s abacus treatise, quite unusually, and (1401–1428), Field remarks that “in treated three-dimensional geometry, as did his fa- any picture it is architectural elements that are mous Libellus de quinque corporibus regularibus, most likely to provide sets of orthogonals and incorporated by Pacioli in De divina proportione transversals” and points to Masaccio’s of (for which incorporation Vasari accused Pacioli of the Trinity in the church of in plagiarism). Piero’s mathematical thinking about as providing “a good example in which to three-dimensional geometry surely influenced his search for indications of the method by which the way of rendering bodies in space. Nevertheless, perspective scheme was constructed.” Neverthe- this need for perspective is clearly not manifest less, “the process of extending lines is in general in all of Piero’s works. In many of them there is to be avoided and its results treated with caution.” “no perspective”: figures stand out against the Finally, as was known to Masaccio, “mathematical background landscapes; color and light are used to correctness was not of the essence in obtaining the make bodies seem corporeal. This is the case in the requiredillusion ofthe third ”(p. 50). absolute masterpiece I have already spoken about, the . Regarding this work, The third chapter describes the early life of Field writes (p. 115): “For reasonsthat may be to do Piero—his family, his education, and his training withitsdelicatecolorandexquisitedegreeoffinish, as a painter—and refers to the problem of dating reproductions of this picture manage to look what Piero’sworks.Thebookisfullofillustrations,many passes for reasonably like it without looking suffi- of them in color. Most of the reproductions are ciently like itto be beautiful.” I canconfirm, as I have of good quality, with the exception of a few that seen the painting dozens of times, that the original are shown so small that details are impossible to is always better than any reproductions, given that read. On page 90 Field writes: “Certain elements in the picture’s texture and its purity are difficult to his style apparently owe a debt to the occasionally getacrossthrough a bookor apostcard. Itbecomes fussy prettiness of detailing found in, say, necessarytoimmerse oneselfinthe paintingandto (1395–1455c.) and Gentile da Fabriano (c.1370– feel part of it in order to capture the sense of space 1427).” Fabriano is a small town in the region of that Piero’s color makes the viewer perceive. That Umbria in the center of Italy, very famous for its an- magicalstillness. Little does itmatterthat“nogreat cient paper industry. Just two weeks before writing subtlety is required to note that the composition this review I visited in Fabriano an exceptional ex- presents us withfigures againsta backgroundfrom hibition of the works of Gentile and other artists of which they are essentially disjoint.” the same period, including Masaccio and Donatello. As she examines Piero’s geometrical construc- I agree with the opinion [13] of the curator of the tions, Field notes his paintings’ symmetrical prop- exhibition that Gentile, who spent part of his life in erties. She adds in a note (p. 126, footnote 38) that I , was the binding ring between the painting am sure will be of interest to , “The of Florence, Umbria, and Venice. He was one of the expertsonthemathematicsofsymmetryinthereal most important and brilliant painters that Italian world tend to be crystallographers. A crystallog- civilization has ever produced, not just a painter of rapher and a specialist in the relevant branch of “occasionally fussy prettiness of detailing”. mathematics have now written an excellently clear Chapter 4 bears the title: “The Sense of Space”. and well-illustrated book on the subject: I. Hargittai “InPierodellaFrancesca’stime,spacedidnotexist” and M. Hargittai, : A unifying concept (p. 95) is how the chapter begins. “In saying that [8].” I must say, knowing the Hargittais quite well, his paintings convey a sense of space, we mean that István is a chemist and Magdolna is not a that each picture conveys a sense of its own pic- mathematician. The book [8] is an elementary one, torial space, that is a particular space. But Piero meant for students. I suggest that the reader look himself would not have thought in these terms. In at Hargittai’s other books [9] and [10] on symmetry his day, space was not considered to be an entity and at what is considered the bible in this , with a separate existence. Space was defined as B. Grünbaum and G. C. Shepard, Tilings and Pat- extension, and measured by body.” As an example terns [7].Beingtheauthorofseveralbooksandfilms

March 2007 Notices of the AMS 375 onM.C.Escher,Idonotwishtocommentonthelast sentence of Field’s note: “In addition to its other virtues this book shows a highly commendable tendency to avoid the work of M. C. Escher.” In the other books of Hargittai, however, there are plenty ofworksby Escher. In Chapter 5 Field goes into the details of [2], Piero’s treatise on per- spective. “Perspective was a form of study in which itwasacknowledgedaslegitimatetousemathemat- ics in the pursuit of natural philosophy. The use of mathematicalmethods to give the illusion of depth in pictures clearly belonged to the mathematical end of the subject” (p. 129). The starting point of Pieroisthedefinitionoftheentities:“Apointisthat which has no parts, accordingly the geometers say it is imagined; the line they say has length without width” (p. 134). Field carefully examines the results on geometry enunciated and proven by Piero. She reconstructs in detail the diagrams, reproducing whenever possible the artist’s original drawings. His drawings become more and more complicat- ed; for example, he shows the construction of the perspective image of an octagonal building. The author states (p. 155) that “the mathematics con-

