In 1464, aged nineteen, Luca Pacioli left the small market town of Sansepolcro and travelled east along ruined Roman roads frequented by outcasts and outlaws to one of the largest and most cosmopolitan cities in Europe—Venice. Writer Jan Morris conjures its moody environs well: ‘This damp expanse, speckled with islets, clogged with mudbanks and half-drowned fields, protected from the sea by its narrow strands—this place of beautiful desolation is the Venetian lagoon.’ Presiding over the realm where lagoon meets Grand Canal is the former power centre of the Venetian Republic, the Doge’s Palace, an architectural fusion of east and west described by Ruskin as the central building of the world. Further along the Grand Canal stands the Rialto, the Wall Street of Pacioli’s age. Here the first state bank of Europe was opened in the twelfth century. For the next three hundred years the Rialto dominated international currency exchanges from England to Egypt. In the late thirteenth century the Venetian ducato—ducat or ‘coin of the dukedom’—usurped Florence’s fiorino to become the monetary standard of trade throughout the known world. In 1472, the Venetian Senate attested to its significance when it declared ‘the moneys of our dominion are the sinews, nay even the soul, of this republic’. The ducat was protected above all else and anyone caught violating it was severely punished: those who debased the ducat had their right hand cut off; men caught coining were blinded, while women had their noses cut off.
It was a commercial dominance Venice still enjoyed when the teenage Pacioli first beheld its magnificence. By the 1460s the population of Venice was about 150,000, making it the third largest city on the Continent, after Paris and Naples. For the moment Venice still held sway over the Mediterranean, despite the relatively recent conquest by the Ottoman Turks of its major trading hub—Constantinople—in 1453. Unlike the other Italian city-states, Venice put the demands of commerce high above the rule of the Church. One year after the Turkish defeat of Constantinople, the Venetian Republic signed a peace treaty with the Ottoman Empire, the enemy of Christendom, and continued business as usual—that is, when they were not embroiled in one of their many clashes. For its forbidden trade with the infidel, Venice incurred the wrath of the Pope and was excommunicated several times during the fifteenth century.
Pragmatism was the guiding principle of Venice and had brought it stupendous wealth and power. Its rise to commercial prominence began in the ninth century, founded on its favourable terms of trade with the Byzantine Empire; its monopoly of salt, a commodity then more desirable than gold for its ability to preserve food; and its rule of the waves. Venice’s relationship with the sea was so essential to its survival that it was celebrated annually in an elaborate wedding ceremony. For eight centuries from 997 the city of Venice, in the person of its ruler the Doge, married the Adriatic Sea; every year on Ascension Day the Doge sailed across the lagoon in full wedding regalia to the entrance to the Adriatic and tossed a diamond ring into its briny depths, vowing: ‘O sea, we wed thee in sign of our true and everlasting dominion.’
The might of Venice was built on such seamless bindings of spiritual ritual with commercial advantage and political pragmatism. Even the tale of her ruling saint, Mark the Evangelist, said to have been shipwrecked in the lagoon, is in fact a story of subterfuge and mercantile opportunism. Saint Mark’s mummified body was stolen from a church in Alexandria in 828 by two Venetian merchants and presented to the Doge in his new palace—and thus became the patron saint of Venice, usurping its original and all but forgotten Saint Theodore. The abduction of Saint Mark, a superior saint to Theodore, signified the city’s growing independence and influence. Its rule was further extended in 1203 when its 88-year-old Doge, Enrico Dandolo, diverted a group of pilgrims heading peacefully to the Holy Land and tricked them into conquering Constantinople for Venice instead. They returned with a haul of Byzantine treasure, including the four bronze horses now in the Museum of San Marco, and so the foundations of the Venetian maritime empire and its prodigious wealth were laid.
Venice controlled the mercantile traffic of Constantinople until 1453. The busy life of one Venetian merchant based in Constantinople before its sacking, Jachomo Badoer, is preserved in the pages of his ledger, the only commercial document to have survived the city’s destruction in its entirety. Just as the books of Francesco Datini of Prato are cutting-edge fourteenth-century commercial practice, so the innovative Badoer is an exemplar of the fifteenth-century Venetian businessman. Written from 1436 to 1439 entirely in the new Hindu–Arabic numerals, Badoer’s ledger is an invaluable record of Venetian mercantile life and of the hectic commercial activity of the Levant.
Badoer was a nobleman who for over three years ran a commercial venture in Constantinople, the meeting place of the trade routes of Europe and Asia, trading for himself and as an agent for Venetian merchants. In the busy bazaars of Constantinople he bought spices, incense, leather, wool and slaves to ship back to Venice for his brother to sell on the Venetian market. The first two weeks of November and June were always Badoer’s busiest times, because it was then that his fleet prepared for its return trip to Italy, in compliance with the Venetian Senate, which required merchants to return to Venice at Christmas time and again in July to ensure a regular marketing of goods in the city. To escape the daily grind of the Byzantine marketplace, Badoer rented a villa in the suburbs of Pera, a Genoese colony on the coast nearby.
Venetian trade was closely controlled by the Senate, and the four galleys Badoer took to Constantinople were armed and probably accompanied by warships. The dangers of sea travel—including natural disasters and the ‘Turkish peril’—led to the development of maritime insurance, an industry new in Badoer’s day and one into which he ventured. Charging a premium of 3 to 19 per cent (depending on the risk factor), Badoer suffered only one insurance loss over three years, a testament to the protection provided by the Venetian navy to its merchants.
On reaching Constantinople, Badoer spent his first week unloading his wares and distributing the bills of exchange he had brought from Venice. The bill of exchange (lettera di cambio) and the bank transfer (scritta di banco) were the two credit instruments of Renaissance businessmen and served their interests very well. One took care of fund transfers between merchants living in different cities and the other of fund transfers within a local market. The bill of exchange was widely used in Renaissance finance. Its primary purpose was to allow the transfer of funds between resident merchants and their foreign agents without the risks involved in shipping gold or other precious assets. It also served as a borrowing medium. Badoer charged a commission of 1 per cent for honouring these bills and made no attempt to disguise the interest payments, as was usual in Renaissance finance due to the Church’s opposition to usury.
