7
The Science of Flight

From the texts that accompany Leonardo’s anatomical drawings we know that he considered the human body as an animal body, as biologists do today. He often transferred what he learned from numerous dissections of animals to the human body (see p. 227). But beyond these pragmatic aspects, Leonardo’s anatomical studies of animals were grounded in a profound respect and compassion for all living creatures.¹ Thus it seemed natural for him, as Domenico Laurenza observes, “to give equal ontological and scientific dignity to humans and animals.”²

Comparative Anatomy

Leonardo used his animal dissections to gain knowledge about human anatomical structures, but was also keenly interested in the many differences between the bodies of animals and human beings. “You will draw for this comparison,” he wrote on a sheet showing the superficial muscles of a man’s legs, “the legs of frogs which have a great similarity to the legs of man, in their bones as in their muscles. Then you will follow with the hind-legs of a hare which are very muscular and have agile muscles because they are not encumbered by fat.”³

Leonardo’s love of horses was well known to his contemporaries. He produced a wealth of magnificent studies of horses that are now assembled in a special volume of the Windsor Collection, and he is said to have written an entire treatise on the anatomy of the horse, now lost.⁴ However, a folio in the Windsor Collection contains a superb study comparing the anatomy of the hind leg of the horse with that of the human leg (fig. 7-1). In both drawings, some of the hip muscles are represented by “cords” to show the exact lines of force. Moreover, and perhaps even more remarkably, the comparison between the bones of the lower leg and foot shows Leonardo’s full appreciation of the fact that, compared with the human posture, the horse stands on the tip of its toe.

FACING Birds in flight, 1505 (detail, see plate 1).

FIG. 7-1. Comparison of the human leg with that of a horse, c. 1507. Windsor Collection, Anatomical Studies, folio 95r (detail).

In several of his comparative studies, Leonardo specifically contrasted the limbs of various animals with those of the human body. In fact, among his very first anatomical drawings there is an exquisite series of four studies of a bear’s foot in dissection.⁵ Over the years, these comparative studies of animal limbs led Leonardo to the momentous conclusion that their different structures should be seen as variations of a single underlying theme:

All land animals resemble each other in their limbs, that is in muscles, sinews and bones, and these do not vary except in length and thickness.⁶

For Leonardo, the ability to recognize such anatomical similarities across a wide range of species constituted the very essence of “becoming universal” (farsi universale). This awareness of “universality” corresponds to what we call systemic thinking in contemporary science.⁷

Leonardo’s “Evolutionary” Thought

Leonardo’s statement about the similarities of the limbs of mammal species foreshadows a way of thinking that would re-emerge in biology in the eighteenth century and would lead, eventually, to the formulation of one of the cornerstones of modern biology: Charles Darwin’s theory of evolution. The German school of Romantic biology, which included the great poet, dramatist, and scientist Johann Wolfgang von Goethe, recognized, as Leonardo had three hundred years earlier, a unity of patterns underlying the differences in animal shapes and sizes. Goethe and other biologists and philosophers of this school saw these patterns as manifestations of fundamental organic types, called “archetypes” (Urtypen).⁸ This concept had a tremendous influence on biological thought in France and England during the nineteenth century. Darwin, in particular, acknowledged that archetype theory played a central role in his early conception of biological evolution.⁹

Another foundational idea for Darwin was the notion of gradual changes of anatomical structures over immense periods of time. As I have mentioned, that idea, too, can be found in Leonardo’s writings; not in the context of gradually changing anatomical structures but of gradual changes in the strata and formations of rocks—the bones of the living Earth (see p. 74). Thus, Leonardo anticipated two key ideas that were important ingredients of Darwin’s early conception of the origin of species.

Since Leonardo’s science is utterly dynamic, it is perhaps not surprising that we can find an “evolutionary” flavor in many of his scientific writings. As I have mentioned, he perceived nature’s forms—in mountains, rivers, plants, and in the human body—as being in ceaseless movement and transformation. The world he portrays, in both his art and his science, is a world in development and flux, in which all configurations and forms are merely stages in a continual process of transformation.

