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PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.
PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.
PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.
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PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.
21 More
PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.

PACIOLI, Luca (Lucas de Burgo S. Sepulchri; c.1445-1517). Somma di arithmetica, geometria, proporzioni e proporzionalità. Venice: Paganinus de Paganinis, 10-20 November 1494.

First edition of Luca Pacioli’s Somma di arithmetica, a book which has profoundly shaped our modern economic world. This copy in strictly original condition.

+ The most important mathematical book of the Renaissance, of direct influence on Leonardo da Vinci

+ The birth of modern business: being the first published description and enthusiastic endorsement of double-entry bookkeeping, the Venetian mercantile practice which still underpins global trade

+ A milestone in technology and computing, containing the first appearances in print of:
*mathematical statements using symbols for plus and minus
*the name and many of the ideas of Fibonacci, the 13th-century mathematician who introduced Hindu-Arabic numerals to Europe

+ The first popular mathematical book: published in the vernacular and intended for use by the professional classes

+ Very rare on the market: the present copy is the only one in original binding, and one of only three complete copies, recorded at auction in over fifty years


Luca Pacioli’s life and career perfectly positioned him as the creator of the Renaissance’s most important mathematics book: he was a teacher, traveler, scholar, and—most importantly—a friend and collaborator of the most respected artistic and scholarly luminaries of the period. Born in Sansepolcro around 1445, Pacioli came of age within the sphere of influence of Florence’s humanism. He was raised by a local merchant family, where he would have been trained in basic trade arithmetic. Sansepolcro was also the birthplace and home of the great painter and mathematician Piero della Francesca. Piero mentored Pacioli, depicting his young friend in two paintings; in turn, the Somma includes Piero’s uncredited work on perspective and other topics—a fact which Vasari censures in his Lives of the Artists.

While still a youth, Pacioli traveled to Venice to tutor the sons of a local merchant in abbaco: the day-to-day mathematics used by the city’s businessmen and traders. This may have been where he first encountered double-entry bookkeeping. His next stop was Rome, where he became a friend and confidant of the great humanist polymath Leon Battista Alberti. Alberti was at the time a Papal secretary and connected Pacioli to the Catholic hierarchy in that city. The young math tutor began to study theology, eventually becoming a Franciscan friar. Alberti must have been a powerful mentor to the young Pacioli—an accomplished Renaissance thinker, but also, like Pacioli, not of high birth. While Alberti’s achievements were across diverse fields, from architecture to cryptography, it was mathematics that was central to how he understood the world—a view he clearly transmitted to Pacioli. In the dedication of the Somma, he writes that “nothing in creation will be found constituted but as number, weight, and measure.” After Alberti’s death in 1472, Pacioli left Rome to begin a career as a teacher, lecturing at universities in Perugia, Pisa, and Bologna.

In the 1490s, Pacioli returned to the town of his birth. He spent his time finishing his long-awaited book: a treatise containing all the math there was to know. His style and scope were informed by a didactic method honed from decades of teaching young boys, his direct observation of merchant practices, and his intimacy with some of the most brilliant minds (and important manuscripts) of his age. That book was the Somma, completed under the patronage of Guidobaldo da Montefeltro, Duke of Urbino, whose excellent library was an important resource and to whom the book is dedicated. Its publication earned Pacioli fame—and also the attention of another luminary of the Italian Renaissance: Leonardo da Vinci.

Two years after its publication, Pacioli was invited to the court of Ludovico Sforza in Milan, where Leonardo was working as an engineer. The two men became fast friends, living together and working on unraveling the further secrets of both linear perspective and divine geometry—the fruits of which can be seen in Leonardo’s paintings from the period, including the Last Supper, and the book they worked on together: Divina proportione, printed in 1509. Leonardo had been teaching himself techniques from Pacioli’s Somma—an education which continued during their intimacy in Milan. A to-do list in Leonardo’s Codex Atlanticus reads: “Learn the multiplication of roots from Maestro Luca” (331r), and both the Madrid and Forster codices contain notes on the Somma—in particular, sections on proportions and proportionality, including a mirror-version of the arbor proportionis et proportionalitatis on folio 82r of Pacioli’s work.

