This Accountant is Fire!
This article is the beginning of a multipart series on the story of mathematicians Niccolò Tartaglia, Gerolamo Cardano and Lodovico Ferrari. Back in the early sixteenth century, the math world was a very competitive place. Solutions to problems were provided through puzzles and poetry, friendships were precarious, and some people even died taking their mathematical secrets to the grave. In this story, a century before the sixteenth century, a friar named Fra Luca Bartolomeo de Pacioli laid down the mathematical gauntlet, which stirred inquiry and curiosity in the competitive world of math.
Pacioli was an important person, not just in mathematics but also in accounting and magic. He was also important to Leonardo da Vinci, the genius overachiever we all know and love. At one point in his life, da Vinci and Pacioli used to be roommates and close friends.
Pacioli was born around 1447 in the Tuscan area of Italy and the city called Sansepolcro. Even though his father was still alive, he lived with a family named Befolci. Growing up, he received an abbaco education, which means his focus was mathematics. In addition to his traditional education, he also studied at the studio of the artist and mathematician Piero della Francesca. Pacioli completely admired della Francesca. And he learned a great deal not only as a mathematician but also as an artist.[1]
In 1464, Pacioli moved to Venice, Italy, and tutored three boys who were the sons of the wealthy merchant Antonio Rompiasi. While tutoring in Venice, Pacioli attended secondary school to continue his math studies while working as a teacher, a tutor, and a business manager for Rompiasi.
During this time in Venice, he began to write his first book, which included some of the material he used while tutoring. When he published this work, he dedicated it to Rompiasi’s sons. However, this work did not survive. After tutoring Rompiasi’s children, Pacioli traveled to Rome and lived with the brilliant polymath Leone Battista Alberti, who was working for the Catholic Church. While living with Alberti, Pacioli became interested in theology, joined the Franciscan order, and became a Friar, hence the title Fra.
As a Friar, he traveled from town to town as he taught mathematics at various universities. He taught at the University of Perugia between 1477 and 1480. While in Perugia, he wrote Tractatus mathematicus ad discipulos perusinos, which means A mathematical treatise for the students of Perugia. It was a 600-page textbook dedicated to his students at the University of Perugia. This text was one of his first works that emphasized his skills as an accountant and bookkeeper and highlighted his ability to teach the values of accounting. It held sixteen accounting-related sections, including bartering, exchange rates, and calculating profits. It also included algebra; however, portions of this subject are missing from the extant manuscript.
After his three years in Perugia, Italy, he traveled to Zadar, which is across the Adriatic sea in Croatia.[2] At that time, in the fifteenth century, Croatia was still part of the Venetian Empire. While in Zadar, he authored another book on mathematics; however, this book did not survive. As a result, of the three books that he wrote on arithmetic, the only one that survived was Tractatus.
After his time in Zadar, he traveled to Naples, Rome, and Venice, where he taught at the universities. While in Venice, in 1494, he published his infamous work Summa de arithmetica geometria et proportionalita, which means A summary of arithmetic, geometry, and proportionality. He had made quite a name for himself. He became the subject of a famous painting controversially attributed to the Venetian artist Jacopo de’ Barbari.
From 1496 to 1499, he taught in Milan, where he held the chair of mathematics. It was during his time in Milan that he met Leonardo da Vinci.[3] Then in 1499, he and da Vinci traveled to Florence, where they lived together until Pacioli assumed a teaching position in 1508 in Venice. While living with DaVinci, he also wrote another work called De divina proportione, which means Of divine proportions. This work also included the first image of a rhombicuboctahedron, illustrated by da Vinci, who also illustrated the entire book.
A rhombicuboctahedron is a solid with twenty-six faces that include six squares, eight triangles, and twelve rectangles. It has twenty-four identical vertices that connect where at each vertex is one triangle, one square, and two rectangles. Also, these rectangles can also be squares depending on the shape of the rhombicuboctahedron — and depending on the math.
What was also remarkable about De divina proportione is that it provides a beautiful illustration of the alphabet and how it should be proportioned.
When Pacioli moved to Venice in 1508, he published his work De viribus quantitatis, which means On the Strength of Quantity. Itwas a treatise on math and magic! Referred to as the “foundation of modern magic and numerical puzzles,” it contained chapters on math, math puzzles, and magic tricks. This book was another groundbreaking work that provided the first directions on how to do card tricks! (As a side note, for a neat card trick, check out my last podcast/show of 2021, called the Holiday Puzzle!) This work, De viribus quantitatis, provided directions on how to juggle, make coins dance, and eat fire! Clearly, this fifteenth-century Friar took fire to a whole new level! ***rim shot!***
After his time in Venice, around the age of 61, he traveled back to Sansepolcro and lived out his life. He passed away around the age of 70 on June 19, 1517.
The Summa is his most prominent work. It became a well-known textbook in the schools of North Italy because, for the first time, it covered algebra using the language of Northern Italy.
