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The Movements of Water

Among the four classical elements, water held by far the greatest fascination for Leonardo. Throughout his life, he studied its movements and flows, drew and analyzed its waves and vortices, and speculated about its role as the fundamental “vehicle of nature” (vetturale della natura) in the macrocosm of the living Earth and the microcosm of the human body. ¹

Leonardo’s notes and drawings about his observations and ideas on the movement of water fill several hundred pages in his Notebooks. They include elaborate conceptual schemes and portions of treatises in the Codex Leicester and in Manuscripts F and H, as well as countless drawings and notes scattered throughout the Codex Atlanticus, the Codex Arundel, the Windsor Collection, the Codices Madrid, and Manuscripts A, E, G, I, K, and L. ² The sheer bulk of Leonardo’s writings on water duly impressed his contemporaries and succeeding generations of historians. In fact, water was the only subject, apart from painting, of which an extensive compilation of handwritten transcriptions from the Notebooks was made. This collection of notes, transcribed in the seventeenth century and comprising 230 folios, was published in 1828 in Bologna under the title Della natura, peso, e moto dell’acque (On the Nature, Weight, and Movement of Water). ³

Carrier and Matrix of Life

Leonardo was fascinated by the nature and movements of water for several reasons. I believe that, ultimately, they all have to do with his persistent quest to understand the nature of life, which informed both his science and his art. Leonardo’s science is a science of living, organic forms, and he clearly recognized that all organic forms are sustained and nourished by water:

PRECEDING A stream running through a rocky ravine, c. 1483 (detail, see fig. 2-5).

FACING “Water falling upon water,” c. 1508–9 (see fig. 1-13).

It is the expansion and humor of all living bodies.
Without it nothing retains its original form. ⁴

The term “humor” is used here in its medieval sense of a nourishing bodily fluid. In another Notebook, Leonardo wrote: “[Water] moves the humors of all kinds of living bodies.” ⁵ Being a painter, he had ample experience with water as a solvent and accurately described this chemical property: “It has nothing of itself, but moves and takes everything, as is clearly shown when distilled.” ⁶

Leonardo’s view of the essential role of water in biological life is fully borne out by modern science. Today we know not only that all living organisms need water for transporting nutrients to their tissues but also that life on Earth began in water. The first living cells originated in the primeval oceans more than three billion years ago, and ever since that time all the cells that compose living organisms have continued to flourish and evolve in watery environments. Leonardo was completely correct in viewing water as the carrier and matrix of life.

One of the fundamental principles of Leonardo’s science is the similarity of patterns and processes in the macro- and microcosm. Accordingly, he compared the “water veins” of the Earth to the blood vessels of the human body (see p. 26). ⁷ As blood nourishes the tissues of the body, so water nourishes the Earth’s vegetation with its “life-giving moisture.” ⁸ And as water expands when it vaporizes in the heat of the sun and “becomes mingled with the air,” so blood by its warmth spreads into the periphery of the body. ⁹ Indeed, we shall see that Leonardo described in great detail how blood carries nutrients to the bodily tissues and that he developed an ingenious, though incorrect, theory of how body heat is generated by the turbulent flows of blood in the chambers of the heart (see p. 296).

In his paintings, Leonardo represented water as the carrier of life not only in the scientific sense but also symbolically, in the religious sense. According to the Christian theology that shaped the culture in which he lived, the faithful receive a new spiritual life in the sacrament of baptism, and water is the medium that conveys this sacrament. In the words of the Bible, baptism is rebirth of water and spirit (John 3:5). Several of Leonardo’s paintings contain variations on this fundamental religious theme, often integrating the religious symbolism with his scientific understanding of the life-giving quality of water.

This integration is already apparent in the very first record we have of Leonardo as a painter, when he was still an apprentice in the workshop of Andrea del Verrocchio in Florence. Around 1473, when Leonardo was twenty-one, Verrocchio let the youth paint one of two angels and parts of the background in his picture of the Baptism of Christ (plate 2). ¹⁰ Leonardo painted a wide, romantic stretch of hills and pinnacles of rocks of the kind that would form the backgrounds in many of his later paintings, and to that he added a long watercourse, flowing from a pool in the far distance all the way to the foreground, where it forms small waves rippling around the legs of Christ. While these ripples in the foreground represent the lifegiving water of the sacrament, the watercourse in the background, cutting through arid rocks and flowing into a fertile valley, portrays water as the carrier of biological life in the macrocosm of the Earth.

This theme is expanded and elaborated in several of Leonardo’s later paintings, in particular, in three of his masterpieces—the Virgin of the Rocks (plate 8), the Mona Lisa (plate 11), and the Madonna and Child with Saint Anne (plate 7). In the Virgin of the Rocks, Leonardo depicts a prophetic meeting of the infant Christ with the infant Saint John long before the Baptism. According to a fourteenth-century legend, this meeting took place during the Holy Family’s flight into Egypt, where they lived in the wilderness after their escape from Herod’s massacre of the innocents. Leonardo has placed the scene in front of a rocky grotto and turned it into a complex meditation on the destiny of Christ, expressed through the gestures and relative positions of the four protagonists, as well as in the intricate symbolism of the surrounding rocks and vegetation. ¹¹ An angel conspicuously points to the Baptist, directing our attention to his spiritual dialogue with Christ, while Mary tenderly protects the children with her outstretched arms.

