5
The Elements of Mechanics

As Leonardo studied, drew, and painted “all the forms of nature,” he investigated not only their external qualities and proportions but also the forces that had shaped and continued to transform them. He saw similar patterns in the macro- and microcosm, but his careful investigations of these patterns of organization made him realize that the forces underlying them were quite different.

In his extensive studies of flowing water, Leonardo recognized correctly that gravity and the fluid’s internal friction, or viscosity, were the two principal forces operating in its movements (see p. 39). In his detailed observations of rock formations, he identified water as the chief agent in the formation of the Earth’s surface (see p. 69). Moreover, he speculated about the nature of the tectonic forces that caused layers of sedimentary rock to emerge from the sea and to form mountains (see p. 89).

In his studies of plants and animals, Leonardo identified the soul as the vital force underlying their formation and growth. Following Aristotle, he conceived of the soul as being built up in successive levels, corresponding to levels of organic life. The first level is the “vegetative soul,” which controls the organism’s metabolic processes. The soul of plants is restricted to this metabolic level of a vital force. The next higher form is the “animal soul,” characterized by autonomous motion in space and by feelings of pleasure and pain. The “human soul,” finally, includes the vegetable and animal souls, but its main characteristic is reason.

The “Noble” Role of Mechanics

The autonomous, voluntary movements of the human body fascinated Leonardo and became a major theme in his anatomical work. From their origin in the center of the brain (the “seat of the soul”), he traced the transmission of the forces underlying various bodily movements through the central and peripheral motor nerves to the muscles, tendons, and bones (see p. 211). Leonardo argued that these muscles, tendons, and bones were nature’s “mechanical instruments,” which were essential to bodily movements and were best analyzed in terms of the laws and principles of “mechanical science” (see p. 131).

FACING Studies of power transmission, c. 1495.
Codex Madrid I, folio 123v (detail).

From the early days of his apprenticeship as artist and engineer in Florence, Leonardo had been familiar with basic principles of mechanics applied to simple machines—levers, screws, wedges, pulleys, balances, and the like. Later on, he applied these principles in his inventions of more complex machines and mechanical devices. But it was in Milan, when he developed a keen interest in mathematics and made the transition from engineering to science, that he became motivated to study mechanics with more sustained effort and in much greater depth.¹

At that time, Leonardo had begun to investigate the movements of the human body and had discovered a broader and more “noble” role for the science of mechanics. “The instrumental or mechanical science,” he would write fifteen years later in his Codex on the Flight of Birds,* “is very noble and most useful above all others, because by means of it all animated bodies that have movement perform all their operations.”² The contrast of this statement with the pronouncements of the architects of the mechanistic worldview in the subsequent two centuries is rather remarkable. For Descartes, Bacon, and Newton, the ultimate value of the science of mechanics was the human domination of nature. For Leonardo it was the understanding and imitation (in the case of the flight of birds) of the animal body in motion.

To understand in detail how nature’s “mechanical instruments” work together to move the body, Leonardo immersed himself in studies of problems involving weights, forces, and movements. He showed how joints operate like hinges, tendons like cords, and bones like levers. While he studied the elementary principles of mechanics in relation to the movements of the human body, he also applied them to the design of numerous new machines, and as his fascination with the science of mechanics grew, he explored ever more abstract topics, often struggling with conceptual problems that would be fully understood only centuries after his death.

The Body—a Machine?

In view of Leonardo’s brilliant achievements in mechanical engineering and his extensive applications of principles of mechanics to the body’s “mechanical instruments,” it is tempting—but, in my view, erroneous—to believe that Leonardo saw the entire human body as a machine. Many Leonardo scholars have, implicitly or explicitly, taken this view. Kenneth Keele, for example, in his thorough analysis of Leonardo’s entire corpus of anatomical studies, published in 1983, repeatedly referred to “the human machine,” seemingly paraphrasing Leonardo.³

A decade later, Paolo Galluzzi echoed this perception in his superb volume on Renaissance Engineers: From Brunelleschi to Leonardo da Vinci, when he titled a section about Leonardo’s anatomical studies “The Human Body as a Wonderful Machine.” In the opening paragraphs of this section, Galluzzi quoted Leonardo’s statement that “nature cannot give movement to animals without mechanical instruments” (see p. 131), and then added the following comments:

Leonardo sought to demonstrate the close analogy between the machine and the body. He saw both as wonderful achievements of Nature, where iron laws govern not only mechanical instruments but also the motions of animals.⁴

I disagree with the assessment by Keele, Galluzzi, and other historians that Leonardo pursued a mechanistic approach in his anatomical studies and saw the human body as a machine. I believe that such a conclusion is based on an unwarranted Cartesian interpretation of Leonardo’s writings. It amounts to projecting a reductionist, mechanistic model of the human body onto Leonardo’s scientific views of the macro- and microcosm—views that were utterly organic and unmarred by the mind-body split introduced by Descartes more than one hundred years after Leonardo’s death.

René Descartes based his view of nature on a fundamental division into two separate and independent realms: that of mind and that of matter.⁵ The material universe was a machine and nothing but a machine. Nature worked strictly according to mechanical laws; everything in the material world could be explained in terms of the arrangement and movements of its parts. Descartes extended this mechanistic view of matter to living organisms. Plants and animals, for him, were simply machines. Human beings were inhabited by a rational soul, which was of divine origin. But the human body was a mere automaton, indistinguishable from an animal-machine. The body was completely divorced from the mind, the only connection between the two being by the intervention of God.

The difference between Descartes’s and Leonardo’s views of living organisms is profound. For Leonardo, nature enables animals to move with the help of “mechanical instruments,” but this does not mean that animals are machines. The crucial difference is that, in Leonardo’s view, the soul, which is inherent in all living organisms, is the origin of bodily movement and is also the body’s “composer” (desso corpo compositore), or formative force.⁶ Leonardo’s concept of the soul is quite different from that of Descartes. For him, the soul forms an indivisible whole with the body, controlling both its voluntary and involuntary movements. The range of the soul’s activities is different in plants, animals, and humans; it increases in stages with increasing biological complexity (see p. 153). I have argued in my previous book that Leonardo’s integrative concept of the soul—as the agent of perception, and as the vital force underlying the body’s formation and movements—while strikingly different from Descartes’s, comes very close to our modern concept of cognition.⁷

“Why nature cannot give movement to animals without mechanical instruments,” Leonardo writes in his Anatomical Studies, “is demonstrated by me in this book on the active movements made by nature in animals.”⁸ It seems quite clear that he refers here to the voluntary movements of animals, which are achieved by means of mechanical instruments but originate in and are controlled by the animal’s soul—a far cry from Descartes’s animal-automata.

In many passages in Leonardo’s manuscripts, he marvels at the beauty and grace that arise from subtle interactions between animals’ bodies and souls. For example, he observes that the delicate cognitive processes (as we would say in modern scientific language) of a bird in flight will always be superior to those of a human pilot steering a flying machine:

It could be said that such an instrument designed by man is lacking only the soul of the bird, which must be counterfeited with the soul of the man … [However], the soul of the bird will certainly respond better to the needs of its limbs than would the soul of the man, separated from them and especially from their almost imperceptible balancing movements.⁹

At times one can even discern a fine sense of humor in Leonardo’s insistence that the movements of animals are not purely mechanical. “Nature does not go in for counterweights when she makes organs suitable for movement in the bodies of animals,” he muses on another folio of his Anatomical Studies, “but she places inside the body the soul, the composer of this body.”¹⁰ More important, this passage is a clear statement on the integrative nature of the soul as both mover and composer of the body.

From the passages quoted above and from the general nature of his science as a science of organic forms, qualities, and transformations, it seems evident that Leonardo’s approach to anatomy was not mechanistic, at least not in the Cartesian sense. He fully realized that that the anatomies of animals and humans involve mechanical functions, and this became the principal motivation for his extensive studies of “mechanical science.” But these studies were always embedded in a broader organic conceptual framework.

Occasionally Leonardo referred to complex anatomical systems as “machines.” On two folios of the Anatomical Studies—one showing the set of muscles controlling the complex movements of the head, and the other the superficial muscles of the thigh—he uses the term “this machine of ours” in sudden outbursts of awe and wonder about the body’s complexity, interjected between detailed anatomical descriptions.¹¹ In both statements the term “machine” is applied to a complex system of mechanical functions, rather than to the body as a whole.

On other occasions, Leonardo uses the term “machine” to refer to phenomena in the macrocosm. He speaks of water as “the vital humor of the terrestrial machine,”¹² and of the ebb and flow of the tides as “the breathing of this machine of the Earth.”¹³ It is clear that here again “machine” should not be given a Cartesian meaning but should be understood to refer to a complex living system, nourished by water, with tides moving rhythmically like the breath and the flow of blood in the human body, and animated by a vital force of growth, or “vegetative soul” (see Leonardo’s description of the living Earth, p. 67).

Leonardo da Vinci created a unique synthesis of art, science, and design.¹⁴ He was a mechanical genius who invented countless machines and mechanical devices, and he maintained a lively interest in the theory of mechanics during most of his mature life. Yet his science as a whole was not mechanistic. He saw the world as an infinite variety of living forms continually shaped by underlying processes, and of patterns of organization recurring in the macro- and microcosm. He formulated mechanical models when he thought they would help him understand natural phenomena, but unlike scientists in subsequent centuries, he never considered the world as a whole, nor the human body, as nothing but a machine.

Leonardo’s Machines

Throughout his adult life, and especially during his years at the Sforza court in Milan,¹⁵ Leonardo was famous not only as an artist but also as a mechanical engineer. His duties as court painter and “ducal engineer” included, in addition to painting portraits and designing pageants and festivities, a variety of small engineering jobs that demanded ingenuity and skills in the handling of materials. Leonardo’s many creative talents were perfectly suited for these tasks. He invented a large number of astonishing devices during this time, which firmly established his reputation as engineer-magician at court.

Among his inventions were doors that opened and closed automatically by means of counterweights; a table lamp with variable intensity; folding furniture; an octagonal mirror that generated an infinite number of multiple images; and an ingenious spit, on which “the roast will turn slow or fast, depending upon whether the fire is moderate or strong.”¹⁶

Leonardo did not limit his engineering skills to these gadgets but invented numerous machines of a more industrial nature. These included a variety of textile machines for spinning, weaving, twisting hemp, trimming felt, and making needles, as well as machines for casting and hammering metal, shaping wood and stone, drawing strip and wire, coining and grinding—in short, machines for the basic industries of his time.

A special type of machine were the measuring instruments Leonardo invented and designed for his scientific experiments.¹⁷ In particular, he made many attempts to improve clock mechanisms for time measurement, which was still in its infancy in his day. In the Codex Madrid I, Leonardo put forth a systematic exposition of the main components of a mechanical clock: the use of the spring as the driving force and of the fusee (a conical drum) to compensate for the lessening force of the spring; power transmissions through gear-trains; and various forms of regulation systems known as escapements. All these elements are discussed in detail and pictured in superb drawings covering several pages.¹⁸

In addition to mechanical engineering, Leonardo was also engaged extensively in civil and military engineering. He was known as one of Italy’s leading hydraulic engineers and during his years at the Sforza court was probably in charge of all hydraulic works in Lombardy (see p. 32). He improved the existing systems of locks, invented special machines for digging canals, and skillfully inserted small dams into rivers to prevent damage to properties along their banks.

One of his most ambitious but unrealized hydraulic projects was a navigable waterway between Florence and Pisa. Leonardo imagined that this waterway would provide irrigation for parched land and could also serve as an “industrial” canal, providing energy for numerous mills that would produce silk and paper, drive potters’ wheels, saw wood, and sharpen metal (see p. 23).