The famous portrait of Pacioli by Jacopo de' Barbari, painted in the wake of the acclaim earned from the Somma, was once thought to be by Leonardo. It portrays Luca Pacioli as a master of all the realms of mathematics—the Platonic mysteries of geometry alongside the everyday calculations of arithmetic, and both as a teacher (the figure in the background may be Guidobaldo or a nameless student) and a writer. His ability to unite all of these realms and communicate them to the public has earned him a place as a founding father of modern mathematics and technology.


Pacioli, who began his career as an abbaco teacher and private tutor, was intimately familiar with the practical world of accounting, ledger books, and economic arithmetic. His Somma di arithmetica not only brought these ideas into print—some for the very first time—but elevated the concerns of the businessman and the practical accountant to the intellectual level of the rest of the humanist curriculum, presented as part of the sum of mathematical learning and an essential part of human knowledge. He confirms the study of economics as a liberal art.

Although double-entry bookkeeping had been known in Italy since at least the 13th century, Pacioli provides its first description in print in any language in book 9 of the Somma, entitled Particularis de computis et scripturis (“Details of Accounting and Recording”). In addition to eschewing the Latin of the scholastics in favor of the Italian of the merchants, Pacioli provides a remarkably clear and concise guide to succeeding in business. In outlining how to maintain account books, he gives examples and templates with easy-to-remember adages and even quotations from Dante.

The description of the “Venetian Method,” comprising book 9 of the Somma di arithmetica, is emblematic of a sea-change that had been developing in the status of mercantile theory and practice in Italy, where a middle class who worked for wages was growing in the cities. For several centuries loans had been forbidden as usury and merchants were generally looked down upon as exploiting the rich and the poor alike. But in the great trading centers of Italy, a new form of “business ethics” was being developed. Scholastic writers like San Bernardino of Siena and Antoninus Florentinus began to articulate a moral philosophy which made room for the practice of ethical trade and banking, for which the monetary landscape of the 15th century was increasingly calling.

The accuracy of account-keeping was a major tenet of these new systems of ethics. As economic historian Jacob Soll notes, “businessmen maintained republics,” and the integrity of account books was vital to the integrity of the businessman (and the state). While the proto-economic theorists of the scholastic universities had begun to address the morality of businessmen, they did not pursue practical questions of how this might be achieved. Pacioli’s work is a major advance in reconciling business with ethics, providing a clear guide to virtuous, as well as effective, accounting and pursuit of profit.


For the mathematical Pacioli, the well-kept account book maintains and preserves the health of the state. He writes that there are three necessities for a successful business:

*Cash or credit
*A good accountant
*Good internal control [“bello ordine”]

The accountant—who must be well-versed in mathematics—is responsible for the internal control of the books. If followed, Pacioli’s system allows for a business man to “pursue profit lawfully” and achieve good results. He also notes that “without double entry, businessmen would not sleep easily at night. Their minds would keep them awake with worry about their business. To prevent this stress, I wrote book 9 of the Somma” (trans. Cripps).

Pacioli details not only the mathematics of accounting, but the practical issues of how many and what kinds of ledgers to use, so that a business owner can understand exactly what they have and what they need to succeed. He writes: “begin with the assumption that a businessman has a goal when he goes into business. That goal he pursues enthusiastically. That goal, and the goal of every businessman who intends to be successful, is to make a lawful and reasonable profit.” The Somma emphasizes practices which are still basic to accounting: the importance of understanding inventory maintenance, cash versus capital, and using a uniform currency to keep accounts. Pacioli is also credited as the first author to describe the “rule of 72”—a method of calculating compound interest which continues to be part of the accounting curriculum today.