In defiance of official Church policy, Venetian attitudes to interest rates were closer to our own: they considered it legitimate to borrow money at interest as long as it was determined by the market and was practised openly. If the interest rate was reasonable, around 5 to 8 per cent, the Venetian courts would enforce the collection of a contract. To honour his bills of exchange, Badoer had at least four accounts with local bankers in Constantinople, where banking was organised along the same lines as on the Rialto: a bank’s primary function was not to lend money, but to transfer the funds of its depositors, who personally presented themselves to authorise the transfer of money to creditor accounts in different cities.
Most importantly, Badoer kept his books using a system that was essentially double entry, despite its inaccuracies (for example, his ledger does not balance exactly). His ledger, which mainly recorded accounts receivable and accounts payable, shows debits on the left-hand page and credits on the facing page, a two-column system that was the hallmark of Venetian double entry and is still used today. But unlike today’s accountants, Badoer was not concerned with problems of valuation: he recorded his acquisitions of merchandise at cost using market value or, when the transaction was not made with money in the marketplace (barter was still common), estimated money values. He also used a profit and loss account to record his net income so he could regularly assess the health of his business. As in many of the early ledgers, the entries in Badoer’s ledger are in paragraph form, with a rough column of monetary values in the right-hand margin denominated in Byzantine money. Badoer’s use of Byzantine currency entailed constant conversion between Venetian ducats and Turkish asperi, calculations he made using the ever-fluctuating exchange rates determined at the Rialto; this invaluable information was conveyed to him via his regular correspondence with Venice. The Rialto’s busy foreign exchange market provided regular exchange-rate updates for the many currency conversions carried out daily in Venice, the centre of international trade.
So successful did the merchants of Venice become that by the fourteenth century, traders were travelling across Europe to the Rialto—and sending their sons—to learn from their expertise in the commercial arts, especially abbaco arithmetic, currency exchange and their famous bookkeeping system. Merchants arrived from Germany in such great numbers that a five-storey complex—known as the Fondaco dei Tedeschi, or ‘trading post of the Germans’—was built near the Rialto to accommodate them. And so it was natural that an ambitious young man such as Luca Pacioli, with a flair for abbaco mathematics, should seek his fortune in Venice.
When Pacioli arrived in Venice in the 1460s, its immense wealth was mired in decadence, the stink of its canals was disguised by incense, perfumes and spices, and its inhabitants were afflicted with malaria, the plague and a pervasive lassitude. A celebrated Venetian physician of the day attributed their ill health to sexual excesses, gluttony, a sedentary life and sudden changes in temperature. The choir of San Marco was known throughout Europe and music filled the streets. Bands as we know them today had just become fashionable and frenzied dancing was common. The Piazza San Marco had been given a new look: a Byzantine-styled entrance had recently been added to the Doge’s Palace to proclaim Venice’s position as the major power between Christendom and the east—and to distract attention from the increasing threat posed by the Ottoman Empire to Venice’s command of the eastern Mediterranean. The renovation included the gradual repainting of 22 frescoes in the palazzo’s Great Council Chambers, work assigned to the Venetian artist Gentile Bellini. His absence from the job in the next decade (in 1479) was a sign of the times: in that year the Doge lent Gentile to the Ottoman Sultan Mehmed II for two years, as part of the terms of a peace treaty between Venice and the Ottoman Empire.
The teenage Pacioli travelled to Venice to take up a post as tutor and abbaco teacher to the three sons of a wealthy fur merchant, Ser Antonio de Rompiasi, and he seized every opportunity that Venice offered. He continued his mathematical studies and gained valuable commercial experience working as an agent for Rompiasi, who was based on the Giudecca, an island south of Venice proper. It had originally been a place of banishment and, later, the first settlement of Jews (hence its name), but by the fifteenth century it had become a fashionable and highly desirable Venetian district. Two decades later in 1487, when he was writing his great mathematical encyclopaedia in Perugia, Pacioli remembered Rompiasi and his sons fondly. Referring to his first written work (now lost)—an abbaco text from 1470 dedicated to the three brothers—Pacioli wrote of his ‘illustrious pupils, the brothers Bartolo and Francesco and Paolo de Rompiasi of the Giudecca, worthy merchants in Venice, sons of Ser Antonio, within their paternal and fraternal shadow, I found shelter, in their own house’.
Pacioli lived with the family on the Giudecca and his primary duty was to teach Rompiasi’s teenage sons arithmetic and bookkeeping to equip them for the family business. He also took advantage of Venice’s pre-eminence in mathematical studies, attending its Scuola di Rialto, a school founded in 1408 which attracted students from across Europe wanting to learn mathematics, astronomy, theology and natural philosophy, and where Pacioli studied under Domenico Bragadino, Venice’s public reader in mathematics. In a practice dating from 1433, official professors and lecturers in Venice were well paid by the state (from rates levied on house rents and business profits) and so academic positions there were highly sought after. Venice was also the first Italian city to endow public lectures in algebra, and Pacioli would himself return to the Scuola di Rialto forty years later to lecture in mathematics. In his role as agent for Rompiasi’s maritime business, Pacioli travelled throughout the Adriatic and perhaps as far afield as Greece and the Holy Land.
Most importantly, while working with Rompiasi, Pacioli learnt bookkeeping Venetian-style, a priceless skill he took with him when he left Venice, and Rompiasi’s service, to travel to Rome with Leon Battista Alberti in 1470. How Pacioli met Alberti, the great Florentine Humanist, history does not tell, but Alberti now became Pacioli’s trusted mentor and guide to the inner sanctum of the Eternal City.
Rome was then a city of ruins with a population of no more than 60,000, a small provincial town whose outlying suburbs had returned to nature. But it was the centre of the most powerful European institution of the day, the Church. And Alberti, as a member of its administration (in 1471 he was Secretary to the Papal Chancery), was in a position to introduce Pacioli to the men who ruled it, including Pope Paul II, his successor Sixtus IV, and Sixtus’s nephew Giuliano della Rovere, who became Pope Julius II in 1503 and who would become Pacioli’s patron. (The Renaissance system of patronage formed one of its primary webs of socio-economic relationships, essential not only for artists but for anyone who aspired to worldly success. Pacioli was fortunate throughout his life in his patrons, who gave him financial support, protection, favours and access to networks of powerful men.)
