The weight of water

Leonardo inherited a vision of the physical world whose basis was set by Aristotle in the treatise On the Heavens, which dates from the fourth century BC. Like Aristotle, Leonardo thought that each of the four elements—earth, water, air, and fire—moves according to its innate tendency to occupy its natural place. The Aristotelian theory states that the natural place of earth and water is the bottom, so those two elements move downward and are deemed heavy. The natural place of air and fire is the top, so they move upward and are designated as light. Earth is considered to be heavier than water, and fire is lighter than air.³ 

Unlike Aristotle, however, who did not suggest an unambiguous way to compare the weights of two different bodies, Leonardo thought that the notions of heavier and lighter implied a comparison via a common property. In a note on folio 33 recto of the Codex Madrid I, for example, Leonardo described a method for comparing the weight of a lead cylinder with that of an identical volume of water, which is obtained by means of a wax mold of the lead cylinder itself.

The core concept of Leonardo’s note—what today we would call specific gravity—probably came to Leonardo via a Latin text that began to circulate in Europe between the 12th and the 13th centuries, the Liber Archimedis de insidentibus in humidum (Archimedean Book on Floating Bodies).⁴ The opus derived from Arab translations and amalgamations of Greek source works belonging to the Archimedean tradition. Scholars still debate whether Leonardo had direct access to the text or had heard of its contents from the engineers and scholars whom he met during his stays at the main Italian courts. In any case, within the treatise, the specific gravity of a body (gravitas secundum speciem) is clearly defined with respect to a reference volume, while the total weight of the body (gravitas secundum numerositatem) refers to the object in and of itself.

The influence of De insidentibus in humidum on Leonardo’s thoughts, however, was not limited to the idea of specific gravity. Traces of at least two passages from the apocryphal Archimedean text appear among the da Vinci notes. The first refers to proposition 5 from book 1 of the authentic On Floating Bodies by Archimedes (see box 1), which states that any floating object displaces its own weight of fluid. The second passage, in contrast with Aristotle, affirms that a body is neither heavy nor light when it is surrounded by a substance made of the same element.

The first passage was quoted by Leonardo in a few places, always with more or less the same words. For example, a note dating from the 1490s on folio 563 recto of the Codex Atlanticus says that “the weight of the ship equals exactly the weight of the water that the ship displaces from its own site.” (The translations here and below are our own.) Leonardo’s few notes, however, do not address the cause of buoyancy, which Archimedes had identified with a force whose intensity precisely equals the weight of the water displaced by a floating body.

In many of Leonardo’s notes, the second passage, too, appears in nearly identical form. The first of those notes dates from the 1490s, when Leonardo observed, “No element weighs in its own sphere” (Codex Arundel, folio 181 recto). Later in the decade he stated that “the weight is a certain push or desire, or wish to escape, that arises when an element is shifted into another one” (Codex Madrid I, folio 145 verso).

In the first decade of the 16th century, Leonardo noted that “no part of an element has heaviness or lightness in its own element. Move a vessel of water in air: All the water weighs in air and no part of this water has heaviness in itself. Move air under the water: All the air is light and it has no lightness or heaviness in itself” (Codex Atlanticus, folio 515 verso). Similarly, he wrote on folio 255 verso of the Codex Arundel that “no element weighs in itself, but it weighs above a lighter one, and it weighs less above a heavier one.” When a body is immersed in a material with a smaller specific gravity, the body does not receive a boost sufficient to prevent it from moving downward; it therefore increases its heaviness. On the other hand, when the same body is immersed in a substance with a greater specific gravity, it is pushed upward and increases its lightness. The absence of heaviness or lightness of a body immersed in itself is a concept similar to the present notion of neutral buoyancy.

