-Fra Luca Pacioli
Critical Essay By
Dr. Daniele Struppa
Professor, President of Chapman University
Schmid College of Science and Technology; Mathematics
View Bio
Fra Luca Pacioli (1447-1517) was a Franciscan friar born in Borgo Santo Sepolcro (now Borgo Sansepolcro) in Tuscany. Early in his life he moved to Venezia as a tutor in the house of the merchant Antonio Rompiasi. In his capacity, he accompanied the merchant in his travels and developed significant knowledge in the practice of commercial accounting. In those years he also attended the lecture on commercial mathematics given at the Rialto School by Domenico Bragadin, a mathematician for the Venetian Republic. In 1468 Pacioli became a lecturer of mathematics and priest in San Marco.
Fra Pacioli is now best remembered for two very important books: the Summa de arithmetica, geometria, proportioni et proportionalita (Venice, 1494) and the Divina Proportione (Venice, 1509).
The Summa, as Pacioli himself writes, is not an original work, but rather offers a summary of the state of the art in Renaissance mathematics and stresses the importance of mathematics in a variety of professions and arts, including the painter, the musician, the astrologist, the cosmographer, the architect, the lawyer and the doctor. But possibly the most important application is to commerce. In his work Pacioli describes rules for investments, sales, profits and the necessity of being able to translate different weights and currencies when commerce takes place across borders. An entire section (Particularis de Computis et Scripturis) is dedicated to a complete and detailed description of the accounting methodology that we now call double-entry bookkeeping (or partita doppia as it is called in Italian). Pacioli acknowledges that this idea was known before him, and he attributes it to the Venetian merchants, from which, he tells us, he acquired what they called the Venetian method. This is, nevertheless, the first time that the methodology is explained in detail and formalized so that it could be used by a larger audience.
The basic idea is to record every transaction in two sections (for debits and for credits) in order to satisfy the fundamental accounting equation that states that the assets of a company are the sum of the liabilities plus the owner’s equity. In double-entry bookkeeping every transaction is therefore recorded both in the debits and in the credits column, and the sum of all debits must equal the sum of all credits. The main advantage of this system is the simplicity that it brings and the ability for the merchant to reconcile all the transactions and identify possible errors. The system also provided a means to deal with possible disputes, and it quickly became adopted throughout the mercantile system.
It is because of this contribution that Fra Luca Pacioli is often called the father of accounting, and this motivates why his bust was chosen to commemorate the establishment of the Jerrold A. Glass Chair in Accounting and Economics.
Another aspect of Pacioli’s intellectual interest, which is the reason for the specific quote that was chosen for his bust, is best represented in his Divina Proportione, a book he wrote in Milano around 1498 while working at the court of the Duke of Milano, Ludovico il Moro. It was eventually published in Venezia in 1509. This book is dedicated to the way in which geometry is present in various aspects of the visible world and how it can be applied to art and architecture. There are many interesting observations one could make on this book, and on its illustrations. There are in fact two sections of illustrations, the first containing twenty-three capital letters drawn, by Pacioli himself, with ruler and compass. The modern reader may be surprised at the inclusion of letters as geometrical objects, but this is in fact a tradition that has been kept alive even in modern times by Donald Knuth in his continuing work on the font creation software Metafont. The second set of illustrations is even more interesting, as it consists of sixty woodcut illustrations, originally drawn by Leonardo da Vinci while taking mathematics lessons from Pacioli himself, at the time they were both in Milano.
The book has, however, great interest independently of the illustrations. It is divided into three parts. The first is dedicated to the so-called golden ratio (or divine ratio), the irrational number that is the positive solution to x2-x-1=0, and whose interesting properties were first studied by Euclid. This number appears in a variety of distinct situations. It is for example the ratio between the diagonal and the side in a regular pentagon; it is also the ratio of the sides of the rectangle with the property that it can be cut into a square and a smaller rectangle with the same aspect ratio. Finally, the golden ratio is the limit of the quotient of two consecutive Fibonacci numbers. For Pacioli, however, this number had an interest beyond its mathematical properties, and it represented the Holy Trinity, God’s omnipresence and the quintessence, the fifth element that in medieval cosmogony filled the region of the universe above the earth. The second part of the book is devoted to the application of mathematics to architecture, and the third part deals with the five regular solids (also known as Platonic solids), and it appears to be a translation, without credit being given, of a work by another great Italian artist-mathematician, Piero della Francesca.
It is because of this second book, and the interest that Pacioli devoted to the intersection between art and mathematics, that we decided to choose, for his bust, a quote that reads “Without mathematics there is no art,” a sentence that captures the essential role of mathematics for painters learning the (at the time new) ideas of perspective.
References
Donald E. Knuth, The Concept of a Meta-Font, https://s3-us-west-2.amazonaws.com/visiblelanguage/pdf/16.1/the-concept-of-a-meta-font.pdf
Mario Livio, The Golden Ratio; The Story of Phi, the World’s Most Astonishing Number, New York, 2002.
Fra Luca Pacioli, Divina Proportione, https://issuu.com/s.c.williams-library/docs/de_divina_proportione
Fra Luca Pacioli, Particularis de Computis et Scripturis, https://aura.abdn.ac.uk/bitstream/handle/2164/18994/Sangster_etal_AH_Lost_in_Translation_VOR.pdf?sequence=2