What is even more remarkable, however, is that Leonardo intuited both of the two contradictory theories of evolution that would dominate nineteenth-century scientific thought: the evolution of closed physical systems from order to disorder, described by the physicists who developed thermodynamics, and the evolution of life from disorder to ever increasing order and complexity, described by Darwin and other biologists.

I have discussed in detail how Leonardo anticipated both the first and second laws of thermodynamics, and how his thorough understanding of the dissipation of energy led him to deep insights about the nature of irreversible processes—the “consumption” of natural forms under the influence of physical forces over long periods of time (see pp. 184ff.).

Leonardo did not fail to notice the contradiction between this phenomenon of gradual transformations from order to disorder and life’s continual creation of ever-increasing diversity, which he also observed. Within modern science, it took more than a hundred years to resolve the contradiction between the two theories of “evolution” developed in the nineteenth century.¹⁰ For Leonardo, who never developed any kind of theory of biological evolution, the contradiction was not inherent in his science. He simply perceived it as two opposing trends in natural phenomena, and he asserted that the evolution toward ever-increasing diversity of living forms always outpaced the opposing trend of evolution toward increasing disorder. “Nature, capricious and taking pleasure in creating and producing a continuous succession of lives and forms,” he wrote in the Codex Arundel, “is eager and quicker to create than time is to destroy.”¹¹

The Dream of Flying

The investigation of the body’s voluntary movements was a major theme in Leonardo’s anatomical studies; he also compared human movements with the movements of various animals. In particular, he analyzed the gait of horses and drew comparisons with the human manner of walking:

The walking of men is always in the manner of the universal gait of four-footed animals; because just as they move their feet cross-wise, in the manner of the trot of the horse, so a man moves his four limbs cross-wise; that is, if he thrusts his right foot forward in walking, he thrusts the left arm forward with it, and so it always continues.¹²

In addition to the gaits of landbased animals, Leonardo studied the movements of fish in water; but what fascinated him more than any other animal movement was the flight of birds. It was the inspiration for one of the great passions in his life—the dream of flying.

FIG. 7-2. Andrea Pisano, The Myth of Daedalus, c. 1343. Sculpted panel on Giotto’s campanile in Florence.

The dream of flying like a bird is as old as humanity itself. But nobody pursued it with more ingenuity, perseverance, and commitment to meticulous research than Leonardo da Vinci. From his early years in Florence, he must have been well acquainted with the legendary flight of Daedalus and Icarus who, according to the Greek myth, escaped from Crete on wings of feathers and wax. Daedalus, the cunning craftsman, is depicted with his powerful wings in one of the sculpted panels that decorate the lower portion of Giotto’s campanile (fig. 7-2). This image may well have been both inspiration and challenge to the young Leonardo, whose earliest drawings of flying birds, insects, and artificial wings date from around 1470, when he had just established himself in Florence as an independent artist.¹³ During the same period, he painted his Annunciation, in which the angel’s entirely realistic wings,* growing from the shoulder blades, were obviously modeled after the real wings of birds.¹⁴

However, Leonardo soon realized that flying like a bird would require more than finely crafted angels’ wings. He would need to understand the subtle details of how birds sustain themselves in the air and be able to absorb that knowledge into the design principles of his flying machine. As his scientific mind matured after his move to Milan, he began to develop a comprehensive approach to this challenge that involved numerous disciplines—from aerodynamics to human anatomy, mechanics, the anatomy of birds, and mechanical engineering. He diligently pursued these studies throughout most of his life, from his early years at the Sforza court in Milan to his old age in Rome.¹⁵ No project is better suited to illustrate Leonardo’s systemic, integrative approach to scientific research, its brilliant application to engineering, and his persistent endeavor to imitate nature than his lifelong quest for a “science of flight.”