In Leonardo’s conception, heaviness or lightness was not an absolute property of an element, as claimed by Aristotle; instead, heaviness and lightness depended on the matter from which a body is made and the substance surrounding the body. Furthermore, Leonardo considered heaviness and lightness as coexisting in a body, “like unequal weights on a balance” (Codex Arundel, folio 264 recto), until an equilibrium is restored when the body is brought back to its own element.

Leonardo’s theory of weight, in brief, incorporates pseudo-Archimedean notions to extend, or at least reinterpret, the Aristotelian concepts of heaviness and lightness. It has much in common with the present understanding of what happens to a body when it is immersed in a substance with a different specific weight, but Leonardo did not provide an explanation of buoyancy similar or equivalent to Archimedes’s principle.

Leonardo’s theory ran into difficulties when he considered the weight of water inside a container. Assuming that water has weight only when it is moved in or rests above an element with a smaller specific gravity—and, in particular, that it does not have weight when surrounded by more water—Leonardo initially concluded that water does not weigh on the bottom of its container, because the container is made of a heavier element (Codex Madrid I, folio 145 verso). Some natural phenomena apparently confirm that hypothesis. In many notes Leonardo recalled that mud does not compact on the bottom of a pond, as it would do if the weight of water were to tamp it down, or that aquatic plants wave freely as if they were not burdened by any weight. (See, for example, the Codex Arundel, folio 266 verso.)

The way some instruments work, however, evidently contradicts those observations. As an example, Leonardo observed that in some kinds of reciprocating pumps, a valve is located at the bottom of the pump cylinder, and it opens only when the external force acting on the pump prevails over the weight of water that rests on the valve (Manuscript H, folio 73 verso). Moreover, experience suggests that water also exerts a push on the walls of its container, as demonstrated by the jets that spring from holes in vessel walls; Leonardo depicted that phenomenon on folio 151 recto of the Codex Madrid I. The notes from the first half of the 1490s reveal that the problem of water’s weight was one that Leonardo found to be difficult and frustrating.

In about 1495 a compromise solution emerged in a note on folio 74 recto of Manuscript H. There Leonardo affirmed that the weight of water resting on the bottom of its container is reduced by the weight of water that rests on the walls. With that idea in mind, he came up with experiments for determining how much of water’s weight rests on the container bottom—he called them “ways of weighing water” (folio 146 recto of the Codex Madrid I). The most significant of them, which is depicted on folio 150 recto of the Codex Madrid I, is a balance attached to the movable bottom of a container; figure 1 is a reproduction of Leonardo’s sketch of the experiment. A similar experiment is depicted by a drawing from around the same time that appears on folio 1023 verso of the Codex Atlanticus. For that drawing, however, the related note is not about the weight of water on the bottom of the container but rather about the weight that rests on a hole in the bottom. Leonardo wrote, “If you want to know whether or not the water that perpendicularly stays over the hole in the bottom of the vessel weighs on the hole itself, that is to say if water partly pushes on the walls of the vessel, [perform the] experiment as illustrated above.”

Historians are still debating whether Leonardo ever quantified the weight of water that rests on the bottom of a vessel or eventually abandoned the idea of a subdivision of weight between the bottom and the walls of a container. In a note on folio 219 recto of the Codex Atlanticus dating from 1510, Leonardo raised the divided-weight hypothesis one more time; he probably never came to a definitive conclusion about its validity.

Notwithstanding his indecision vis-à-vis the role of a container’s walls, Leonardo stated in many notes that the magnitude of the action that water exerts on a point at the bottom of a container depends only on the vertical distance between the point and the free surface of water. Leonardo probably came to that conclusion, which shows an understanding of what today is called the hydrostatic paradox, by means of actual experimentation.⁵ After all, the experiment depicted in figure 2 (from folio 152 recto of the Codex Madrid I) is rendered with such great accuracy that it almost appears to be a photograph of an experiment in progress. With time, investigations of that sort give Leonardo a great mastery of the hydrostatic paradox. For example, in a note on folio 26 recto of the Codex Leicester dating from 1508, Leonardo considered two pipes with the same diameter: The lengths and inclinations of the pipes may be completely different, but as long as their tops and bottoms are delimited by two parallel planes, they “generate the same weight of water” on their bottoms when filled.

