S is for Summa di Arithmetica

Pacioli also created a font. This Majuscule seems especially appropriate.

Having just spent the past two months analyzing a 1494 accounting textbook, it seems natural to devote one my alphabetic letters to the greatest math teacher of the Renaissance age–Luca Pacioli. I stumbled upon him and his work last year for the letter “P,” so I’m not going to rehash his biography.

Luca Pacioli woodcut from Summa, his 1494 600-page math textbook.

Nor will I tell you the secrets of my 35-page treatise on how this chapter on double-entry bookkeeping for Florentine wool merchants reveals their pious contract with heaven and the Catholic church. Feather Beds and Jesus may just be my next book, who knows? What I will talk about is why this work was so revolutionary, despite accusations of plagiarism and critics calling it of “little or no value.” Boo on them!

Free the Numbers!

If numbers give you a headache, I apologize in advance. But we have to talk about numbers. Perhaps you aren’t crazy about multiplying large numbers, like 9876 * 6789. That’s what calculators are for. Now imagine that it’s the year 1490 CE, and you’re still using Roman numerals, and you don’t have a calculator. You have to multiply IXDCCCLXXVI * VMDCCLXXXIX. Can you imagine? There were, apparently, ways to multiple Roman numerals that involved writing them in columns, doubling and halving, then crossing out odds and evens. You would be desperate to find an easier way. Welcome to Hindu-Arabic numerals.

Pacioli’s explanation on how to multiply 9876 & 6789, from Summa.

Long story short, Hindu-Arabic numerals are the ones we use, even though 56% of Americans recently polled claim they’d never use Arab numerals. The Hindus likely originated them, the Arabs adopted them widely, and key scholars like Pope Sylvester II and Leonard of Pisa (also known as “Fibonacci”) helped get them adopted in Europe. Fibonacci–yes, the guy with the sequence–wrote a book called Liber Abaci which was all about how to use this new system to solve problems. A few copies were circulating by the time of the Renaissance in the monasteries and businesses because merchants and engineers were using it in their calculations.

While it’s rumored today that the church banned their use, apparently that wasn’t true. It was the case that some places, like the Florentine banking guild, banned their use because they thought they were too easily altered, e.g. making 150 into 1500. They worked around that. Meanwhile, if you’re the son of a wealthy merchant in the 1490s, you didn’t learn math in grammar school. You learned rhetoric and grammar, and lucky you learned to read at all, because that meant you weren’t a farmer. Math wasn’t widely taught, not even in college. In the year 1500 at the University of Rome, there were 31 law, 15 medicine, 16 philosophy/theology, and 2 math professors. Fra Luca Pacioli might have been one of them.

Examples of accounting debit and credit entries in Summa.

What Pacioli did was teach Fibonacci’s arithmetic, a style specifically aimed at business problems called abaco, which was how to do problems without an abacus and with these newfangled numbers. Pacioli’s 600 page textbook is all about business arithmetic including weights and measures, mining assays, practical geometry for engineering, and accounting. It was written in the vernacular–Italian–and it uses Hindu-Arabic numerals.

“Envious and Malignant”

Pacioli has been called “the father of accounting” and the inventor of double-entry bookkeeping. Let’s set the record straight: he didn’t invent it. The debit/credit style of accounting was in fairly wide use in Italy by the 1300s, had probably come from the Arabs, who maybe learned it from the Indians or the Chinese. He called it the Venetian system.

Summa is a textbook, a compilation of tons of problems and examples that Pacioli had collected in 25 years of teaching mathematics. He cobbled pieces together from other people’s teaching notes. Hence, he’s also been widely accused of plagiarism. Remember yesterday how Vasari invented the idea that Raphael died of too much sex? Vasari had an active imagination.

Vasari, in his biographies of painters, wrote about Piero della Francesca, an early user of geometric perspective. Piero was from Sansepolcro, which was where Pacioli grew up. Biographers say Pacioli learned his abaco arithmetic and geometry from the artist. Decades later, Piero wrote a book about geometry. Pacioli wrote two books which included geometry, and one of them was fairly identical to Piero’s. So Vasari cast Pacioli as the most scurrilous villain and said he was “envious and malignant” and claimed the “master’s work of his own.”

Evidence suggests otherwise. They knew each other for years, and Pacioli went on to study mathematics extensively, while Piero was off painting. The work that Pacioli allegedly stole was written in the 1480s, near when Pacioli was working on Summa. Historians in recent years have showed convincingly that they probably worked on it together and both published it, which wouldn’t have been plagiarism.

Piero della Francesca paints his friend Luca, with a bunch of other people. Photo from wikimedia.

Regardless of what Vasari thought, Piero della Francesca thought highly enough of his friar buddy, who was always good at sums, to include him a painting. This is Piero’s Montefeltro altarpiece, commissioned by the Duke of Urbino in honor of the birth of his son. That son, Guidobaldo, is shown as Jesus with mother Mary fashioned after the Duke’s wife while the duke kneels in front on the right. Baby Guidobaldo grows up to be Raphael’s patron, by the way. These guys all knew each other.

Behind the duke on the right are St. Francis and St. Andrew, and peering between them is another friar: that’s Luca! His head is bleeding because he’s playing the part of Peter of Verona, who was killed by rival Christian (Cathars) assassins with an axe to the head.

The Father of Printed Math in the Vernacular

What Pacioli did do all by himself was get this giant math textbook into a newfangled German technology known as printing. Summa became the first business math textbook, with the bookkeeping section, printed in the vernacular, i.e. Italian. The printers created 2000 copies, 61 of which are still in existence.

The De Scripturis section on accounting had its copyright renewed several years later and went through a second printing. It helped the spread of Arabic numerals and let billions forever after find an easier way to multiply IXDCCCLXXVI * VMDCCLXXXIX. That’s why Luca Pacioli is roundly considered to be the father of accounting as well as a key contributor to the history of mathematics. The Sri Lankans, who appreciate multiplication, honored him with a postage stamp on the 500th anniversary of Summa’s publication.

Not to mention that Pacioli invented a font. It’s what the Met used to use in their advertising.

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