Historian Jacob Burckhardt called Alberti the first universal genius—and his was a genius of both body and mind. Alberti was intellectually brilliant, physically beautiful and an outstanding athlete. According to his autobiography, which he wrote in the third person around 1438, he could ‘with feet tied, leap over a standing man; could in the great cathedral [the Basilica di Santa Maria del Fiore in Florence] throw a coin far up to ring against the vault’ and he ‘amused himself by taming wild horses and climbing mountains’. Such self-praise was common in the Renaissance and Alberti’s was well founded. A leading Humanist thinker accomplished in mathematics and passionately devoted to its application to art, architecture and daily life, he was also one of the first to understand the enormous potential of the printing press (invented in Europe in the 1450s) and of writing in the vernacular, an understanding he passed on to Pacioli. Alberti was the first to formulate the mathematics of perspective to explain to painters how they could achieve a naturalistic illusion of depth in their work—or, in other words, convey on a flat surface the impression of three dimensions. In 1435 he published his method in Latin as De pictura (‘On Painting’), the first ever treatise on the theory of painting. (It was published in the vernacular the following year.) The new method of perspective painting would be taken up by Piero della Francesca and Leonardo da Vinci, taught by Pacioli, and would revolutionise Italian art.
Departing radically from medieval thought, Alberti also valued material wealth. He expressed a respect for money that was new in Europe and would characterise his century, distinguishing it from the Middle Ages, an era when money was scarce, peasants were the majority, barter was the primary mode of exchange, people lived largely on what they or their village could produce, and wealth was seen as an obstacle to salvation. Money, wrote Alberti in the 1430s, is ‘the root of all things’: ‘with money one can have a town house or a villa; and all the trades and craftsmen will toil like servants for the man who has money. He who has none goes without everything, and money is required for every purpose.’ As historian Fernand Braudel argues, something new enters European consciousness in Alberti’s writing—along with his celebration of money went thriftiness and a concern with the value of time, ‘all good bourgeois principles in the first flush of their youth’. This radically new attitude towards wealth in the Renaissance is rarely remembered today, when we celebrate almost exclusively its artistic flowering.
Already contemplating his mathematical encyclopaedia, Pacioli was drawn to Rome by the prospect of working in the Vatican library. Built by Pope Nicholas V (1397–1455) with Alberti’s guidance, by 1455 the Vatican library had the largest collection of manuscripts in Europe, greatly enhanced by the fall of Constantinople two years earlier, which prompted an influx into Italy of Greek scholars and manuscripts, especially the works of ancient Greek science and mathematics. As a result, in the 1470s Pacioli could access most of the texts of Greek and Arabic mathematics, which were available for the first time to scholars in Italy in the celebrated Renaissance libraries of Rome, Florence, Venice, Milan and Urbino.
When Alberti died in 1472, Pacioli left Rome for Naples, another large centre of learning and Greek scholarship. He found work as a merchant and an abbaco teacher before leaving for Perugia two years later—and thus began his life as an itinerant teacher. Pacioli became a travelling salesman for Hindu–Arabic mathematics and spent the rest of his life wandering across Italy, teaching first as an abbaco master and later at universities as professor of mathematics.
Because Italy was a series of warring city-states at the time, such extensive travels were dangerous unless you travelled with the protection of the Church. With its sanction, monks could journey unmolested and find accommodation almost anywhere. Perhaps for this reason—and for the career opportunities offered by the Church—soon after leaving Rome Pacioli took the vows of a Franciscan friar. By 1475 he had joined the Conventual Franciscans, the division to which his new patron Giuliano della Rovere and the then Pope (Sixtus IV) both belonged, and which was influential in Sansepolcro. The Conventuals were the most liberal variety of Franciscans and allowed Pacioli to teach mathematics and travel about the countryside with almost as much freedom as a layman. And because of his mathematical accomplishments and powerful friends in the Church, Pacioli was also granted special exemptions from several rules of his brotherhood, especially those regarding the ownership of property (according to the will he left behind, Pacioli died a wealthy man).
In Perugia, Pacioli embarked on his great work, the Summa de arithmetica, geometria, proportione et proportionalità, the first encyclopaedia of all the mathematics known in Europe at the time, which synthesised the three major mathematical traditions he had inherited: medieval European mathematics, Arab mathematics and the ancient Greek sources, which had recently arrived in Europe from Constantinople. To support himself he worked as an abbaco teacher. Perugia’s city council appointed Pacioli first as a public lecturer in abbaco arithmetic and later in geometry as well. He gave his lessons in Latin to a class of about one hundred and fifty students who spoke a range of Italian dialects and foreign languages.
In December 1477 Pacioli began work on the second (and only surviving) of his three unpublished abbaco textbooks for his pupils, Tractatus methematicus ad discipulos perusinos (‘Mathematics Treatise for the Students of Perugia’), which he finished the following April. It includes a section on mercantile tariffs which Pacioli copied from elsewhere without acknowledgement. This provides an instructive context in which to see his borrowings in the Summa and subsequent charges of plagiarism, such as those levelled by Vasari: borrowing without attribution was a regular and acceptable practice in the abbaco tradition in which Pacioli worked. There were no readily available texts for abbaco teachers and so they wrote their own, copying other texts which were all ultimately sourced in Fibonacci’s Liber abaci. Only when these texts began to be printed following the spread of the printing press in Italy in the late 1470s did questions of copyright—and therefore of plagiarism—begin to arise.
In 1481 Pacioli left Perugia for Zara, a city now in Croatia but then the capital of the Venetian territory of Dalmatia, where he wrote his third and most advanced abbaco text for his students. When he returned to Italy, Pacioli took his masters degree in theology, a course which at the time included mathematics, and by 1486 he had attained the highest academic rank of the day, ‘magister’, or master. This qualified him to teach mathematics at university level, which was better paid than abbaco teaching. There were thirteen universities in Italy in Pacioli’s day and each one employed only one or two mathematicians, but Pacioli would be appointed the first chair of mathematics at two of them (those of Perugia and Rome). Pacioli’s combined training in both abbaco and university mathematics was extremely rare. Only two other mathematicians in the whole of Renaissance Italy are known to have possessed similar expertise and training.