As in Verrocchio’s Baptism, a mountain stream emerges in the far distance from the misty atmosphere surrounding pinnacles of rocks and breaks through the rocky landscape, flowing all the way to the foreground of the painting where it runs through a small pool—an allusion to the Baptism. However, the rocks are rendered here in much more detail and with astonishing geological accuracy (see pp. 77ff.), and the luxuriant vegetation in the grotto’s moist environment is clear testimony to the generative powers of water, presented by the artist in a subtle synthesis of scientific knowledge and religious symbolism (see pp. 102ff.).

The Mona Lisa is Leonardo’s deepest meditation on the mystery of the origin of life—the theme that was foremost in his mind during his old age. The central theme of the artist’s most famous painting is life’s procreative power, both in the female body and in the body of the living Earth. Essential to this power is the fundamental role of water as the life-giving element (see pp. 318ff.).

The theme of the origin of life is taken up again in the Saint Anne, which Leonardo painted around the same time as the Mona Lisa. Here the artist returned once more to exploring the mystery of life within a religious context. The painting shows Mary, her mother Saint Anne, and the Christ child together with a lamb in a highly original composition. Its theological message can be viewed as a continuation of Leonardo’s long meditation on the destiny of Christ, which began with the Virgin of the Rocks. ¹²

Once more, the familiar mountain lakes and jagged rocks rise high into the background, although they are less imposing than those behind the Mona Lisa. In both paintings, the central theme is the mystery of the origin of life in the human body and in the body of the Earth. In the Saint Anne, this is rendered even more complex by the presence of three generations and by the myth of the virgin birth. There is a double mystery here: the immaculate conception of Mary by Saint Anne and that of Christ by Mary. To emphasize the analogy between human nature and the Earth, Leonardo has mirrored the three generations in the painting’s foreground by three tiers of mountain lakes, interlinked by small waterfalls, in the background.

What these four paintings—the Baptism, the Virgin of the Rocks, the Mona Lisa, and the Saint Anne—have in common is Leonardo’s extended reflection on water as the life-giving element in the macrocosm of the Earth and the microcosm of human existence. Drawing on his scientific understanding, his artistic genius, and his great familiarity with religious symbolism, Leonardo expressed this meditation in a series of masterpieces that have become enduring icons of European art.

Nature’s Fluid Forms

Another reason Leonardo was so fascinated by water is that he associated it with the fluid and dynamic nature of organic forms. Ever since antiquity, philosophers and scientists had recognized that biological form is more than shape, more than a static configuration of components in a whole. There is a continual flux of matter through a living organism, while its form is maintained; there is growth and decay, regeneration and development. This dynamic conception of living nature is one of the main themes in Leonardo’s science and art. ¹³ He portrayed nature’s forms—in mountains, rivers, plants, and the human body—in ceaseless movement and transformation. And, knowing that all organic forms are sustained by water, he sensed a deep connection between their fluidity and the fluidity of water.

As Leonardo observed the flow of the life-giving element, he marveled at its endless versatility and adaptability. “Running water has within itself an infinite number of movements,” he noted in Manuscript G, “sometimes swift, sometimes slow, and sometimes turning to the right and sometimes to the left, now upwards and now downwards, turning over and back on itself, now in one direction and now in another, obeying all the forces that move it.” ¹⁴ In the Codex Atlanticus he wrote: “Thus, joined to itself, water turns in a continual revolution. Rushing this and that way, up and down, it never rests, neither in its course nor in its nature. It owns nothing but seizes everything, taking on as many different characters as the places it crosses.” ¹⁵

In addition, Leonardo carefully studied the actions of water in the erosion of rocks and river banks, its transformations into solid and gaseous forms (known in science today as phase transitions), and its properties as a chemical solvent. He never divided these diverse properties into separate categories but saw them all as different aspects of the fundamental role of water in nourishing and sustaining life:

Without any rest, it is ever removing and consuming whatever borders upon it. So at times it is turbulent and goes raging in fury, at times clear and tranquil it meanders playfully with gentle course among the fresh pastures. At times it falls from the sky in rain, snow, or hail. At times it forms great clouds out of fine mist. At times it moves of itself, at times by the force of others. At times it increases the things that are born with its life-giving moisture. At times it shows itself either fetid or full of pleasant odors. Without it nothing can exist among us. ¹⁶

For Leonardo, the fluid and ever-changing forms of water were extreme manifestations of the fluidity that he saw as a fundamental characteristic of all the forms of nature. He also noticed, however, that certain flows of water can produce forms that are surprisingly stable: eddies, vortices, and other forms of turbulence known to scientists today as coherent structures (see p. 55). He observed and sketched a great variety of these relatively stable turbulent structures, and I believe that his lifelong fascination with them came from his deep intuition that, somehow, they embodied an essential characteristic of living, organic forms.