This view of business—underwritten by double-entry bookkeeping—was so successful that it has established itself in the language of vice and virtue writ large. While the Good Book may be scripture, God’s book is surely a ledger. During the Reformation, we begin to see the adoption of bookkeeping metaphors into the language of Heaven and Hell, where Heavenly credits and debits are weighed against each other in a final reckoning. David Wootton writes that “The account book was thus something more than a metaphor: it was an organizing principle through which all human behavior could be interpreted. And that principle was one of endless pursuit, of insatiable acquisition, of limitless aspiration. Double-entry bookkeeping comes, we may say, to substitute itself for virtue.”


The Somma di arithmetica is often regarded as the first printed book on algebra, making it an early milestone in the history of the technologies that most powerfully shape our lives today: computing and programming. While the word computus in the Middle Ages largely referred to the computations required to understand and calculate the church calendar, the concepts of computing and calculation were important to many aspects of Renaissance life. As Arielle Saiber writes, “there were computers in Renaissance Italy. Excellent and varied computers: they were the people who calculated quantities, formulated algorithms, proposed new mathematical objects and equations, tested proofs.” Techniques for number crunching were vital not only to accountants and clerics determining the date of Easter, but to a variety of other professions and pursuits—including architects, artists, and astronomers. Efforts to find ways to solve complex equations quickly were what would eventually lead to the development of Charles Babbage’s analytical engine and then the modern computer. Some of these ideas find their origin in the manuscript works of Fibonacci, whom Pacioli cites as a source for much of the contents of the Somma—as discussed by Keith Devlin in the foreword to this catalogue.

While famous in our time for his eponymous sequence and the “golden ratio,” in the 13th century Fibonacci was known for his arithmetic textbook, the Liber abbaci, which circulated in manuscript form for several centuries. The son of a trader who spent time in Muslim North Africa, he studied arithmetic in Algeria where he encountered Hindu-Arabic numerals and the work of the Persian mathematician al-Khwarizmi (from whose name our word “algorithm” is derived). The Liber abbaci became the basis of the curriculum of Italy’s abbaco schools, which gained momentum in Northern Italy as a way to train children in the practical business applications of math. Hindu-Arabic numerals are a place-value number system, meaning that the value of each digit depends on its place within the larger number. To the user of an abacus or counting board, this system based on positions within columns would have come very naturally.

Pacioli’s Somma is the first printed work to illustrate the finger symbolism of numbers—a relationship between our bodies and the number system which dates back to the development of the base-10 system and the word “digitus” itself, which means both finger and number (hence our current “digital age”). It is also the first place in print where plus and minus appear as symbols, alongside the root sign.

Whereas up until then, mathematical questions were usually expressed as word problems as a matter of course, Pacioli’s book enshrined in print the possibility of a symbolic representation of equations. His use of p~ and m~ to represent “piu” (plus) and “meno” (minus) would eventually morph into the symbols that we use today, with “+” deriving ultimately from the ampersand and “–“ from the m~. This was the first step in a long process of mathematical development that was continued by the next generation of Pacioli’s readers. The emergence of symbolic algebra in the following century was the beginning of the expression of mathematics in a purely symbolic language—specifically, the coding and the expression of algorithms which eventually could be read by the modern computer.

The Somma was a major influence on Niccolò Tartaglia, who was the first to solve cubic equations—a task which Pacioli discusses the difficulty of in the Somma. Girolamo Cardano published Tartaglia’s solution in his own Ars Magna, a work on algebra which uses variants of Pacioli’s symbols and is one of the mathematical masterpieces of that century. The search for a purely symbolic language was driven not only by practical concerns but also by philosophical and religious desires to uncover a “perfect” language—ideas which would eventually come together in the work of Gottfried Leibniz and the creation of calculus and symbolic logic.


The breadth of the applications of Pacioli’s work are borne out in the trajectories of his students and colleagues. He was the teacher of Domenico Novara da Ferrara, who would go on to teach Copernicus—whose mathematical astronomy proved the heliocentric model. Evidence also suggests that Pacioli likely met and taught perspective to the artist Albrecht Dürer in the early 1500s. While the Somma represents the pinnacle of mathematical knowledge in the Renaissance, it also provided a firm foundation for the mathematics that would come—providing training and guidance not only for future generations of mathematicians, astronomers, and artists but also businessmen, merchants, and economists.