Continuing his research for the Summa, Pacioli spent much of the early 1480s in Florence, immersing himself in the mathematics available in the splendid Medici Public Library. He praised the library for its excellent mathematical manuscripts, including Witelo’s Perspectiva, a key thirteenth-century treatise on optics based on the work of the eminent Arab mathematician Ibn al-Haytham, which was an essential source for the mathematics of artificial perspective used by Renaissance painters such as Piero della Francesca. In Florence, Pacioli also met and befriended many of the leading artists and sculptors of the day, including Botticelli, who had recently returned from Rome, where he had been among the artists painting the frescoes on the walls of the Sistine Chapel.
In 1486, Pacioli was appointed professor of mathematics in Perugia. Like any twenty-first-century academic, he writes of these years as burdened with the demands of teaching while he attempted to write: ‘If I do not seem to have treated these questions properly, I pray that they may correct my way of speaking and have pity on one who feels other worries, as I feel the burden of daily reading, lecturing and teaching, here in this beloved august City of Perugia.’ When his contract at the University of Perugia expired, Pacioli returned to his birth town for this first time since he had left it aged nineteen.
Home with his fellow Franciscans, Pacioli immediately fell out with his monastery’s authorities: in 1491 he had a heated disagreement with the head of the Conventual Franciscans and was nearly excommunicated from the order. Exactly what this disagreement was about is not known, but it may have related to a complaint made about Pacioli to the order in the same year, which led the Franciscans to forbid the monk to teach the young men of Sansepolcro. This hazy episode has prompted speculation about Pacioli’s possible homosexuality, fuelled by his long intimacy with Leonardo da Vinci during the 1490s.
Otherwise, Pacioli spent this time in Sansepolcro quietly finishing his manuscript on everything that was known about mathematics in 1490s Italy. He was lucky to have one of the best libraries in Europe nearby, over the hills to the east in Urbino. The library had been built up by the bookish Duke of Urbino, Guidobaldo, a friend of Pacioli and the son of Federico da Montefeltro (he who commissioned the famous Montefeltro altarpiece from Piero della Francesca mentioned earlier). A renowned intellectual, Guidobaldo had collected most of the important mathematical manuscripts of the age, including Jacobus Cremonensis’ translation of Archimedes, the Algebra of al-Khwarizmi, and Piero della Francesca’s De quinque corporibus. These works—along with Fibonacci’s Liber abaci—formed the basis of Pacioli’s two bestselling books, the Summa and De divina proportione (‘The Divine Proportion’, on the golden ratio).
In 1494, encouraged by his friends and his new patron Marco Sanuto, Fra Luca Pacioli was ready to publish the sum of his learning.
And so, aged 49, Pacioli travelled to Venice for the second time, to take advantage of the new opportunities opened up by the recently arrived printing press for those with written material to offer the public. In Pacioli’s case, the material was the culmination of his life’s work so far: his mathematical encyclopaedia, Summa de arithmetica, geometria, proportione et proportionalità. As we have seen, Pacioli had by now achieved a distinction that was almost unique in fifteenth-century Italy: he was an experienced teacher in both commercial (abbaco) and speculative (university) mathematics. He had also spent six years as a merchant’s assistant in the busiest trading centre in Europe, and he had for almost twenty years been studying the entire body of mathematics known to the Mediterranean of his day: the rediscovered work of the ancient Greeks, the Latin mathematics of the medieval schoolmen and the advances of the Arabs. The hefty manuscript he brought with him to Venice contained this collected mathematical knowledge, based largely on Euclid’s Elements and the work of Fibonacci. His manuscript would become the first printed book to deal with Hindu–Arabic arithmetic and its offshoot, algebra, and contain the first printed treatise on Venetian bookkeeping. These two great contributions to the scientific and commercial life of Europe—its transmission of algebra and of double-entry bookkeeping—make the Summa the work for which Pacioli is now remembered.
As the leading mathematician of the moment, Pacioli was able to find both a patron to fund the printing of his enormous manuscript and a printer in Venice who was willing to publish it. The printer was Paganino de Paganini, who had set up his printing shop in Venice in 1483 when the new communications technology was becoming well established in the city. Although printing had been invented three decades earlier in Germany—probably by a Mainz metalworker, Johann Gutenberg—the art of printing numbers and figures accurately and efficiently had only been invented the year before Paganini opened his press. The German printer Erhard Ratdolt, who was based in Venice—the up-and-coming centre of the new industry—had noticed that while many works of the ancients were printed in Venice, almost nothing mathematical had appeared because there was no way of reproducing figures. And so Ratdolt devised a way to reproduce tables of figures and other mathematical symbols accurately and in 1482 printed the first mathematics text of the Humanist programme—Euclid’s Elements.
Four years earlier in 1478, the Venetian Republic had produced one of the earliest known printed books on mathematics, an abbaco treatise in the vernacular known as the Treviso Arithmetic. It is a telling moment in the history of printing: Humanists had been demanding a printed edition of Euclid, but the first printed mathematics book was for merchants, not scholars. Commercial imperatives and practical necessity drove the early printers’ decisions about which books to publish as much as the competing agenda of scholars and the Church, much to the outrage of Humanists such as the celebrated Erasmus of Rotterdam. While making extensive use of printing to disseminate his own work, Erasmus was scandalised by the fact that so soon after its invention, the press had escaped the control of scholars and fallen into the hands of merchants and businessmen. Pacioli’s mentor Leon Battista Alberti was one of the first to understand the revolutionary significance of the new technology. Writing in Rome in the 1460s, Alberti gives a sense of the quantum leap in communications that printing provided: ‘we greatly approved the German inventor who in these times has made it possible, by certain pressings down of characters, to have more than two hundred volumes written out in a hundred days from an original, with the labour of no more than three men; for with only one downwards pressure a large sheet is written out.’