Today, from our modern perspective of complexity theory and the theory of living systems, we can say that Leonardo’s intuition was absolutely correct. The fundamental characteristic of a water vortex—for example, the whirlpool that is formed as water drains from a bathtub—is that it combines stability and change. The vortex has water continuously flowing through it, and yet its characteristic shape, the well-known spirals and narrowing funnel, remains remarkably stable. This coexistence of stability and change is also characteristic of all living systems, as complexity and systems theorists recognized in the twentieth century. ¹⁷

The process of metabolism, the hallmark of biological life, involves a continual flow of energy and matter through a living organism—the intake and digestion of nutrients and the excretion of waste products—while its form is maintained. Thus, metaphorically, one could visualize a living organism as a whirlpool, even though the metabolic processes at work are not mechanical but chemical.

Leonardo never used the analogy between the dynamic of a water vortex and that of biological metabolism, at least not in the Notebooks that have come down to us. However, he was well aware of the nature of metabolic processes. Indeed, we shall see that his detailed description of tissue metabolism in connection with the flow of blood in the human body must be seen as one of his most astonishing scientific insights (see p. 312). Thus, it seems not too far-fetched to assume that he was so fascinated by whirlpools and vortices because he intuitively recognized them as symbols of life—stable and yet continually changing.

A Source of Power

Leonardo saw water not only as the life-giving element but also as the principal force shaping the Earth’s surface and as a major source of power, which could be harnessed by human ingenuity. In his time, three hundred years before the Industrial Revolution, the windmill, the water wheel, and the muscles of beasts provided the only power to drive human technologies, and among those Leonardo thought that water had the greatest potential. At the age of fifty, when he was famous as a painter throughout Europe and known as one of Italy’s leading military and hydraulic engineers, he dreamed of a grand scheme for a kind of “industrial” canal along the river Arno between Florence and Pisa. ¹⁸ He imagined that such a waterway would provide irrigation for the surrounding fields as well as energy for numerous mills that could produce silk and paper, drive potters’ wheels, saw wood, forge iron, burnish arms, and sharpen metal. ¹⁹ Leonardo’s ambitious project was never realized, but it was a prophetic vision. Centuries later, the powers of steam and hydroelectricity would indeed transform human civilization.

As an engineer, Leonardo was also well aware of the destructive power of water. In the plains of northern Italy, at the foot of the Alps, an elaborate system of canals had been built for irrigation and for commercial navigation, and one of the main challenges faced by hydraulic engineers was how to protect these canals from the flooding of their tributaries (see p. 32). This flooding happened periodically during heavy autumn rains and after a sudden spring melting of the Alpine snows. Leonardo paid great attention to these inundations, which could be very violent. He had witnessed a catastrophic flooding of the Arno in his native Tuscany at the age of fourteen. This childhood experience must have left a deep impression on him and perhaps was the cause of his morbid fascination with floods, which he considered the most frightening of all cataclysmic events. ²⁰ “How can I find words to describe these abominable and frightening evils, against which there is no human defense?” he wrote in the Codex Atlanticus. “With swollen waves rising up, it devastates high mountains, destroys the strongest embankments, and tears out deeply rooted trees. And with voracious waves, laden with the mud of plowed fields, it carries off the fruits of the hard work of the miserable and tired tillers of the soil, leaving the valleys bare and naked with the poverty it leaves in its wake.” ²¹

As a hydraulic engineer, Leonardo invented special machines for digging canals, improved the existing systems of locks, drained marshes, and modified the flows of rivers to prevent damage to properties along their banks. As an architect, he designed elaborate landscape gardens with splendid fountains, running water for cooling wine, sprinkler systems for refreshing guests during the hot summers, and automatic musical instruments played by water mills. ²²

He decided early on that his reputation and skills in hydraulic engineering and landscape design would be grounded in a thorough understanding of the flow of water. In his science and his art, Leonardo never tired of observing, analyzing, drawing, painting, and studying how water moves through the air, the blood vessels of the human body, the vascular tissues of plants, and the seas and rivers of the living Earth.

The Water Cycle

Since Leonardo’s science was based on repeated observations of natural phenomena combined with meticulous analysis, ²³ it is not surprising that he had an accurate understanding of the evaporation and condensation of water and was able to describe it clearly. “Readily it rises up as vapors and mists,” he wrote in Manuscript A, “and, converted into clouds, it falls back as rain because the minute parts of the cloud fasten together and form drops.” ²⁴ A slightly more detailed description can be found in the Codex Arundel:

At times it is bathed in the hot element and, dissolving into vapor, becomes mingled with the air; and drawn upward by the heat, it rises until, having found the cold region, it is pressed closer together by its contrary nature, and the minute particles become attached together. ²⁵

He was also well aware of the fact that water continually cycles through the earth and atmosphere: “We may conclude that the water goes from the rivers to the sea, and from the sea to the rivers, thus constantly circulating and returning.” ²⁶ Taken together, these statements seem to indicate that Leonardo had a clear understanding of the essential phases of the water cycle—how water in the oceans, heated by the sun, evaporates into the air; how it rises into the atmosphere until cooler temperatures cause it to condense into clouds; how minute particles in the clouds coalesce into larger drops that precipitate as rain or snow; and how this precipitation eventually flows into rivers that carry it back into the oceans.

In actual fact, however, Leonardo’s views of the water cycle were far from clear. He considered several different explanations, struggled for many years because none of them satisfied his critical mind, and arrived at the correct view only in his old age, in his early sixties. How are we to understand that? What prevented a man of his genius from understanding a natural process that seems so evident to us today?