Air Pressure and Lift

Leonardo’s first intense period of research on flying machines began in the early 1490s, about a decade after his arrival in Milan. By that time, he had been fully accepted as artist and “ducal engineer” at the Sforza court and had become Prince Ludovico’s favorite court artist.¹⁶ He had living quarters and a large space for his workshop in the Corte Vecchia, where Ludovico housed important guests, and was engaged in a flurry of intellectual and artistic activities. These included creating the molds for the gran cavallo (a large equestrian statue honoring Ludovico’s father), painting the Last Supper, and carrying out experiments in optics and mechanics, as well as designing and testing his first flying machines.

From his early observations of birds in flight, Leonardo recognized the compression of the air under the bird’s wing during the downward stroke as a critical element in the generation of lift. The detailed aerodynamics of a bird in flight is very complex and was understood by Leonardo only many years later (see pp. 261ff.). However, even his early explanations of aerodynamic lift in the 1490s contained some important insights into the physics of flight. As science historian Raffaele Giacomelli has noted, these partial insights are impressive in view of the fact that before Leonardo, no natural philosophers had bothered to wonder how birds sustain themselves in the air.¹⁷ Following Aristotle, it was simply believed that birds were supported by air as ships are by water, their wings and tails being analogous to the ships’ oars and rudders.

When Leonardo observed birds in flapping flight, he recognized in this process two phenomena of mechanics he had discovered in other circumstances. One was the fact that, unlike water, “air has the ability to compress and rarefy.”¹⁸ The other was the principle, now known as Newton’s third law of motion, that for every physical force there is an equal and opposite reactive force (see p. 200). In his Notebooks, Leonardo mentioned both of these phenomena many times, and very early on he used their combined effect in his attempts to explain aerodynamic lift.

In his very first Notebook, the Codex Trivulzianus, begun in the late 1480s, we find a concise exposition of the basic idea, illustrated with two well-known examples from everyday life:

When the force generates more velocity than the escape of the resisting air, that air is compressed in the same way as bed feathers when compressed and crushed by the weight of the sleeper. And the object that pressed on the air, meeting resistance in it, rebounds in the same way as a ball striking against a wall.¹⁹

In other words, aerodynamic lift is explained by the compression of air under the bird’s wings during the downward strokes and the resulting upward rebound of the bird’s body.*

In a passage in the Codex Atlanticus, written around the same time, Leonardo adds another important principle of mechanics to his explana-tion—the relativity of motion—which he also recorded many times in various manuscripts (see p. 177). He argues that, since the motion of an object against still air is equivalent to the motion of air against a fixed object, the force sustaining an eagle in the air during flapping flight is the same as the force of the wind pushing a sailing ship:

As much force is made by the thing against the air as by the air against the thing. See how the wings striking against the air sustain the heavy eagle high up in the thin air…. See also how the air moving over the sea strikes against the swelling sails and makes the loaded and heavy ship run fast.²⁰

From this observation, Leonardo derives a momentous conclusion. “Therefore, by these reasons, asserted and demonstrated,” he continues in the same paragraph, “you will know that a man with his assembled and great wings, exerting force against the resisting air and conquering it, will be able to subjugate it and raise himself above it.” What this means is that Leonardo’s belief in the possibility of human flight was established during his earliest investigations. In his entire life, he never lost this belief. His firm conviction that, some day, human beings would be able to fly like birds was not based on hope, but was grounded in sound scientific principles.

FIG. 7-3. Experiment to test the human capacity to efficiently flap an artificial wing, c. 1487–90. Ms. B, folio 88v (detail).

Having convinced himself that the critical challenge for human flight was to flap artificial wings with enough force and velocity to compress the air underneath and be lifted up, Leonardo set out to systematically test this strategy. In a series of experiments at the Corte Vecchia that combined mechanics and human anatomy, he carefully measured the body’s capacity to generate various amounts of force in different bodily positions.²¹ In addition, he designed a large membrane-covered wing (fig. 7-3) to test the possibility of flapping it efficiently enough to lift a heavy plank attached to its base. The aim of all these studies was to find out how a human pilot might be able to lift a flying machine off the ground by flapping its mechanical wings.

