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Placjiary in the Renaissance Kim Williams

Does your hometown have any

mathematical tourist attractions such

as statues, plaques, graves, the cafd

where the famous conjecture was made,

the desk where the famous initials

are scratched, birthplaces, houses, or

memorials? Have you encountered

a mathematical sight on your travels?

I f so, u,e invite you to submit to this

column a picture, a description of its

mathematical significance, and either

a map or directions so that others

may follou, in your tracks.

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: dirk.huylebrouck@ping.be

E ven the smal les t towns of central Tuscany are do t t ed with monu-

ments that a t tes t to the cultural weal th

of f i f teenth-century Italy. One such town not only conta ins monuments , but was the b i r thp lace of two mathe-

mat ic ians who are "monuments" in themselves. The town is Sansepolcro,

or, as it was known 500 years ago, Borgo San Sepolcro; the mathemat i - cians are Piero del la F rancesca (Fig. 1) and Luca Pacioli (Fig. 2). Piero 's skill

as a mathemat ic ian has been admirably laid out in the Mathematical Intel]i- ge~wer by Mark Pe te r son [26], who also touched on the con t roversy that sur-

rounds Luca's publ icat ion, under his own name, of Piero ' s work.

Reading Mark 's article, I was sur-

pr i sed at the p lagiary charge against Luca. The Franc i scan ' s por t ra i t adorns the cover of one of the books in my li- brary; looking at it, I doub ted he could

be so guilty. Here, I sa id to myself, is a case not only for the Mathematical Tourist, but for the Mathemat ical Pri-

vate Detective!

The Scene of the Cr ime: Borgo San Sepolcro Two men born in the same town in the

santo century, pursuing the same math- emat ical interests: Was Luca an ac- compl ished mathemat ic ian , or a fraud- ulent plagiaris t? Did a scheming Franc iscan aching to leave his mark on pos ter i ty mean to consign Piero to ob-

scuri ty as a mathemat ic ian , though highly prais ing him as an art ist? I de- c ided to visit Sansepolcro , to bring Piero and Luca to life and provide a his-

tor ical contex t f rom which to ref lect on plagiary in the Renaissance. Besides, Sansepolcro offers a r ich i t inerary for

the mathemat ica l tourist: a free mu- nicipal art museum, the house of Piero and the monas te ry where Luca lived, two public pa rks with s ta tues of the

pro tagonis ts of this story, and com- memora t ive p laques ded ica ted to each.

The most ly regular, or thographic layout of the s t ree ts suggests that

Sansepolc ro or ig inated as a Roman

castram. Popular legend has it that Borgo San Sepolcro was founded by pi lgr im saints Egidio and Arcano, who,

on their re turn from the Holy Land at the end of the tenth century, buil t a

chapel to house rel ics they had re- t r ieved from the Holy Sepulchre

(hence the name). The town grew up a round the chapel, which has long s ince disappeared. Today Sansepolcro is a thriving, p leasan t town of some

20,000 inhabitants. It l ies in the upper Tiber valley, on the winding road ($73)

that connects M e z z o in Tuscany to Urbino in Umbria. The his tor ic center, su r rounded by high walls and entered

th rough formal arches, re ta ins much of its Renaissance ambience. Thanks to its being off the bea ten tour is t path, it

is unc rowded and s o m e h o w private. Fifty years ago, Kenneth Clark wrote

of it, "Borgo San S e p o l c r o . . . p resen t s today very much the same aspec t as it d id in Piero 's t ime, and everyone who has vis i ted it has felt i ts affinity with

his work" [3, p. 2].

The Vict im: Piero del la Francesca Piero del la F rancesca (also known as Piero Franceschi or Piero dei Fran-

ceschi) was born in Borgo San Sepol- cro in the early years of the f if teenth

century, pe rhaps in 1406 but poss ib ly as late as 1412. His family were well-

to-do merchants . He might have be- come a successful bus inessman, as did his b ro the r s Marco and Antonio, had

he not become one of the mos t signif-

icant painters of Renaissance Italy. Al- though it would have been cus tomary

for the son of a merchan t to s tudy com- merc ia l mathemat ics in an abacus school, Piero s tudied in Sansepolcro ' s munic ipal g rammar school; here he p robab ly encounte red the geomet ry of Eucl id for the first t ime [18, p.624].

Piero lived most of his first for ty years in Sansepolcro, until more and more pres t ig ious commiss ions for paint ings and f rescoes cal led him to the cour ts of pr inces like Mala tes ta in Rimini,

�9 2002 SPRINGER-VERLAG NEW YORK, VOLUME 24, NUMBER 2, 2002 4 5

Figure 1. Piero della Francesca.

Este in Ferrara , and Montefel t ro in Urbino. Still, he con t inued to accep t commiss ions f rom local pa t rons in

Sansepolcro, a l though he somet imes s t ipula ted enormous lead t imes in the cont rac ts for them, reflect ing the fact that they were done in spare moments .

Piero 's t ies with Sansepolcro were so great that t radi t ion c la imed he de- p ic ted his b i r thplace in the background of his St. Jerome and a Disciple, now

in the Galleria deU'Accademia in Venice, pa in ted in about 1450. The affect ion

that Piero felt for Sansepolc ro was re- turned; he was a mos t r e spec ted citi- zen of the town. F rom 1477 on, he was a member of the city council. He was

priore, a very honorab le posit ion, of the chari table Confraterni th di S. Bar-

to lomeo and was appointed an accoun- tant to oversee city finances, evidence of his sldlls in commercial mathematics. He was also, of course, the town's most fa-

mous artist. Most of the frescoes he did in Borgo are lost today. Fortunately for the mathematical tourist, the Mnseo Civico 1 contains three of his works, the

Resurrection, the Polittico della Miseri- cordia, and S. Gialiano. It also contains

a full-length portrait of Piero painted by Santi di Tito around 1630. He is depicted as a young man, wearing the formal

robes appropriate to the posi t ions of honor that he held in Sansepolcro. Next to the portrai t is a case displaying a fac- simile of the archive containing the reg-

istration of Piero's death.

