P is for Pacioli

Portrait of Pacioli attributed to Jacopo de’Barbari at the Capodimonte Museum in Naples.

Luca Pacioli wrote the book on accounting. He also wrote the book on algebra, geometry, mathematics, and arithmetic. It’s a musty old book by now, fragrant with the smell of parchment, dust, museum purifiers, and centuries passing. But Summa de arithmetica, geometria, proportioni et proportionalita contains the world.

Pacioli’s Summa de Arithmetica, photo from the Smithsonian.

Da Vinci’s Buddy, Della Francesca’s “Admirer”

Pacioli was born in Tuscany, studied in Venice, taught in Rome, Croatia, Naples, and eventually settled in Milan under the patronage of Ludovico Sforza. Sforza supported another famous scholar: Leonardo.

By age 50, in 1497, Pacioli was working steadily on his multi-volume series about all things numerical. Da Vinci was interested, and Pacioli shared his knowledge of mathematics and proportionality with the great artist and scientist. Da Vinci, in turn, illustrated the Divina proportionae section.

DaVinci illustration: a Rhombicuboctahedron. Photo at Wikipedia.

Of course, it was written in Italian, since Pacioli was a resident of all Italy. Some of his work was explanation, while other parts were translation. Biographers believe that Pacioli likely studied under another learned artist and geometer, Piero della Francesca, when he was younger. Della Francesca himself had written a treatise on proportions and geometry. Pacioli fans describe the third section of Summa as a translation of Della Francesca’s work into Italian, the language of the great Renaissance minds.

Admirers of Della Francesca later claimed Pacioli plagiarized the work, by not explicitly crediting Piero’s work. But Summa doesn’t claim to be original–geometry builds on Euclid, algebra originated with the Arabs, and the bookkeeping section reflected merchant practices that were hundreds of years old. I don’t ever remember a textbook writer mentioning that Al-Khwarizimi invented algebra; they just completed the square as if it fell out of the clouds. Summa’s a textbook, and a pretty significant one at that.

Detail from Particularis de computis at the Smithsonian. My favorite part is the “typo” 16/17, middle of the page left.

Father of Accounting

There were other textbooks, especially for geometry. Summa is the first known work to describe contemporary business mathematics, in the section: Particularis de computis et scripturis (Details of calculation and recording). Even in this, it was especially comprehensive, explaining:

  • Double-entry bookkeeping (remember T-accounts?)
  • Bills of exchange
  • Credit
  • Float
  • International Sales Representatives
  • Taxation
  • Trial Balances
  • Balance Sheets
  • Cost Accounting
  • Accounting Ethics

Books should be closed each year, especially in partnership because frequent accounting makes for long friendship.

One of Pacioli’s “proverbs.”

Dreaming of a Future Filled with Logarithms

There’s nothing in Summa about bitcoin, mortgage-backed securities, Ponzi schemes, or social media marketing, so I suppose it doesn’t anticipate every crazy business notion that ever existed. But it does explain the basic idea behind the Rule of 72, anticipating logarithms by several centuries. The Rule of 72 is a simple way of estimating the impact of compound interest rates. The time an investment will double in value is roughly equal to 72 divided by the interest rate.

Even in 1497, Pacioli understood that compound interest was practically a miracle.

A pristine copy of Summa Arithmetica sold at Christie’s in 2019 for $1.2 million. The parchment and condition of the 500-year-old manuscript alone would have raised the value for a collector. But the ideas inside, whether for the business world or for the dreamers of art and science?


A woodcut of Pacioli used throughout Summa, photo from wikipedia.


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