Leonardo’s first design of a “flying ship,” based on these early experiments, is a rather strange contraption (fig. 7-4). It shows an upright craft with four flapping wings, placed inside a vessel that is shaped like a bowl and is accessible via a ladder and a hatch. The pilot, crouched down in the center of the craft, generates the necessary force by pushing two pedals with his feet while simultaneously turning two handles with his hands. As Domenico Laurenza points out, “There is no note, no mention to be found … of how the pilot will steer the machine in flight. He becomes almost an automatic pilot: he simply has to generate the force to lift off the ground.”²²

FIG. 7-4. Leonardo’s “flying ship,” c. 1487–90. Ms. B, folio 80r.

FIG. 7-5. The “aerial screw,” c. 1487–90. Ms. B, folio 83v (detail).

Another flying machine designed by Leonardo around the same time is his famous “helicopter” or “aerial screw.” It is based on the same idea of lift being achieved by means of proper compression of air, this time by a helical surface rotating rapidly through it. On a folio in Manuscript B, Leonardo drew a small sketch of such a device (fig. 7-5), and next to it he provided a succinct description of how it would work:

Let the outer extremity of the screw be of steel wire as thick as a cord, and from the circumference to the center let it be 8 braccia [about 16 feet]. I find that, if this instrument, in the shape of a screw, is well made—that is, made of linen cloth with its pores stopped up with starch—and is turned swiftly, it will make a female screw in the air and will rise up high.²³

In the same passage, Leonardo suggested trying out his aerial screw with “a small model made of paper, whose axis will be made of a fine steel blade, bent by force, and when released it will turn the screw.” It is quite likely that Leonardo actually built a working model along those lines, which would have been similar to children’s toys known in his time and still used today.²⁴ However, it is doubtful that the full-sized aerial screw could have been turned fast enough by human muscle power to provide sufficient lift. Be that as it may, we can easily recognize the aerodynamic principle by which Leonardo’s craft was meant to rise into the air as the same principle underlying the functioning of the modern helicopter.

During the those years, Leonardo also designed a series of quite realistic machines for flapping flight in which the pilot is placed horizontally and controls a variety of subtle movements with his hands and feet. In addition, certain movements are achieved automatically by means of springs. Figure 7-6 shows an example from this series of designs. It is a highly finished technical drawing, accompanied by three sets of explanatory notes laid out neatly on the page. The plank on which the pilot is supposed to lie and the two foot pedals to operate the flapping of the wings are clearly visible. The pilot’s hands and arms are used for maintaining balance and changing direction, not unlike in a modern hang glider.

Closer examination of Leonardo’s drawing and text shows that during the downward stroke, the wings not only flap but also fold backward and inward, their tips moving toward the pilot’s feet.²⁵ This elegant movement, imitating the actual wing motion of birds, is achieved by means of a complex system of joints, pulleys, and springs—a masterpiece of delicate mechanical engineering. The accompanying text includes the prudent reminder: “You will experiment with this machine over a lake and you will wear as a belt a long wineskin, so that if you fall in, you will not drown.”²⁶

FIG. 7-6. Design for a flying machine, c. 1487–90. Ms. B, folio 74v (detail).

FIG. 7-7. Working model of the flying machine, wood, 1988. Museum of the History of Science, Florence.

The drawing shown in figure 7-6, together with similar drawings from the same period in Manuscript B and the Codex Atlanticus, represents Leonardo’s most sophisticated design of flying machines. These drawings became the basis of several models built by modern engineers.²⁷ Figure 7-7 shows one of these models, built from materials that were available in the Renaissance. The limitations of these materials—wooden struts, leather joints and thongs, and skin of strong cloth—make it evident why Leonardo could not create a viable model of his flying machines, even though they were based on sound aerodynamic principles. The combined weight of the machine and its pilot was simply far too heavy to be lifted by human muscle power.