When Piero ' s father died in 1464,

Piero became head of his family and began a pro jec t to recons t ruc t the fam- ily house, now known as the Casa Piero

della F rancesca (Fig. 3), located at Via Aggiunti 71. He is c redi ted with hax4ng

done the archi tec tura l design himself, in the 1470s, which reflects the archi-

tecture of the Ducal Palace in Urbino. A plaque on the faqade of the house (Fig. 4) reads:

In honor OF PIERO DELLA FRANCESCA

in the fifteenth century

[the] sovereign painter

from whose mastery Perugino learned the marvels of ar t

and Italy the geometr ic pr inciples of perspec t ive

Here where her great son lived and where at 82 years he a scended

into the heavens [this memorial] is p laced in the year

1876 [by a] grateful and reverential country.

It was while living in this house that Piero wrote his mathemat ica l t reat ises.

He also deco ra t ed the house with his own frescoes, though these are now most ly lost (a de tached fresco from the

house depic t ing a young Hercules is now housed in the Isabella S tewar t Gardner Museunl in Boston). Piero 's

house is today the home of the Fon-

1The Museo Civico, Via Aggiunt~ 65, is open seven

days a week from 930 to 1300 and 1400 to 1800.

Telephone +39-0575-732218, e-mail museocivico@-

technet.it Figure 2, Luca Pacioli.

4,6 THE MATHEMATICAL INTELLIGENCER

Figure 3. Piero's house at Via Aggiunti, 71, Sansepolcro, the architectural design of which is

credited to Piero himself. It is now the home of the Fondazione Piero della Francesca.

Piero was bur ied on 12 October 1492 (the day Columbus d iscovered America) in the family tomb in the chapel of S. Leonardo in the clois ter of

a church known as the Badia, as he had reques ted in his will. The Badia no

longer exists, having been abso rbed into the Cathedral . There is no prec i se evidence of where Piero 's tomb is.

Parco Piero del la Francesco, a pub- lic park across the s t ree t f rom the Fon-

dazione, conta ins a s ta tue sculpted by Arnaldo Zocchi in 1892 on the occas ion of the 400th anniversary of Piero ' s

dea th (Fig. 6).

P i e r o ' s A r t a n d H i s M a t h e m a t i c s

Piero 's paint ings and frescoes, which have an undef inable crystal l ine quality,

hold a special fascinat ion and enduring popular i ty today. Some ar t his tor ians

a t t r ibute the specia l charac te r of his works to the se rene express ions on the faces of the charac te rs in the paintings,

o thers to their pecul ia r quality of ap- pear ing frozen in a par t icu la r moment . The more mathemat ica l ly incl ined crit-

ics cite the influence of geometry on Piero 's art. There are at least three as- pec ts of his paint ing that have earned him the reputa t ion as the most mathe-

mat ical of painters: the use of underly- ing geometr ic cons t ruc t ions to govern his compositions3; the use of perspec-

Figure 4. The plaque commemorating Piero della Francesca.

dazione Piero della Francesca , 2 a s tudy

cen te r and col lect ion of documenta- t ion re la ted to Piero. The Fondaz ione also conta ins a small r esearch l ibrary (Fig. 5), where the mathematical tourist can browse books on Piero, and look at the anastat ic and facsimile works of

Hero ' s and Luca's mathematical trea- tises [6-8, 22-25], as I did while doing the research for this article.

In the last years of his life, Piero went blind, a l though he cont inued his intel lectual act ivi t ies with the help of a

r eader and a secretary. Picturing him being read to, medi ta t ing on what he heard, and dictat ing his thoughts to a secre ta ry is a happ ie r picture than that

depic ted in the memoirs of one who re- called, as a child of 10, leading Piero by the hand a round the Borgo.

2The Fondazione Piero della Francesca is normally open to the public from 915 to 1400 Monday through Fn-

day. I recommend calling Signora Serena Magnani for an appointment, +39-0575-740411, or e-mail fpdf@ats.it.

3For studies that point out the underlying geometry in Piero's paintings, see Charles Bouleau, La geometria segreta dei pittori, Milan, Electa, 1996:101-115: Robert Lawlor, Sacred Geometry: Philosophy and Practice, London, Thames and Hudson, 1992, p. 63; �9 Del Buono and P~erluigi De Vecchi, L'opera completa di Piero della Francesca, Milan, Rizzoli, 1967:106,

Figure 5. The research library of the Fon-

dazione Piero della Francesca, inside Piero's

house.

VOLUME 24, NUMBER 2, 2002 4"7

Figure 6. Kim Will iams with the statue of Piero (photograph by Mark Reynolds).

tive constructions to create the illusion of pictorial space4; the translation of all of his forms, both architectural and fig- ural, into resemblances of regular geo- metric solids.