As a military engineer, Leonardo was frequently consulted about strategies of warfare, and he often responded with ingenious designs of new fortifications and grandiose plans to dam up or divert rivers to conquer enemy troops.¹⁹ Most of his work for military rulers consisted in designing structures to defend and preserve towns and cities.²⁰ However, he also designed extravagant machines of destruction—bombards, explosive cannonballs, catapults, giant crossbows, and the like. At the same time, paradoxically, he was vehemently opposed to war, which he called a “most beastly madness” (pazzia bestialissima). Various explanations of this apparent contradiction can be put forward.²¹ Leonardo was in constant need of a stable income that would allow him to pursue his scientific research, and he shrewdly relied on his great skills in mechanical engineering to secure financial independence by offering designs of impressive war machines. Moreover, he may have been aware that most of these fanciful designs would never be realized.

However, it is also clear from Leonardo’s Notebooks that he was fascinated by the destructive engines of war, perhaps in the same way that natural cataclysms and disasters fascinated him. We may not be able to resolve the contradiction between his pacifist stance and his services as military engineer, but may have to accept it as one of many contradictions in the complex personality of a great genius.²²

Leonardo’s outstanding contributions to mechanical, civil, and military engineering are discussed extensively in several books, including the beautiful volume Renaissance Engineers: From Brunelleschi to Leonardo da Vinci, by science historian Paolo Galluzzi,²³ and the lavish catalogue of an exhibition at the Musée des Beaux-arts de Montréal, edited by Galluzzi, which covers both Leonardo’s engineering and architecture in great detail.²⁴ His technical drawings are frequently exhibited around the world, often supplemented by wooden models that show in impressive detail how the machines work as he had intended.²⁵

The combination of artist-engineer was not unusual in the Renaissance.²⁶ Leonardo’s teacher Verrocchio, for example, was a renowned goldsmith, sculptor, and painter as well as a reputable engineer. The great Renaissance architect Filippo Brunelleschi first gained notice in Florence as a sculptor and later on, when he was famous as an architect, was also acclaimed for his inventive genius as an engineer. The young Leonardo admired him greatly and declared his indebtedness to the great architect by drawing several of Brunelleschi’s renowned lifting devices and architectural plans.

What made Leonardo unique as an engineer, though, was that many of the novel designs he presented in his Notebooks involved technological advances that would not be realized until several centuries later. Even more important, he was the only one among the famous Renaissance engineers who made the transition from engineering to science. To know how something worked was not enough for him; he also needed to know why. Thus an inevitable process was set in motion that led him from technology and engineering to pure science. As art historian Kenneth Clark notes, we can see the process at work in Leonardo’s manuscripts:

First, there are questions about the construction of certain machines, then … questions about the first principles of dynamics; finally, questions which had never been asked before about winds, clouds, the age of the earth, generation, the human heart. Mere curiosity has become profound scientific research, independent of the technical interests which had preceded it.²⁷

Leonardo’s passage from the study of medieval empirical technology to theoretical mechanics began with the emergence of a strong interest in mathematics during his first period in Milan, when he was in his late thirties. An important event was a visit to the nearby city of Pavia in 1490. Leonardo went there together with the architect Francesco di Giorgio on behalf of the duke of Milan to inspect work on the city’s cathedral.

In Pavia, Leonardo met the mathematician Fazio Cardano, a specialist in the “science of perspective,” which in the Renaissance included geometry and geometrical optics. Leonardo’s discussions with Cardano ignited a passion for mathematics that would remain with him until his old age.²⁸ While his fellow architect and engineer Francesco returned to Milan as soon as their work was completed, Leonardo stayed in Pavia for another six months to consolidate his understanding of geometry with studies in Pavia’s magnificent, world-famous library.²⁹

Immediately after his return to Milan he began two new Notebooks, now known as Manuscripts A and C, in which he applied his new knowledge of geometry to a systematic study of perspective and optics as well as to elementary problems of mechanics. Leonardo’s application of geometrical reasoning to the analysis of machines was highly original. Inspired, most likely, by his discussions of Euclid’s celebrated Elements of Geometry with Cardano in Pavia, he began to separate individual mechanisms, or “elements,” from the machines in which they were embedded. This conceptual separation did not arise again in engineering until the eighteenth century.³⁰

In fact, Leonardo at that time planned (and may even have written) an entire treatise on Elements of Machines, in which he would use geometry to analyze basic mechanisms in terms of elementary principles of mechanics—the transmission of power and motion, measurement of forces, and so on. Such analysis was important to Leonardo not only for understanding and improving upon existing mechanisms but also for the “very noble” purpose of understanding the individual actions of muscles, tendons, and bones in generating bodily movements (see pp. 153–54).

Leonardo’s treatise on Elements of Machines, if ever written, has been lost. But his Codex Madrid I contains extensive preparatory studies for such a treatise. In this manuscript, written in the late 1490s while he was also completing The Last Supper, Leonardo analyzed over twenty elementary mechanisms in countless variations—screws, levers, hinges, springs, couplings, gears, pulleys, and so on. As historian of technology Ladislao Reti has shown in his thorough and beautiful analysis of the Codex Madrid, Leonardo’s elements of machines include all the mechanical devices described in the work of early nineteenth-century French scholars of the Ecole Polytechnique, which was traditionally considered to be the first systematic study of elementary mechanisms.³¹

The mechanisms described in the Codex Madrid were well known to Renaissance engineers, although Leonardo invented many new versions and combinations of them. However, none of his predecessors or contemporaries had analyzed in detail how they worked. For centuries, animals had been attached to carts or traction devices; men had turned cranks, worked mechanical tools by hand, and operated treadmills. Simple machines were built and used according to tradition without asking how friction could be reduced or the transmission of muscle power improved.

Leonardo’s approach was profoundly different. He never followed traditional solutions without questioning them, but analyzed them according to mechanical rules and principles deduced from observation and experiment. He paid special attention to the transmission of power and motion from one plane to another, which was a major challenge of Renaissance engineering. An extremely complex example is his design of a water-powered mill for simultaneously drawing and rolling cannon-barrel segments. Leonardo’s illustration of his design (fig. 5-1) shows a machine assembly of fifteen links in which the initial power of the water turbine (at bottom left of the assembly) is transmitted three times between vertical and horizontal axes with the help of a combination of toothed wheels and worm gears. Each time the power increases twelvefold while the turning speed successively decreases until it reaches the sturdy solid wheel (at top right of the assembly) that presses on the cannon segment beneath it.³² The transfer of power is clearly indicated by Leonardo in a small diagram below the main drawing in which the numerical power ratios are indicated for each gear.

This drawing is a splendid illustration of Leonardo’s extraordinary capacity for coordinating complex mechanical functions, analyzing them precisely, and presenting them visually with great clarity. From the time when his interest in mathematics was kindled at the University of Pavia, his work in mechanical engineering became inextricably linked to the analysis of his machines in terms of geometry and principles of mechanics.

The Science of Weights

Leonardo began his theoretical studies of mechanics with the “science of weights,” known today as statics, which is concerned with the analysis of loads and forces of physical systems in static equilibrium, such as balances and levers. In the Renaissance as today, this knowledge was very important for architects and engineers, and the medieval science of weights comprised a large collection of works compiled in the late thirteenth and fourteenth centuries, which Leonardo studied extensively. Some of these were treatises and fragments translated from Greek or Arabic, usually ascribed to Euclid or Archimedes, while others were original writings of medieval authors.³³

The mathematical foundations of statics were established in antiquity by the great mathematician and scientist Archimedes in a treatise titled On the Equilibrium of Planes, which contains his exact determinations of centers of gravity and his proof of the general law of the lever. Archimedes’s mathematical proofs were purely geometrical, which delighted Leonardo and led him to declare enthusiastically that “mechanics is the paradise of the mathematical sciences.”³⁴

FIG. 5-1. Water-powered mill for rolling and drawing cannon-barrel segments. Codex Atlanticus, folio 10r.

Of the medieval authors, Leonardo drew most heavily upon two works by Jordanus de Nemore, a thirteenth-century mathematician about whom almost nothing is known but who left several treatises on mathematics and mechanics that show considerable skill and originality. Leonardo read the two principal works by Jordanus on statics, Elementa de ponderibus (Elements of the Science of Weights) and De ratione ponderis (On the Nature of Weight), the latter being a greatly expanded and improved version of the Elementa.

In his usual fashion, Leonardo absorbed the key ideas from the best and most original texts in the corpus of the medieval science of weights, commented on some of their postulates in his Notebooks, verified them experimentally, and refuted some incorrect proofs. The Codex Atlanticus, in particular, contains several pages of his Italian translation of various postulates from Jordanus’s Elementa and De ratione ponderis, probably from a single manuscript that contained both works.³⁵

The centerpiece of Leonardo’s mathematical treatment of statics is the classical Archimedean law of the lever. It states that a lever, or balance, will be in equilibrium when the ratio of the two weights (or forces) is the inverse of the ratio of their distances from the fulcrum. This law appears repeatedly in various forms in Leonardo’s Notebooks. In the Codex Atlanticus, for example, he states:

The ratio of the weights that hold the arms of the balance parallel to the horizon is the same as that of the arms, but is an inverse one.³⁶

In the Codex Arundel, Leonardo expresses the law in terms of a formula that in modern algebraic notation would be written as w₂ = (w₁d₁)/d₂:

Multiply the longer arm of the balance by the weight it supports and divide the product by the shorter arm, and the result will be the weight which, when placed on the shorter arm, resists the descent of the longer arm, the arms of the balance being in equilibrium at the outset.³⁷

Leonardo used the law of the lever to calculate the forces and weights necessary to establish equilibria in numerous simple and compound systems involving balances, levers, pulleys, and beams hanging from cords. In addition, he carefully analyzed the tensions in various segments of the cords, probably for the purpose of estimating similar tensions in the muscles and tendons of human limbs.

FIG. 5-2. Pivoting plank in equilibrium with two forces acting at different angles. Ms. E, folio 65r (as reconstructed by Clagett, “Leonardo da Vinci: Mechanics”).

Science historian Marshall Clagett has discussed Leonardo’s diagrams and analyses of the principles of statics in great detail.³⁸ Clagett emphasizes that Leonardo applied the law of the lever not only to situations where the forces act in a direction perpendicular to the lever arms, but also to forces acting at various angles. The Codex Arundel and Manuscript E contain numerous diagrams of varying complexities with weights exerting forces at different angles via cords and pulleys. Leonardo recognized that in such cases the relevant length in the law of the lever is not the actual length of the lever arm but the perpendicular distance from the line of the force to the axis of rotation. He called that distance the “potential lever arm” (braccio potenziale) and marked it clearly in many diagrams.

In a diagram in Manuscript E (fig. 5-2), for example, Leonardo shows a bar that is pivoted at one end at point a with a weight m suspended from its other end at point t. A second weight n exerts a horizontal pull via a cord running over a pulley. The problem is to determine the weights m and n necessary to keep the bar in equilibrium. In his solution, Leonardo identifies the two potential lever arms as ab and ac, and he states correctly that, at equilibrium, the weights m and n will be inversely proportional to the distances ab and ac.

In modern statics, the potential lever arm is known as the “moment arm” and the product of moment arm and force is called the “moment of force,” or “torque.” Leonardo clearly recognized the principle that the sum of the moments about any point must be zero for a system to be in static equilibrium. According to Clagett, this discovery was his most original contribution to statics, going well beyond the medieval science of weights of his time.

Fluids in Equilibrium

While Leonardo experimented with balances, levers, and pulleys to explore the laws governing mechanical systems in static equilibrium, he also studied the equilibrium of fluids, known today as hydrostatics. In the Codices Madrid, which contain most of his early investigations of the “science of weights,” we also find comments on water pressure and references to the principle of Archimedes, as well as drawings of scales measuring the buoyancy of weights submerged in water.

