How double-entry bookkeeping changed the world

By Keith Devlin @profkeithdevlin

Examining a first edition copy of Pacioli’s book  Summa  (1494)

Examining a first edition copy of Pacioli’s book Summa (1494)

What’s your reaction when you see the term “double-entry book-keeping”? Do you associate it with cool, societal-changing innovations like the Internet, Google, social media, laptops, and smartphones? Probably not. Neither did I—until I was asked to write a brief article about the fifteenth century Italian mathematician Luca Pacioli, to go into the sale catalog for the upcoming (June) Christie’s auction of an original first edition of his famous book Summa de arithmetica, geometria, proportioni et proportionalita (“Summary of arithmetic, geometry, proportions and proportionality”), published in 1494, which I referred to in last month’s column. (I also gave a talk at a public showing Christie’s organized in San Francisco on April 24, which gave me an opportunity to examine the book myself.)

Sure, I knew what double-entry book-keeping was. Indeed, I spent several days in March going through my QuickBook records as I prepared my annual tax filing. Though by no stretch of the imagination am I in the big-income category, my tax-filing situation is not simple. I have several sources of income from around the world—from my university position and my ed tech startup company BrainQuake; fees from writing, speaking, and consulting; royalties from books; and more recently income from a number of pensions and annuities. As a result, I long ago started keeping meticulous records, using Excel spreadsheets to keep track of individual activities and QuickBooks to bring it all together. Excel is a digital implementation of ninth century commercial arithmetic and algebra, as laid out by al-Khwārizmī in his two famous books on the subjects; QuickBooks an implementation of the book-keeping methods described by Pacioli in Chapter 9 of Summa.

Having spent time studying both al-Khwārizmī and Leonardo of Pisa (a.k.a. Fibonacci) while I was researching my three books on Leonardo and his hugely influential mathematics text Liber abbaci (the project began with a single book in mind), I had long ago come to appreciate the magnitude of the mathematical developments those two authors had catalogued—and thereby contributed to—in terms both of human intellectual progress and the impact of those methods on the way people around the world go about their daily lives. Double-entry book-keeping, on the other hand, the one topic Pacioli covered that those previous authors had not, simply never caught my attention. It’s just keeping records, right? What’s the big deal?

Given my experience with al-Khwārizmī and Leonardo, I should have known better. But there is a reason why people hardly ever give any thought to just how revolutionary, in their time, were numbers (and the associated innovation of money), arithmetic, the Hindu-Arabic representation, the classical arithmetic algorithms, and algebra. Each of those innovations changed human life in such fundamental ways that, once humanity had them we incorporated them (and the products and activities they brought in) into our daily lives to such an extent that we no longer gave them any more thought. Their fundamental role became no more remarkable than the presence of air and water. 

The same is true of more recent innovations, such as radio, the telephone, computers, the Internet, laptops, and mobile phones. To those of us who lived through at least some of those innovations, we still think of them as life-changing developments. But ask anyone of school age and it is clear that to them, all of those technologies are just part of the everyday environment. Nothing remarkable. It is a measure of the greatness of any innovation that completely transforms the way we live, that before long we no longer recognize how profound and remarkable it is.

Time, then, to take a fresh look at double-entry bookkeeping.

The benefit of keeping detailed records of financial transactions was recognized back in ancient times. For example, in ancient Rome the first emperor, Augustus, created imperial account books and established a tradition of publishing data from them. While Augustus’ primary purpose may have been propaganda—to publicize his personal spending—he made use of the accounts to plan projects and think about how the empire was managed. According to historian Jacob Soll in his excellent book The Reckoning, Augustus’ attention to the accounts enabled Rome to flourish. 

But the beginnings of modern bookkeeping came much later, in the emerging city-states of northern Italy in the eleventh century, where the Crusades sparked a massive growth in commercial activity. As trade flourished, merchants in Florence and Venice, in particular, developed a method of accounting that became known as bookkeeping alla veneziana (“the Venetian method”). 