An example of Piero's use of un- derlying geometric construct ions to govern his composit ions is found in the fresco of the Resurrection in the Museo Ci~4co in Sansepolcro. In about

1458, Piero was commissioned by the high magistrates of Borgo San Sepol- cro, the ConservatoH, to paint a Res- un'ection to adorn the meeting room in which public audiences were held. Moved by 1480 (that is, during Piero's lifetime) to its present location in what was at that time the Palazzo Munici- pale, this splendid fresco was an im- portant symbol for the town (the Holy Sepulchre). In the Resun'ection, Christ's head is at apex of an isosceles triangle, the other angles of which lie in the lower corners of the painting. (Piero also included a self-portrait in the fresco; the sleeping soldier on the left, propped against Christ's tomb with his head back, is identified with the artist.)

As a mathematician, Piero produced three treatises, De prospectiva pin- gendi [7], Trattato d'abaco [6], and De co~79oHbus regularibus [8] 5, all three probably written between 1480 and 1490, while Piero was in residence in Sansepolcro and already more than 70 years old ( though Trattato d'abaco is sometimes dated as early as 1450 [18, p. 635]). De p~vspectiva pingendi was dedicated to artists, as its title, On painting in perspective, implies. But, as Mark Peterson has shown, Piero did- n't limit himself to simply laying out techniques for painting, but treated perspective as a mathematical tool, de- veloping the first European theorem in geometry after Fibonacci [26, p. 34].

Piero's second treatise, the Trattato d'abaco, deals with commercial math- ematics and the gauging of volumes. I have mentioned that Piero came from a mercantile family, and that his skills in commercial mathematics were used in his capacity as a controller of fi- nances in Borgo San Sepolcro. One of the greatest skills of the merchant was the ability to gauge volume, in order to judge quantities and costs [1, p. 86-89.] Piero's interest in gauging volumes is ex4dent in the volumes represented in his paintings. One example is the Dream of Constantine, part of the fres- coes in the Cathedral of Arezzo that il-

4For studies of the underlying perspective constructions, see Laura Geatti and Luciano Fortunati, "The Flagellation of Chnst by Piero della Francesca: a Study of its

Perspective" in The Visual Mind, Michele Emmer, ed., Cambridge MA, MIT Press, 1994:207-213; Marilyn Aronberg Lavin, Piero della Francesca's Baptism of Christ,

New Haven and London, Yale University Press, 1972; Rudolf Wittkower and B.A.R. Carter, "The Perspective of Piero della Francesca's Flagellation," Journal of the Warburg and Courtauld Institutes XVI (19531:292-302.

SPiero's work was published in 1915 by Mancin~ under the name De corporibus regularibus; in 1995 it was republished under the name Libellus de qulhque corporibus

regularibus. Since Luca's copy bears a similar name, to distinguish the two in this article, I am maintaining the 1915 title for Piero's work and Libellus for Luca's copy.

48 THE MATHEMATICAL PNTELLIGENCER

lustrate the Legend of the True Cross,

in which the pavil ion where Constan- t ine s leeps is depic ted as a combina-

t ion of cyl inder and cone. As Michael

Baxandal l poin ts out:

�9 there is a cont inui ty be tween the

mathemat ica l skil ls used by com-

mercia l people and those used by the pa in ters to p roduce the pic tor- ial p ropor t iona l i ty and lucid sol idi ty

that s t r ike us as so remarkab le to- day. Piero 's De abaco is the token of

this continuity. The s ta tus of these

skills in his socie ty was an encour- agement to the painter to asser t

them playfully in his pictures. As we

can see, he did. It was for conspic- uous skil l his pa t ron pa id him [1, p.

101-102].

Piero 's in teres t in gauging vo lumes is c losely re la ted to his in teres t in the solids. His third t reat ise was De cor- poribus reg~darib~s, in which he de- scr ibes the cons t ruc t ions for the sol ids (at the cen te r of the p lagiar ism con-

t roversy involving Luca). In turn, his in- terest in solids appears to be closely re- lated to his interest in perspective, for in

order to represent objects correct ly in a perspect ive construction, they must be made somehow to confornl to a more or

less regular solid. In De pmspectit, a pin- gendi, for instance, Piero takes into con-

siderat ion the hunlan head, an object that has no straight lines that can dis- appear toward vanishing points. If the

head can be abs t rac ted and t rea ted as s imilar to a regular solid, then it can be made to conforn~ to a perspec t ive con-

struction. Kenneth Clark, compar ing one of Piero ' s drawings of a head in

perspec t ive with one of his f rescoes, recognized "the deep reserve of hu- mani ty with which Piero in his f inest

work could cover the Eucl idean frame-

work of his forms" [3, p.54]. Thus, though apparent ly Piero ' s

three t rea t i ses deal with three separa te subjects , that is, the prac t ica l mathe- mat ics of volume, the regular solids, and the geomet ry of perspect ive , each

re la tes to the other, for as Robin Evans

wrote:

Piero ins inuates their [the regular bodies ' ] geomet ry into everything,

and perspec t ive is the medium of in- sinuation. Piero ' s paint ing is seen

therefore as the i l lustrat ion of one

kind of geomet ry by means of an- o ther [10, pp. 142-143].