cerned need not to go beyond the methods we find Courtesy of Yale University Press. in Piero’s Trattato d’abaco.” Why not, if it can help Perspective drawing of an octagon building us understand? plan. The horizontal-vertical-diagonal- Piero defends the knowledge of perspective that horizontal line segments from P to P ′ is necessary for painting: “Therefore it seems to implement the geometric construction of the me that I should show how much this science is perspective transformation. From the Parma necessary to painting. I say that perspective liter- MS of De Prospectiva Pingendi. ally means things seen at distance, represented as enclosed within given limits [that is, on the picture plane] and in proportion, according to the quantity of the distances, without which nothing and in particular regarding the identity and the can be degraded correctly [Piero calls degradations significance of the characters, was recently pub- the deformations performed on the figures]...I lished in Italy by Ronchey: L’enigma di Piero [17] say it is necessary [to employ] perspective, which (following the big success of books like The Da distinguishes all quantities proportionately, as a Vinci Code, the words “code” and “enigma” have true science, demonstrating the degradation and become very appreciated by publishers). Pilate, the magnification of all quantities by means of lines” character on the left with purple boots, would have (p. 163). beenJohnVIIIPalaeologos,the nexttolastemperor of Byzantium. The character with his back turned The Flagellation of Christ to the viewer would have been the Turkish sultan. When talking about the true science of perspective, Jesus represents Western Christianity flagellated one cannot overlook The Flagellation of Christ. by the Turks. One of the characters to the right Field writes (p. 174): “It would be perhaps more would represent Johannes Bessarion, who later be- accurate to describe Piero’s Flagellation not as came a cardinal; the gentleman in brocade is Nicolò an example of correct perspective but as the ex- III d’Este of Ferrara, the site of the 1439 Council, ample. Despite longstanding disputes about the in which John VIII participated; the young man overall significance of the picture, and in particular betweenthem is Thomas Palaeologos, the lastlegit- the identification of the three figures in the right imate heir of the Eastern Empire. Thus the scene, foreground, the Flagellation is almost invariably painted in 1459, would have referred to events that chosen as an illustration to elementary lectures took place in 1439. Field’s volume also reproduces on perspective and has had a long career as the an image of a three-dimensional reconstruction of subject of reconstructions.” An essay exploring Piero’s painting carried out by Antonio Criminisi. new hypotheses on the meaning of this painting, Field comments (p. 177), “Apart from perspective,