The first book printed in Venice (Cicero’s Epistolae ad familiares, an ancient Roman classic) had been published in 1469. But the printing houses of Venice struggled to find a market for their unwieldy printed ‘manuscripts’ of classics and religious works, and within five years nine of Venice’s twelve printers had gone bust. It seemed the new technology was not commercially viable. But the merchant bankers of Venice thought otherwise. They soon realised the commercial potential of printed books and invested the large sums required to keep the printing presses running. To the merchants of Venice, the printed book was simply a commodity like any other and could be sold along the trade routes of Europe like pepper, silk, wax and other luxury goods. Venice became the centre of the new communications technology, the Silicon Valley of the Renaissance, and many of the first printed works on business and commerce were published in the city on the lagoon.
By the time Pacioli returned in 1494, Venice had become the publishing capital of southern Europe, with more than 268 printing shops run mostly by experts from Germany and France. They came to Venice because of its favourable business conditions: its large labour force, low printing costs, stable liberal government run by merchants for merchants, readily available patronage, and its vigorous intellectual community which could provide the translators, proofreaders and scholarly advisors that a successful printing press required. The time- and labour-saving advantages of the printing press were huge compared to the old medieval communications technology of manuscripts handwritten by scribes. For example, in 1483 the Ripoli Press charged three times as much for setting up and printing a translation of Plato’s Dialogues as a scribe did for duplicating the same work. But the press produced 1025 copies, the scribe one copy, making printed books more than 300 times cheaper than manuscripts. Books rapidly became widely available and affordable to a new class of readers; in 1500 the price of a book in Venice was about a week’s salary for a teacher or a skilled artisan, equivalent to the price of a good desktop computer today.
The printing press created an explosion in demand for multiple copies of instruction manuals and texts for students and teachers—and their sudden availability to a wide audience helped to break down the culture of secrecy that had prevailed in medieval Europe, spawned by the guilds, who were more concerned with guarding their trade secrets than with publicising their knowledge and technical skills. With the printing press came an ‘avalanche’ of how-to books (similar to the many hundreds of DIY books published each year today), explaining the previously arcane arts of everything from playing musical instruments to keeping accounts in double entry.
This enthusiasm for the new technology was particularly pronounced among Renaissance mathematicians, who, according to historian Paul Lawrence Rose, were possessed of an ‘almost missionary faith’ in the power of the printing press to spread knowledge. Pacioli was one who led the way into this new world. Far from considering vernacular translations beneath him, and despite being fluent in Latin, Pacioli broke with scholarly tradition and wrote his encyclopaedia in Italian, the language of the people. He also encouraged the use of the new mathematics and its Hindu–Arabic numerals, lamenting those merchants who still used Roman numerals and the old methods in their arithmetic. His Summa was emblematic of the new printing programmes.
As was typical in these early days of printing, Pacioli stayed in Venice during the printing of his encyclopaedia, visiting Paganini’s print shop daily to correct and add new material to his manuscript as it went to press. The initial print run, estimated to have been around two thousand copies, would have taken from nine to twelve months to complete, at a rate of one sheet of paper (or two pages) per day of the 615-page book. Pacioli writes in the Summa of its production, saying that he worked day and night with ‘industry in the workshop of that clever man Paganino de Paganini’, correcting his manuscript with his own hand.
The printing of the Summa was funded by Pacioli’s patron Marco Sanuto, a professor of mathematics from a famous patrician family whose intellect, virtue and thoughtful generosity Pacioli praises at length in his introduction. With an eye to eternity, he says that Sanuto had made ‘this volume of mine possible so that it may be handed down to posterity’.
But Pacioli dedicated the Summa to the Duke of Urbino, Guidobaldo da Montefeltro, whose library he had used for his mathematical researches and who may also have been Pacioli’s student at one time. In his dedication to the duke, Pacioli outlines his intentions for his work, which were revolutionary at a time when books were written in Latin and destined for a cloistered scholarly elite. Rather, the Summa is written in the language of the people so it can be read by ‘each and every man’ and used in everyday life. Pacioli tells the duke that he has written this book because he ‘desire[s] to be of use’ to Guidobaldo’s subjects. And he takes great pains to stress that although ‘I am not ignorant of eloquent style, and realise that you should be addressed with a wave of eloquence since you are so learned in Ciceronian eloquence’, he has decided to write his encyclopaedia in the vernacular ‘because if this were written in Latin each and every man could not understand it. I have written it so that it may bring advantage and pleasure to those who in literature are learned or not.’
Mathematics, says Pacioli, applies to almost every human activity, from astrology, cosmography and theology, architecture, painting, sculpture and music, to business, law and military strategy. ‘Why, the citadels of states, the walls of cities, the towers, trenches, ramparts, mounds, and all of the other defensive and offensive weapons of war are nothing else but geometry and proportion,’ says Pacioli, giving the example of Archimedes’ famous defence of Syracuse from the Romans with his mathematical knowledge, which he used to build weapons including a huge crane. Known as the ‘Claw of Archimedes’, it allegedly lifted enemy boats out of the harbour and upturned them, drowning all their warriors. Pacioli concludes by saying that ‘if you examine carefully each one of the other sciences and liberal arts there is not one which does not use in some way harmony, measure, and proportion’. According to him, without these three mathematical properties ‘everything ceases to exist’.
The Summa’s title and introduction are in Latin but the main text is in the Florentine vernacular, an Italian dotted with Latin and Greek words and abbreviations, and expressions from the local dialect, which was comprehensible to a large audience of merchants, businessmen, students, artists and technicians, as well as the more broadminded scholars otherwise used to reading in Latin.
Pacioli deliberately championed the vernacular not only because ‘the subject matter will bear more fruit if there are more people to read it’ but also because, as he was the first to understand, Latin was not at all suited to explicating mathematics. To make his prose as clear as possible for his readers, Pacioli also took his images and analogies from daily life. For example, he explains eight different ways of multiplying. The sixth method was generally known as the square, cell, sieve or net, because of the way the numbers were set out on the page. But Pacioli called it the ‘gelosia’, because, as he says, ‘the arrangement of the work resembles a lattice or “gelosia”. By gelosia we understand the grating which it is the custom to place at the windows of houses where young ladies or nuns reside, so they cannot easily be seen. Many such abound in the noble city of Venice.’ This passage, with its image and anecdote inspired by the everyday sights of Venice, is typical of Pacioli’s mathematics.