Since antiquity, hydrostatics had been an independent discipline, unrelated to the study of the flow of water (now known as hydrodynamics). Its general principles had been enunciated clearly by Archimedes in his classical text On Floating Bodies. This treatise contains, in particular, the famous principle that now bears Archimedes’ name. It states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object.

Most of the theoretical work of Archimedes was so advanced that it was poorly understood by his contemporaries and succeeding generations.³⁹ Translations of various fragments on hydrostatics were reproduced in several medieval texts, generally without a clear understanding of the underlying principles. These texts could have given Leonardo only a very sketchy knowledge of Archimedean hydrostatics.

FIG. 5-3. Experiment for determining the force of buoyancy. Codex Madrid I, folio 181r (detail).

Leonardo certainly had a general knowledge of buoyancy. He knew that objects weigh less in water than in air and he even tried to determine the difference experimentally. Two similar drawings in the Codices Madrid show weights hanging from a scale, one weight in the air and the other submerged in water contained in a vessel.⁴⁰ In the Codex Madrid I (fig. 5-3), the text accompanying the illustration mentions several experiments of that kind and lists quantitative results. Leonardo also knew the basic principle of floating bodies. An earlier folio in the same Codex Madrid I shows an elegant little sketch of a floating boat (fig. 5-4) together with the following comment: “As much weight of the water leaves the place where a boat floats, as the weight of that boat itself.”⁴¹ However, as far as we can tell from his existing manuscripts, Leonardo never fully stated the principle of Archimedes. He knew the formula for floating objects but seems to have been unaware that a similar formula holds for submerged objects. Leonardo’s numerous notes on hydrostatics make it evident that he reached only a partial understanding of the Archimedean laws of buoyancy.

FIG. 5-4. Illustration of the amount of water displaced by a floating boat. Codex Madrid I, folio 123v (detail).

It is interesting to examine in detail what prevented Leonardo from fully understanding the principle of Archimedes. To understand the origin of the force of buoyancy, one needs to know that water pressure increases with depth, and that the increased pressure is exerted in all directions. As a consequence, there is a net upward force on the bottom of a submerged object, as illustrated in figure 5-5.

FIG. 5-5. Buoyant forces on a “water ball” in equilibrium (a) and on a solid spherical object of equal volume (b). From HyperPhysics.com, Georgia State University.

FIG. 5-6. Measurement of the variation of water pressure with depth by means of a series of moveable plates, sustained by counteracting forces that are generated by weights and transmitted to the plates via strings and pulleys. Codex Leicester, folio 6r (detail).

A hypothetical ball of water anywhere in the vessel will be in equilibrium because its weight is supported exactly by the force of buoyancy, or net pressure (see fig. 5-5a). If the water ball is replaced by a solid spherical object of equal volume, the distribution of pressure on the object will be the same (see fig. 5-5b). Hence, the buoyant force on the solid object is equal to the weight of the water displaced, as stated by Archimedes.

Leonardo knew, at least in his later years, that water pressure in a reservoir acts on vertical walls and increases from the surface to the bottom. In the Codex Leicester, which he wrote when he was in his late fifties, he depicts a very ingenious experiment designed to measure the increase of pressure with depth (fig. 5-6). As described in the accompanying text, the experiment involves a water tank “in which one of the walls is a loose parchment, sustained by a row of plates, as the drawing shows, at which plates you put so much opposite weight as to sustain precisely those plates in contact with the front of the aforementioned water tank.”⁴² If Leonardo had actually carried out this experiment, it would have shown him the correct linear increase of pressure with depth.

In spite of his knowledge of the variation of water pressure, it would have been difficult for Leonardo to reason that the weight of the water ball in figure 5-5a is balanced by the net upward pressure. Archimedes’s sophisticated understanding of the distribution of water pressure had been lost over the centuries and would not be rediscovered by Blaise Pascal until 150 years after Leonardo. In addition, the concept of the weight of a portion of water, immersed in the surrounding water filling the vessel, was foreign to Leonardo. During most of his life, he accepted the Aristotelian view that “all the elements are without weight in their own sphere but possess weight outside their sphere.”⁴³ This Aristotelian theory of gravity, which was commonly held in the Middle Ages and the Renaissance, made it difficult, if not impossible, to understand the hydrostatic equilibrium of a portion of water as the balance between its weight and the net upward pressure of the water surrounding it.*

The Aristotelian view of a natural weightless state of the elements also prevented Leonardo from developing a sophisticated conception of water pressure, which is essential for a full understanding of the principle of Archimedes. He was aware of the pressure generated by a piston, but he had great difficulties conceptualizing a pressure distribution of water in its “natural” state.

Pressure in fluids is a concept of modern hydrostatics where it is defined as force (or weight) per unit of area. As physicist and Leonardo scholar Enzo Macagno pointed out in his detailed studies of Leonardo’s writings on hydrostatics,⁴⁴ this definition was hard to accept in the Middle Ages and the Renaissance because it requires the division of two quantities of different dimensions—a force by an area. Such divisions, or multiplications, were fully accepted only in the seventeenth century when geometry was replaced by algebra—a much more powerful mathematical language, capable of expressing relationships between physical quantities in terms of abstract equations.

Because of these mathematical limitations, Leonardo and his contemporaries never reached more than a qualitative understanding of fluid pressure. The first to give a full account of the pressure distribution in a fluid in hydrostatic equilibrium was the mathematician and philosopher Blaise Pascal in the seventeenth century. His formulation is now known as Pascal’s principle. It states that pressure applied to a confined fluid at any point is distributed undiminished through the fluid in all directions and acts upon every part of the vessel’s confining surface at right angles.

Leonardo struggled with the concept of pressure for many years without ever fully understanding it. As historian of science Constantino Fasso has documented, some pages of the Notebooks show anticipations of essential insights that would be formulated clearly in subsequent centuries, while others contain confused and contradictory statements.⁴⁵ For example, the Codex Madrid I, in which Leonardo recorded his early investigations of hydrostatics, already contains an evident, though imprecise, anticipation of Pascal’s principle. Next to a sketch of a weight placed on a wine skin, there is a marginal note: “Any part of the skin feels equally the pressure of the weight.”⁴⁶ But elsewhere in the same Notebook, Leonardo gives a wrong description of a similar situation. He depicts a vessel filled with water on which pressure is exerted by a weight through an air cushion, and he comments that the increase in pressure “pushes … in all parts of that vessel in the way and proportion exerted before by the water alone.”⁴⁷ In other words, he assumes that the increase in pressure is not constant throughout the vessel (as stated by Pascal’s principle), but varies with depth in proportion to the hydrostatic pressure that existed before the application of the weight.

Even Leonardo’s mature writings on hydrostatics in the Codex Leicester are not free from such contradictions. The description of his brilliant experiment to measure the variation of water pressure with depth (see fig. 5-6) is a good example. After stating correctly that the pressure on the vertical walls increases from the surface to the bottom of the vessel, and describing how to measure the increase, Leonardo adds a few lines in which he confuses the issue again. “The same rule,” he asserts, “may be used at the bottom to see in which part of the bottom of the vessel water presses more on that bottom.”⁴⁸ Clearly, at least at the time of writing these lines, Leonardo was not aware that the water pressure is distributed equally across horizontal planes.

On a folio in the same Codex Leicester, a few pages after the contradictory statements mentioned above, we find Leonardo’s most mature discussion of hydrostatic pressure. Once again, he considers a vessel filled with water to which pressure is applied through an opening at the top:

Water, pressed through the mouth of the vessel,” he writes, “acquires in its contact with that vessel a uniform pressure [potentia]. I intend that the water, because it is pressed, acquires that uniform pressure in addition to the unequal pressure that existed in this water before, since it is evident that water by itself exerts more weight on an orifice at the bottom of the vessel than on the surface, and for each degree of depth it acquires degrees of weight.⁴⁹

Comparing this statement to the one in the Codex Madrid mentioned above, which describes the same experiment and was written about ten years earlier, makes it evident that Leonardo’s thoughts on hydrostatic pressure had matured significantly during the intervening years. In the Codex Leicester, he states unequivocally and correctly that the pressure exerted on the water by a weight is distributed uniformly throughout the vessel, and that the resulting total water pressure is composed of two parts: the original hydrostatic pressure that is unequal, increasing from the surface to the bottom, and the constant pressure that is added through the application of the weight. Moreover, he correctly states that the original hydrostatic pressure increases linearly with depth.

It seems that the only flaw in Leonardo’s analysis is the lack of an explicit definition of pressure as force (or weight) divided by area, and hence he shows a slight confusion between pressure and weight. Otherwise, this statement in the Codex Leicester is a clear anticipation of Pascal’s principle. Since the passage is, as far as we know, chronologically Leonardo’s latest discussion of hydrostatic pressure, we may take it as his definitive pronouncement on the subject.

Another hydrostatic phenomenon that puzzled Leonardo a great deal was the equilibrium of liquids in communicating vessels, which was well known during his lifetime. “The surfaces of all liquids at rest that are joined together below are always of equal height,” he noted correctly in the Codex Atlanticus.⁵⁰ In the medieval “science of weights,” questions of statics had always been treated by applying the laws of the equilibrium of the balance, and so it was natural for Leonardo and other Renaissance engineers to use the same approach in trying to explain the law of communicating vessels. The analogy of a balance in equilibrium with equal arms, loaded by the weights of the water in the two vessels, is correct only when the two vessels are equal. When one vessel is larger, the hypothetical balance can be in equilibrium only if the water surface is higher in the smaller vessel. This is contrary to the experimental evidence, as Leonardo did not fail to notice.

The resolution of this paradox—how a small amount of water on one side can balance a large amount on the other side—requires the full understanding of hydrostatic pressure reached by Pascal in the seventeenth century. According to Pascal’s principle, the hydrostatic pressure must be the same at all horizontal planes for the water to be in equilibrium. This means that the height under the surfaces must be the same, but their areas are irrelevant because pressure equals weight per unit area.

Lacking this sophisticated conception of pressure, Leonardo never succeeded in completely solving the paradox of communicating vessels. However, he found an ingenious explanation during his first reflections on hydrostatics. In the Codex Madrid I, there is a sketch of communicating vessels of unequal size with water levels at equal height indicated correctly (fig. 5-7). In this drawing, Leonardo has divided the water in the larger vessel into several columns. In the accompanying text, he explains that not all the water in the larger vessel is active in counterbalancing the weight of the water in the smaller vessel, but only one column (a–n) with the same cross section as that of the smaller vessel (m–r). In view of Leonardo’s very limited understanding of water pressure at the time, this reasoning is remarkable. “Leonardo’s approach to the problem of communicating vessels,” comments Fasso, “seems to me the most advanced that could be achieved at a time in which the concept of pressure had not yet been devised.”⁵¹

FIG. 5-7. Leonardo’s pictorial explanation of the law of communicating vessels. Codex Madrid I, folio 150r (detail); reconstructed by Macagno, “Mechanics of Fluids in the Madrid Codices.”