The answer to the puzzle provides a fascinating example of the tremendous power of the philosophical framework known today as a scientific paradigm—the constellation of concepts, values, and perceptions that form the intellectual context of all scientific investigations. ²⁷ One of the foundations of the medieval worldview was the conviction that nature as a whole was alive, and that the patterns and processes in the macrocosm were similar to those in the microcosm. This analogy between macro- and microcosm, and in particular between the Earth and the human body, goes back to Plato and had the authority of common knowledge in the Middle Ages and the Renaissance. ²⁸ Leonardo fully embraced it as one of the guiding principles of his science and discussed it repeatedly (see pp. 65ff.). Whenever he explored the forms of nature in the macrocosm, he also looked for similarities of patterns and processes in the human body, and so it was natural for him to compare the “water veins” of the Earth to the blood vessels of the body.

Our modern systemic conception of life fully validates Leonardo’s method of exploring similarities between patterns and processes in different living systems, and his view of the Earth as being alive has reappeared in today’s science, where it is known as Gaia theory. ²⁹ However, Leonardo ran into difficulties with his comparisons between the living Earth and the living human body because he extended them beyond the similarity of patterns to comparisons of forces and material structures. One of the important insights of modern systems and complexity theories has been that, even though patterns of relationships between the components and processes of two different living systems may be similar, the processes themselves and the forces and structures involved in them may be quite different. ³⁰ It took Leonardo the better part of his life to realize this, but he clearly did so in his old age.

Since the total amount of water on Earth is finite, Leonardo argued, the water carried into the sea by the rivers must somehow cycle back to their sources, “thus constantly circulating and returning.” ³¹ Since he conceived of water as a “humor” that nourishes the Earth just as the blood nourishes the human body, he imagined that there must be water veins inside the body of the living Earth corresponding to the blood vessels in the bodies of animals and humans:

The body of the Earth, like the bodies of animals, is interwoven with a network of veins which are all joined together, and are formed for the nutrition and vivification of that Earth and of its creatures. They originate in the depths of the sea, and there after many revolutions they have to return through the rivers, created high up by the bursting of these veins. ³²

This was the traditional view, put forward by philosophers from Aristotle to the Renaissance: inside the living Earth, there is a system of water veins, in which the water circulates like the blood in a living body, until the veins eventually break in the high mountains. There, the water emerges from mountain springs, is collected by the rivers, and flows back into the sea. Leonardo realized, of course, that rivers are also fed by rainwater and melting snow. But for many years he maintained that their principal sources were the internal veins of the Earth. Even though he encountered many logical inconsistencies, he was unwilling for the longest time to abandon the powerful analogy between the circulation of water in the Earth and that of blood in the human body. “The water that rises within the mountains,” he wrote in his early forties, “is the blood that keeps these mountains alive.” ³³

Leonardo’s scientific mind was not content with the beautiful metaphorical description of water as “the blood that keeps the mountains alive.” He needed to explain how the water actually rises up to the mountain springs through internal channels. It was clear to him that some forces counteracting gravity had to be at work:

The water which sees the air through the broken veins of the high mountain summits is suddenly abandoned by the power which brought it there, and when it escapes from these forces that elevated it to the summit, it freely resumes its natural course. ³⁴

But what exactly were these forces? To find an answer, Leonardo used a method that was characteristic of all his investigations. Understanding a phenomenon, for him, always meant connecting it with other phenomena through a similarity of patterns. In this case, he identified two similar phenomena—how the blood in the human body rises to the head and how the sap in a plant rises up from its roots—and he assumed that the same forces were acting in all three examples:

The same cause which moves the humors in all kinds of living bodies against the natural course of gravity also propels the water through the veins of the Earth, wherein it is enclosed, and distributes it through small passages. As the blood rises from below and pours out through the broken veins of the forehead, and as the water rises from the lower part of the vine to the branches that are cut, so from the lowest depth of the sea the water rises to the summits of the mountains where, finding the veins broken, it pours down and returns to the low-lying sea. ³⁵

Having established this similarity of patterns, Leonardo then set out to identify the common forces underlying them. Over the years, he tried and then rejected several explanations. ³⁶ At first, he thought that the water was drawn up inside the mountains as steam by the heat of the sun, and he suggested that this process was similar to blood rising to a man’s head when it is hot:

When the sun warms a man’s head, the blood increases and rises so much with other humors that, by pressuring the veins, it often causes headaches. ³⁷

“The heat of fire and sun by day,” Leonardo argued, “have the power to extract the moisture from the low places of the mountains and draw it up high in the same way as it draws the clouds and extracts their moisture from the bed of the sea.” ³⁸

Subsequently, however, he discovered two reasons why this explanation could not work. He noted that on the highest mountain tops, closest to the heating sun, the water remains cold and is often icy. Moreover, with this mechanism the greatest amount of water should be drawn up in summer when the sun is hottest, but mountain rivers are often lowest at this time.