T h e P e r p e t r a t o r : L u c a P a c i o l i

Luca Pacio]i (also known as Paciolo or

Paciuolo, and somet imes as Luca di

Borgo after his hometown) contrasts with Piero in a lmost every way. Where

Piero remained closely tied to his home- town, Luca left Borgo as a young man and lived a peripatet ic life. Where Piero

was a respected, honored citizen, Luca was difficult to get along with and in-

volved in frequent quarrels with his fel- low Franciscan friars. Where Piero, in

Figure 7. Luca Pacioli as depicted in his Summa.

addit ion to being one of the most im- por tant artists of the Renaissance, made original contributions to mathematics [4, 24], Luca was mainly a compiler, even

stooping, as we shall see, to plagiary. Luca's ample figure, c lothed in his Fran-

ciscan robes, will be almost as familiar to most readers as his mathematical works. The most famous portrai t of him,

painted by Jacopo dei Barbari in 1495

and conserved in the Museo di Capodi- monte, in Naples, has come to represent the archetypal Renaissance mathemati-

cian. Luca even included his own por- trait in the printed edition of the great-

est of his books (Fig. 7). Luca was born in Borgo San Sepol-

cro in 1445. His pa ren t s p robab ly died

VOLdME 2.~, NUMBER 2. 2002 4 9

when Luca was a child, as he was ra ised in a foster family. In 1464, at abou t the age of 19, he went to live in

the house of Sig. Rompiaci , a Venetian merchant , as a tu tor for the master ' s children. He a t tended lessons given by

Domenico Bragadino, Venice 's public l ec turer in mathemat ics . He also ac-

compan ied Rompiaci on many busi- ness trips, acquiring great skill in com- merc ia l mathemat ics a long the way.

His first mathemat ica l t rea t i se was wr i t ten for his young pupi ls in this pe- riod.

Luca donned the robes of the Fran-

c i scans in 1470, at the age of 25. In doing so, he jo ined the ranks of o ther fr iars who dis t inguished themselves

in mathemat ics (among them, John Pecham, 1230-1292, au thor of Perspec-

t i w comm~nis, an important work on

optics, and Robert Grosse tes te , 1168- 1253, appl ied mathemat ic ian and trans- la tor of Nicomachus and Dionysius the

Areopagite) . It was from then on that Luca was known as "fra Luca" (fra is

an abbrevia t ion of frate-- fr iar) . It is s ignif icant that Luca chose to jo in the Franc i scan order, for Franc iscans

were among the few rel igious orders that opera ted outs ide the monas te r ic context . As a Franciscan, Luca was

free to travel, and he taught mathe- mat ics in many cities: Perugia, Venice, Florence, Rome, Naples, Bologna, Mi-

lan, Pisa, Paris, and Zara (presen t day Zadar in Dalmatia, then under the do- minion of Venice).

Luca 's choice to jo in an o rde r may indicate not only rel igious fervor but also persona l ambit ion. To a young

man with no family connec t ions and no wealth, belonging to a rel igious order secured a degree of privi lege that he

o therwise could not have enjoyed. But s imply belonging to the o rde r was not enough to sat isfy his asp i ra t ions to

greatness . He t r ied to buy a cardinal- ship, "offering first 30,000 ducats , then 40,000, to Pope Alessandro VI Borgia to make him a cardinal" [25, p. 79]!

Apparently, Luca was someone you e i ther loved or hated. Franc iscans take a vow of happiness when they join the

order; i t 's hard to sow d iscord among them, but Luca managed to do so at leas t twice. In 1491, he was res ident in

a Franc iscan monas te ry a t t ached to

the Chiesa di San Francesco (Fig. 8) in

Borgo San Sepolcro when he b e c a m e involved in a major conflict with the

general minis te r of the Franc i scan or- der. On 29 June the general minis te r is-

sued an o rde r forbidding Luca to t each mathemat ics to young laymen, threat- ening him with excommunica t ion if he

disobeyed. Disobey he did, and on 3

August the general minis ter i ssued an order to the abbot of the Borgo San Se- po lc ro monas te ry not to receive Luca.

But eventual ly Luca was forgiven. In May 1492 he was allowed to reenter the

monastery, and in March 1493 he was once again a l lowed to p reach during Lent. Later, in 1509, his fellow friars

were outraged when Luca was awarded privileges by the Pope, especially a large expense account. The rift healed only

when Luca agreed to renounce these privileges [ 13].

In 1496 Luca was called to the Sforza court of Ludovico il Moro in Milan as

public lecturer in mathematics. There he met Leonardo da Vinci, with whom he

would remain in contact until the last years of his life. While in Milan, in 1498,

Luca wrote De divD~a propo~'tione, ded- icated to li suoi caHss imi discipuli . . .

del BoTyo Sa~ Sep~dchro (his deares t disciples of Borgo San Sepolcro). As is well known, it was Leonardo who sup-

pl ied the 59 i l lust ra t ions of the sol ids

Figure 8. The monastery and church of San Francesco in Sansepolcro, where Luca lived when

in residence in Borgo, with the statue of Luca placed in front.

5 0 THE MATHEMATrCAL INTELLIGENCER

for Luca's De divina proportione (imagine having Leonardo illustrate your book!). In Luca's unpublished manuscript De rip,bus quantitatis (On the Fo~ve of Mathematics) [25], he praises Leonardo's unsurpassed mas- tery in drawing the geometric solids. Forced to flee Milan in 1499, he and Leonardo lived in Florence for the next few years, during which time Luca as- sumed teaching positions in both Bologna and Pisa.

Luca specified in his will that he be buried in the Franciscan church in whatever city he died in. He probably died in Sansepolcro in 1517, and is probably buried in the church of San Francesco in Sansepolcro, next to which is the monastery where he lived, but there is no tomb for him in the church, and no documents record his burial there. In the piazza in front of the Franciscan church and monastery stands his statue, the back of which will be, for mathenmticians, perhaps more interesting than the front. It indi- cates what Luca is best remembered for: his treatise on the diuina propor- tione and his geometrical methods for constructing the capital letters of the alphabet (Fig. 9).