Forces and Motion—A Conceptual Maze

During his studies of the “science of weights” and of hydrostatics, Leonardo became interested in the general relationships between forces and motion. In his attempts to outline what in subsequent centuries would be called a “science of motion,” he encountered conceptual difficulties that were far greater than those in his works on statics. The geometrical reasoning he used for his analyses of machines was much harder to apply to the dynamic phenomena of bodies moving under the influence of forces and colliding with one another. Besides, the concepts required to describe these phenomena mathematically—concepts like energy, momentum, force, acceleration, and so forth—had not yet been fully developed. In fact, it would take another two centuries to clearly identify and define these basic concepts of mechanics. As science historian Domenico Bertoloni-Meli points out, even Galileo’s early speculations on motion, a hundred years after Leonardo’s, were entangled in “a conceptual and terminological maze at the intersection between Aristotelianism and a new science.”⁵²

According to the Aristotelian four-element theory, commonly held in the Middle Ages and the Renaissance, the movements of the elements arise from their natural tendencies to return, when disturbed, to their proper places within concentric spheres around the Earth.⁵³ Leonardo held on to this teleological explanation of forces and motion for most of his life, but on several occasions he questioned its basic premises, realizing that they were obstacles in his attempts to understand mechanical phenomena. For example, in a small notebook written during his early studies of mechanics and now known as Manuscript I, he listed a series of questions about motion that he intended to explore. Besides being a lively testimony to Leonardo’s relentless curiosity, these questions clearly indicate his doubts about the Aristotelian scheme:

What is the cause of motion? What is motion in itself? What is it that is most adapted for motion? What is impetus? What is the cause of impetus, and of the medium in which it is created? What is percussion? What is its cause? What is rebound? What is the curvature of straight motion and its cause?⁵⁴

A few years later, he expressed similar doubts about the Aristotelian view of gravity in connection with the flow of water (see p. 40). In these struggles with the conceptual maze of classical and medieval mechanics, Leonardo sometimes showed an intuitive grasp of abstract concepts and relationships that was far ahead of his time, while at other times he could not free himself from the constraints of the traditional Aristotelian ideas.

In reviewing Leonardo’s achievements in kinematics and dynamics in the following pages, I shall a few times compare them to those of Galileo, the other great scientist from Tuscany, born more than a century after Leonardo. Galileo published his early speculations on movement, De motu antiquiora (Older Works on Movement), in 1592, about a hundred years after Leonardo’s early work on mechanics; and Galileo’s mature work, the Discorsi (Discourses), was published in 1638, about 125 years after Leonardo’s mature writings. The comparison between Leonardo’s and Galileo’s mechanics is fascinating because they performed similar experiments, struggled with similar conceptual problems, and used similar mathematical language, stating the regularities they discovered in terms of proportions and geometrical laws rather than algebraic equations. Galileo’s mature work marks a kind of conceptual halfway point between his and Leonardo’s early speculations and the publication of Isaac Newton’s Principia (Principles) in 1687. Newton’s grand opus, in which he used algebraic notation and his newly invented calculus, was the triumphant completion of the new “science of motion.”

The Four “Powers” of Nature

In his attempts to bring some clarity to classical and medieval mechanics, Leonardo identified four basic variables—motion (or velocity), weight, force, and percussion (or “impact,” as we would say today). He did not use the modern term “variable” but instead used the term “power” (potentia), which appears frequently, with a wide range of meanings, in his Notebooks. In his writings on hydrostatics, for example, potentia means “pressure”; in other texts it corresponds to our modern term “energy”; and Leonardo also describes the force exerted by living organisms as an “invisible power.”

The fact that Leonardo’s use of potentia is rather vague compared to modern scientific terminology is not surprising for the fifteenth century. Today we know that his four “powers” of nature all have different dimensions, and hence I believe that potentia in this context is best understood in the sense of our modern term “variable.” This interpretation is reinforced by Leonardo’s insistence that the outstanding common characteristic of the four “powers” is that they vary continuously (like the variables in our modern mathematical functions). Among Leonardo’s early notes on mechanics in the Codex Madrid I is this emphatic statement:

We will be telling the truth by affirming that it is possible to imagine all the powers capable of infinite augmentation or diminution…. They can grow from nothing to infinite greatness by equal degrees. And by the same degrees they decrease to infinity by diminution, ending in nothing.⁵⁵

Having defined the basic variables of mechanics, Leonardo then tries to establish quantitative relationships between them. For most of his life, he believed that all such relationships could be represented as direct or inverse proportions—more generally, as what we now call linear functions. He called such proportional, or linear, algebraic relationships “pyramidal,” using pyramids and isosceles triangles (i.e., triangles with two equal sides) to represent them geometrically.⁵⁶

On a folio in the Codex Atlanticus, Leonardo illustrated the actions of the powers of nature with various examples, heading the page prominently with the words: “All natural powers … are to be called pyramidal inasmuch as they have degrees in continuous proportion toward their diminution as toward their increase.”⁵⁷ In other words, all powers vary continuously and their variations can be expressed in terms of linear relationships.

Leonardo’s belief that “pyramidal” (linear) relationships were universal in nature was derived from his familiarity with linear proportions in perspective. Like most mathematicians of his time, he frequently used geometrical figures to represent algebraic relationships, and he saw the pyramid, or isosceles triangle, as a powerful symbol of a conceptual link between optics and mechanics.

This belief in the universal nature of linear relationships prevented Leonardo at times from recognizing other types of algebraic relationships that are not so easily expressed in terms of geometrical figures. For example, he failed to recognize that the distances traversed by falling bodies increase with the squares of the times instead of linearly. Galileo made the same error in his early work, and for the same reasons. In a letter written in 1604, he stated as an “indubitable principle” that the speeds of falling bodies increase in proportion to the distances traversed, which is equivalent to saying that both speeds and distances increase linearly with time. According to science historian Bertoloni-Meli, “Galileo may have been led to (incorrectly) assuming the proportionality between speeds and distances by the intuitively privileged role … of direct proportionality over more complex relations.”⁵⁸

Galileo eventually corrected his error, and Leonardo, too, discovered more complex functional relationships between physical variables late in his life. But by then he was too busy with other projects to revise his statements on the universality of pyramidal relationships between the four powers of nature.⁵⁹

Leonardo’s writings on motion, weight, force, and percussion are widely dispersed in his Notebooks, from the beginning of his scientific notes in the Codex Trivulzianus and the early Codices Forster and Manuscript A to his notes in Manuscript E, compiled in old age, and throughout the large collections of notes in the Codices Arundel and Atlanticus.* In the following pages, I will be able to touch upon only a few highlights of Leonardo’s observations, discoveries, and speculations in the field of mechanics.

The Nature of Motion

When Leonardo extended his studies of mechanics beyond statics, he realized that the nature of motion and its relationships to various forces would now be the central subject of his investigations. Thus, it is not surprising that he considered motion to be the most fundamental of his four powers of nature. “Speak first of motion,” he wrote in a note to himself, “then of weight because it arises from motion; then of force, which arises from weight and motion; then of percussion, which arises from weight, motion, and often from force.”⁶⁰

The causal links between the four powers established in this statement seem somewhat arbitrary, and indeed Leonardo soon realized that motion, weight, force, and percussion were so tightly interconnected that it was impossible to single out any one of them as primary:

Gravity, force, and percussion are of such a nature that each by itself alone can arise from each of the others … and all together and each by itself can create motion and arise from it.⁶¹

Nevertheless, it seemed to Leonardo that motion would be the best starting point to analyze the relationships between the four powers of nature. In the Codex Arundel, he reasserts that motion, like the other three powers, varies continuously, and in the same passage he correctly identifies the basic relationship between velocity, distance, and time:

That motion is slower which covers less distance in the same time. And that motion is swifter which covers more distance in the same time…. It is in the power of motion to extend to infinite slowness and likewise to infinite velocity.⁶²

Even more remarkable is the fact that Leonardo recognized the relativity of motion. “The motion of the air against a fixed thing is as great as the motion of the moving thing against the motionless air,” he noted in the Codex Atlanticus. “And the same occurs in water, which in a similar circumstance has shown me the very same nature.”⁶³ Indeed, on an earlier page in the same Codex, Leonardo accurately observed: “The action of a pole drawn through still water resembles that of running water against a stationary pole.”⁶⁴

Leonardo must have recognized the importance of this discovery, because he recorded it many times in various manuscripts. All these statements are clear and beautiful expressions of an important principle of modern mechanics. The relativity of motion was rediscovered and formulated mathematically in the late seventeenth century by the renowned physicist and mathematician Christiaan Huygens in connection with the laws of collision. It is the basis of the wind tunnel, the principal experimental tool of modern aerodynamics.

As we do in kinematics today, Leonardo distinguished between “straight” (linear) and curved motion, and he listed circular, spiral, and “irregular” movements as special types of curved motion.⁶⁵ In view of his great fascination with spirals (see p. 62), it is not surprising that he identified four distinct spiral movements corresponding to convex, plane, concave, and “columnar” (helical) spirals.

In his extensive work with machines, Leonardo had ample opportunity to study wheels rotating about various axes. He correctly distinguished between angular and linear velocity,* as we would say today. “That part of a revolving wheel moves with less motion which is nearest to the center of that revolution,” he noted in Manuscript E.⁶⁶ As mentioned before, Leonardo contrasted the circular motion of wheels with the spiral motion of water vortices in his pioneering studies of turbulent flows of water and air (see p. 47). “The spiral or whirling motion of every liquid is so much swifter as it is nearer to the center of its revolution,” he observed with complete accuracy, “[whereas] the circular motion of a wheel is so much slower as it is nearer to the center of the revolving object.”⁶⁷

Leonardo fully realized that with the distinction between angular and linear velocity he had discovered a property that was characteristic not only of rotating wheels but of all circular motion. We can find several illustrations of this insight in his manuscripts, including the following, involving hunting tools, in his early notes on mechanics in Manuscript I:

Field-lances or hunting-whips have a greater movement than the arms … because, in moving the arm, the hand describes a much wider circle than the elbow; and in consequence, moving in the same time, the hand covers twice the pathway covered by the elbow; therefore, it may be said to be of a speed double that of the motion of the elbow.⁶⁸

Leonardo did not fail to recognize the centrifugal force generated by circular motion, and he also observed correctly that, when an object rotating on a string is released from its rotation, it will fly off tangentially:

The weight that moves around the fixed point of a string where it is joined, pulls and stretches this string with great power, and if such a string is separated from its fixed point, the weight carries with it the said string along that line into which it was drawn at its separation from its fixed point.⁶⁹

It took another 150 years for these characteristics of circular motion to be rediscovered by Robert Hooke and Christiaan Huygens.

Force and Motion

In his analysis of the relationships between force and motion, Leonardo stayed largely within the confines of the Aristotelian framework. He distinguished between “natural” motion—the spontaneous movement of an element toward its natural state—and “violent” or “accidental” motion, in which an element is displaced from its natural state by some force. “Gravity and levity are accidental powers,” he explained in the Codex Atlanticus, “which are produced by one element being drawn through or driven into another. No element has gravity or levity within its own element.”⁷⁰

In this Aristotelian context “gravity” did not refer to a force, as it does in Newtonian physics, but rather to the “heaviness” that is created by displacing a solid object upward, away from the Earth and thus out of its natural place. Similarly, “levity” was thought to be created when air is displaced downward and submerged in water. In both cases, the displacements were “violent” motions for which forces were required, while the return of the element to its natural place was due to an inherent tendency rather than an external force.

Aristotle’s distinction between natural and violent motion, and his assertion that these two types of motion were fundamentally different and could not be mixed, were accepted by natural philosophers throughout the Middle Ages and the Renaissance. In the early seventeenth century, the distinction between natural and violent motion was abandoned by Galileo but was still debated among his contemporaries.

The situation clarified gradually when the concept of inertia came into focus. Indeed, according to science historian Robert Lenoble, “Modern mechanics was born with the principle of inertia.”⁷¹ The final decisive step was made by Newton, who clearly recognized inertia as the tendency of a massive body to preserve its state of rest, or uniform straight motion, unless acted upon by a force. A consequence of this fundamental insight, now known as Newton’s first law of motion, was that force henceforth was no longer associated with motion in general, but specifically with changes of a body’s state of motion; in other words, with acceleration.

Like all medieval and Renaissance scholars, Leonardo accepted Aristotle’s assertion that an object in violent motion would continue to move only as long as there was a force acting on it. “No inanimate thing can push or pull something without going along with it,” he wrote in his early Manuscript A, “and what pushes it can only be force or weight.”⁷² An obvious problem with this Aristotelian position was the difficulty in explaining why a thrown stone, for example, continues to move after losing contact with the hand exerting a force on it, or an arrow after losing contact with the bow that propelled it. The medieval philosophers were well aware of this difficulty, and they found an ingenious solution. They postulated that the moving force impressed, or infused, an impetus into the moved object that kept it in motion until the impetus eventually dissipated, like heat in an iron after it is removed from the fire.

