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Geometry Lessons: Mathematics and Optics

Leonardo surely learned some commercial arithmetic in his youth in Vinci, perhaps from his grandfather Antonio da Vinci, a Tuscan merchant, and he owned multiple abacus books. As Verrocchio’s apprentice in Florence, Leonardo would have been exposed to medieval theories of optics and learned the rudiments of geometry as part of his mastery of perspective.

In Milan, Leonardo inquired more deeply into these subjects. He encountered the physician and mathematician Fazio Cardano (1444–1524), who published John Peckham’s Common Perspective during Leonardo’s first year at the Sforza court and probably became Leonardo’s first science tutor. By 1490, Leonardo was actively hunting for the most important works of medieval Islamic and European optics in Milan and the university town of Pavia. His desire to understand vision led him to learn more mathematics and to begin to study anatomy and natural philosophy, dissecting and even creating an artificial model of the eye. Ultimately, Leonardo developed a critical understanding of the pupil as a pinhole.

As part of his investment in these subjects, in 1495 Leonardo purchased the Franciscan mathematician Luca Pacioli’s Summa arithmetica (1494), one of the most expensive books in his library. One year later, Pacioli (ca. 1446/48–1517) arrived in Milan and became Leonardo’s geometry tutor and most important collaborator. Leonardo created sixty drawings of semi-regular polyhedra for Pacioli’s Divine Proportion, completed in Milan in 1498 and published in Venice in 1509. Leonardo’s approach to mathematics was more visual than analytical, though he triumphantly claimed to have bested Archimedes by squaring the circle.

[Paula Findlen]

Euclid, fl. ca. 300 BCE

Elementa geometria

Venice: Erhard Ratdolt, 1482

Barchas Collection KA1482 .E88 F

Erhard Ratdolt (1442–1528) issued the first printed edition of Euclid’s Elements in Venice on May 25, 1482. The most innovative feature of this milestone publication was the inclusion, for the first time, of mathematical diagrams alongside the text. The book’s success was contingent upon the quality of the diagrams. As Ratdolt pointed out in his dedicatory letter, “hardly any of the important mathematical works had been published due to the difficulties other printers had in printing diagrams.” Having solved this challenge “by inventing a method to print diagrams like letters,” he paved the way for more readers to have access to the diagrams and, thus, to the Elements.

Leonardo was among the beneficiaries of this technological innovation. When he studied Euclid’s Elements, he might have needed help from maestro Luca Pacioli (ca. 1446/48–1517), including his unpublished Italian translation of Euclid. Diagrams, however, would have enabled Leonardo to engage with the work. If the diagram was the recourse for mathematical novices or illiterati whenever they lost their path in the text, still more so for this visual genius. Leonardo learned geometric language through diagrams. This learning resulted in his unique visualization of geometric figures as visible and tangible bodies (see his cuboctahedron in Pacioli’s De Divina Proportione), which contrasts with the conventional grammar of abstract diagrams in the Elements. Leonardo also expressed mathematical ideas through diagrams. His mathematical sketches, including squaring curvilinear figures and cubing solid figures, demonstrate the ultimate expression that one could reach through diagram-based inquiries. In sum, surrounded by more than 480 exquisite diagrams in this book, Leonardo was not an omo sanza schema (man without diagrams).

[Eunsoo Lee]

John Peckham, ca. 1230–1292

Perspectivae communis libri tres

Cologne: Arnold Birckmann, 1580

Barchas Collection QC353 .P4

Gift of Samuel I. & Cecile M. Barchas

John Peckham was a thirteenth-century English Franciscan theologian who wrote fifty books on various topics from science and philosophy to Franciscan spirituality and poverty. Peckham’s Perspectiva communis was by far the most popular and comprehensive of all medieval treatises on geometrical optics. Completed in 1279, it was used as a standard introduction to the subject in the universities of the late Middle Ages (including universities in Paris and Oxford, where Peckham taught theology and astronomy). The book had a wide scope, touching upon all the principal topics of medieval optics: the properties of light, the anatomy and physiology of the eye, reflection, refraction, visual perception, and errors and psychology of vision, to name a few. Peckham’s Perspectiva was of great importance to Renaissance artists and scholars, such as Alberti and Ghiberti, who formulated linear perspective, a representational system also used by painters to create an illusion of a three-dimensional space on a flat surface.

Peckham’s treatise was also significant for Leonardo, whose notebooks reveal ample familiarity with it. Leonardo used his friend Fazio Cardano’s edition, published in Milan in 1482–1483, just after he arrived there. Intending to write a treatise on optics, Leonardo roughed out a translation from Latin of Peckham’s introduction and created some of his own accompanying geometrical diagrams. Leonardo’s engagement with this book demonstrates his lifelong exploration of vision, in which the eye serves as a window to learning, an instrument enabling direct contact with the natural environment, and a tool for uncovering obscured truths.

[Ron Reichman]