In their ledgers, the Venetian merchants listed debits and credits in two separate columns. As Pacioli subsequently explained in Summa, this was the key to the new form of bookkeeping: “All the creditors must appear in the ledger at the right-hand side, and all the debtors at the left. All entries made in the ledger have to be double entries—that is, if you make one creditor, you must make someone debtor.” Today, we call this “double-entry bookkeeping.” 

One important benefit of this system is it provides a built-in error detection tool; if at any moment in time the sum of debits for all accounts does not equal the corresponding sum of credits for all accounts, you know an error has occurred.

In Florence, in the fifteenth century, the bank run by the Medici family adopted double-entry accounting to keep track of the many complex transactions moving through accounts. This enabled the Medici Bank to expand beyond traditional banking activities of the time. It started opening branches in different locations, offered investment opportunities, and made it easy to transfer money across Europe using exchange notes that could be bought in one country and redeemed in another. This growth allowed them to dominate the financial world at a time when Florence was the center of the world for trade and education.

This then was the environment in which Pacioli grew up and lived. As a result, when he set out to write an account of all the commercial mathematics known at the time, his list of contents included one topic that could not be found in Leonardo Pisano’s Liber abbaci : book-keeping.


Portrait of Luca Pacioli, generally attributed to Jacopo de’ Barbari, ca.1500

Portrait of Luca Pacioli, generally attributed to Jacopo de’ Barbari, ca.1500

Luca Bartolomeo de Pacioli was born between 1446 and 1448 in the Tuscan town of Sansepolcro, where he received an abbaco education, the package of commercially-oriented Hindu-Arabic arithmetic, practical geometry, and trigonometry that had been common in Italy since Leonardo published Liber abbaci, on which the schooling was based. (I tell that story in my 2011 book The Man of Numbers.) With texts written in the vernacular rather than the Latin used by scholars, abbaco focused on the skills required by merchants. 

Around 1464, Pacioli moved to Venice, where he worked as a tutor to the three sons of a merchant. It was during this period that he wrote his first book, a short text on arithmetic for the boys he was tutoring.

In 1475, he started teaching in Perugia, first as a private teacher, then, in 1477, becoming the holder of the first chair in mathematics at the university. In 1494, he published his book Summa, which made him famous. In 1497, he accepted an invitation from Duke Ludovico Sforza to work in Milan, where he met Leonardo da Vinci, with whom he worked and taught mathematics to until their paths diverged around 1506. Pacioli died at about the age of 70 on 19 June 1517, most likely in Sansepolcro where it is thought he spent his final years.

In addition to Summa, published in Venice in 1494, Pacioli wrote a number of other mathematics books:

Tractatus mathematicus ad discipulos perusinos  (Ms. Vatican Library, Lat. 3129) is a nearly 600-page textbook dedicated to his students at the University of Perugia, where Pacioli taught from 1477 to 1480. It covers merchant arithmetic (barter, exchange, profit, mixing metals, etc.) and algebra.

De viribus quantitatis  (Ms. Università degli Studi di Bologna, 1496–1508), a treatise on mathematics and magic. 

Geometry  (1509), a Latin translation of Euclid's Elements.

Divina proportione  (written in Milan in 1496–98, published in Venice in 1509). Two versions of the original manuscript are extant, one in the Biblioteca Ambrosiana in Milan, the other in the Bibliothèque Publique et Universitaire in Geneva. The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio and its supposed potential application in architecture. Leonardo da Vinci drew the illustrations of the regular solids in Divina proportione, while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletal solids, which allowed an easy distinction between front and back. The work also discusses the use of perspective by painters. 

Two of Pacioli’s books were to have a lasting impact. One was his book on the golden ratio, which gave the initial impetus to a cottage industry of writings that continues to this day, claiming to have identified the number Euclid referred to as the “extreme and mean ratio” in any manner of worldly objects and human artistic creations. While a few of those claims have substance (mostly the ones about the botanical world), the vast majority are entirely spurious, originally no doubt inspired in part by the use of the adjective “divine,” plus the proximity (to Pacioli) of Leonardo da Vinci, and thereafter driven by a dangerous intellectual mix of wishful thinking and mathematical naiveté.