The medieval theory of impetus was formulated in its most elaborate form in the fourteenth century by the French scholastic philosopher Jean Buridan. Leonardo studied Buridan’s theories, including his imaginative theory of tectonic movements in geological cycles, through the writings of Albert of Saxony (see p. 89). As he progressed with his investigations of mechanics, he used the concept of impetus with increasing frequency. In his very first Notebook, the Codex Trivulzianus, he described the phenomenon of impetus but did not use the term:

Every moved or percussed body retains in itself for some time span the nature of that percussion or movement; and it will retain it so much more or less as the power and the force of that blow, or motion, is greater or smaller.⁷³

Around the same time he carefully analyzed, again without using the term “impetus,” how the disturbance caused by a stone thrown into a still pond is transported outward in circular ripples:

The water, though remaining in its position, can easily take this tremor from neighboring parts and pass it on to other adjacent parts, always diminishing its power until the end.⁷⁴

In subsequent years, definitions of “impetus” along the lines of Buridan appear frequently in Leonardo’s Notebooks. In Manuscript G, for example, he notes:

Impetus is the impression of motion transmitted by the motor to the moved object. Impetus is a power impressed by the motor in the moved object.⁷⁵

Leonardo applied the concept of impetus to many mechanical phenomena that we now associate with inertia, such as the stability of a spinning top, various oscillating motions, and collisions.⁷⁶ However, while Buridan described impetus quantitatively as being proportional both to the quantity of matter and the velocity of the object, no such quantitative treatment is evident in any of Leonardo’s extant Notebooks.

In his early work on mechanics, Leonardo maintained the medieval imagery of impetus as a power that was infused into a moving object and that subsequently dissipated by itself. But as he became increasingly interested in friction, he became keenly aware of the effects of viscosity in water and air (see p. 41) and realized that it was air resistance that gradually diminished the impetus of a projectile:

The power of the motor … attaches itself to the moved body, and over time it is consumed in the penetration of the air, which is always compressed in front of the moving object.⁷⁷

In addition, Leonardo recognized that in the motion of projectiles there was a continuous interplay between violent and natural motion. “The natural motion, conjoined with the motion of a motor, consumes the impetus of that motor,” he observed late in his life in Manuscript E.⁷⁸ With this statement describing the interplay between the projectile’s inertia and the force of gravity (as we would say today), Leonardo clearly transcended the Aristotelian framework in which natural and violent motions could never be mixed.

In the centuries after Leonardo, the medieval concept of impetus gradually evolved into the modern concept of momentum, defined as the product of an object’s mass and its velocity and as a vector, that is, a quantity having both a magnitude and a direction. At the end of the sixteenth century, Galileo still used impetus in the sense of a self-dissipating entity.⁷⁹ Descartes abandoned this image by introducing the term “quantity of motion” to replace “impetus,” but he saw it as being independent of direction. In the seventeenth century, finally, Huygens was the first to state the conservation of the quantity of motion and gave it the full meaning of our contemporary concept of momentum.

Conservation of Energy

Among the basic concepts of mechanics, energy was the one that took longest to be identified and precisely formulated. It is much more abstract than the concepts of mass, force, or momentum, and has become one of the most important concepts of modern physics. Its importance is due to the fact that the total energy in any physical process is always conserved. Energy can change into many different forms—gravitational, kinetic, heat, chemical, and so on—but the total amount of energy in a particular process, or set of processes, never changes. There is no known exception to the conservation of energy. It is one of the most fundamental and most far-reaching laws of physics.

The famous German philosopher and mathematician Gottfried Wilhelm Leibniz, a contemporary of Newton, is usually credited with being the first to recognize the conservation of energy. He called it a “living force” (vis viva) and defined it as the product of the mass of an object and its velocity squared.* Newton accepted the conservation of energy but is said to have disliked conservation principles and did not attach great significance to them. Interestingly, the discussion of conservation as a fundamental law of nature involved philosophical and theological concerns for both Newton and Leibniz.⁸⁰

The term “energy” in its modern sense was first used in the early nineteenth century, and scientists and philosophers argued for many years about whether energy was some kind of substance or merely a physical quantity. It was only with the formulation of thermodynamics in the midnineteenth century that energy was defined as the capacity of doing work, and the conservation of the total amount of energy through multiple transformations in mechanical and thermodynamic processes was clearly formulated.

Considering the very gradual emergence of the concept of energy over more than three centuries, and the reluctance even by Newton to attach importance to energy conservation, it is truly remarkable that Leonardo da Vinci had an intuitive grasp of it as early as the late fifteenth century. In his science of living forms that undergo continual changes and transformations, Leonardo paid special attention to the conservation of certain quantities—mass and volume in particular—and developed his own original “geometry done with motion” to express these principles of mathematically.⁸¹ He extended the concept of conservation even to the motion of solid objects in space. “Of everything that moves,” he noted, “the space which it acquires is as great as that which it leaves.”⁸²

Leonardo saw the conservation of volume as a general principle governing all changes and transformations of natural forms, whether solid bodies moving in space or pliable bodies changing their shapes. He applied it to the flow of water and other liquids (see p. 42), as well as to various movements of the human body, especially the contraction of muscles.⁸³

In view of Leonardo’s special perspective on principles of conservation, it is perhaps not surprising that he intuitively recognized the conservation of energy in his studies of mechanics. A striking example of this intuition is presented in the Codex Madrid I, where he describes and sketches an experiment involving a container from which water is tapped at various heights (fig. 5-8).

Leonardo begins the discussion of this experiment by clearly stating the problem. “Here the question is asked,” he writes, “which of these four waterfalls has more percussion and power in order to turn a wheel: fall a or b, c or d?” In the subsequent analysis, he notes that the initial spouting speed of the jets increases as the tapping level is lowered because of the increase in water pressure. However, he then makes the following conjecture.

I have not yet experimented, but it seems to me that [the four jets] must have the same power … [for] where the force of percussion is lacking, it is compensated by the weight of the waterfall.⁸⁴

In modern terminology, we would say that Leonardo reasons as follows. As the kinetic energy (or “force of percussion”) generated by the free fall of the water particles diminishes with the decreasing height of the spouts, this decrease is compensated for by the increasing potential energy (or “weight of the waterfall”), which results in the increase of water pressure and initial spout velocity. As a result of this compensation, the total energy (or “power”) of the jets at their impact on the ground remains constant regardless of the height of their spouts.

FIG. 5-8. Water jets falling from a container at four different heights. Codex Madrid I, folio 134v (detail).

It is worth noting that Leonardo’s sketch of the four “waterfalls” is not accurate. Had he actually carried out the experiment, he would have noticed that the horizontal distances traveled by the jets are pictured incorrectly. Although the jets’ spouting speed does increase with decreasing height of the taps, the time it takes for them to hit the ground becomes shorter, which affects the horizontal distances they can reach.

FIG. 5-9. Correct relationships between horizontal distances reached by jets and heights of spouts. Adapted from Fasso, “Birth of Hydraulics During the Renaissance Period.”

Today we can easily calculate the time it takes the freely falling water particles in each jet to reach the ground from the height of the spout, and the horizontal distance they can reach during that time with their initial spouting speed.⁸⁵ This calculation shows that the horizontal distance increases as the tapping level is lowered, reaches a maximum for the tap at half the height of the container, and then decreases again symmetrically for the taps in the container’s lower half. The curve obtained by plotting the horizontal distances reached by the jets against the heights of their spouts is an ellipse (fig. 5-9).

Without our modern terminology and quantitative formulations, such a calculation was far beyond Leonardo’s reach. These limitations make it all the more impressive that he intuited and correctly formulated the conservation of energy for flowing water. This important conservation law was rediscovered and precisely formulated only in the mid-eighteenth century by the mathematician Daniel Bernoulli, and is now known as Bernoulli’s theorem.

Movements of Consumption

When physicists in the nineteenth century developed the science of thermodynamics they formulated two fundamental principles, known today as the first and second laws of thermodynamics. The first law is that of the conservation of energy. The second law states that, while the total energy involved in a process is always conserved, the amount of useful energy diminishes, dissipating into heat, friction, and so on. It is truly amazing that Leonardo da Vinci anticipated both of these fundamental laws of physics and that his thorough understanding of the dissipation of energy led him to deep insights about change, transformation, and the nature of time, foreshadowing similar insights in modern physics by more than three centuries.

FIG. 5-10. Rotary ball bearing. Codex Madrid I, folio 20v (detail); model by Muséo Techni, Montreal, 1987.

The loss of machine power through friction was well known to Renaissance engineers. Their hoists, cranes, and other large machines were made of wood, and the friction between movable parts was a major problem. Leonardo invented numerous sophisticated devices for reducing friction and wear, including automatic lubrication systems, bearings made of semiprecious stones, and mobile rollers of various shapes—spheres, cylinders, truncated cones, etc.⁸⁶ Figure 5-10 shows an elegant example of a rotary bearing composed of eight concave-sided spindles rotating on their own axes, interspersed by balls that can rotate freely but are prevented from lateral movements by the spindles. When a platform is put on this ball bearing, friction is reduced to such an extent that the platform can be turned easily even when it is carrying a heavy load.

All the great Renaissance engineers were aware of the effects of friction, but Leonardo was the only one who undertook systematic empirical studies of its nature and properties. He investigated the frictional forces between various solids, as well as those involving water and air, and he designed experimental equipment for these studies that was far ahead of his time.⁸⁷

Leonardo found by experiment that when an object slides against a surface, the amount of friction is determined by three factors: the roughness of the surfaces, the weight of the object, and the slope of an inclined plane:

In order to know accurately the quantity of the weight required to move a hundred pounds over a sloping road, one must know the nature of the contact which this weight has with the surface on which it rubs in its movement, because different bodies have different frictions….

… Different slopes make different degrees of resistance at their contact; because, if the weight that must be moved is upon level ground and has to be dragged, it undoubtedly will be in the first strength of resistance, because everything rests on the earth and nothing on the cord that must move it…. But you know that, if one were to draw it straight up, slightly grazing and touching a perpendicular wall, the weight is almost entirely on the cord that draws it, and only very little rests upon the wall where it rubs.⁸⁸

Leonardo’s conclusions are fully borne out by modern mechanics. Today, the force of friction is defined as the product of the frictional coefficient (measuring the roughness of the surfaces) and the force perpendicular to the contact surface (which depends both on the object’s weight and the slope of the surface). Leonardo not only analyzed the forces of friction correctly but also obtained reasonably accurate quantitative results two centuries before the modern study of friction began and three centuries before the subject was fully elaborated by the physicist Charles Coulomb.

Leonardo extended his keen interest in friction to his extensive studies of fluid flows. The Codex Madrid contains meticulous records of his investigations and analyses of the resistance of water and air to moving solid bodies, as well as the resistance of water and fire moving in air.⁸⁹ Well aware of the internal friction (viscosity) of fluids, he dedicated numerous pages in the Notebooks to recording its effects on fluid flow (see p. 41). “Water has always a cohesion in itself,” he wrote in the Codex Leicester, “and this is the more potent as the water is more viscous.”⁹⁰

Air resistance was of special interest to Leonardo because it played an important role in one of his great passions—the flight of birds and the design of flying machines (see pp. 250ff.). “In order to give the true science of the movement of birds in the air,” he declared, “it is necessary first to give the science of the winds.”⁹¹

Careful reading of Leonardo’s notes on mechanics makes it evident that he recognized that all the different kinds of friction he studied—the grinding of axles and pivots in machines, the friction between colliding objects, and the resistance encountered by bodies moving in water and air—had the same net effect. They all resulted in a “consumption of power,” or dissipation of energy, as we would say today. The fundamental observation that in any physical process some energy is dissipated and cannot be recovered, which would become a cornerstone of the science of thermodynamics 350 years later, is illustrated repeatedly in Leonardo’s manuscripts.