Additionally, for my math history buffs, there were rumblings that Pacioli plagiarized others’ works, specifically with his two works, the Summa and the Tractatus mathematicus. Some math historians note that he copied from other works to write his math texts verbatim. And though some science historians may consider that plagiarism, others refer to it as appropriation. Dr. Albrecht Heeffer, a professor of philosophy at Ghent University, in his 2009 paper titled Algebraic partitioning problems from Luca Pacioli’s Perugia manuscript, notes that Pacioli’s application and style of argumentation show an evolution from his older work Tractatus to his later work Summa.
Additionally, Heeffer notes that it is evident that Pacioli’s twenty years of teaching math across Italy and Croatia influenced his pedagogy and writing style. Since we cannot fully immerse ourselves into the fifteenth century to understand what Pacioli was presenting in his works, it is essential to note that we must look at the existing manuscripts to understand whether he plagiarized others’ works. Additionally, we must look at the style of his writing. And for Summa, Heeffer notes that “Pacioli introduces a new style of argumentative reasoning which was absent from abacus algebra.”[4]
For example, referencing my research, if we evaluate the commentaries written by Hypatia of Alexandria in the fourth century, she also pulled material from Euclid’s Elements and the Almagest. However, these were commentaries, and the one element that defines them as Hypatia’s work is that she provided a different pedagogy, style, and argumentative reasoning. Even though the equations are the same, the delivery and the explanations are different. Hypatia’s style and pedagogy are how we can define the difference between plagiarism and appropriation.
So, his work Summa was one of the first works that outlined a set of tools for accountants and bookkeepers. This textbook provided instructions on how to complete accounting ledgers. It described the importance of outlining assets, capital, expenses, receivables, inventories, and liabilities. It showed how to balance a ledger and conduct year-end entries. And he also impressed the importance of the Ricordanze, which means Rememberances. This work was similar to a managerial to-do list and included a list of “promises, obligations, and conditional agreements.”[5]
Furthermore, and most notable, this work was the first math textbook to describe the process of double-entry bookkeeping. This procedure became the standard for merchants across Italy. The method requires that you record your business transactions twice. The first entry is recorded as a debit. The second entry is recorded as a credit. At the end of the day, the accountant could compare values to ensure that the sum of the debts equals the sum of the credits. It was a whole new process for the world of accounting!
What is most notable in the Summa is that Pacioli inadvertently challenged the math world by proclaiming that there is no way to solve a cubic equation, much like there is no way to square the circle. If you are interested, the story on squaring the circle is in my previous article called Pseudomathematics.
This concept of not being able to solve a cubic equation came from when he worked with DaVinci, who showed that one could not obtain Euclidean irrationals of the form
\sqrt{a+\sqrt{b}}when working with the roots of the cubic equation x3+2x2+10x=20. [6]
And so Pacioli concluded his book Summa de Arithmetica with the statement that “the means [for solving cubic equations] by the art of algebra are not yet given, just as the means for squaring the circle are not given.”[7] In other words, he declared that it is an impossible problem to solve. This very mathematical statement roused the sixteenth-century competitive math world in ways unimaginable.
In two weeks, I will post my next article and will elaborate on this equation and begin the story of Niccolò Tartaglia, Gerolamo Cardano and Lodovico Ferrari by introducing the next key player Scipione del Ferro. Until next time, carpe diem!
[1] Onians, John. “Luca Pacioli.” In Bearers of Meaning, 216–24. The Classical Orders in Antiquity, the Middle Ages, and the Renaissance. Princeton University Press, 1988. http://www.jstor.org.ezproxy.snhu.edu/stable/j.ctv173f208.21.
[2] O’Connor, John J., and Edmund F. Robertson. “Luca Pacioli.” MacTutor History of Mathematics, July 1999. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Theon.html.
[3] Onians, John. “Luca Pacioli.” In Bearers of Meaning, 216–24. The Classical Orders in Antiquity, the Middle Ages, and the Renaissance. Princeton University Press, 1988. http://www.jstor.org.ezproxy.snhu.edu/stable/j.ctv173f208.21.
[4] Heeffer, Albrecht. “Algebraic Partitioning Problems from Luca Pacioli’s Perugia Manuscript (Vat. Lat. 3129),” 10:1–45, 2009.
[5] Sangster, Alan, Greg Stoner, Paul DeLange, Brendan O’Connell, and Giovanna Scataglini-Belghitar. “Pacioli’s Forgotten Book: The Merchant’s ‘Ricordanze.’” The Accounting Historians Journal 39, no. 2 (December 2012): 27–44.
[6] Gindikin, Semyon. Tales of Mathematicians and Physicists. Translated by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988.
[7] Gindikin, Semyon. Tales of Mathematicians and Physicists. Translated by Alan Shuchat. 3rd ed. Birkhäuser Boston, 1988