Pacioli’s other influential book was Summa. Yet, in many respects, Summa is little more than an updated, vernacular version of Liber abbaci, which itself was an updated Latin translation of al-Khwārizmī’s Arabic books on arithmetic and algebra. But two factors resulted in Summa having a degree of impact that greatly exceeded those two earlier works. 

First, thanks to the recent invention of the printing press, Summa was the first major printed mathematics text, a format that could be duplicated and sold on a wide scale. In the days when manuscripts were hand-written, authors of mathematics texts avoided any use of the abstract symbols they used to do calculations—other than the basic numerals—because they could not rely on accurate copying of formulas and equations by the scribes who made copies. But with print, there was nothing that prevented them having entire pages consist of little else than formulas and equations. (The reason people today associate mathematics with symbols is a result of the printing press. Before then, mathematics was a subject presented in prose.)

Indeed, as I recounted in The Man of Numbers, we would today not know about Leonardo’s work and the major role it played in the development of the modern world, were it not for Pacioli’s acknowledgement that Summa was based largely on Leonardo’s teachings. The thirteen whole or fragmentary handwritten Liber abbaci manuscripts that are now treasured items in the libraries lucky enough to have them would likely still be gathering dust in archives, unseen by modern eyes. 

That they are not is due to Pietro Cossali, an Italian mathematician in the late eighteenth century, who came across the reference to Leonardo while studying Summa in the course of researching a mathematical history book he was writing. Intrigued by Pacioli’s brief reference to an unknown “Leonardo Pisano” as having been the source for most of the contents of Summa, Cossali began to look for the Pisan’s manuscripts, and in due course learned from them of Leonardo’s important contribution. (A French historian subsequently invented a surname for the newly re-discovered Leonardo: “Fibonacci,” and thereby helped give rise to a modern-day mathematics legend.)

To return to my main theme: because it was a print book, Summa achieved a far wider readership than Liber abbaci, or any of the other handwritten manuscripts that were based on Leonardo’s work. And so its impact was far greater. For that, Pacioli was simply lucky that he wrote his book after the printing press became available.

On the other hand, we can definitely credit Pacioli for the other factor that made Summa unique: his inclusion of a chapter on accounting.

As with Liber abbaci, Summa was more than a business person’s “how to” manual. Both were scholarly mathematical texts, written in the rigorous logical fashion of Euclid’s Elements.

Summa consists of ten chapters covering essentially all of Renaissance mathematics. The first seven chapters form a summary of arithmetic; chapter 8 explains contemporary algebra (initiating the use of logical argumentation and theorems in studies of the subject); chapter 9 covers various topics relevant to business and trade (including barter, bills of exchange, weights and measures, and bookkeeping); and chapter 10 describes practical geometry and trigonometry. As I noted earlier, none of the methods described are due to Pacioli himself; his contribution, which was significant, was the comprehensive, comprehensible exposition.

Significantly, Summa was also the first printed book to codify and give a comprehensive explanation of modern, double-entry bookkeeping, a system of accounting with a long history going back to Jewish bankers in Cairo in the eleventh century (maybe earlier), and used by Italian merchants and bankers, including the Medicis in Florence, throughout the fourteenth and fifteenth centuries. 

Pacioli clearly viewed chapter 9 as significant, devoting 150 pages to its coverage of mathematical techniques for business. It is in the section titled Particularis de computis et scripturis  (“Details of calculation and recording”) that he describes the accounting methods then in use among northern-Italian merchants, including double-entry bookkeeping, trial balances, balance sheets and various other tools still employed by professional accountants today. (The chapter also introduces the “Rule of 72” for predicting an investment’s future value, a technique that anticipated the developmentof the logarithm over a century later.)

In deciding to include a substantial chapter on business mathematics, Pacioli was simply reflecting the local needs of the time, just as al-Khwārizmī wrote his algebra book in response to the changing practices of the merchants around him in ninth century Baghdad, who were developing ways to “scale up” arithmetic to handle multiple trades where the same calculation was being repeated often with different numbers. 