376 Notices of the AMS Volume 54, Number 3 Courtesy of Yale University Press. March e epopciapned iPeodlaFacsa n[2] xxix–xlii. in Francesca, pp. della Piero di pingendi prospectiva De del 4 criticaleditionof htaecerymc oemdr hnthose than paper modern short a more In Piero. by much used clearly are that gendi emotions. strong stirs such always picture admire this at to closely very Looking order canconfirm. in I meant This finish.” picture was inits the details one to that close go seems a to It is this picture. century, pretty fifteenth very the of standards the by using for lation reason good another is there Leonardo. to attributed polyhedra regular of Proportione double of of and account book-keeping, its entry for known best (1478), Proportionialitate et Proportioni the Geometrica, was book He the Pacioli. of Luca author second is figure right the the that from conjectured been has It lt,i hc o h rttm h ue fdrawing of rules the time first the for which in plete, com- however manual, a as treatise Piero’s consider “To writes Ghione mathematician the Nicco-Fasola, h otflr . Montefeltro The .Gin,Beeitouin u otnt matematico contenuto sul introduzione Breve Ghione, F. h ahmtc sdin used mathematics The a enaaye sn ahmtcltools mathematical using analyzed been has sa lutain t odru oor Even colour. wonderful its illustration: an as 2007 10) nwihaefudimages found are which in (1509), epopciapingendi prospectiva De um eArithmetica, de Summa eDivina De epopciapin- prospectiva De 4 21x125cm) 172.5 x (251 ulse ihthe with published h Flagel- The dtdby edited oie fteAMS the of Notices , p 324): concludes followed (p. She legacy. that Piero’s period by played the role the in and place took that develop- ments to scientific addresses Francesca Field della Galilei”, Piero Galileo “From chapter, last the In Galilei Galileo Towards this in Piero of works the context. of importance , the and Johannes and Cusanus Nicolaus particular in history considering , the mathematical on of section in long taught a is arts There learned universities.” the to relation in and in its time, particular his whole of life a intellectual the as of “The 7 context work art?” the his Chapter it considers is in chapter “But present question, Then crucial right). the asks the she on characters the Alterpiece aon dell’Ovo Piero’s, of Madonna masterpiece important another ines craftsmen first oneofthemainenginesofthescientificrevolution.” constituted very that situation the a , of became to who one reality as historic to Piero true more consider is it that believe We mathematics. of and painting both in specialist a “is fapvlo al,tesraeo rs vault, cross degree of a two. equations than of algebraic greater surface of solution the the vault, and pavilion a volume of the Francesca”: “The della Italy, Piero in of mathematics published recently in [6] considered essay are another treatises Piero’s from Three view. examples the this with in agreement in Ge- up is Projective Field end ometry.” modern of would flow that great century’s streams 19th first the gave road, to winding life and long a following ideas after mathematical that, also but which expressed, ideas, clearly Francesca’s are della Piero only not mate underesti- to be would schemes, systematically graphic correlated through given are perspective with 5 Treatises SeealsoM.D.Davis, nCatr6(Otc n luin)Fedexam- Field Illusion”) and (“ 6 Chapter In ie”i i nrdcint the to introduction his of “force in the lines” to Though reference Francesca. Galileo Piero’s della say, Piero by, di- to owned a debt of to recognition rect the skein—intended for tangled argue strands a some in trace than more does no com- that a sketch of story—a part plicated of revolution. sketch the Scientific is Nor the of ori- the gins of still account and complete a itself, less in a story not complete is of centuries seventeenth revival to the fifteenth a the from of mathematics description This og dtr,Rvna1977. Ravenna editore, Longo , Lc ail rbbyapasamong appears probably Pacioli (Luca 5 ir el rnec’ Mathematical Francesca’s della Piero locle the called also , h uhr ocueta Piero that conclude authors The h Montefeltro The La 377 third book of De prospectiva pin- [6] E. Gamba, V. Montebelli, and P. Piccinetti, gendi does indeed seem like a La matematica di Piero della Francesca, Lettera pre-echo of Galileo’s later advocacy Matematica 59 (2006), pp. 49–59. for mathematics, Piero proposed [7] Milton Glaser, Piero della Francesca (A. Rauch, this “force” as establishing the ed.), Marsilio, Venezia, 1991. [8] B. Grünbaum and G. C. Shephard, Tilings and truth of artificial perspective, that Patterns, W. H. Freeman and Co., New York, 1987. is he was concerned with a matter [9] I. and M. Hargittai, Symmetry: A Unifying in which the use of mathematics Concept, Shelter Publ. Inc., Bolinas, CA, 1994. was entirely accepted. Galileo was [10] I. Hargittai, ed., Symmetry: Unifying Human extending the use of mathematical Understanding, Pergamon Press, New York, 1986. methods into areas where they had [11] , ed., Symmetry 2: Unifying Human Under- not been used by his predecessors standing, Pergamon Press, New York, 1989. and were not generally accepted by [12] M. Kline, Mathematical Thought from Ancient to contemporaries. Modern Times, Oxford University Press, New York, My purpose is not to attempt 1972. [13] M. L. Lamberti and M. Fagiolo dell’Arco, eds., to establish a direct connection Piero della Francesca e il novecento, Marsilio, between such later work and Piero, Venezia, 1991. butrather toshowwhatdeveloping [14] L. Laureati and L. Mochi Onori, eds., Gentile story Piero’s work as a whole, both da Fabriano e l’altro Rinascimento, Electa, Milano, his paintings and his writings, tak- 2006. en together as a contribution to the [15] R. Longhi, Piero della Francesca (1927), reprinted cultural history of his time, should by Sansoni, Firenze, 1963, pp. 9–111. English quo- be seento belong to.Pierois a good tations from the translation by David Tabbat, Sheep example of the learned craftsman, Meadow Press, Riverdale-on-Hudson, NY, 2002. and the activities of his intellectual [16] L. Pacioli, De divina proportione, Venezia, 1509. [17] S. Ronchey, L’enigma di Piero, Rizzoli, Milano, kin were to make notable contribu- 2006. tions not only to the development [18] L. Venturi, Il gusto dei primitivi (1926), reprinted of mathematics but also to the Einaudi, Torino, 1981. emergence of a mathematical natu- ral philosophy in the two centuries following Piero’s death. I believe that Field’s long essay has reached the objective it set for itself, and I believe that it can make mathematicians become interested in Piero’s scientificworks.Butaboveall,itatteststothestrong aesthetic and emotional impact of Piero the artist. I would like to conclude by inviting readers to go and seePiero’soriginalworks, togo toUrbinoto admire atleasttheThe Flagellation of ChristandLa Madon- na di Senigallia. No reproduction (in Piero’s case this is absolutely true) can do justice to the physical beauty of the actual works and to the unexplainable “stillness”.

References [1] M. G. Ciardi Dupré and G. Chesne Dauphiné Griffo, eds., Con gli occhi di Piero: abiti e gioiel- li nelle opere di Piero della Francesca, Marsilio, Venezia, 1992. [2] Piero della Francesca, De prospectiva pingendi, critical edition by G. Nicco-Fasola, Casa editrice Le lettere, Firenze, 1984. [3] P. del Poggetto, ed., Piero e Urbino, Piero e le Corti rinascimentali, Marsilio, Venezia, 1992. [4] M. Fagiolo dell’Arco, La grande ombra: Piero del- la Francesca e il novecento, in Piero della Francesca e il novecento, M. L. Lamberti and M. Fagiolo dell’Arco, eds., Marsilio, Venezia, 1991, p. 35. [5] J. V. Field, Piero della Francesca. A Mathematician’s Art, Yale University Press, New Haven, 2005.

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