The Summa is divided into two volumes, but all known surviving copies—99 copies of the 1494 edition and 36 of the 1523 edition—are bound into one book. Volume I contains nine chapters: chapters 1 to 7 cover arithmetic; Chapter 8 is the first systematic exposition in the vernacular of algebra; Chapter 9, on commerce, is divided into twelve sections, the first ten on matters such as barter and bills of exchange, the eleventh on bookkeeping, the twelfth on exchange rates and weights and measures. Much of the first volume was derived from Fibonacci. Volume II of the Summa contains one chapter only, which is the first printed vernacular text on geometry, summarising and updating the work of Archimedes, Euclid, Fibonacci and Piero della Francesca. During his years in Venice in the 1460s, Pacioli had discovered a copy of Fibonacci’s long-neglected Liber abaci in a monastery and immediately understood its value. By translating large sections of Fibonacci into the vernacular and including them in the Summa, Pacioli restored this seminal work to European mathematics.
Pacioli’s famous bookkeeping treatise—Volume I, Chapter 9, Part 11 of the Summa—is so central to any history of double-entry bookkeeping that it requires detailed attention and will be discussed separately in the next chapter. But before we leave the Summa behind, it is worth noting its most significant contributions to European mathematics.
Pacioli introduced Hindu–Arabic numerals and their basic arithmetic to a wide audience in Italy for the first time. He explained the rules for working with these new numerals—for example, for addition, subtraction, division, fractions and roots—as well as including multiplication tables and examples of the many different ways of multiplying, such as the ‘gelosia’ method mentioned above. In the fifteenth century, multiplication with numerals was considered to be extremely difficult, and division almost impossible, an art to be attempted only by experts. Pacioli also invented two new symbols, one for plus and one for minus, which became standard notation in Italian Renaissance mathematics (although they are not the symbols we use today).
Most importantly, the Summa was the first vernacular book printed in Europe to contain algebra—and it marks a dramatic departure from the algebra of the abbaco tradition. The Summa moves from using algebra as a means of solving specific problems to algebra as a language for making abstract arguments; Pacioli generalises algebraic derivations and formulates them as universally valid theorems. The extensive use of the Summa by sixteenth-century algebraists helped to lay the foundations of the scientific revolution and thus of modern science. Among the mathematicians who drew on the Summa were Nicholas Tartaglia (1500–57), who, with just fifteen days of schooling, wrote a treatise on arithmetic (1556) and numbers (published posthumously in 1560), and the notorious Girolamo Cardan (1501–76), a gambler and possible murderer obsessed with scandal, astrology and philosophy, whose Ars magna (1545) on algebra was the most advanced work of its day.
The Summa also contains the first printed text on the mathematics of linear perspective for Renaissance artists and architects, based on the work of Piero della Francesca. Pacioli acknowledges this in his dedication, when he mentions Piero’s ‘copious work which he composed on the art of painting and on the force of the line in perspective’ which is in the duke’s library in Urbino.
Pacioli’s Summa de arithmetica, geometria, proportione et proportionalità was published in Venice on 20 November 1494, becoming the first mathematical encyclopaedia of the Renaissance and one of the earliest books to be printed on the Gutenberg press. Given that it includes the entire mathematical and commercial knowledge of his age, Pacioli’s encyclopaedia is correspondingly massive. Even for its time, when big books were popular with wealthy bookbuyers for their apparent gravitas and resemblance to manuscripts (which were considered more valuable than printed books), the Summa was an exceptionally large book. It measures 25 by 30 centimetres and runs to 615 densely printed pages, the equivalent of a 1500-page textbook if it were typeset today. The Summa is also widely considered to be one of the most beautiful of all the early printed books. The 1494 edition in the library of Sansepolcro is leather-bound and metal-studded. Inside, its pages are tissue thin, its print is stark and the capital letters of its gothic font are decorated with woodcuts of Fra Luca dressed in monk’s robes and holding a compass. In its margins are explanatory graphs, diagrams and computations.
Priced at 119 soldi, the Summa was expensive (the popular Aesop’s Fables was only two soldi) but well within the means of the wealthy merchants, artisans and nobles of Venice, Florence, Milan and other Italian cities. It was a commercial success, selling steadily over several decades and making its author famous. Highly unusually in those early days of print when intellectual copyright was a new concept, Pacioli was given a ten-year copyright on the initial publication and in 1508 he petitioned the Venetian Senate for a twenty-year copyright on any reprint of the original 1494 edition, which he was granted. This made him one of the first writers to be granted literary copyright. Ten years after the Summa was first published, Pacioli’s bookkeeping treatise was extracted and published separately by Paganino de Paganini in Tuscany as La scuola perfetta dei mercanti (‘The Perfect School of Merchants’), under Pacioli’s name.
A second edition of the complete Summa was published in 1523 at Toscolano on Lake Garda and paid for by the printers. This edition was greeted by an even more receptive public—and only then was Pacioli widely lauded, posthumously (he died in 1517), for having dared to write in the language of the people. The Summa became the most widely read mathematical work in Italy for a century and trained several generations of readers in mathematics and bookkeeping.
The publication of the Summa brought Luca Pacioli fame throughout Italy and he was given one of the greatest honours of the age: his portrait was commissioned. Pacioli became the first mathematician in Europe of whom we have an authentic portrait, and probably the first mathematician ever to have a portrait painted. The Portrait of Fra Luca Pacioli was painted in Venice, probably in 1495 by Jacopo de’ Barbari, and now hangs in the Galleria Nazionale di Capodimonte in Naples. It shows Pacioli dressed in grey Franciscan monk’s robes, demonstrating a mathematical problem. One hand points to a diagram drawn in chalk on a slate, the other rests on the page of an open manuscript, next to which is a big red book, probably the Summa, and a wooden model of a dodecahedron (one of the five Platonic solids, solid polyhedra whose faces are all identical regular polygons, such as a cube).
Beside Pacioli stands a young auburn-haired man long believed to have been Guidobaldo, the Duke of Urbino, especially as the portrait is dedicated to him. But recently an English mathematician, Nick Mackinnon, analysed the portrait, which he believes depicts a real geometry lesson, to claim that the young man is in fact the German artist and future mathematician, Albrecht Dürer, who was in Venice in 1495 seeking the secrets of the new Italian painting. Mackinnon thus argues—persuasively—that Pacioli’s portrait ‘captures one of the greatest moments of the Renaissance, the transmission to Albrecht Dürer, and hence to the world north of the Alps, of the geometry of Ancient Greece and the basis of the new art of Italy’. If he is right, then Pacioli played a pivotal role not only in the history of mathematics and commerce, but also in the history of European art.