In a second explanation, Leonardo suggested that the water might be drawn up in a process of distillation, fueled by the Earth’s internal heat. He was aware of the presence of fire within the Earth from observations of hot springs and volcanoes, and he had also experimented with several types of distillation apparatus. ³⁹ Perhaps, he suggested, the interior fires of the Earth boil water in special caverns until it rises as vapor to the roofs of those caverns, “where, coming upon the cold, it suddenly changes back into water, as one sees happen in a retort, and goes falling down again and forming the beginnings of rivers.” ⁴⁰ Again, Leonardo found an argument against his own explanation. Such extensive distillation, he realized, would keep the roofs of internal caverns wet from the rising steam, but he remembered from his explorations of mountain caves that they were often bone dry.

A third proposal was based on the observation that water rises in a vacuum within an enclosed space. Leonardo was quite familiar with this phenomenon. One of his early inventions, when he was still working in Verrocchio’s bottega, had been a method of creating a vacuum to raise water by means of a fire burning in a closed bucket. ⁴¹ Now he hypothesized that the internal fires might rarefy the air in the Earth’s caverns and thus raise the water to the top. However, he soon realized that this would not work, because additional air would enter the cavern through the openings of the mountain springs and would thus stop the siphoning action of the vacuum.

On a folio of the Codex Leicester, Leonardo summarized both the distillation model and the siphoning model together with their counterarguments. ⁴² He illustrated the discussion with a drawing showing the cross section of a mountain with interior veins running from the sea all the way up to the top where they connect with two large caverns. Right below, we see clear sketches of the two processes of distillation and siphoning (fig. 1-1). An accompanying folio contains numerous drawings illustrating experiments with various siphons. ⁴³

As yet another alternative, Leonardo suggested that the water might be drawn up inside the mountains by some process similar to the action of a sponge, but that vague idea did not satisfy him either. “If you should say that the Earth’s action is like that of a sponge,” he countered, “the answer is that, even if the water rises to the top of that sponge by itself, it cannot then pour down any part of itself from this top, unless it is squeezed by something else, whereas with the summits of the mountains one sees the opposite, for there the water always flows away by itself without being squeezed by anything.” ⁴⁴

FIG. 1-1. Models of water circulation by distillation and by siphoning action. Codex Leicester, folio 3v (detail).

After many years of considering various explanations and finding counterarguments to all of them, Leonardo finally realized that his analogy between the blood vessels of the human body and the water veins of the Earth was too narrow; that in the water cycle, the water does not circulate inside the mountains but rises as vapor through the air, drawn up by the heat of the sun, and then falls as rain on the mountaintops. On a folio of the Windsor Collection, written after 1510 when he was around sixty, Leonardo stated unequivocally that “the origin of the sea is contrary to the origin of the blood,” because the rivers “are caused entirely by the aqueous vapors raised up into the air.” ⁴⁵

Around the same time, in a note in Manuscript G about water as the carrier of minerals, Leonardo stated quite casually, as a matter of fact, that the rivers are produced by clouds:

The saltiness of the sea is due to the numerous water veins, which in penetrating the earth find the salt mines, and dissolving parts of these carry them away with them to the ocean and to the other seas from whence the clouds, originators of the rivers, * never raise them up. ⁴⁶

From our contemporary perspective, Leonardo’s long intellectual struggle to understand the water cycle is extremely interesting. His successive theoretical formulations are quite similar to the theoretical models that are characteristic of modern science. ⁴⁷ Like scientists today, he continually tested his models and was ready to replace them when he found that they contradicted some empirical evidence. Moreover, as he progressed, he kept in mind the analogies and interconnecting patterns to phenomena in other areas, and revised his theories about those other phenomena accordingly. Thus, as he modified his explanations of the water cycle, he also modified similar models of the functioning of the heart and the flow of blood in the human body (see pp. 284–85).

In the end, Leonardo came to realize that, although water and blood both carry nutrients to living systems (as we would say today) and both cycle continually, the pathways of the two cycles and the forces driving them are quite different. During the years of 1510–15, when he finally reached a clear understanding of the water cycle, he also came to the conclusion that the blood in the human body is moved by the pumping action of the heart (see pp. 290ff.). That Leonardo was able, in his old age, to abandon the narrow analogy between the circulation of blood in the human body and the circulation of water in the body of the Earth, which had been firmly established in medieval philosophical thought, is an impressive testimony to his intellectual integrity, his perseverance, and the power of his scientific method.

When he was in his early sixties and reached his full understanding of the water cycle and the movement of blood in the human body, Leonardo also produced his most sophisticated writings in botany, in which he described the transport of “vital sap” through the vascular tissues of plants (see pp. 120ff.). It would be fascinating to know how his insights into the circulation of water and blood affected his ideas of how water rises through the plant tissues from the roots to the top. Today we know that this is a consequence of the evaporation of water from the leaves and of its intermolecular forces—the “cohesion in itself,” as Leonardo called it (see p. 41). Unfortunately, we are not likely to ever know Leonardo’s last thoughts on these matters since the manuscript that may have contained his definitive treatise on botany has been lost. ⁴⁸

From Hydraulic Engineering to the Scientific Study of Flow

The majority of Leonardo’s extensive collections of notes and drawings on the flow of water were concerned with problems of hydraulic engineering and with the phenomenon of flow itself. In the Renaissance, the latter was a subject unique to Leonardo. The movement of rigid bodies had been studied since antiquity. In contrast, although hydraulic engineers had produced magnificent works—from the great aqueducts and luxurious thermal baths of the Romans to the ingenious navigation locks of the early fifteenth century—it had not occurred to any of them to wonder how flowing water could be described mathematically. Nor did they attempt to explore the fundamental laws of fluid flow, the subject of our modern discipline of fluid dynamics. Leonardo did both. His investigations, drawings, and attempted mathematical descriptions of flow patterns in water and air must be ranked among his most original scientific contributions, leading him to discoveries that would reappear only centuries later.