In the loggia of the Palazzo dei Laudi is a plaque dedicated to Luca Pacioli (Fig. 10), which says:

To Luca Pacioli Who was friend and consultant to

Leonardo da Vinci and Leon Battista Alberti who first gave to algebra

the language and structure of science and whose was the great discovery

of applying it to geometry He discovered the double accounting

of commerce gave to works of mathematics principles and norms invariant

in lucubration for posterity The people of San Sepolcro

Upon the initiative of the society of workers

ashanmd of 320 years of oblivion to the great fellow citizen

placed [this memorial in] 1878.

L u c a ' s M a t h e m a t i c s

Luca's prime occupat ion was teaching mathematics. In this capacity, he wrote

Figure 9. The back of the base of the statue of Luca, depicting the "divina proportione" and

the geometric construction for the capital A from Luca's alphabet.

one didactical treatise in Perugia (Trattato di aritmetica e algebra, 1478, a manuscript copy of which sur- ~dves in the Vatican Library), and oth- ers in Venice and Zara, both now lost. In addition, Luca delighted in collect- ing puzzles and other mathematical amusements, compiling them in a work titled De ludis or Sehifanoia (Boredom Vanquisher), also lost. He displayed his skills in commercial and practical mathematics in his unpublished De viHbus quantitatis [25].

But it was his printed works that earned him his place in the history of mathematics. In 1494, two years after Piero della Francesca 's death, Luca's Summa de Arithmetica geometria proportioni et proportionalit# [24] was published in Venice by Paganino dei Paganini (it was reprinted in 1523). The Summa is a compendium of four fields of mathematics: arithmetic, alge-

bra, Euclidean geometry, and double- entry bookkeeping. The importance of the Summa can hardly be overesti- mated. It became the starting point for the study of mathematics in the Ren- aissance, relegating the manuscripts that preceded it to obscurity. Though the authors of the publications that fol- lowed on the heels of the Summa pointed out errors in Luca's text, its pri- macy was unchallenged.

In 1509, Luca's second major publi- cation, De divina proportione [22, 23], was issued. Written in 1496, while Luca was in Milan with Leonardo, it was published in a combined edition with two other works, the Trattato deU'ar- chitettura and a treatise on the five regular solids, with the unwieldy name LibeUus in tres partiales tractatus di- visus quinque corporium regolarium (Book divided into three parts on the

f i ve regular bodies, hereinafter re-

VOLUME 24. NUMBER 2. 2002 51

Figure 10. The plaque at the loggia of the Palazzo dei Laudi commemorating Luca.

fe r red to as Libelhts). The t rea t ise on archi tec ture was no doubt the resul t of Luca 's t ime spent in Rome in 1471 with Leon Batt ista Alberti, au thor of De re aedificatoria (Te~ Books on Architec- ture). The Libelhls, we now know, is Luca 's t ransla t ion into the vernacular of Piero 's Latin De corporibus regzt- laribus. Luca's third book, his able

t rans la t ion from Latin into the vernac- ular of Euclid 's Elements, was also publ i shed in 1509 in Venice.

P i e r o a n d L u c a

There is no documen ta ry evidence to show what kind of re la t ionship Piero della Francesca and Luca Pacioli had,

whether a friendship, a master-and-pupil relationship, or even a correspondence between professionals. Piero knew Luca, for they were both presen t in the cour t

of Montefel t ro in Urbino in the pe r iod 1472-1474. It has been sugges ted that when Piero was losing his sight in his la ter years, it was Luca who was his as-

s is tant [2, p.107], but there is no p roo f of this�9 The age difference of some 40

years might seem to suppor t the idea that Piero needed a co l labora t ion with Luca, but is it af ter all so imposs ib le

that a man of 70 could and would ded- icate himself with such great effect to

s tudies of mathemat ics? It may be that after a busy life devoted to his art, ad- vanced age finally gave Piero the time

for mathematics! Piero hinlself provides a clue, writing in the dedicatory letter of

De corporibus regularibus to Duke Guidobaldo Montefel t ro that he under-

took this work in his old age so that "his wits might not go torpid with dis-

use" [20, p.488; 3, p.53]. Each man created, in his own way,

a tes t imonial to the other. Piero 's trib-

ute to Luca appea r s as a cameo por t ra i t in the Madom~a and Child with Saints and Angels Adored by Federigo da Montefeltro (cal led the Pala di Brera), pain ted a round 1475 and now in the

P inacoteca di Brera in Milan. Kenneth Clark identif ied it wi th this comment: " . . . St. Pe te r Martyr, who looks over

the shoulder of St. Franc is to the right � 9 is a por t ra i t of Luca Pacioli, the fa- mous mathemat ic ian , who was a nat ive

of Borgo and a friend, though ulti- mate ly a t r eacherous friend, to Piero" [3, p.49]. Luca 's tes t imonia l to Piero ap-

pears in the dedica t ion to the Summa, where he calls Piero, famously, the "monarch of paint ing of our times."

Fur ther along in the Summa, on page 68 verso, he credi ts Piero with having wri t ten a "worthy" b o o k on perspect ive

in which he speaks "est imably of paint- ing, set t ing forth in an es t imable way the technique and i l lustrat ion of the

method." However , Luca a lways pra ises Piero as a painter , or at mos t as an author on pa in ter ly subjects, but

not as a mathemat ic ian .