Soon after the portrait was painted, Pacioli left Venice for Milan, where a 42-year-old engineer and master of theatrical spectaculars had bought a copy of the Summa in 1494 for its arithmetic and treatise on linear perspective. At the insistence of this engineer, Pacioli was summonsed to the Court of Milan by its ambitious ruler Ludovico Sforza. Inspired by the Medici in Florence, Ludovico was in the midst of transforming his realm into a true Renaissance city, a centre of the arts and learning complete with a court of intellectuals. As part of his modernisation, he had recently introduced mathematics lectures, and in 1496 he invited Luca Pacioli to take up Milan’s first Chair of Mathematics.
The engineer responsible for Pacioli’s move to Milan was Leonardo da Vinci, who had arrived in the city fourteen years earlier. Vasari says that Leonardo was first presented to the Milanese court not as an engineer nor even as a painter but as a musician, and ‘took with him a lyre that he had made himself, mostly of silver, in the shape of a horse’s skull, a very strange and novel design which made the sound fuller and more resonant’. But Leonardo was keener to promote himself to Ludovico as a military engineer, a more highly esteemed profession, writing in his letter of introduction: ‘In short, I can contrive an infinite variety of machines for attack or defence’, including cannons, armoured cars, siege-machines, tunnel borers and bridges. Only as an aside does Leonardo mention that ‘in painting I can do everything that it is possible to do’.
When Pacioli met Leonardo in Ludovico’s Castello Sforza in 1496, the artist was preoccupied with mechanics, hydraulics, architecture and engineering. According to an anonymous source, Leonardo was ‘of a fine person, well proportioned, full of grace and of a beautiful aspect. He wore a rose coloured tunic, short to the knee, although long garments were then in fashion. He had, reaching down to the middle of his breast, a fine beard, curled and well kept.’ The two men were obsessed with arithmetic and geometry, believing, as Leonardo put it, that they embraced ‘all the things in the universe’, that without them ‘nothing can be done’.
At the time, Leonardo was working on two ambitious art projects: a huge equestrian sculpture in bronze to honour Ludovico’s father (which was never cast), and a giant fresco in the church of Santa Maria delle Grazie, about ten minutes’ walk from the Castello Sforza. The church and its buildings were being completely remodelled by Ludovico to glorify the Sforza dynasty, and Leonardo had been commissioned to paint the monks’ dining room with the subject traditionally used for refectories: the Last Supper. As with everything he did, Leonardo’s approach to the fresco was scientific. He had taught himself arithmetic from Pacioli’s Summa and now Pacioli himself was in Milan to help Leonardo with the mathematics of linear perspective for the creation of his hyper-real Last Supper. On its completion in 1498, the Last Supper caused an immediate sensation, with Pacioli one of the first to praise it. Leonardo’s fresco was widely copied in a range of sizes and, then as now, visitors to Milan (including the King of France) rushed to see it. In his own treatise on art, Leonardo calls perspective ‘the subtlest investigation and invention of the mathematical studies which, by force of lines makes remote that which is near and large that which is small’.
During his three years in Milan, Luca Pacioli collaborated with Leonardo da Vinci on several projects, including his next book, De divina proportione, which he dedicated to Ludovico Sforza. Although Pacioli finished De divina proportione in 1498, he did not have it printed for another eleven years: his busy life at the court of Milan was brought to an abrupt end with the invasion of the city in October 1499 by the army of Louis XII of France, who took the Castello Sforza. It is said that French soldiers dragged Pacioli from his lodgings at the nearby San Simpliciano monastery and destroyed his mathematical models, believing they were the work of the devil. Pacioli escaped Milan with Leonardo and together they made their way to Mantua, where Isabella d’Este (whose sister Beatrice was married to Ludovico) was receiving refugees from the Milanese court. In gratitude, Pacioli dedicated a second book he had been working on in Milan to Isabella and her husband, the Marquis of Mantua. The book—called De viribus quantitatis (‘On the Powers of Numbers’)—was a compendium of magic, recreational mathematics and proverbs. It was not published during Pacioli’s lifetime and remains unpublished, having languished for five hundred years in the Bologna University Library.
Like most of Pacioli’s work, it was written in the vernacular. The first of its three sections is the earliest known comprehensive collection of mathematical games and problems, such as the famous conundrum faced by a man with (in Pacioli’s version) a ‘wolf, a goat and a bundle of cabbage’ who wants to cross a river in his boat, which is only big enough to take himself and one other item at a time. The second section is a range of puzzles and jests, from card tricks and instructions on how to write a sentence on the petals of a rose to how to wash your hands in molten lead—Pacioli explains that you must soak your hands in cool well water, shake them, and then you are ready to put them in a pan of molten lead over a flame: it will not cook you and ‘it will appear to be a miracle’. (This trick was tested by Adam Savage and Jamie Hyneman in the 2009 season finale of MythBusters and amazingly they found it worked with lead heated to 450°C.) The third section is a collection of proverbs and verses, including 22 riddles.
After leaving Isabella d’Este’s refuge in Mantua in late 1499, Pacioli shared a house with Leonardo da Vinci in Florence and worked as a professor of mathematics at its university. He then returned to Venice in 1508 to supervise the printing of De divina proportione and his Latin translation of Euclid’s Elements. But before he embarked on his second great printing programme, on 11 August 1508 Pacioli gave an introductory lecture on the fifth book of Euclid at the church of San Bartolomeo near the Rialto Bridge. Some five hundred people came to hear the celebrated mathematician speak, including architects, printers, ambassadors, magistrates, theologians, artists and philosophers. The famous Venetian printer Aldus Manutius was there and may have brought along Erasmus, who was staying with him near the Rialto while supervising the printing of his translations of Euripides and a collection of ancient proverbs. Intriguingly, after leaving Italy in 1509 Erasmus wrote his famous satire, In Praise of Folly, in which he mocks scientists who use maths to bamboozle their audience. His description of these boffins rather accurately parodies the methods used by Luca Pacioli in his talk on Euclid: ‘When they especially disdain the vulgar crowd is when they bring out their triangles, quadrangles, circles, and mathematical pictures of the sort, lay one upon the other, intertwine them into a maze, then deploy some letters as if in line of battle, and presently do it over in reverse order—and all to involve the uninitiated in darkness.’ In his book, Erasmus set out to deflate the pretensions of anyone who claimed special knowledge or importance, whether they were philosophers, merchants or clerics.