In Florence, the Medicis were using double-entry accounting to keep track of the many complex transactions moving through accounts. As a result, the Medici Bank was able to expand beyond traditional banking activities of the time, setting up branches elsewhere and offering customers investment opportunities, as well as making it easy to transfer money across Europe using exchange notes that could be bought in one country and redeemed elsewhere. The Medicis’ mathematically-driven financial expertise enabled them to dominate the financial world at a time when Florence was the center of world trade.

Pacioli’s Summa showed others how it was done. He was surely writing a book for which he knew there was a great need. In short, Summa did for accounting what Liber abbaci had done for Hindu-Arabic arithmetic: it made it go mainstream, presenting it in a way that enabled ordinary people (at least those with some facility with numbers) to master the mathematical techniques required for finance and commerce. That’s the reason Pacioli is sometimes referred to as “the father of accounting.”

When we look back at the development of human society, we tend to see major leaps forward, initiated by a single individual or a small group, with longer periods of more steady progress. The bold initiators who launch society on a new path seem like superhuman geniuses, made of different mental stuff than the rest of us. But, in fact, each major leap forward is always a cumulative effect resulting from many individuals, each making small steps over years, decades, and even centuries. The intellectual giants we see absolutely deserve credit for what they did, but they are still mortals. As one of the greatest “giants” of all, Isaac Newton, famously and revealingly said, “If I have seen further than others, it is by standing upon the shoulders of giants.”

What those giants did that resulted in their names being prominent in the history books is bring together many accumulated small advances, interpret and synthesize them into a whole, and then package that whole in a fashion that is readily accessible to others less immersed in the details and the history. In some cases, among them Archimedes (around 250 BCE), Newton (seventeenth century), and Einstein (twentieth century), the influencers brought their own originality into the synthesis. 

With others, although their own original work was in some cases significant, it was solely their synthesis and packaging of the work of others that they are known for. Such was the case for Euclid (whose mammoth text Elements, ca 350 BCE, established the modern canons of geometry and number theory), al-Khwārizmī (the author of the ninth century text that established algebra as a widely used tool in commerce and then later engineering and science, who we met earlier), Leonardo of Pisa (Fibonacci—also encountered earlier—whose 1202 book Liber abbaci brought Hindu-Arabic arithmetic and algebra to the West), and Pacioli with his Summa. Though each of these authors produced other books where they presented their own work, it was the breadth and accessible quality of their expository works that changed the course of human history.

The same is true today, with technology. Two giants who changed the world in the 1980s are Steve Jobs and Bill Gates. But neither made any breakthroughs in the design of computers or the creation of software systems. Rather, they took the best of what was available, and packaged it in a way that millions of others could use. In the worlds of science, mathematics, and engineering, the professional cultures direct their admiration towards the innovators who come up with new ideas, and tend to downplay or even dismiss the individuals who package those new ideas into an accessible form that others can use. But both invention and packaging/marketing are required in order to change the world.

It is then, as a “packager” that we must view Pacioli in order to recognize the major impact he had on the course of history.

Because of the power of the recently invented printing press to spread multiple copies of identical texts relatively cheaply and quickly, Pacioli’s book-keeping treatise, as the first printed synthesis of the method, led to a rapid adoption of Venetian book-keeping, and by 1800, use of the system was standard across Europe. But that was not the end of the revolution.

Not long afterwards, the business world found another, far-reaching use for “bookkeeping alla veneziana.” It came about as a result of the desperate efforts of an English potter to prevent his company going bankrupt.

Josiah Wedgwood 

Today, the name Wedgwood is synonymous with fine pottery, sold all around the world. Less well known, is the major influence this eighteenth century English potter had on mass-market manufacturing in the early days of the industrial revolution.

Born in Staffordshire, England in 1730, Josiah Wedgwood was a highly talented potter and, it turned out, a skillful entrepreneur. Having learned the basic skills of pottery from his father, also a potter, he founded his own company while still very young. That company (the Wedgwood Company) was one of the first to adopt an industrial, mass-production approach to manufacture (and the first to do so for the manufacture of pottery). 

By the late 1760s, his traditionally produced, expensive classical designs had found a ready market among the nobility, among them Queen Charlotte (the wife of George III), who he persuaded to grant him permission to refer to his crockery sets as the “Queen’s Ware”. (A smart marketing move.) But Wedgwood wanted more.