Biomimicry

Eventually, Leonardo became aware that he could not achieve the required power-to-weight ratio for successful flapping flight. It would take him ten years to reach this conclusion, perhaps because those were years of frequent travels in central Italy with neither the sufficient time, nor the required ample workshop space, to test new designs of flying machines.²⁸ However, during his travels Leonardo continued to record his observations of birds in flight in two pocket-sized notebooks, now known as Manuscripts K and L.*

When he finally settled down in Florence, Leonardo intensified his ornithological studies, engaging in careful and methodical observations of birds in flight, down to the finest anatomical and aerodynamic details. He spent hours in the hills surrounding Florence, near Fiesole, where he could see eagles, swans, and other large birds in gliding and soaring flight. Leonardo intended these large birds to be his models for new designs of a flying machine that would imitate nature ever more closely, maneuvering with agility, keeping its balance in the wind, and moving its wings like a real bird.

He summarized his observations and analyses in a small Notebook called Codex on the Flight of Birds (Codex Sul Volo). Looking through the pages of this elegant manuscript, one almost has the impression that Leonardo wanted to become a bird himself. Not only does he call his flying machine uccello (bird) but he also uses anatomical terms for its parts, speaking, for example, of its “fingers” (wing tips) and the “tendons” (tierods) to move them. In some passages he shifts effortlessly back and forth between the third person (describing a bird in flight) and the second person (addressing himself or the pilot of his flying machine), for example, in the following series of instructions about how to keep one’s equilibrium in the wind:

If the bird should wish to turn quickly on one of its sides.… And if you wish to go west without flapping the wings. … That bird will rise up high … always turning on its right side or on its left side…. If in your straight rise the wind should be likely to upset you, then you are at liberty to bend by means of the right or left wing …²⁹

FIG. 7-8. Study for a mechanical wing imitating the wing of a bird, 1505. Codex Sul Volo, folio 7r.

This fusion of identities of the real and the mechanical bird in the text is matched, if not surpassed, by a similar fusion in Leonardo’s drawings. His designs of mechanical wings sometimes mimic the anatomical structure of a bird’s wing so accurately (for example, in fig. 7-8) that even experts find it hard to tell the difference. As I shall discuss in more detail, Leonardo’s attitude of seeing nature as a model and mentor is now being rediscovered in the discipline of ecological design, and especially in the practice of biomimicry (see p. 324).

The Flight of Birds

To appreciate the significance of Leonardo’s insights into the physics of flight, it is useful to review how modern scientists explain the flight of birds. Even a cursory glance at the relevant literature shows that this is an exceedingly complex subject that is still not fully understood. The science of flight is based on aerodynamics, which itself is part of the more general, and notoriously difficult, discipline of fluid dynamics (see pp. 33ff.). Even with powerful supercomputers at their disposal, scientists today are still not able to accurately model the turbulent flows of air around the surfaces of an airplane (or a bird).³⁰ In spite of these mathematical difficulties, however, the basic features of “animal aerodynamics” are now well known.³¹

FIG. 7-9. Forces on a gliding bird.

To understand how birds fly, it helps to start with gliding flight, because a gliding bird is subject to the same forces as a gliding airplane with fixed wings. Two pairs of forces need to be balanced in steady flight: weight and lift, as well as thrust and drag (fig. 7-9). The direction of the lift is always perpendicular to the motion of the wing; it will be vertical only if the wing moves exactly horizontally (which flapping wings hardly ever do).