If Erasmus did have Pacioli in his sights here, then in this particular instance it was probably not without reason. Although Pacioli generally aimed to bring mathematics to the widest possible audience in the most accessible way, he was also prone to flights of fancy, especially when he attempted to conflate mathematics with Christianity. He told the crowd that Euclid’s Book V on proportion was one of the most difficult sections of the Elements and that proportion ‘is the quality which alone penetrates the inmost being of the most high and undivided Trinity’, just the sort of remark that might have prompted a deflating barb from Erasmus.
The following year, Pacioli was back in the printing shop of Paganino de Paganini, printing his Latin edition of Euclid and De divina proportione. According to Pacioli, De divina proportione was inspired by the fervent discussions which regularly erupted in Ludovico Sforza’s court about the application of mathematics and natural science to art, a subject of crucial importance in the Renaissance. In response, Pacioli had decided to write a book which explained the mathematical basis of the arts for ‘everyone who loves to study philosophy, perspective, painting, sculpture, architecture, music and other mathematical disciplines’.
In his dedication of De divina proportione to Ludovico, Pacioli assures him that his esoteric mathematical speculations are ‘no old women’s tales, no false and ludicrous jestings, no lying and unreliable poetic imaginings, which please the ears only with empty vapours’. But despite Pacioli’s declaration of its practical purposes, De divina proportione is also his most mystical work. It concerns the ‘divine proportion’—better known today as phi, or the golden mean or ratio—and its relation to the five Platonic solids. The golden ratio—which results when a line is divided so that the short portion relates to the longer portion as the longer relates to the whole—is intimately interconnected with the Fibonacci sequence and likewise recurs with uncanny frequency throughout the natural world, including in the human body (for example, the navel divides the body according to the golden ratio); in the spiral growth of shells; the proportions of a dolphin’s eye, fins and tail to its body; and the seed heads of a sunflower.
Leonardo made a set of 60 beautiful geometric drawings for De divina, which Pacioli acknowledges in his dedication. Thanks to Leonardo’s illustrations, its powerful patrons and its accessibility, De divina proportione became the best known and most successful of all Pacioli’s books in his day.
Pacioli claimed to have written one more book—De ludo scacchorum, rumoured to have been the first book on chess—which many doubted ever existed because no copy had been found. But Pacioli’s claim was confirmed in 2006 when a copy was discovered in a library in northwestern Italy. While Pacioli was travelling from city to city teaching abbaco in the 1470s a new style of chess was spreading through southern Europe, characterised by the dramatically enhanced powers of the queen and bishop and extended powers for the pawns. Because it made the queen the most powerful piece on the board, it was called ‘mad queen’s chess’—scacchi alla rabiosa—and quickly became popular in the courts of Italy. Pacioli’s book on the subject contains over a hundred chess problems and instructions, including how the new powers of the queen, bishop and pawn worked on the board.
The rediscovery of Pacioli’s De ludo scacchorum (‘Of the Game of Chess’) excited interest mostly because of its possible connection to Leonardo da Vinci (who probably made its futuristic black and red illustrations). But the book is significant in the context of Pacioli’s life because it clearly shows the vast range of his interests as a mathematician as well as the depth of his intellect. Some of the puzzles it contains are so complex and sophisticated that they have been attributed to Leonardo, a known genius. While it is most likely that Leonardo was involved in this project—it was probably compiled around 1500 when the two men were together in the court of Isabella d’Este, a noted chess player—it is more likely that Pacioli himself was the intellect behind the puzzles.
After seeing his Euclid and De divina proportione through the press in Venice, Pacioli returned to Sansepolcro at the end of 1509. In welcome, the attentive friars of his monastery filed a complaint against him to the general of the Franciscan order, arguing that not only had he failed to abide by the vows of poverty required by the Franciscans, but that ‘this Master Luca’, far from being a model monk fit to direct others, ‘according to what we understand and see daily, is a man who ought to be corrected’. The head of the Franciscan monastery in Sansepolcro asked that Pacioli be deprived of his papal favours and all administrative duties. History does not record the details of Pacioli’s misdemeanours and much has been speculated on their nature, for example, that with his cosmopolitan lifestyle, fame and mathematical genius, Pacioli had lorded himself over his less learned brothers in his provincial home town. Or, that his reprehensible behaviour was of a more scandalous nature, pertaining to sexual and other worldly indulgences. Whatever it was about Pacioli that had caused the complaint to be made, it was soon dropped—and in reply, less than three months later, on 22 February 1510, Pacioli was appointed head of the monastery in Sansepolcro by the general of the Order of Conventual Franciscans.
In 1514, Pacioli accepted his final academic appointment, as professor of mathematics at the University of Rome. His listing on the university’s faculty roll is the last record we have of him. He died a few years later, probably in 1517.
Pacioli’s contemporary Daniele Caetani of Cremona expresses an early sixteenth-century view of the monk’s achievements, focusing on his exceptional gift for collecting, organising, simplifying and making available the mathematical knowledge of the Renaissance:
The science of mathematics, after being hidden in speculation and conjecture, what was not practical because so many of the bodies had been reduced to various and complex figures, Lucas alone of many rendered simple by explanation, so that even the very ignorant could understand just as if he had set it forth under their very eyes. And who in former times ever dared to make this very notable beginning? Surely no one, not even the most learned mathematician before Lucas Paciolus, who was in this respect a man of the rarest pattern and almost unique.
Pacioli became famous in his day for his knowledge of mathematics, his gift for systemising, formulating and updating it, and his passion for disseminating its secrets via the printing press to the widest possible audience in their own tongue. But his lasting fame would rest on his 27-page bookkeeping treatise, Particularis de computis et scripturis—and it is to this work that we now turn.