In Leonardo’s thinking, the “action” that water exerts on the bottom of a container is related to the action that water exerts on the walls of the same container. With that understanding, he imagined an indirect way to determine how much of water’s weight rests on the bottom. Leonardo used the idea of a mobile surface, as illustrated in figure 1, but that time he adapted it to a wall of the container.

Leonardo’s interesting and novel experiment is depicted in a drawing (figure 3) dating from around 1495 on folio 149 verso of the Codex Madrid I. The mobile wall of the water-filled tank is kept in place by the weight of an object that hangs from a pulley; the weight of the object measures the total push that the water exerts on the wall. Leonardo stated that by means of “well-done experiments,” water’s weight on the bottom can be determined by subtracting the four identical weights on the walls from the weight of the whole volume of water.

There is no evidence that Leonardo actually performed the experiment, and his instruction to subtract the four weights shows that in dealing with water, he did not understand what we now call the vectorial nature of forces. Moreover, Leonardo incorrectly positioned the balancing force that acts on the mobile wall. Apparently, at the end of the 15th century, he was not fully aware that the intensity of the action exerted by water on a vertical surface increases according to water depth. To balance the action of the water without torquing the movable wall, the balancing force would need to be applied at one-third of the height of the wall from the bottom, not at the wall’s center.

By around 1508, however, Leonardo was aware of the height dependence of water’s thrust on a wall. Folio 6 recto of the Codex Leicester deals with an actual hydraulic engineering problem: how to build banks and levees that are strong enough to prevent collapse. In his notes there, Leonardo identified the “level” of water as the only parameter that affects the intensity of water’s action. Water indeed “pushes in the height of the bank, from the surface to the bottom, with varying power,” and that “variation” is actually “caused by the difference, or inequality, of water height” from point to point. Leonardo correctly deduced that “water gains degrees of power in each degree of its depth,” and as a consequence, the bank must also “increase its resistance from its top to its bottom,” so that the “resistance” of the bank equals the “power” of water. For more of Leonardo on the power of water, see box 2.

Leonardo did not develop any calculational technique to theoretically determine how the power exerted by water varies, but on the same folio he sketched one of his most original techniques (figure 4) for measuring the “inequality of this power.” With such a setup, he could not only measure the magnitude of the whole action that water exerts on the wall, as he proposed to do with the experiment depicted in figure 3, he could also evaluate the magnitude and distribution of the actions that water exerts on all the bands making up the wall. No evidence exists to suggest that Leonardo actually performed the experiment, but in conceptualizing a mechanical discretization, he was probably the first in the modern era to come close to envisioning the present notion of the hydrostatic pressure distribution.⁶ 

The extant manuscripts of Leonardo da Vinci contain a remarkable and detailed record of a 15th-century scholar’s approach to some basic concepts in hydrostatics. Leonardo’s intellectual journey through the field, previously unexplored in the modern era, may appear hesitant and affected by sudden transitions between correct and incorrect notions. But a chronological examination of his works reveals an evolving understanding.^7,8 That evolution is the result of Leonardo’s very personal research method—one that balanced received wisdom against intuitions, observations, and experiments—that allowed him to move from the technical and theoretical knowledge of his time and to perform his own genuine investigation of the physical nature of water. And it is probably the inventiveness of his method, even more than the remarkable results that he obtained with it, that makes Leonardo the outstanding Renaissance man who, half a millennium later, still fascinates us.

We are deeply grateful to the late Enzo Macagno (1914–2012), whose advice and support were of fundamental importance as we began this article.

Paolo Cavagnero is a postdoc and Roberto Revelli is an associate professor in the department of environment, land, and infrastructure engineering at the Polytechnic University of Turin in Turin, Italy.