In all mechanical engineering, Leonardo points out, “you have to deduct as much … from the power of the instrument as that which is lost by the friction in its bearings”;⁹² and his experiments with rebounding objects led him to this conclusion: “I have learned from percussion that the falling movement exceeds the reflex movement.”⁹³ A particularly elegant demonstration of energy dissipation is recorded in Manuscript A, which contains some of Leonardo’s earliest investigations of mechanics. In a simple sketch (fig. 5-11), he compares the trajectory of a ball flying freely through the air with one where the ball bounces repeatedly, losing a certain amount of “power” in each bounce.

In his studies of mechanical engineering, Leonardo early on investigated the medieval belief that power could be harnessed through perpetual motion machines. At first he accepted this idea. He designed a host of complex mechanisms to keep water in perpetual motion by means of various feedback systems. But as his understanding of the dissipation of energy matured, he realized the impossibility of such a task.

FIG. 5-11. Comparison between the trajectories of a freely thrown and a bouncing ball. Ms. A, folio 24r (detail).

“Descending water,” he concluded, “will never raise from the place where it comes to rest to the height from where it started an amount of water equal to its weight.”⁹⁴ In the end, Leonardo scoffed at attempts to build perpetual motion machines: “I have found among the excessive and impossible delusions of men the search for continuous motion, which is called by some the perpetual wheel.”⁹⁵

In the nineteenth century, the second law of thermodynamics was first formulated in terms of the dissipation of energy in thermal engines, but was soon recognized to be of much broader significance.⁹⁶ It introduced into physics the idea of irreversible processes, of an “arrow of time,” as it came to be called. According to the second law, there is a certain trend in physical phenomena. As mechanical energy is dissipated and cannot be recovered, physical processes proceed in a certain direction—from order to disorder. To express this direction mathematically, physicists introduced a new quantity, called “entropy,” which measures the degree of disorder, and hence the degree of evolution of a physical system. In its most general formulation, the second law of thermodynamics states that any isolated physical system will proceed spontaneously in the direction of ever-increasing entropy, or disorder.

Leonardo not only had a clear understanding of energy dissipation but also intuited its broader significance. He always paid special attention to the “consumption” of forms under the influence of physical forces over long periods of time. For example, his detailed description of the erosion of rocks carried by rivers and streams—from sharply angled fragments to smaller and rounder stones that eventually turn into gravel and fine sand—is a perfect illustration of an “entropic” sequence (as we would say today) toward ever increasing disorder (see p. 70). The entire passage would not look out of place in a modern textbook on thermodynamics. The same is true for Leonardo’s vivid description of rock weathering: “Water wears away the mountains and fills up the valleys, and if it could, it would like to reduce the Earth to a perfect sphere” (see p. 70).⁹⁷

Astonishingly, Leonardo associated these irreversible processes with a specific conception of time, as the founders of thermodynamics would do 350 years later. Physicists in the nineteenth and twentieth centuries discussed the idea of a direction of time, manifest in the evolution of physical processes from order to disorder. They called it the “arrow of time” to distinguish it from the reversible time coordinate in Newtonian physics. Leonardo, using only slightly different language, introduced the notion of a physical quality of time. On a folio in the Codex Arundel, he jotted a brief reminder to himself: “Write of the quality of time as distinct from its geometry.”⁹⁸ The quality of time he had in mind was that of “the consumer of all things”; in other words, time’s irreversibility in the physical processes of transformation and decay.

The conception of time as the consumer of all things can be found already in Leonardo’s early writings. A folio in the Codex Atlanticus, dating from around 1480, contains an evocative passage inspired by Ovid’s Metamorphoses, in which Leonardo imagines the beautiful Helen of Troy as an old woman, her face ravaged by the passing of time:

O time, consumer of all things! O envious old age, you destroy all things and consume all things with the hard teeth of the years, little by little, in slow death. Helen, when she looked in her mirror and saw the withered wrinkles that old age had made in her face, wept and wondered to herself why she had twice been carried away.⁹⁹

Leonardo applied his qualitative conception of time and, accordingly, the conception of “movements of consumption,” to three major domains: the transformations of the human body in the course of its life, those of the body of the Earth in the course of geological time, and to the consumption by attrition of the moving parts of machines.¹⁰⁰ His vision of the transformation and consumption of forms in these three domains as different manifestations of one universal process anticipated evolutionary thought in physics by more than three centuries and must be ranked as one of his greatest scientific achievements.

Weight, Force, and Motion

Weight was the second of Leonardo’s four powers of nature. The relationships between weight, force, and motion were at the center of his attention when he began his theoretical studies of mechanics with a long series of empirical investigations of the medieval “science of weights.” The main instrument of these investigations was the balance, and the basic theoretical framework was the classical Archimedean law of the lever (see p. 164).

Between 1490 and 1500, while he was painting the Last Supper in Milan, Leonardo made detailed studies of all the parts of a balance, experimenting with different kinds of suspensions or supports of the beams, different cords and weights, and so on. He systematically altered each variable in turn so as to get a clear understanding of the underlying principles. Not only that, he discussed possible errors arising from the differences between the mathematical treatment of a balance with the actual physical construction:

The science of weights is led into error by its practice, which in many instances is not in agreement with this science, nor is it possible to bring it into agreement. This arises from the axes of the balances through which science is made from such weights. These axes, according to the ancient philosophers, were treated as having the nature of mathematical lines, and in some places as mathematical points. These points and lines are incorporeal, whereas practice treats them as corporeal, because this is what necessity demands for supporting the weight of these balances together with the weights on them that are to be judged, … These errors I set down here below.¹⁰¹

Leonardo then proceeded to list several possible errors. For example, he observed that the central line of the beams can run below, through, or above the fulcrum of a balance. “Only the one through the middle is perfect,” he explained. “The one above is the worst; that below is less bad.”¹⁰²

These meticulous studies of the balance not only allowed Leonardo to calculate the forces and weights needed to establish equilibria in numerous compound systems, including balances, levers, and pulleys (see pp. 164–65), but also made him realize that the weight of a body is identical to the force of gravity acting on it. “The force is always equal to the weight that produces it,” he noted in the Codex Atlanticus.¹⁰³ In view of his general acceptance of the Aristotelian view of gravity, which did not include the concept of an actual force, Leonardo’s correct association of weight with a gravitational force, arrived at empirically, is rather remarkable.

The relationship between gravity and motion was of great interest to Leonardo in his anatomical studies because he wanted to understand the exact sequences of bodily movements in walking, running, jumping, and other activities (see pp. 213ff.). In preparation for detailed analyses of such movements Leonardo carried out many calculations to locate the center of gravity (called “center of mass” in modern mechanics) in a variety of geometric figures.¹⁰⁴

The rules and principles for the exact determination of the center of gravity of triangles, squares, and other geometric figures had been established in antiquity by Archimedes in his treatise On the Equilibrium of Planes. Leonardo followed Archimedes closely in his presentation of proposals and proofs on centers of gravity, citing several passages from the great classic. Like Archimedes, he frequently divided geometric figures into triangles and used the law of the lever to show that pairs of those triangles balanced about a certain point. He may even have made physical models of his triangles and put equal weights at the angles to determine their centers of gravity.

Leonardo did not limit his investigations to plane figures, as Archimedes did, but also determined the centers of gravity of several solids. Most important, he discovered the exact location of the center of gravity of the tetrahedron, the regular solid composed of four equilateral triangles. The argument of Leonardo’s proof is based on the tetrahedron’s symmetry and sounds very modern to us.¹⁰⁵ He draws the height, which he calls an “axis,” from the center of the figure’s base to the opposite vertex (fig. 5-12) and reasons that, since the base is an equilateral triangle, the distribution of mass around this axis is symmetrical in all directions. Therefore, the center of gravity must lie somewhere on the axis. He then notes that, because of the tetrahedron’s symmetry as a regular solid, the same argument can be made for any of its four axes. Hence, the center of gravity must lie at the point where they intersect. Finally, he constructs that point of intersection and finds that it lies at a distance of one fourth of the height’s length from the base.

FIG. 5-12. Construction of the center of gravity of the tetrahedron. Codex Arundel, folio 218v (detail).

On another folio in the same Codex Arundel, Leonardo writes down a concise summary of his result:

The center of gravity of the body of four triangular bases is located at the intersection of its axes and will be in the fourth part of their length.¹⁰⁶

The proof of Leonardo’s theorem is concise and elegant, certainly one of his most significant discoveries in geometry.

Falling Bodies

During the last years of his first period in Milan, Leonardo not only immersed himself in detailed studies of statics—the “science of weights”—but also examined the motion of falling bodies. According to the Aristotelian view of gravity, this was an example of the natural motion of “weights” toward the Earth, and Leonardo did not fail to notice that free fall under the influence of gravity is an accelerated motion. He explained this fact in terms of the medieval impetus theory. In most cases, the concept of impetus had been applied to “violent” motion, that is, motion in which an object is forced to move against its natural tendency. Indeed, Leonardo himself defined impetus as “a power impressed by the motor in the moved object” (see p. 180). But since he had already established the identity of force and weight, it was natural for him to apply the concept of impetus also to motion under the influence of gravity, viewing the acceleration of a falling body as the continuous impression of impetus by the body’s weight. “Impetus arises equally from weight as from force,” he explained.¹⁰⁷

Leonardo believed during most of his life that “pyramidal,” or linear, relationships were universal in nature, as I have discussed (see p. 175),¹⁰⁸ and so it is not surprising that he asserted that the velocity of a falling body increases in direct proportion to time:

The natural motion of heavy things, at each degree of its descent acquires a degree of velocity. And for this reason, such motion, as it acquires power, is represented by the figure of a pyramid.¹⁰⁹

We know that the phrase “each degree of its descent” refers to units of time, because on an earlier page of the same Notebook he writes, “A weight that descends freely in every degree of time acquires … a degree of velocity.”¹¹⁰ In other words, Leonardo is affirming the mathematical rule that for freely falling bodies there is a linear relationship between velocity and time.

Leonardo’s statements are entirely correct. In today’s mathematical language, we say that the velocity of a falling body is a linear function of time, and we write it symbolically as v = gt, where g denotes the constant gravitational acceleration. This language was not available to Leonardo. The concept of a function as a relation between variables was developed only in the late seventeenth century. Even Galileo described the functional relationship between velocity and time for a falling body in words and in the language of proportion, as did Leonardo 140 years before him.¹¹¹

Leonardo anticipated another discovery for which Galileo is famous. Instead of trying to verify his assertion about gravitational acceleration experimentally with falling bodies, which would have been almost impossible with the primitive clocks of his time, he had the same brilliant idea that Galileo had a century later—that a ball rolling down an inclined plane would accelerate in the same way as a freely falling object, only more slowly, which would allow one to measure the accelerated motion with reasonable accuracy even with simple instruments. “Although the motion is oblique,” he reasoned, “it observes in each of its degrees an increase in motion and in velocity in arithmetic progression.”¹¹²

Leonardo’s sketch next to this statement is most ingenious (fig. 5-13). He has drawn the inclined plane as an isosceles triangle (ebc) that is meant, at the same time, to represent the arithmetic progression of the velocity with time. The triangle forms one face of an irregular pyramid whose opposite face is a vertical triangle (identical to the triangle abc), representing the linear relationship between velocity and time for the corresponding vertical fall. The correct relationships between the relevant variables can easily be recognized in Leonardo’s diagram. The velocities at the end of both the vertical and the oblique descents are equal (both represented by bc), while the times involved in acquiring the velocities (represented by the edges of the two triangles) are clearly seen to differ. In addition, Leonardo marked the midway velocities (mn = op) to show that the same relationships hold for all intermediate velocities along the vertical and oblique descents.