In order to grow his company beyond that limited market, he looked for ways to manufacture cheaper sets to sell to the rest of society. This involved both experimenting with different materials and developing ways to produce and sell at scale. 

By staying abreast of scientific advances, he was able to adopt materials and methods to both revolutionize the production and improve the quality of his pottery. In particular, his unique glazes began to distinguish his mass-produced wares from anything else on the market. 

He also proved to have a flare for marketing, and today he is credited as the inventor of modern marketing techniques such as illustrated catalogues distributed by direct mailings, money-back guarantees, traveling salesmen carrying samples, self-service, and free delivery. 

In 1764, he received his first order from abroad. Just three years later, he was able to write of his pottery, “It is amazing how rapidly the use of it has spread all most [sic] over the whole Globe.”

Unfortunately, however, that rapid growth brought problems of finance, and by late 1769, Wedgwood and his partner, Thomas Bentley, had serious cash-flow problems and an accumulation of stock. Like many entrepreneurs, too much early success brought him to the edge of bankruptcy.

In response, in 1772 Wedgwood decided to use double-entry book-keeping to examine his firm’s accounts and business practices to see if there was a way for his company to survive. The results proved enlightening and, for the business world, far reaching.

He found that the firm’s pricing was haphazard, its production runs too short to be economical, and that it was spending unexpectedly large amounts on raw materials, labor and other costs, without collecting its bills fast enough to finance expanding production.

Statue of Josiah Wedgwood at the Wedgwood factory in Staffordshire, UK

Statue of Josiah Wedgwood at the Wedgwood factory in Staffordshire, UK

He also made an important discovery: the distinction between fixed and variable costs. He immediately understood the implications of their difference for the management of his business.

He told Bentley that their greatest costs—modelling and molds, rent, fuel and wages—were fixed: “Consider that these expenses move like clockwork, and are much the same, whether the quantity of goods made be large or small.”

He realized that the more their factory produced, the cheaper these fixed costs would be per unit of production.

In other words, by scrutinizing his books using double entry, Wedgwood had uncovered the commercial benefits of mass production.

To take advantage of his observation, Wesgwood had to take Pacioli’s book-keeping system and apply it beyond its mercantile origins in an exchange economy, to the world of manufacturing, where the emphasis is on the production of goods. That was a major shift, with enormous consequences, both for his company and for the world.

The need to incorporate new elements—labor and materials per unit of production—into an enterprise’s accounting system so that managers could calculate the cost of each unit of production posed significant conceptual difficulties. (That today we don’t give this a moment’s thought, is further indication of how fundamentally Wedgwood’s revolution changed the world.)

The challenge was that the transactions needed to incorporate the manufacturing of products into the existing double-entry system were not financial; they did not involve the exchange of goods, rather activities such as adding the cost of labor acquired or of materials bought. These “non-financial” transactions were new, and to fit them into the 300-year-old accounting system was not easy. Only after a century of factory production had such accounting problems become better understood. 

Meanwhile, and not altogether unconnected, the rise of the joint-stock company brought double-entry bookkeeping center stage, giving birth to a new profession: accounting.

The huge amounts of capital expenditure required to build railways—raised from private investors on stock exchanges and managed by joint stock companies—also generated new issues of accounting and accountability.

As a result of all these advances, by the 1860s, accountants in Britain were legally required at every phase of a company’s life. Financial statements had gone from being an incidental product of an enterprise’s book-keeping system in 1800, to being bookkeeping’s raison d’être a century later.

Looking back, we see that Venetian bookkeeping proved to be an ideal system for generating the financial statements that were required for the modern industrialized world. It could accurately record capital and income (as required by law and investors), it could distinguish between private expenses and corporate costs, and it could produce data that helped to evaluate past investment decisions.

It doesn’t get more relevant and important to today’s world than that.

For additional photos from the Christie’s event where I spoke about Pacioli, see the April 27 blogpost on the Stanford Mathematics Outreach Project site.

Read the Devlin’s Angle archive.