Mathematical Treasure: Luca Pacioli’s Divina Proportione

Luca Pacioli was an Italian mathematician and friend of Leonardo da Vinci. He is especially known for Sūma de Arithmetica Geometria Proportioni & Proportionalita, a 600-page collection of the mathematics known at the time, published in 1494. The second work for which Pacioli is known is Divina Proportione (Linda Hall Library call number QA464.P32 1509). Although not stated explicitly, it is generally believed that Leonardo da Vinci did the illustrations for this 1509 book. Pacioli taught da Vinci mathematics and the two became close friends. In fact, Pacioli is the first to describe da Vinci’s famous The Last Supper painting in print, praising it in the dedication page of Divina Proportione. (Pedretti, p. 180) Below is the title page of Divina Proportione.

The book contains geometric studies of capital letters, including M, which was the iconic logo of the Metropolitan Museum of Art in New York from 1971 to 2016.

There are 59 illustrations of polyhedra; some of which are solid, and some, like the one below, which are skeletal, making it easier to see edges and vertices.

The 1509 publication actually contains 3 works that were completed in 1498. The first is Divina Proportione; the second contains geometry related to architecture, including the golden ratio; and the third is an Italian translation of a work by Italian artist Piero della Francesca. There are two surviving manuscript versions, and it is possible to purchase facsimile editions.

For more information and images of Pacioli’s Divina Proportione, visit the following pages:

• "Mathematical Treasures - De Divina Proportione, by Luca Pacioli" by Frank J. Swetz in Convergence, with two images from the copy at Columbia University’s Butler Library.
• "Mathematical Treasure: De Divina Proportione" by Frank J. Swetz in Convergence, with three images from the copy at the University of Oklahoma Library.
• “Leonardo da Vinci's Geometric Sketches” by Frank J. Swetz in Convergence, which displays illustrations of the Platonic solids by Leonardo Da Vinci from a facsimile of Pacioli's manuscript version of De Divina Proportione.

Images in this article were taken by the author at the Linda Hall Library of Science, Engineering & Technology and are used with permission. The Linda Hall Library makes available all existing digital images from its collection that are in the public domain to be used for any purpose under the terms of a Creative Commons License CC by 4.0. The Library’s preferred credit line for all use is: “Courtesy of The Linda Hall Library of Science, Engineering & Technology.”

References

Medieval Manuscript Facsimiles for Book Collectors & Libraries. https://www.facsimilefinder.com/.

O'Connor, J. J., and E. F. Robertson. “Luca Pacioli.” MacTutor History of Mathematics archive. https://mathshistory.st-andrews.ac.uk/Biographies/Pacioli/.

O'Connor, J. J., and E. F. Robertson. “Piero della Francesca.” MacTutor History of Mathematics archive. https://mathshistory.st-andrews.ac.uk/Biographies/Francesca/.

Pedretti, Carlo. Leonardo da Vinci: The Complete Works. David & Charles Books, 2013.

Index to Mathematical Treasures