Both lift and drag are generated by the flow of air across the bird’s wings. The principles are the same as for the flow around the wings of an airplane. Figure 7-10a shows a cross section of an airplane wing, also known as an “airfoil,” with four streamlines indicating the flow of air across it. This could be a picture either of a wing moving through stationary air, or of air flowing over a stationary wing in a wind tunnel. The two points of view are entirely equivalent, as Leonardo da Vinci was the first to recognize (see pp. 265–66).

When the oncoming air hits the leading edge of the wing, it separates into two parts, one streaming above the wing and the other below. Because of the particular shape of the airfoil, this separation is not symmetrical. Above the wing, the streamlines are compressed; below the wing, they expand, as can be seen in figure 7-10a. In terms of flow velocities, this means that the velocity of the air flow increases above the wing while it decreases below the wing.

FIG. 7-10. Airflow over the cross section of a wing, showing (a) streamlines and (b) distribution of pressures on the wing. From Anderson, A History of Aerodynamics.

Now there is a general theorem in fluid dynamics, known as Bernoulli’s theorem (after its discoverer, the eighteenth-century mathematician Daniel Bernoulli), according to which any increase in the velocity of a flowing fluid is accompanied by a corresponding decrease in its pressure, and vice versa.* Consequently, the distribution of pressures exerted on the surface of the airfoil will be such that the pressures on the bottom surface (where the air flow is slower) will be higher than the pressures on the top surface (where the air flow is faster), as shown in figure 7-10b, in which the longer arrows indicate areas of higher pressure. The net effect is an upward lifting force on the wing.

This lifting force is always accompanied by a drag on the airplane (or bird) due to air resistance. More precisely, the drag is caused by the friction between the body of the flying object and the streaming air, resulting in shear stresses on the body surface. This “viscous drag” is not the only type of drag on a flying object. There is also a “pressure drag” produced by a turbulent wake behind the object, and the “induced drag” generated by trailing vortices behind the tips of the wings. In practice, engineers usually measure only the total drag.

The crucial quantity in wing design is the ratio of lift to drag. Since the same thrust produces both lift and drag, the drag on the wings determines an airplane’s efficiency. A higher lift-to-drag ratio means that less thrust (and hence less fuel) is needed to produce the necessary lift. In modern aerodynamics, it is relatively easy for scientists to calculate lift but very difficult to calculate drag, because of the critical role of air turbulence. In addition to the turbulent wake behind the flying object and the vortices behind the wing tips, the flow of air close to the body surface, in a region known as the boundary layer, is also highly turbulent. All these turbulences make calculations and mathematical modeling of the drag on an airplane extremely difficult.

FIG. 7-11. Angle of attack, a, of a cambered airfoil.

The lift that can be achieved with a given amount of thrust depends not only on the shape of the airfoil but also on its angle relative to the oncoming air (see fig. 7-11). As the airfoil is tilted upward, the lift will increase with an increasing “angle of attack.” Even a flat wing generates some lift when tilted upward. However, a convex, or “cambered,” airfoil will produce more lift and less drag. We shall see that birds can adjust both the angle of attack and the camber of their wings to move effectively in various situations.

Naturally, a wing cannot be tilted upward indefinitely to obtain more and more lift. When the angle of attack reaches a certain critical value, a stall will occur: the airflow over the top surface of the wing will generate a strong turbulent wake, which causes a sudden loss of lift and increase of drag. Most airplanes will stop flying and will start to fall when their wings stall. Birds, amazingly, use the process of stalling very effectively in their landing maneuvers. Just before touching down on a perch, a bird will often sharply increase its angle of attack and, in a well-timed stall, will lose its speed and drop down safely on the perch.

Gliding is the simplest form of flight, and some birds glide most of the time they are airborne. In addition, eagles, hawks, and many other birds use a special form of gliding called soaring, which allows them to stay aloft for long periods of time without flapping their wings, by using rising air currents to increase their gliding time. Storks and albatrosses are masters of soaring. They may soar over the sea for days at a time, sometimes even crossing an entire ocean without flapping their wings.³²

High lift-to-drag ratios are essential for soaring. Albatrosses and other large birds achieve these high ratios with their long and narrow wings. Smaller terrestrial gliders, like hawks and falcons, are able to adjust their wing area by bending the joints in their wings, reducing the area for fast gliding and expanding it for soaring. In addition, many soaring birds conspicuously spread the large “primary” feathers at the tips of their wings. This reduces the trailing vortices at the wing tips and thus increases the lift-to-drag ratio.