FIG. 5-13. Accelerated motion on an inclined plane. Ms. M, folio 42v (as reconstructed by Clagett, “Leonardo da Vinci: Mechanics”).

Leonardo’s diagram of the inclined plane is an impressive example of his great capacity for analyzing the elements of complex phenomena and presenting them visually with great clarity. However, it is doubtful that he actually experimented with balls rolling down inclined planes, as Galileo did. Had he done so, Leonardo would certainly have observed that the distances of falling bodies increase with the squares of the times and not linearly. Instead, he maintained erroneously that “in each doubled quantity of time, the length of the descent is doubled.”¹¹³

As mentioned above, Galileo made the same error in his early work, but corrected it after experimenting with inclined planes (see pp. 175–76). The fact that Leonardo held on to his belief in a linear relationship between the distances traversed by falling bodies and the times elapsed seems to indicate that he never carried out the experiments he had designed so brilliantly. Marking off the distances covered by balls rolling down inclined planes during successive time intervals would have been relatively easy, whereas measuring their ever-increasing velocities during the same time intervals would have been a considerable challenge. Indeed, Galileo attained an understanding of the linear relationship between instantaneous speeds and times only with great effort and several years after he realized that the distances traversed by falling bodies increase with the squares of the times.¹¹⁴

In addition to the inclined plane, Galileo used the pendulum as a major tool for analyzing the effects of gravity on motion. Leonardo, too, studied the motion of the pendulum and used it as a regulator in clocks.¹¹⁵ But he failed to recognize one of its most important properties: the period of the oscillation is independent of the mass of the bob. Galileo famously used this property of the pendulum to demonstrate that in a vacuum all objects will fall with the same acceleration and therefore will reach the ground at the same time, regardless of their mass.

FIG. 5-14. Energy loss in pendulum motion. Codex Madrid I, folio 147r (detail).

FIG. 5-15. Approximate regularity of pendulum swings for small arcs. Codex Madrid I, folio 147r (detail).

Still, Leonardo did make some significant discoveries about pendulum motion. He described it as the interplay between natural motion (the downswing) and accidental motion (the upswing), and he realized that, because of the inevitable energy loss through friction, “the accidental motion will always be shorter than the natural.”¹¹⁶ This statement, on a folio of the Codex Madrid I, is illustrated with a simple sketch (fig. 5-14) that clearly shows the loss of energy on the upswing. It is another of Leonardo’s many illustrations of the dissipation of energy.

On the same folio, Leonardo analyzes the rate at which the arc of the pendulum diminishes under the influence of friction and notes that for small arcs the oscillations will be more uniform: “The smaller the natural motion of a suspended weight, the more the following accidental motion will be equal in length.” Again, the observation is clearly illustrated in a small drawing (fig. 5-15). With this discovery of the increased regularity of pendulum swings for small arcs, and hence of the approximate regularity of their beats, Leonardo anticipated the theoretical formulations of Galileo and the practical applications to the development of pendulum clocks by about a century and a half.

Finally, still on the same folio of the Codex Madrid, Leonardo juxtaposes the trajectory of a pendulum and its interplay of natural and accidental motion with the trajectory of a stone thrown in an arc. Even though his detailed comparison of the two trajectories contains some errors, his observation that in both cases there is a similar interplay between gravity and impetus is prescient. In the seventeenth century, Galileo would make the same juxtaposition in his celebrated Dialogue on the Two Chief Systems of the World.

Ballistic Trajectories

The study of ballistic trajectories was of special interest to Leonardo in his work as a military engineer. In 1502, when he was fifty and had acquired great fame as an artist and engineer, he was hired by the papacy to travel throughout central Italy to inspect ramparts, canals, and other fortifications and make suggestions for their improvement.¹¹⁷ He designed ingenious new fortifications. Instead of castles with high vertical walls he envisioned low bastioned fortresses arranged in a series of concentric curves so as to minimize the impact of cannonballs.¹¹⁸

To develop these effective military designs, Leonardo needed to have accurate knowledge of the trajectories of projectiles—a subject that was poorly understood and replete with erroneous assumptions at the time. Throughout the Renaissance and up to the late seventeenth century, the trajectory of a cannonball was pictured in all military treatises dealing with artillery as rising at an angle along a straight line, followed by a short curved section, and then falling in a perfectly vertical line.¹¹⁹ Several variations of this picture were proposed by mathematicians in the sixteenth and seventeenth centuries. All of them—even Galileo in his early work—pictured the rising portion of trajectories as a straight line (fig. 5-16).

Leonardo, by contrast, did not fail to recognize (without naming it) the parabolic nature of ballistic trajectories. On the folio in Codex Madrid I cited above, his juxtaposition of the trajectory of a pendulum with that of a stone thrown in an arc is accompanied by a beautiful sketch of a series of ballistic trajectories for different launch angles (fig. 5-17).¹²⁰ The parabolic shapes of these trajectories are clearly visible.

Leonardo did not have the mathematical tools to calculate these shapes, and with the experimental equipment available at the time, he could not have determined them by monitoring the paths of actual cannonballs. But he used his powers of scientific analysis and his systemic way of thinking to find an ingenious solution to the problem. Instead of studying the trajectories of stones or cannonballs, he studied jets of water, where the trajectories can actually be seen. He realized that these jets were composed of water particles subject to the same forces of natural and accidental motion (of gravity and inertia, as we would say today) as the stones and cannonballs.

FIG. 5-16. Ballistic trajectories according to (1) Niccolò Tartaglia, 1537; (2) Girolamo Cardano, 1550; (3) Bernardini Baldi, 1621; and (4) Galileo Galilei, 1592. From Bertoloni-Meli, Thinking with Objects.

In Manuscript C, written shortly before Codex Madrid I, Leonardo describes how he systematically studied the trajectories of water jets: “Test in order to make a rule of these motions. Make the test with a leather bag full of water with many small pipes of the same inside diameter, installed along one line.”¹²¹ The accompanying drawing shows such a bag with water pouring out from four small spouts arranged at different angles, including one in a vertical direction (fig. 5-18). It is evident from this drawing that Leonardo’s sharp eye perceived not only the correct parabolic shapes of the water jets but also, impressively, their slight distortions due to air resistance. The characteristic flattening of their ascending portion and the steepening of their descending portion are clearly visible in the drawing.

FIG. 5-17. Parabolic trajectories for different launch angles. As usual in Leonardo’s diagrams, the direction of motion is from right to left. Codex Madrid I, folio 147r (detail).

The accuracy of Leonardo’s drawing of these trajectories is truly astonishing. After Leonardo, the parabolic form of ballistic trajectories was observed by Galileo in 1609 and was proven mathematically by his most famous disciple, Evangelista Torricelli, in 1644. Torricelli also rediscovered, 150 years after Leonardo, the distortion of ballistic trajectories by air resistance, while Galileo failed to take this effect into account. The calculation of the exact ballistic curve with air resistance had to wait for Newton, who published it in his Principia in 1687.

FIG. 5-18. Parabolic water jets with air resistance. Ms. C, folio 7r (detail).

The Origin of Physical Forces

Having established the identity of a body’s weight with the force of gravity acting on it, it was natural for Leonardo to consider force as his third power of nature, after motion and weight.

The origin of physical forces was one of the most persistent and perplexing questions in the development of classical mechanics. Galileo did not address the problem, limiting his investigations to the motion of material bodies under the influence of various forces. He was criticized for failing to do so by Descartes, who vigorously promoted a thoroughly mechanistic view of the world in which both living and nonliving phenomena were reduced to the motions and mutual contacts of small material particles. The force of gravity, in particular, was explained by Descartes in terms of a series of impacts of tiny particles contained in subtle material fluids that permeated all space.¹²²

Descartes’s theory was highly influential throughout most of the seventeenth century, until Newton replaced it with his conception of gravity as a fundamental force of attraction between all matter, acting at a distance and diminishing with the square of that distance. Newton’s conception, in turn, was criticized by many of his contemporaries, who were shocked by the idea that a force of attraction should act at a distance without being transmitted by any medium. The famous architect and mathematician Christopher Wren, for example, was reported to have “smiled at Mr. Newton’s belief that [gravity] does not occur by mechanical means, but was introduced originally by the Creator.”¹²³ The definitive solution of this vexing problem had to wait until the development of the field concept by Michael Faraday and James Clerk Maxwell in the nineteenth century and of Albert Einstein’s theory of gravity (his general theory of relativity) in the twentieth.

Leonardo, who remained largely within the Aristotelian framework and was unencumbered by the fundamental division between mind and matter to be introduced a century later by Descartes, looked at the question of the origin of physical forces very differently. Since the movements of falling bodies, flowing water, rising air, and blowing wind were thought to be caused by the natural tendencies of these elements to move toward their proper places, the problem was reduced to explaining the origin of the “accidental” or “violent” forces that disturbed the balance of the elements.

For Leonardo, the principal accidental force came from the muscle power of animals and humans, which was indeed one of the main sources of energy in his time. The origin of this muscle power (for both humans and animals) was in the soul, from where it was transmitted to the body’s muscles by invisible, nonmaterial nervous impulses that traveled through the sensory and motor nerves in the form of waves.¹²⁴ In other words, the origin of force was nonmaterial. “Weight is corporeal,” he explained, “and force is incorporeal; weight is material and force is spiritual.”¹²⁵ As I have mentioned, Leonardo often used the word “spiritual” in the sense of being immaterial and invisible, and this is how he described the ultimate nature of accidental forces:

Force is nothing but a spiritual power, an invisible potency, which is created and infused, through violence from without, by sentient bodies in non-sentient ones.¹²⁶

Similar clear and articulate definitions of accidental force appear repeatedly throughout Leonardo’s Notebooks. But because of his use of the term “spiritual,” there has been confusion about his conception of force among many historians. From their mechanistic perspectives, Leonardo’s statements seem to reveal a spiritual, or even esoteric, dimension of his thought. Such a dimension may be discerned in some of his philosophical statements but not in his conception of physical forces, in my opinion. Leonardo’s definition of force is unambiguous, derived from empirical evidence, and consistent with the overall framework of his scientific thought.

Force, Motion, and Work

In his work in mechanical engineering, Leonardo had ample opportunity to observe the effects of various forces in machines like the pulley and the lever. In particular, he paid special attention to the transmission of power and motion from one plane to another (see p. 162). It is typical of his scientific mind that he not only used this empirical knowledge to improve existing machines but also tried to derive general principles of mechanics from his observations.

One of those principles was the conservation of work, a special case of the conservation of energy. Several years after his comments on the conservation of energy in the Codex Madrid I, Leonardo stated in the pocketsized Notebook known as Manuscript F:

If a power moves a body a certain distance in a certain time, the same power will move half of this body in the same time twice that distance … [or] the whole distance in half that time.¹²⁷

This statement of the conservation of work is in complete agreement with our modern definitions of work as “force times distance” and of power as “work over time.”

Leonardo also discovered a principle that would become known as Newton’s third law of motion two hundred years later. It states that “for every action, there is an equal and opposite reaction.” In other words, physical forces always come in equal and opposite pairs. Leonardo did not formulate this observation as a general principle, but he clearly stated it in terms of many concrete examples.

