Ivars Peterson's MathTrek

December 9, 1996

Leonardo, Water, and Math

"... the wave in a vessel runs several times from the circumference to its center and from the center to the circumference."

These words, translated from Renaissance Italian, represent the observations of Leonardo da Vinci, based on experiments he performed using a large circular vessel filled with water. Touching the water at the vessel's center produced a ripple that moved outward in an ever-widening circle to reach the rim, then was reflected back as a circle contracting toward the center. From the center, a circular wave would again propagate outward to renew the cycle.
 Sample from the Codex Leicester showing the English translation of a portion of the page. Leonardo also noted that the shape of the object generating a ripple had a very little effect on the wave's eventual shape, which invariably became circular. "An object of triangular shape dropped flat on the water will produce an almost circular motion," he observed. "An elongated object, in its contact with water, will also cause circular waves on the percussed surface." Leonardo da Vinci had a special interest in water, both practical and theoretical. Between the years 1506 and 1510, as he shuttled between projects in Milan and Florence, Leonardo jotted down his thoughts in a scientific notebook. His intention was to write a treatise on the effects produced by moving water, not only on Earth but also in the sky. Among his notes, tucked into the margin of a page, was a tentative table of contents, with chapter headings that covered many different aspects of water, including its characteristic behavior, its motion in rivers and the sea, the effect of waves along its surface, and the use of dams and canals to control its flow.
The 72 pages of Leonardo's notebook contain a wide range of comments. For example, he studied how waves interact when they meet, noticing that two sets of circular ripples run into each other, overlap, then continue on without affecting each other's shape.

"Water is of such agility that the motion of any wave ... will never be hindered by the motion of another wave which were to come its way," he concluded.

Leonardo understood the difference between the motion of water and the motion of a wave on its surface. "Circular surface waves penetrate one another as impulses, not as a body of water, for the water does not move from its previous place on account of the waves, but only the impulses are transmitted," he wrote.

Waves breaking on beaches, obstacles affecting the flow of rivers, the shape of air bubbles trapped in water, the generation of whirlpools, devices for traveling under water, along with why the sky is blue and why seashells can be found on mountaintops all attracted his attention.

I don't normally get a chance to browse the mind of a genius -- to follow the threads of Leonardo's musings and to admire the quick, deft sketches he drew to illustrate his ideas. However, Leonardo's scientific notebook, known for historic reasons as the Codex Leicester, is now available on a remarkable CD-ROM. The CD provides not only images of the pages, scrawled in Leonardo's cryptic mirror script, but also an ingenious overlay mechanism -- a kind of translating "glass" -- that can be positioned anywhere on a page to reveal the English translation of the written words within its rectangular bounds. It's like having a magic eye to read Leonardo in his own words.

The "Leonardo da Vinci" CD-ROM, produced by Corbis of Bellevue, Wash., also provides a beautifully illustrated outline of Leonardo's life and times and offers an evocative glimpse into the mind of a true Renaissance man. It includes reproductions of Leonardo's paintings, detailed commentaries on his writings in the Codex Leicester, and other features, embellished by music of the period.

I have spend hours wandering from one thought to another through the Codex Leicester, packed as it is with information of all sorts. Interestingly, this work represents just a fraction of Leonardo's voluminous writings, which might have run to more than 13,000 manuscript pages intotal. Only about 7,000 of those pages have survived.

The Codex Leicester includes very little explicit mathematics, though the influence of mathematics on Leonardo's thinking is certainly evident in the way he approached the interactions of waves and bubbles and in other topics. In an earlier manuscript, Leonardo himself declared, "He who does not know the supreme certainty of mathematics is wallowing in confusion." In the same work, he also insisted that "no one read me who is not a mathematician."

To Leonardo, mathematics meant an emphasis on rigor, coherence, and logic. Like most painters, architects, and engineers of his time, he had a thorough grounding in practical geometry. He had trouble with algebra, and he was not very good at doing arithmetic, perhaps out of absentmindedness. For example, when he cataloged his writings in 1504, he calculated as follows: "25 little books, 2 bigger books, 16 bigger again, 6 bound in vellum, 1 book bound in green chamois: total 48."

Nonetheless, mathematics was very important to Leonardo. For theory, he often turned to experts for advice and explanation. It's quite possible that his collaboration with mathematician Luca Pacioli and their joint investigations of geometry strongly influenced the composition of Leonardo's painting The Last Supper.

References:

Bramly, Serge. 1991. Leonardo: The Artist and the Man. New York: HarperCollins.

Peterson, Ivars. 1995. Off the beach. Science News 148(Aug. 19):120-121.

______. 1982. Paint by Digit. Science News 122(Nov. 13):314-315.

Information about the "Leonardo da Vinci" CD-ROM is available at http://www.corbis.com/.

Illustration courtesy of Corbis, 15395 SE 30th Place, Bellevue, WA 98007.