T h e I n d i c t m e n t

The charge of p lagiar i sm was first and virulently al leged by Giorgio Vasari, the

ar t is t and archi tec t who is pe rhaps bes t known for his col lec t ion of b iographies of the pr incipal art ists , sculptors , and archi tec ts of the I tal ian Renaissance [30]. Indeed, Vasari was so outraged by

what he saw as Luca 's theft of Piero 's work that he begins the Life of Piero

with it:

52 THE MATHEMATICAL INTELLIGENCER

Truly unhappy ave those u,ho, after labouring over their studies to give pleasure to others and to leave behind a name for themselves, are not per- mitted either by sickness or death to bring to perfection the works they have begun. And it often happens that when such a person leaves behind him works which are not quite f lnished or that are at a good stage of develop- m.ent, they are usu~7)ed by the pre- sumption of those who seek to cover their own ass's hide with the noble skin of the lion. And i f Time, which is sa id to be the fa ther of T~th , sooner or later reveals what is true, it is none the less possible that for some period of time the man who has done the work can be cheated of the honour due his labours; this is what happened to Piero della Francesca from Borgo San Sepolcro.

He was regarded as an uncommon master of the problems of regular bod- ies in both arithmetic and geometry, but the blindness which overtook him in old age and f inally his death kept him. from completing his brilliant ef- forts and the many books he w~vte which are still preserved in Bo~yo, his native town. The man who should have tried his best to increase Piero's glo~nj and reputation (since he leavened everything he kn.ew fi'om him), instead wickedly and mali- ciously sought to ~n.ove his teacher Piero's name and to usu~79 for himself the honour due to Piero alone by pub- lishing under his own name--that is, Fra Luca del Bo~yo--all the efforts of that good old man who, besides ex- celling in the sciences mentioned above, also excelled in painting [30, p. 163].

And later, he adds:

As I mentioned earlier, Piero was a most diligent student of his art and

frequently practised drawing in per- spective; he possessed remarkable knowledge of Euclid, to the extent that he comprehended better than any other geometrician all the curves in regular bodies, and thus he shed the clearest light yet upon these matters with his pen. Master Luca del Bo~yo, the Franciscan monk who wrote about

regular bodies in geometry, was his student. And when Piety reached old age and died after having written many books, the said Master Luca usu~oed them for his own purposes and had them printed as his own work once they had fallen into his hands af- ter his master's death [30, p.167].

Vasari is frequently caught telling in-

terest ing anecdo tes lacking basis in fact. One such er ror in his life of Piero

concerns Piero 's bl indness, which Vasari says came on Piero at the age of 60 (circa 1470), though it is known that

Piero could see as late as 1482, because he personal ly made notes to his will,

and because his t rea t i ses were wri t ten be tween 1480 and 1490 [18, p. 634]. Even given the inclusion of errors,

however, Vasari 's pen could make or b reak the reputa t ion of an artist. As the

t rans la tors of his Lives of the Artists note, "Few art is ts he cri t icized have been definit ively rehabi l i ta ted, and al-

most all the figures he se lec ted for par- t icular praise have renmined those most popula r with col lectors , scholars

and visi tors to the major nmseums of the world [30, p. x]."

But in spite of Vasari 's authority, the case against Luca was general ly dis- missed for centuries, mainly for lack of

proof. H e r o ' s books were supposed to have been in the l ibrary of Montefel t ro in Urbino, but by the end of the eigh-

teenth century they were no longer to be found. With the passage of time, Vasari 's c laims grew weaker and

weaker , while Luca 's reputa t ion grew ever more sterling.

The Damning Proof Finally, 400 years af ter the accusat ion, came the first damning discoveries. J. Deunis toun indica ted the p resence of

the De co~:poribus regularibus manu- scr ipt in the Vatican Library in the 1850s; in 1903 Guglielmo Pittarell i

found and identif ied the manuscript ; Pit tarell i verif ied in 1908 that the Vati- can manuscr ip t was ident ical to Luca 's Libellus; in 1915 Giro lamo Mancini as-

cer ta ined that marginal notes to the manuscr ip t were in Piero ' s own hand, and finally publ i shed the work in Piero 's name [20]. Proving that the

manuscr ip t in the Vatican Library was

Piero della F rancesca ' s is t an tamount

to proving that the Libellus publ ished as par t of Luca's 1509 edi t ion of De di- vina propovtione is a copy. Nowhere

does Luca give credi t to Piero as hav- ing au thored or even cont r ibu ted to the

work. This effectively cons t i tu tes liter- ary theft.

As if the truth that Luca usurped the

De corporibus regulaHbus of Piero were not enough, in 1989 Enrico Picutti

demons t ra t ed that this was not the only plagiary commit ted by Luea, but one of several (may we call him a "se-

rial plagiarist"?). First, in his Summa [24], Luca included 54 p rob lems from

Piero ' s Trattato d'abaco without cred- iting Piero. Later he included the other 138 p rob lems in the Libellus [27, p.

76-77]. Finally, according to Picutti, Piero was not the only ~ictim of Luca's

ambit ion. The sect ion t i t led Geometria in the Summa is a t ranscr ip t ion of the manuscr ip t of another mathemat ic ian,

a maes t ro Benedet to of F lorence [27, p.76].

The discovery that Luca copied the

work of Benedet to of F lorence helps expla in one aspec t of Luca 's work that his readers often decry: his uneven writ ing style (Bernardino Baldi in the

s ix teenth century wrote that Luca's style "makes one nauseous") . Bear in

mind that in the Renaissance wor ld of independen t city-states, d i f ferences in

d ia lec t and accent were quite marked be tween cit ies and regions, as they are to a lesser extent today. It is difficult to bel ieve that in the few years that

Luca lived in F lorence he would have come to speak a perfect Florent ine, but in copying word-for-word the writ ings

of Benedetto, he would natural ly be copying the Florent ine style as well.