But while Pacioli taught many illustrious students, it is this printed book that has allowed his influence to truly take hold. Although he composed the Somma in his hometown, Pacioli brought the finished manuscript to be printed in Venice—at that time, the center of the publishing world. It was also a center of scientific publishing: a decade earlier, Ratdolt’s Venetian editio princeps of Euclid was the first book with technical diagrams printed alongside the text.

The publisher whom Pacioli chose, and would continue to work with throughout his career, was Paganino Paganini. Paganini had entered the world of printing by marrying the daughter of the German printer Franz Renner. Somma di arithmetica was one of the first major products of the press, executed with the help of Paganino’s typecutter virtuoso son Alessandro—who would later go on to print the first Arabic Qur’an.

Pacioli’s Somma was a large and serious undertaking, which took about a year to print in its entirety. It was sold for 119 soldi—about a week’s wages for a scholar. It was clearly intended as an investment for savvy merchants and businessmen, who would make a return through its use. Olschki writes that it was the most widely used mathematical text in Italy for 50 years after its publication, and Pacioli applied for a 20-year copyright extension in 1508. This is a testament to the importance and influence of Pacioli’s work, which informed generations of merchants on the basic tenets of bookkeeping, fast calculation, and arithmetic. Ingrid Rowland writes that it “may be the most elegant and compendious of all vernacular manuals.” Neither a simple primer nor a tome for the learned, the Somma was a serious work made to meet the demand of the burgeoning classes of businessmen, artists, artisans, and merchants in Renaissance Italy.


This copy, uncut in its unlined vellum wrapper, is the only one in original condition recorded at auction in over fifty years. A large and beautiful work prized for its usefulness, Pacioli’s Somma di arithmetica has had a place for centuries in the world’s ancient institutional libraries but has been rare on the market in the last century. Our copy, which was previously in a Piedmontese noble library, is in superlative condition but has not gone unused throughout its long life. Indeed, it bears evidence of having been in the hands of those engaged with the worlds of mathematics, science and business. A contemporary owner has written on the flyleaf a mnemonic device for a math-based divination game—one very similar to the games discussed in Pacioli’s manuscript work De viribus quantitatis (“On the Powers of Numbers”). Seventeenth-century marginalia record both astronomical calculations and several receipts for rented rooms.

Super-chancery folio (317 x 220mm). 308 leaves. Collation: p8 a-z8 8 ?8 10 AA14 A-H8 I-K6 (p1r title and contents; p1v author’s address to Marco Sanuto, epigrams by Fa Pompilius and Giorgio Sommariva, quire register, colophon; p2r dedicatory letters to Guidobaldo da Montefeltro; p4v summaries of part one; p4v table of contents of part one; a1r preface, text of part one; AA14v second colophon; A1r text of part 2 (Geometria); K5r-6r table and register of Geometria; K6v blank). 56 lines and headline. Types: 9/12:130G, 10:92Ga, 8:84G, 13:68Gb, 16:87G. Dedication and incipits on a1r printed in red. Many woodcut diagrams, one full-page diagram printed in red and black; woodcut initials from several sets, including one (repeated) historiated initial L depicting Pacioli and the Somma.

Typographic evidence shows the continued demand for the Somma in the decade and more following its publication in 1494. A deficiency in stock—perhaps due to an original miscalculation of required copies or merely a mishap at the printer—necessitated a small number of sheets to be re-set in type and printed again, not once, but twice. These sheets (a-c, d1, 2, e1, A1) indicate three issues of the first edition were available for sale to contemporary purchasers over a short period. The present copy conforms to the second issue, with these reprinted sheets set with type material which may date to c.1502. A third setting of these sheets is dated c.1507-1509. Clarke interprets this as evidence of an important book selling very well on first publication, followed by renewed demand justifying reissues until the stocks are finally exhausted. D.A. Clarke, “The first edition of Pacioli’s ‘Summa de Arithmetica’ (Venice, Paganinus de Paganinis, 1494),” Gutenberg-Jahrbuch (1974), pp. 90-92.

