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Mathematical Treasure: Della Francesca's Archimedes

Frank J. Swetz (The Pennsylvania State University)

Piero della Francesca (ca. 1416-1492) is best remembered as a Renaissance painter whose murals adorn some of the great churches of Italy and whose portraits and outdoor scenes convey a sense of depth. But in his time, he was also recognized as a mathematician who composed several mathematical treatises, of which only a few survive today. These include the following three works: 

  • Trattato d’abaco, a simple arithmetic book;
  • De prospectiva pingendi, the first text to describe the use of geometric perspective in drawing and painting; and
  • Libellus de quinque corporibus regularibus, a study of the five regular solids.

Modern scholarship has ascertained that Piero, in the mid fifteenth century, also wrote and illustrated a collection of works of Archimedes (Banker, 2005). This manuscript of 82 folios is housed at the Biblioteca Riccardiana in Florence, Italy; however, the whole manuscript can be viewed via the World Digital Library. The following excerpts are provided through the courtesy of the World Digital Library; they include pages from the following works of Archimedes:

  • On the Sphere and Cylinder;
  • On the Measurement of the Circle;
  • On Conoids and Spheroids;
  • On the Equilibrium of Planes;
  • On the Quadrature of the Parabola; and
  • The Sand Reckoner.

Figure 1. The first page of the manuscript and the beginning of On the Sphere and Cylinder.

In the mid fifteenth century, only a few Latin copies of Archimedes’ work existed in Europe. In the early 1450s, Pope Nicholas V commissioned Jacobus Cremonensis to make a Latin translation of the available Greek copies of Archimedes’ work into Latin. Piero transcribed Cremonensis’ work into his own manuscript, probably in the late 1450s.

Figure 2. A page from On the Sphere and Cylinder with supporting diagrams. Piero added over 200 such illustrations, clarifying and proving statements in the text.

Figure 3. The first page of On the Measurement of the Circle.


Figure 4. An illustrated page from On Conoids and Spheroids.


Figure 5. The title page for the On Spirals section.

Figure 6. Illustrations of the construction of spirals. PIero used a compass to construct these diagrams; the holes from the points of his compass remain as evidence of the drawing process.


Figure 7. More illustrations of the construction of spirals. As in Figure 6, it is evident that Piero used a compass to construct these diagrams, as the holes from the points of his compass can still be seen.


Figure 8. The beginning of Archimedes’ On the Equilibrium of Planes.


Figure 9. The opening pages of On the Quadrature of the Parabola. In his writing, Piero used the tight humanistic script of this period.


Figure 10. The page above demonstrates the partitioning of a parabola into basic geometric shapes, allowing for area approximation.


Figure 11. The beginning of The Sand Reckoner, a theoretical tract in which Archimedes attempted to approximate the number of grains of sand that would fill the universe. It was Piero della Francesca’s work on this manuscript that convinced many scholars that he should also be recognized as a mathematician (J.V. Field, 2005).

To fully appreciate this manuscript, it is suggested that viewers see the complete work at the World Digital Library site. The images above may be used in your classroom and for private study; for all other purposes, please seek permission from the Riccardiana Library of Florence (or Biblioteca Riccardiana Firenze).


James R. Banker. “A manuscript of the works of Archimedes in the hand of Piero della Francesca,” The Burlington Magazine, vol. 147 (March 2005), pp. 165-169.

J. V. Field. Piero della Francesca, A Mathematician’s Art, Yale University Press, 2005.

World Digital Library:

Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Della Francesca's Archimedes," Convergence (January 2014)