When he was first employed as painter and “ducal engineer” at the Sforza court in Milan in 1490, Leonardo had already spent eight years in the capital of Lombardy, which was a vibrant trading center of tremendous wealth and a major seat of political power in northern Italy. During those years, he had not only painted the Virgin of the Rocks and a highly original portrait of the mistress of Ludovico Sforza, but had also undertaken an extensive program of self-education during which he systematically studied the principal fields of knowledge of the time. ⁴⁹

From his first years in Lombardy, Leonardo was fascinated by the engineering problems involved in the region’s elaborate system of canals. During the previous three centuries, hydraulic engineering in northern Italy had reached a level of considerable sophistication. ⁵⁰ The wealth of the Lombard region was dependent on the control of water and on land reclamation from marshes. Hydraulic engineering was needed to reduce damage from the periodic flooding of Alpine rivers, to supply the cities with water, to keep ports working, for irrigation, and for commercial navigation. The great canals of Lombardy, wide enough to let two large barges pass, interconnected the principal rivers of the area and featured a series of sophisticated locks for overcoming differences of water levels.

As ducal engineer at the Sforza court, Leonardo was probably in charge of all hydraulic works and thus became thoroughly familiar with the existing technologies and the problems that needed to be solved. ⁵¹ Indeed, the Codex Leicester contains a vast number of observations on practical hydraulic problems in rivers and canals. Before Leonardo, such knowledge had been transmitted mostly orally, and the approach of the Lombard engineers was purely empirical: all their practices and rules were based on the success or failure of previous similar works. This did not satisfy Leonardo’s scientific mind. He needed to know the reasons behind the empirical rules, and so he embarked on his lifelong studies of the laws of fluid flow, beginning with the basic dynamics of the flow of rivers and proceeding to complex patterns of turbulent flow.

Even in the midst of his theoretical studies, Leonardo always kept their practical applications in mind. For example, during a discussion of the flow of water around immersed obstacles, he noted: “The science of these objects is of great usefulness, for it teaches how to bend rivers and avoid the ruins of the places struck by them.” ⁵² In Manuscript F, written during the same period as the Codex Leicester, we find the following admonition: “When you put together the science of the movements of water, remember to put beneath each proposition its applications, so that such science may not be without its uses.” ⁵³

In Leonardo’s time, the scientific study of flow phenomena, now known as fluid dynamics, was entirely new. It was a field of study he himself created single-handedly. However, in view of his dynamic conception of the world and his practice of portraying nature’s forms in his drawings and paintings as being in ceaseless movement and transformation, such a study must have seemed completely natural to him. Indeed, flow was one of the dominant themes in his science and art. In the words of hydraulic engineer and Leonardo scholar Enzo Macagno, “To Leonardo, if not everything, almost everything was flowing or could be in one state of flow or another.” ⁵⁴

In early Greek philosophy, the idea that everything in the world is in a process of constant change was expressed in the famous saying by Heraclitus of Ephesus, “Everything flows.” There is no evidence that Leonardo was familiar with the philosophy of Heraclitus, but in an intriguing double portrait by the famous architect Donato Bramante, who was a close friend of Leonardo, Bramante represented his friend as Heraclitus and himself as Democritus. ⁵⁵

Since he saw movement and transformation as fundamental characteristics of all natural forms, Leonardo assumed that the basic properties of flow were the same for all fluids, and he found this confirmed by his observations. He emphasized especially the similarity between flows of water and air. “In all cases of motion, there is great conformity between water and air,” he noted in Manuscript A, ⁵⁶ and in the Codex Atlanticus: “The movement of water within water acts like that of air within air.” ⁵⁷ However, Leonardo was well aware that air differs from water in being “infinitely compressible,” * whereas water is incompressible. ⁵⁸

As far as flows of liquids were concerned, Leonardo experimented not only with water but also investigated the flows of blood, wine, oil, and even those of grains like sand and seeds. ⁵⁹ His experiments with granular materials are especially remarkable. He realized that he could learn something about the flow of water by observing a similar but somewhat simpler phenomenon—the flow of grains in which the individual flowing particles are actually visible. This method of using simplified models to analyze the essential features of complex phenomena is an outstanding characteristic of our modern scientific method. ⁶⁰ The fact that Leonardo used it repeatedly is truly remarkable. In his view of flow as a universal phenomenon of gases, liquids, and granular materials, and his attempts to use the latter as models of the former, Leonardo’s thought showed a level of scientific abstraction that was centuries ahead of his time.

Modern Fluid Dynamics

In modern science, the field of fluid dynamics (also referred to as “fluid mechanics,” “hydrodynamics,” and “hydraulics” †) is notoriously difficult because of the pervasive appearance of turbulent flows that have so far eluded a comprehensive mathematical analysis. ⁶¹ In an oft-quoted phrase, physicist and Nobel laureate Richard Feynman called turbulence “the last unsolved problem of classical physics.”