An extremely fine copy, in strictly original condition. The survival of this volume in this condition is remarkable, especially given that it retains its original, unlined vellum wrapper binding. The binding is its first, thereby preserving nearly the entire original sheet size with numerous deckle edges, and there have been no interventions, aside from two minor marginal paper repairs; the type impressions remain fresh. We know of no finer copy. Over the past five centuries the volume has acquired some minor signs of wear; a detailed condition report is available upon request.

Contemporary Italian unlined vellum wrapper with BAL written on front cover, sewn on white leather thongs, missing two fore-edge ties, title lettered along lower edges in a contemporary hand, a pair of original flyleaves at front (flyleaves defective, later paper spine label, minor staining, minor wear at edge of lower cover, minor wear around thongs at spine).

“BAL” (initials lettered on front cover in an early hand) – “Aperi Premati Magister Flamine Peia Bispane Bispena” (contemporary mnemonic device for an arithmetic-based divination game written on front flyleaf) – astronomical calculations and rental annotations dated 1648-56 written on title-page; occasional annotations elsewhere – repeated illegible hand-stamped roundels on front flyleaf – A noble Piedmontese library (sold: Bolaffi, 16 December 2014, lot 637).

Bibliographical References
BMC V 457; Bod-inc L-167; BSB-Ink P-2; Essling 779; Goff L-315; Goldsmiths’ 5; HC(+Add) 4105; IGI 7134 (re-issue I); ISTC il00315000; Klebs 718.1; Riccardi ii, 226; Sander 5367; Smith, Rara Arithmetica, p. 54.


Brooks, Richard. Bean Counters: The Triumph of the Accountants and How They Broke Capitalism. London: Atlantic Books, 2018.
Clarke, D.A. “The first edition of Pacioli’s ‘Summa de Arithmetica’ (Venice, Paganinus de Paganinis, 1494),” in Gutenberg-Jahrbuch (1974), pp. 90-92.
Cripps, Jeremy. Particularis de computis et scripturis, 1494, Luca Pacioli: A Contemporary Interpretation. Seattle: Pacioli Society, 1994.
De Roover, Raymond. San Bernardino of Siena and Sant’Antonino of Florence: The Two Great Economic Thinkers of the Middle Ages. Boston: Harvard Graduate School of Business Administration, 1967.
Devlin, Keith. The Man of Numbers: Fibonacci’s Arithmetic Revolution. London: Bloomsbury, 2011.
Eco, Umberto. Serendipities: Language and Lunacy. New York: Columbia University Press, 1998.
Gleeson-White, Jane. Double-Entry: How the Merchants of Venice Created Modern France. New York: W.W. Norton & Company, 2011.
Heefer, Albrecht. “The Genesis of the Algebra Textbook: From Pacioli to Euler,” in Almagest (May 2012), pp. 27-61.
McCarthy, Patricia with Alan Sangster and Greg Stoner. “Pacioli and Humanism: pitching the text in Summa Arithmetica,” in Accounting History, vol. 13 (2008), pp. 183-206.
Olschki, Leo. Geschichte der Neusprachlichen Wissenshaftlichen Literatur. Heidelberg: Carl Winter, 1918.
Rowland, Ingrid. “Abacus and Humanism,” in Renaissance Quarterly, vol. 48, no. 4, pp. 695-727.
Saiber, Arielle. Measured Words: Computation and Writing in Renaissance Italy. University of Toronto Press, 2018.
Sangster, Alan. “The Printing of Pacioli’s Summa in 1494: How Many Copies were Printed,” in The Accounting Historians Journal, vol. 32, no. 1 (June 2007), pp. 125-145.
Soll, Jacob. The Reckoning: Financial Accountability and the Making and Breaking of Nations. New York: Basic Books, 2014.
Wootton, David. Power, Pleasure, and Profit: Insatiable Appetites from Machiavelli to Madison. Cambridge: Harvard University Press, 2018.


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