A c c i d e n t or I n t e n t ?

By now I think there is no doubt that

Luca copied Piero 's work as well as the work of the hapless Benedet to of Flo- rence. Was he aware that this was wrong? One thing we know is that he

took s teps to ensure that the same thing didn ' t happen to him.

In the paral le ls be tween the cultural

con tex t of the late f if teenth and the ear ly twenty-first centur ies are lessons to be learned, for the advent of the print ing press c rea ted at that t ime a

VOLUME 2,4, NUMBER 2, 2002

cul tural divide similar to that c rea ted by the advent of e lectronic publishing

today. Piero and Luca s tood on ei ther side of the gulf, Piero in the wor ld of the manuscript , Luca in the world of the press, jus t as scholars today are divided

be tween traditional print publ icat ion and the new electronic publication.

Piero p roduced his treatises, both text and figures, by hand, consigning them to the l ibraries of his patron, Montefel-

tro of Urbino. Luca, on the o ther hand, embraced the new medium, establish- ing a f irm relat ionship with the Vene-

tian publ i sher Paganino de ' Paganini who was to publ ish all three of his b o o k s in print, thereby assur ing them

wide distr ibution. Bear in mind that when we talk

about manuscr ip ts we don ' t use the

te rm in today 's sense of a draft vers ion of a text. Manuscripts in the f if teenth cen tury were f inished documen t s writ-

ten by hand, sometimes by the author himself (an autograph manuscript), sometimes by a professional copyist.

The new medium

of print had enormous

consequences for the promulgation

of knowledge from the late

1400s forward. They were sometimes lavishly illus- trated, described as "illuminated man- uscripts." Figure 11 shows the first

page of the manuscr ip t of Piero 's De co~'potqbus regularibus in the Vatican

Library. Figure 12, on the o ther hand, shows a page from the print edit ion of

Luca's Summa. The new med ium of print had enor-

mous consequences for the promulga-

t ion of knowledge f rom the late 1400s forward. One consequence was that

scholar ly works became nmch more accessible. Enrico Giusti writes,

Of the work of mathematics enjoying the greatest diffusion, the Elements of Euclid, about 200 manuscript codices are known today; even discounting the inevitable losses, it can be calcu- lated that not more than 300 copies of the Elements were at the disposition of medieval scholars, and those were neither easily accessible nor equally reliable. In contrast, in the century be- tween the f irs t printed edition (Venice

Figure 11. The manuscript copy of Piero's De corporibus regularibus in the Vatican Library.

Figure 12. A page from the print edition of Luca's Summa.

1482) and the end of the sixteenth ce~- tu~j about 70 editions of the Eleme~ts were printed, for a total, calculati~g p~ldently 300 copies per edition, of about 20,000 volumes that ci~vulated th~vugho~lt Europe [13, p.17].

Another consequence was that pr in ted works in t roduced a uniformity

that was previously unknown. Scholars in d i spara te a reas of Europe, for ex- anlple, could now possess uniform vol- umes that a l lowed them to cite page

numbers and figure numbers , sure that fel low scholars could correc t ly identify

what was being discussed. (Ironically, this is a feature that is lacking in some

of today 's e lect ronic publicat ions. Doc- uments p roduced in hyper tex t markup language (html) do not include page breaks; because the inser t ion of page

breaks is de te rmined by the configura- t ion of each individual 's persona l com- puter, it is not poss ib le to cite pages se- curely.) Yet ano ther consequence of the advent of pr int was reliability. In

nmnuscripts , e r rors commi t ted by a copyis t could a l ter the meaning of a text; an und iscovered er ror could be

t ransmi t ted unwit t ingly by future copy-

ists. Of course, we all know that er rors

occur in pr in ted documen t s as well, but er rors in pr int could be easi ly cor- r ec ted in an er ra tum or in future edi-

tions. For all the new advantages of print,

it ra i sed new prob lems as well. It is at this poin t that "intel lectual property ,"

which we hear so much about today, becomes an issue. Today we talk about how easy it is to go on the Internet and

"grab" an image and some text and put it up on a web page as one ' s own. In Luca 's day, it was easy to take the text

or f igures from a manuscr ip t and pub- lish them in print under one ' s own

name; the fact that many manuscr ip ts were one-of-a-kind and tucked away in

pr ivate col lect ions mean t that such theft would be hard to bring to light.

Another p rob lem w a s that, in the- ory, any publ i sher could pr int anything

he liked. The first ma themat ics b o o k in print, Arithmetica, publ i shed in Turin

in 1478, was in fact an anonymous work, requiring no pe rmiss ion from an author. Was the need of a mechan i sm

for the pro tec t ion of au thors and pub- l ishers recognized, and did a means of p ro tec t ion exis t in the late 1400s? The

answer to both quest ions is yes; the means of p ro tec t ion was cal led a pri~,- ilegio. The first-ever privilegio was granted in Venice in 1469 to a single

publisher , granting him the exclusive rights to all publ ica t ions in the whole te r r i tory of the Republic of Venice.