Turbulent flows are composed of eddies, also known as vortices, in a broad range of sizes, continually forming and breaking down—those swirling and randomly moving patches of fluid that so fascinated Leonardo. The problem in fluid dynamics is that turbulence is not the exception but the rule. At low velocities, the fluid’s internal friction, or viscosity, prevents turbulence from occurring, but as the flow velocity increases turbulence sets in, first at very small scales and eventually spreading throughout the entire fluid.

In the nineteenth century, the mechanical engineer Osborne Reynolds discovered that the onset of turbulence can be characterized in terms of a single parameter, now known as the Reynolds number, which is proportional to the flow velocity, the fluid’s viscosity, and the dimensions of the physical object containing the flow. At low values of the Reynolds number, the flow remains smooth, or laminar, and at a certain critical value it becomes turbulent. This discovery marked a major advance in fluid dynamics. Since different values of the Reynolds number correspond to different types of turbulence, this parameter allows scientists and engineers to compactly characterize turbulent flows.

In spite of this advance, however, scientists have so far not been able to formulate a comprehensive theory of turbulence. The mathematical difficulties arise from the fact that the basic equations of motion governing fluid flow, known as the Navier-Stokes equations, are nonlinear and notoriously difficult to solve. This nonlinearity is the mathematical equivalent of the chaotic nature of turbulence. Any turbulent flow contains a very large number of interrelated variables. The value of any one of these at a particular point depends on the flow at many other points, so that solutions must be obtained at many points simultaneously. This is made even more complicated by the fact that turbulent flows display a broad spectrum of scales. The size of the largest eddies may be over a thousand times that of the smallest, which makes their simultaneous mathematical description exceedingly difficult.

The use of powerful computers to simulate and analyze turbulent flows has recently made it possible to find some solutions for special cases. However, even with the new concepts and techniques of complexity theory, or nonlinear dynamics, scientists and mathematicians have had only limited success. ⁶² They are able to simulate the onset of turbulence and some simple flow patterns—for example, slow, two-dimensional flows around an obstacle—but full turbulent flows are still frustrating all efforts of comprehensive analysis.

Thus, fluid dynamics today consists of a multitude of theories, each valid only for special cases and each invariably invoking some heuristic hypotheses based on experimental observations. ⁶³ In view of this patchwork of theories and of the tremendous difficulties faced by today’s engineers, physicists, and mathematicians in their attempts to solve the enigmas of turbulent flows, Leonardo’s achievements in this field are truly impressive. At a time long before the development of sophisticated mathematical techniques and powerful computers, Leonardo was able to gain many remarkable insights into the nature of fluid flow.

Leonardo’s voluminous notes on the movements of water remained hidden for several centuries after his death, and hence had no influence on the development of science and engineering. The first theoretical analyses of fluids were undertaken only in the eighteenth century when the great mathematician Leonhard Euler applied the Newtonian laws of motion to an idealized “perfect” fluid (that is, a fluid without viscosity), and the physicist and mathematician Daniel Bernoulli discovered some of the basic energy relations exhibited by liquids. Unlike Leonardo two centuries earlier, however, the hydraulic engineers of that time were not interested in theory, and the theorists did not compare their models with observations of the actual flow of fluids. ⁶⁴ Real progress in fluid dynamics had to wait until the nineteenth century when Claude-Louis Navier and George Stokes generalized Newton’s equations for the description of the flow of viscous fluids and Reynolds discovered the parameter that now bears his name. It was only then that physicists and mathematicians rediscovered many of the theoretical ideas about fluid motion that Leonardo had clearly formulated more than three hundred years earlier.

Rivers and Tides, Waves and Flows

In his lifelong studies of “the movements of water,” Leonardo observed the flows of rivers and tides, drew beautiful and accurate maps of entire watersheds, and investigated currents in lakes and seas, flows over weirs and waterfalls, the movement of waves, as well as flows through pipes, nozzles, and orifices. A thorough analysis of all his observations, drawings, and theoretical ideas would fill an entire book, and indeed such a book ought to be written. In this chapter I shall concentrate on Leonardo’s discussions of the main characteristics of fluid flow and of turbulence.

Leonardo was well aware of the difference between flow and wave motion. As I discussed at some length in my previous book, ⁶⁵ he recognized from precise observations of circular ripples in a pond that the water particles do not move along with the wave but merely move up and down as the wave passes by. What is transported along the wave is the disturbance causing the wave phenomenon—the “tremor,” as Leonardo called it—but not any material particles. To make it easier for the eye to follow the precise movements of the water particles, Leonardo threw small pieces of straw into the pond, and he also compared their motion to waves in a wheat field:

The wave flees from the place of its creation without the water changing its position, like the waves which the course of the wind makes in wheat fields in May, when one sees the waves running over the fields without the ears of wheat changing their place. ⁶⁶

The comparison between water waves and waves in a wheat field was quite natural for Leonardo, because he saw wave motion as a universal form of propagation of physical effects in all four elements—earth, water, air, and fire (or light). ⁶⁷ Moreover, he masterfully portrayed the effects of “waves of emotion” in his paintings. This is especially apparent in the ebb and flow of movements in his most grandiose work, The Last Supper. ⁶⁸

As several art historians have pointed out, the internal dynamics of the painting can be perceived as a wave movement, emanating from the figure of Christ in the center, spreading outward in both directions, and then being reflected at the end of the table and the edge of the fresco to return once more to the center. ⁶⁹ A few years before painting the Last Supper, Leonardo had carefully observed interpenetrating waves generated by pebbles thrown into the still water of a pond. ⁷⁰ Now he portrayed a similar effect in the realm of human emotions. The words of Christ, “One of you will betray me,” are dropped into the solemn silence of the assembled company, where they generate powerful waves of emotion that propagate and interpenetrate through all the figures in the composition.