One of his earliest Notebooks, Manuscript A, written around 1490 during his first years in Milan, contains a discussion of the rebounds of a small glass ball on a smooth polished stone. In his analysis, Leonardo stated that the force of the ball’s rebound is equal to the force of its impact.¹²⁸ Around the same time, in 1485, he studied the flight of birds and developed his first designs of flying machines. Observing the wing movements of an eagle, he noted: “As much force is exerted by an object against the air, as the air exerts against the object.”¹²⁹ Some twenty years later, he recognized the same principle in relation to the force of water on an oar: “The amount of movement made by an oar against still water equals the amount of movement made by water against a motionless oar.”¹³⁰

Finally, there is a principle of classical mechanics that is much more abstract and more general, which Leonardo anticipated by several centuries. Manuscript G contains a concise formulation, derived from observations of falling bodies: “Every natural action is made by the shortest way.”¹³¹ A more elaborate and effusive formulation is found on a folio of the the Codex Atlanticus where Leonardo discusses experiments with light rays:

O marvelous necessity, with supreme reason you compel all effects to be linked to their causes, and by supreme and irrevocable law every natural action obeys you by the shortest operation.”¹³²

Leonardo’s enthusiasm about this principle was fully justified. Nearly two centuries later it would turn out to be one of the most important principles of classical mechanics. It was formulated in the seventeenth century by the great mathematician Pierre de Fermat, who observed it in connection with geometrical optics, as Leonardo had done before him. Known today as Fermat’s principle, or the “principle of least time,” it states that light always follows the path of least time. In the nineteenth century, the principle was reformulated by the physicist and mathematician William Rowan Hamilton in much more abstract mathematical language. Hamilton’s principle, also known as “principle of least action,” is valid for all physical systems.*

Some of Leonardo’s statements seem to imply that the path of least time for a natural process is also the shortest path, which is not always the case. Strictly speaking, therefore, Leonardo’s principle cannot be viewed as an early formulation of either Fermat’s or Hamilton’s principle. But what it shares with both is the idea of a certain efficiency in natural phenomena, called “necessity” by Leonardo, which can be measured by observing that the value of some quantity, or variable, becomes a minimum. The level of abstraction of this idea, which is clearly articulated in Leonardo’s statements, is truly exceptional for his time.

Percussion—The Fourth Power of Nature

Leonardo’s concept of “percussion,” his fourth power of nature, corresponds to what we now call “impact” and refers to a broad range of phenomena. They include the striking of bells and tapping of vibrating plates; the impacts of hammers on nails and other surfaces; the impact of colliding balls, as well as the rebound of a ball from a firm surface; and even the destructive effects of mortars from the initial explosions to the impacts of their cannonballs and blast waves. In addition, Leonardo uses “percussion”—or, more frequently, “force of percussion” or “power of percussion”—to denote the energy transferred in the process of impact; in other words, as the equivalent of the modern concept of kinetic energy.

In all his studies of impact phenomena, Leonardo paid special attention to the transfer of energy from one body or medium to another, as well as to the eventual dissipation of that energy. He noted that energy transfer takes place in the process of impact when an initial motion comes to a sudden halt: “The blow is born of the death of motion.”¹³³

In a series of experiments, he struck various objects with a hammer and analyzed how much of the “force of percussion” is absorbed by the object and how likely the object is to break, depending on its supports and on the location of the blow. In one case, he showed how “the hand holding a stone that is beaten does not suffer as much as it would if it received the blow directly.”¹³⁴ In another example of percussion with a hammer, he demonstrated how the kinetic energy of the impact is transformed into heat:

If you beat a thick bar of iron between the anvil and hammer with frequent blows upon the same place, you will be able to light a match at the beaten place.¹³⁵

For the extreme case of explosions, Leonardo gave vivid and detailed descriptions of the destructive effects of the blast wave, as in the following passage from the Codex Atlanticus.

If you discharge a small bombard in a courtyard surrounded by a convenient wall, any vessel that is there, or any window covered with cloth, will be instantly broken; and the roofs will be lifted off slightly from their supports; the walls and ground will shake as in a big earthquake; all the spiders’ webs will fall down; small animals will perish, and every air-containing body nearby will suffer instant harm and some damage.¹³⁶

Percussion also played an important role in Leonardo’s studies of acoustics. As I discussed in my previous book, he observed from experiments with bells, drums, and other musical instruments that sound is always produced by “a blow on a resonant object,” and he correctly deduced that this percussion causes an oscillating motion in the surrounding air.¹³⁷ Moreover, he described the phenomenon of resonance in detail:

The blow given to the bell will make another bell similar to it respond and move somewhat. And the string of a lute, as it sounds, produces response and movement in another similar string of similar tone in another lute. And this you will perceive by placing a straw on the string which is similar to that sounded.¹³⁸

The observations of resonating bells and lute strings suggested to Leonardo the general mechanism for the propagation and perception of sound—from the initial percussion and the resulting waves in the air to the resonance of the eardrum. In Manuscript C, he illustrated his discovery with a charming little sketch (fig. 5-19) in which the generation, propagation, and perception of sound waves is represented symbolically by three little hammers striking a bell, a reflecting wall, and an ear.

FIG. 5-19. Propagation of sound waves from a bell to the ear. Ms. C, folio 16r (detail)

Percussion also lies at the origin of Leonardo’s experiments of dropping pebbles into the still water of a pond. His perceptive analysis of the subsequent phenomena includes detailed descriptions of the impact of the pebble, the generation of an up-and-down motion of the water particles, the spread of the “force of percussion” in circular waves, and its gradually diminishing “power” as the kinetic energy of the impact is dissipated by the water’s viscosity.¹³⁹

In his studies of percussion, Leonardo not only analyzed the transfers and transformations of kinetic energy in various impact phenomena but also tried to determine quantitative relations between the mass and velocity of a colliding object and the damage done by the “power of percussion” (kinetic energy) at impact. To do so, he dropped various weights from different heights on a slab of lead and measured the size of the dents produced by their impact.¹⁴⁰

In another series of experiments, he shot arrows of different weights up to various heights and measured the penetrations of their shafts into soft soil, “the soil being of uniform resistance and the shafts of the same shape.”¹⁴¹ To achieve this uniformity, he produced arrows with identical hollow shafts and gave them different weights by placing stones inside the shafts. From all these experiments, Leonardo concluded correctly that the “power of percussion” is proportional to the weight of the falling object and to the height of its fall.

Collisions of billiard balls and similar objects were another major focus in Leonardo’s investigations of percussion. He distinguished collisions between two moving objects from those between a ball and a firm wall or very heavy object (i.e., rebound phenomena):

There are two kinds of percussion: when the object flees from the projectile that struck it, and when such a projectile rebounds back from the object struck.¹⁴²

Leonardo studied both types of collision in great detail and in many variations. On a folio in Manuscript A (fig. 5-20), he sketched more than a dozen examples of both solid and breakable balls of different masses colliding at various speeds and angles of incidence. The brief verbal descriptions accompanying each sketch make it evident that these are records of systematic experiments with elastic and inelastic collisions, as they are called today.

FIG. 5-20. Studies of elastic and inelastic collisions. Ms. A, folio 8r.

From these experiments, Leonardo tried to derive quantitative relations between the masses, velocities, and angles of the colliding balls. Sometimes he would pose collision problems in clear and concise language:

Ball a moves with three degrees of velocity, and ball b moves with four degrees of velocity. It is asked how such a percussion differs from one in which the ball [b] were to be at rest instead of approaching it [a ] with the said four degrees of velocity.¹⁴³

In other passages, Leonardo stated general rules about elastic collisions. A folio in the Codex Leicester, for example, contains the following statements.

If the percussor is equal and similar to the percussed, that percussor leaves its power completely in the percussed, which flees with fury from the site of the percussion, leaving its percussor there. But if the percussor—similar but not equal to the percussed—is greater, it will not lose its impetus completely after the percussion but there will remain the amount by which it exceeds the quantity of the percussed. And if the percussor is less than the percussed, it will rebound back through more distance than the percussed by the amount that the percussed exceeds the percussor.¹⁴⁴

It is instructive to examine the three statements in this passage in some detail. The first is an accurate description of an elastic collision of two balls of equal mass, one moving and the other stationary, in which the “power” (kinetic energy) of the moving ball is completely transferred to the stationary target during the collision. In fact, Leonardo’s description is the correct answer to the first problem posed in the preceding quotation.

The other two statements in the passage from the Codex Leicester describe two collisions between balls of unequal mass. In the first, the moving ball is heavier than the stationary target; in the second, it is lighter. Here Leonardo’s descriptions are qualitatively correct, but he obviously struggles in his attempts to formulate precise relationships between quantities like mass, velocity, and momentum, which have different dimensions. This would remain a major challenge for scientists for another two centuries after Leonardo (see p. 173).

Indeed, the precise mathematical analysis of impact phenomena had to wait until the seventeenth century, when Huygens used the laws of both the conservation of energy and the conservation of momentum, formulated in terms of algebraic equations, to derive the exact rules for elastic collisions.

Rebound phenomena, in which an object collides with a firm wall, presented an easier problem to natural philosophers because they involve fewer variables. Central to Leonardo’s descriptions of percussion and rebound was the rule that the angle of incidence equals the angle of reflection. From his early investigations of mechanics on, he repeatedly stated that fundamental rule, now known as the law of reflection. A succinct statement, recorded around 1500, can be found in the Codex Arundel:

The angle made by the reflected motion of heavy bodies is equal to the angle made by the incident motion.¹⁴⁵

In Manuscript A, written ten years earlier, we find a similar passage: “The line of percussion and that of its rebound are placed in the middle of equal angles.”¹⁴⁶ The meaning of this statement is perhaps less evident, but it is clearly illustrated with a drawing (fig. 5-21) and accompanied by a more elaborate description:

If the ball b is thrown to c, it will turn back through the line cb, necessarily making equal angles on the wall fg. And if you throw it through the line bd, it will turn back through the line de, and thus the line of percussion and the line of rebound will make an angle on the wall fg situated in the middle between two equal angles, as d appears in the middle between m and n.

The law of reflection was first formulated in optics by the great Arab mathematician Alhazen (Ibn al-Haitham) whose seven-volume work Kitab al-Manazir (Book of Optics) was available during the Renaissance in Latin translation and was discussed by several European philosophers. Leonardo became familiar with Alhazen’s work through these authors and he used the law of reflection extensively in his explorations of spherical and parabolic mirrors, producing a series of precise and beautiful diagrams.¹⁴⁷

But Leonardo went further. He was the first to recognize the broad generality of the law of reflection, applying it not only to mechanics and optics but also to acoustics and hydrodynamics. On the same folio of Manuscript A where he discusses the rebound of a ball thrown against a wall, Leonardo adds a brief note: “The voice is similar to an object seen in a mirror.”¹⁴⁸ In other words, the law of reflection holds equally for light and sound. Several years later, he applied the same reasoning to the rebound of a jet of water from a wall, noting, however, that some of the water peels off as an eddy after the reflection.¹⁴⁹

FIG. 5-21. Illustration of the law of reflection. Ms. A, folio 19r (dcetail); reconstructed by Clagett, “Leonardo da Vinci: Mechanics.”

As I have emphasized, this kind of systemic thinking is typical of Leonardo’s scientific approach as a whole. In the preceding pages, I have discussed about a dozen of his discoveries and anticipations of abstract principles of mechanics that were centuries ahead of his time. They include his understanding of the relativity of motion, his intuitive grasp of the conservation of energy, and his emphasis on principles of conservation in general; his anticipation of the law of energy dissipation (the second law of thermodynamics), and his association of irreversible processes with a physical quality of time; his juxtaposition of the interplay between gravity and inertia in the swing of a pendulum with a similar interplay in the trajectory of a stone thrown in an arc; his experiments with an inclined plane to study gravitational acceleration and with jets of water to study ballistic trajectories; his discovery of the principle now known as Newton’s third law of motion, and his intuitive anticipation of Fermat’s principle; and, last but not least, his derivation of the center of mass of the tetrahedron from symmetry arguments.

All these examples show a level of abstract thinking that makes so many of Leonardo’s scientific statements sound utterly modern, in spite of the evident Aristotelian roots of his science. This level of abstraction is evident throughout Leonardo’s “elements of mechanics,” as well as in all other branches of his science.