Upon his death, o thers were granted s imilar privilegi. After 1480 privilegi were granted for individual works [17]. Our copyright today der ives from the

p~'ivilegi of 500 years ago. Ei ther au- thor or publ i sher could apply for a p~;vilegio by writ ing a le t ter to the rul- ing body of the state; in Venice where Luca Paciol i ' s works were published,

the ruling body was the Senate of the

Republ ic of Venice. The old saying goes, if you want to

p ro tec t your house f rom burglars, ask a thief what to do. Luca wrote to the Senate on 19 December 1508 request- ing a privilegio so that, for the next 20 years, no one could pr int wi thout

Luca 's express pe rmiss ion ei ther his Summa or De divina p~vportione, which was to be publ i shed the follow- ing year. The pt'ivilegio was granted,

VOLUME 24, NUMBER 2, 2002 5 5

and the announcement of it was

p r in ted on the last page of De d iv ina propor t ione publ ished in 1509 as well

as on the last page of the second edi- t ion of the S u m m a publ i shed in 1523. Ironically, Luca's c laim on the mater ia l

mean t that not even the heirs of the le- gi t inmte authors, Piero and Benedetto, could have publ ished the mater ia l that

was rightfully theirs [27, p. 76].

But Was It Really Plagiary? Surprisingly, the t endency is not to

condemn Luca for his usurpa t ion of Piero ' s work, but ra ther to excuse him. In par t this seems to s tem from an un- wil l ingness to pro jec t mode rn ideas on

the past . Gino Loria 's commen t s are representa t ive of the usual arguments:

[L~lca's] beha~,ior, inconceivable to-

day, is new con f i rmat ion that scien- t i f ic hones ty is a s e n t i m e n t wi th mod- e ~ origins; the anc ien t s commi t t ed

witho~tt scruple eve~?4 sort o f p lag iary and, when it came to ind ica t ing sou~ves drawn on, they were overcome

wi th an im~incible amnes ia ; it is thus

no marvel that such a f i 'ee-and-easy systenn should be adopted by a man

who was not an original th inkeh bat an indefatigable compi ler [quoted in

6, p. 32].

Loria 's comments appea r to ring true initially. For instance, cons ider the

case of Euclid and Eudoxus. Euclid is said to be, like Luca, a great compiler . And like Luca, Euclid used the work of

ear l ier mathemat ic ians such as Eu- doxus without giving credit . Yet no- body th inks of Euclid as a plagiarist .

And yet, Vasari 's accusa t ions seem to belie that jus t i f icat ion in the Re- naissance: if copying wi thout identify-

ing the original au thor were indeed

Figure 13. Via Piero della Francesca, Sanse-

polcro.

Figure 14. Via Luca Pacioli, Sansepolcro.

thought innocuous, why would Vasari

have been so vehement? Further , the maes t ro Benedetto, Luca's o ther vic-

tim, lived at the same t ime and in the santo social contex t as Luea, yet when he included in his own t reat ise the

work of a cer ta in maes t ro Antonio, he faithfully c red i ted the original author.

If copying were so widespread, why did maes t ro Benedet to not behave as

Luca did? I am forced to the conclus ion that Luca did it to arrogate credit .

In part, the tendency to excuse Luca

may arise from a reluctance to at t r ibute mathemat ical originality to Piero, an artist, ra ther than to Luca, a mathemat-

ics teacher. An example of this is Fletw Richter contest ing Vasari's accusat ion of Luca: "Piero knew [Luca] well, in-

deed, the two must have worked to-

gether, for how else could Piero, with- out the instruction of a teacher, have handled such complicated nmterial?"

[28, p. 42]. Such a s ta tement shows a fundanmntal misjudgment of Piero 's mathemat ical skills.

Conclus ion Luca a lmost managed to pull it off. The

truth is finally out that he was a schem- ing, ambit ious man, capable of lying and plagiarism in hopes of going down in

history as a great mathematician. There are still those today who don' t want to recognize the "hide of an ass covered

with the noble skin of a lion" as jus t that. The impor tant thing is that Piero has fi- nally been given the credit he deserves

as one of the original mathemat ic ians of the Renaissance, in addit ion to being one of the era 's great painters.

A final odd twist: we owe a deb t of grat i tude to Luca, for if his De d iv ina proport ione inc luded whole sec t ions

of Piero 's Trattato d'abaco and De cor- por ibus regularibus, it ensured the ideas conta ined there in a wider diffu-

sion than they would ever have enjoyed in manuscr ip t form. It may even be that

Piero 's manuscr ip t in the Vatican li- brary would have gone unidentif ied

and Piero 's contr ibut ion to Renais- sance mathemat ics unrecognized had the mater ia l not been so familiar as

par t of Luca's publicat ion.

Whatever the re la t ionship be tween Piero and Luca in life, certainly they

are forever l inked in history. A paint- ing by Angiolo Tricca, execu ted in 1927 and on display in the mayor ' s office in

the municipal i ty of Sansepolcro, illus- t rates the myth we all want to believe, of a learned Piero willingly passing on

to a reverent Luca the principles of the solids. Titled Piero della Francesca dic-

tates the rules o f geomet~y to Luca Pa- cioli, the painting depicts Piero as an old

man seated in a large chair, cane in hand, gesturing at Luea, who, dressed in his Franciscan habit, wri tes formulas on an

easel, while three young men look on, fascinated by the discussion between

the two great men. In Sansepolcro there are two paral-

lel s treets , Via Piero del la F rancesca (Fig. 13) and Via Luca Pacioli (Fig. 14). The mathemat ica l tour is t can walk up

one and down the other, and perhaps by jouruey ' s end will have sor ted out

for him- or herse l f a compl ica ted s tory begun 500 years ago.

Note I was working on this paper when I learned about the destruct ion of the

World Trade Center and part of the Pen- tagon on 11 September 2001. I dedicate

it to the memory of those who lost their lives in the horrifying chain of events.

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VOLUME 24, NUMBER 2, 2002 57