During his extensive travels in central Italy, Leonardo often spent time at the seashore, both on the Adriatic and the Ligurian seas. ⁷¹ During these visits, he invariably observed and analyzed the nature and types of waves, the breaking of surf upon the shore and the generation of mist, the deposits of debris on the beaches, the ebb and flow of the tides, the impacts of waves on rocky coastlines during sea storms, and other phenomena associated with waves. However, by far the largest part of his work on the movements of water was concerned with the nature of fluid flow.

Leonardo not only observed smooth and turbulent flows of water in rivers and canals and over weirs and waterfalls but also carried out flow experiments in controlled laboratory settings. For example, he would fill vessels of different shapes with water, disturb their surfaces with his hand, and observe the effects. In a simple experiment, he produced a rotational vortex. “When the hand is turned in circular movement in a vase halffilled with water,” he noted, “it generates an artificial eddy that will expose the bottom of this vase to the air.” ⁷² With the same method, he also generated more complex forms of turbulence: “The hand drawn frequently back and forth across the vase produces strange movements and surfaces of different heights.” ⁷³

In a series of more sophisticated experiments, Leonardo built special flow channels for observing fine details of water movements. Here is how he described his experimental setup:

Make one side of the channel of glass and the remainder of wood; and let the water that strikes there have millet or fragments of papyrus mixed in it, so that you can better see the course of the water from their movements. And when you have made the experiment of these rebounds, fill the bed with sand mixed with small gravel; then smooth this bed and make the water rebound upon it; and watch where it rises and where it settles down. ⁷⁴

These careful and detailed experiments are typical of the empirical method Leonardo used in all his scientific investigations. In his studies of fluid flow, he designed and tested several experimental methods that are still routinely employed in our modern laboratories five hundred years later. Foremost among them were his techniques of flow visualization. Even though he had exceptional powers of observation, he must have realized how difficult it is to accurately perceive the streamlines of turbulent flows. He repeatedly described how he added millet seeds and other types of grains to the water to make it easier for the eye to follow these complex movements:

This experiment you will make with a square glass vessel, keeping your eye at about the center of one of these walls; and in the boiling water with slow movement you may drop a few grains of millet, because by means of the movement of these grains you can quickly know the movement of the water that carries them with it. And from this experiment you will be able to proceed to investigate many beautiful movements. ⁷⁵

To analyze turbulent flows in rivers, Leonardo observed the movements of leaves or pieces of wood, and he also used sawdust for making complex streamlines visible:

If you throw sawdust down into a running stream, you will be able to observe where the water, turned upside down after striking against the banks, throws this sawdust back toward the center of the stream, and also the revolutions of the water, and where other water either joins it or separates from it, and many other things. ⁷⁶

In other experiments, he used dye to stain one of two coalescing water currents to find out how exactly they merge. These methods of flow visualization were rediscovered in the nineteenth century and have become standard practice in modern fluid dynamics to observe turbulent flows. ⁷⁷ Leonardo applied similar methods for visualizing the streamlines in flows of air, observing the movement of clouds, snowflakes, smoke, dust, leaves, and other things carried by the currents of the wind. “The air moves like a river,” he explained, “and drags the clouds with it just as running water drags all the things that float upon it.” ⁷⁸

On a folio of the Codex Atlanticus, we find a particularly clear and elegant description of flow visualization with millet seeds (fig. 1-2). Leonardo has sketched a glass vessel filled with water and with millet seeds dropping in from above, and he explains that the purpose of the experiment is to find out how exactly the water empties out through a central hole at the bottom. He notes that the seeds drift in from the sides, as the pathlines in the drawing indicate. In another similar experiment, he uses black seeds at the center and white seeds at the periphery in order to see which parts of the water pass through the hole first. ⁷⁹ It is noteworthy that Isaac Newton drew a diagram similar to Leonardo’s in his famous Principia (Proposition XXXVI, Problem VIII) but got the pathlines wrong because he had no means to visualize them. ⁸⁰

FIG. 1-2. Flow visualization with millet seeds. Codex Atlanticus, folio 219r (detail).

The Fundamentals of Flow

After many years of studying and analyzing the movements of water, Leonardo’s sharp observations and methodical experiments gave him a full understanding of the main characteristics of fluid flow. He recognized the two principal forces operating in flowing water—the force of gravity and the fluid’s internal friction, or viscosity—and he correctly described many phenomena generated by their interplay. He also realized that water is incompressible and that, even though it assumes an infinite number of shapes, its volume is always conserved. In addition, he recognized that water changes its properties when materials are suspended or dissolved in it and also when its temperature chan