Luis A. Orozco, Ph.D., Professor Emeritus – University of Maryland, Dept. of Physics
Florence has many treasures, and sometimes it takes a little stimulation to find and enjoy them. It helps enormously to be part of ISI where its director Dr. Stefano U. Baldassarri is well connected with the people who study, classify, care, and show these treasures.
The visit to Florence of a good friend (Dr. Denise Caldwell, the director of Physics at the National Science Foundation in the United States), allowed me to look for something special to show her and the STEM UMD students at ISI. Talking with Dr. Baldassarri I mentioned that it would be wonderful to see one of the manuscripts of Piero della Francesca in Florence. He immediately jumped at the suggestion and said, “You mean manuscript 106 at the Riccardiana? The copy della Francesca made for his personal use of some of Archimedes Treatises on geometry and mathematics?”. “Yes” was my answer and the process started.
Piero della Francesca, the enormously respected painter of the 15th century, came from a family of merchants in Borgo Sansepolcro near Arezzo, south-east of Florence. As such, he was expected to learn basic arithmetic, including the use of the abacus. He apprenticed as a painter in Florence in the workshop of Domenico Veneziano at the end of the 1430s, but probably also met Alberti, Brunelleschi, and many others discussing the formal problem of perspective. I imagine that he became interested in it, not only from the painting point of view but as a mathematical problem; that is: how does one project a three-dimensional object into two dimensions? He continued to work for the rest of his life on those questions; his notebooks attest to that and show his superb mathematical mind and his mastery of geometry. His treatises, of which three survive (Abacus Treatise, On the Five Regular Solids, and On Perspective in Painting), may have been for his patron in Urbino, Federico da Montefeltro, also interested in perspective.
Probably his greatest contribution to geometry was on the study of Archimedean Solids finding a sixth one beyond the so-called five Platonic Solids. Much of Piero’s work was later absorbed into the writing of others, notably Luca Pacioli. His work on solid geometry was incorporated without acknowledgment in “Divina Proportione” by Pacioli. Some may consider this a well-documented case of plagiarism which was only uncovered in the 20th century. Piero also figured out how to calculate volumes of solids in ways that astonish for their elegance and simplicity. One should keep in mind that analytic geometry and calculus were not yet part of mathematics, as they would be incorporated more than two hundred years later.
ISI has an important relationship with the Biblioteca Riccardiana; a few years ago a group of students under the supervision of Dr. Baldassarri translated the Visitor’s Guide into English. The book is carefully made and tells the history of the library and some of the highlights of the collection. Dr. Baldassarri contacted Dr. Francesca Gallori, the director of the library, to inquire about seeing the manuscript. She agreed that we could visit the library in mid-September 2023. The students gathered at Palazzo Rucellai with Prof. Simone Lolli, who also teaches STEM at ISI, and other friends of mine who were visiting from San Francisco. They walked with Dr. Baldassarri to Palazzo Medici-Riccardi. A masterpiece I see as the other cufflink of the beautiful pair Palazzo Rucellai by Alberti, and Palazzo Medici-Riccardi by Michelozzo. Fortunately for all of us, Dr. Baldassarri had invited Dr. Rita Comanducci, one of the professors of Art History at ISI.
We were welcomed by the director Dr. Gallori at the entrance to the library where we signed in and left our backpacks. Dr. Baldassarri gave us an introduction to the Riccardiana, told us how this was the work of some of the Riccardi family (one could say it was their personal library), and showed us the main reading room with the frescoes of Luca Giordano. Then we proceeded to a special room where the book was presented to us for our enjoyment, delight, and awe.
The manuscript Riccardiano 106 contains 82 folio leaves with paper of superb quality. The handwriting has been identified as his, and as Dr. Baldasarri and Dr. Comanducci explained to us, he made ample use of symbols, such as tildes, to shorten the words he was writing, saving time and paper along the way. Although the calligraphy is superb, it requires a well-trained paleographer to directly read the text.
The manuscript is his personal copy made in the late 1450s of the translation into Latin by Iacopo da San Cassiano of some of the works of Archimedes who lived and worked in Syracuse around 250 years before the beginning of the Christian Era. As the Renaissance was happening Archimedes was considered the pinnacle of mathematics and physics. This manuscript dates from a time when there was no printing press producing mathematics books. Piero left ample margins to illustrate what the text proposed and demonstrated. The parts he copied are: On the Sphere and Cylinder (Archimedes considered this his greatest work), On Conoids and Spheroid, On Spirals, On the Equilibrium of Planes, The Quadrature of the Parabola, and The Sand Reckoner.
As I was highlighting the work of Piero della Francesca on going beyond the five Platonic solids, Dr. Comanducci stressed how important these were for the thinkers of antiquity and the Renaissance. Piero added one more to the Archimedean solids, but the rest, a total of thirteen, were found by Kepler, who for a long time tried to fit the orbits of the known planets on a set of them without success. He gave up and successfully tried the elliptical orbits.
For me, it was fascinating to imagine how Piero made those drawings with only a compass and a ruler following the arguments of Archimedes. My hands changed the pages to find the section on “The Quadrature of the Parabola”. This is the section where Archimedes describes how to calculate the area of a parabola intersected by a straight line using approximations that we now clearly identify with calculus. Piero drew a square outside the parabola and inscribed a triangle inside the parabola, clearly limiting the area of the parabola to a value between two that were well-known in the 15th century. This bounding the answer is key to all the STEM disciplines.
The other section I showed the group is the one with the spirals. Not only are they aesthetically beautiful, but they also form the beginnings of the use of what we now call polar coordinates. How Piero could do all that is beyond my understanding, but as a true Renaissance man, he not only pursued the individual beauty but that very peculiar beauty/truth, which is the truth of mathematics.
The visit ended and we wandered on the exhibit of works related to the painter Luca Giordano. There were books, drawings, and manuscripts, including some by Galileo Galilei, the founder of modern physics here in Florence.
Useful Bibliography beyond the Wikipedia page in English about Piero della Francesca.
Mark A. Peterson “The Geometry of Piero della Francesca” The Mathematical Intelligencer 19, 33–40 (1997).
Frank J. Swetz (The Pennsylvania State University) Convergence 2014, Mathematical Association of America. “Mathematical Treasure: Della Francesca’s Archimedes“.
The digitalized version of “Manuscript Riccardiano 106“.