With this understanding of the basic aerodynamics of gliding and soaring, we can now examine the much more complex actions of flapping flight. The most important, and somewhat counterintuitive, insight of modern animal aerodynamics has been the recognition that birds and other flying animals do not flap their wings to maintain themselves in the air, but do so only to produce thrust. Once they move forward against the air, their wings also generate lift owing to their aerodynamic shape, just as in gliding flight. This means that, in order to produce the required forward thrust, the flapping of a bird’s wings must be much more complex than a simple up-and-down motion.

Indeed, slow-motion films of birds in flight have revealed that they do not flap their wings vertically but follow complex patterns that combine several types of motion.³³ The path of the wing tip is a curve, somewhat resembling the butterfly stroke of a swimmer, but the curve is tilted from the vertical by about 30 degrees—down and forward on the downstroke, up and backward on the upstroke. This has the effect that the lift produced by the wings is tilted forward, so that the flapping generates both lift and thrust. For large birds like the albatross, the path of the wing tips is an oval; for some smaller birds it follows a figure eight, and flying insects trace out all kinds of complex loops.

In addition to moving their wings along a tilted curve, birds increase the angle of attack on the downstroke and decrease it on the upstroke; and many birds also change the wing area by folding the primary flight feathers on the outer part of the wing like a fan and pulling the wings closer to the body during the upstroke to reduce their effective surface area. All these movements combine to produce as much thrust and lift as possible on the downstroke and as little drag as possible on the upstroke.

As the bird’s wing sweeps down and forward, the part near the wing’s base (close to the bird’s body) will experience a relative wind mostly from the forward motion, and hence the lift there will be more or less vertical. The wing tip, by contrast, will experience a flow of air that is caused by both the forward motion and the wing’s flapping movement. It will be faster than at the base and will approach the wing from below. Consequently, the lift near the tip will be stronger and tilted forward. In flapping flight, the strength and direction of the air flow change gradually along the wingspan, producing most of the lift along the inner part of the wing and most of the thrust near its tip.

Understanding how birds fly involves many additional aspects. Just as terrestrial animals use different gaits for different speeds, so birds use different flapping patterns for slow and fast flights. The motions of their wings differ not only for different speeds but also from species to species. Then there are the numerous subtle movements birds use for takeoff and landing; for turning, climbing, and descending; for staying on course in the wind; and, in the case of small birds (as well as insects), for hovering in still air. With a keen eye, Leonardo observed the fine details of many of these movements, described them with great accuracy, and sketched them in lively and charming drawings. The critical question is: how much did he understand about the flight of birds?

Leonardo’s Aerodynamics

Leonardo’s first great achievement in formulating a proper science of flight was the recognition that such a science must be grounded in aerodynamics, to use our modern term. “To give the true science of the movement of birds in the air,” he wrote late in his life, summarizing more than thirty years of research, “it is necessary first to give the science of the winds, and this we shall prove by means of the movements of water within itself.”³⁴ In this passage, Leonardo asserts not only that the science of flight must be based on sound aerodynamics (“the science of the winds”), but also that the flows of air can be compared to flowing water, both being described by the same discipline of fluid dynamics, as we would say today.

Since the discovery of the basic principles of fluid dynamics was one of Leonardo’s greatest scientific achievements (see p. 39), it is not surprising that his studies of the aerodynamics of flight led him to many pioneering insights. As mentioned above, he was the first to recognize and clearly formulate the principle of the relativity of motion, according to which a body moving through stationary air is equivalent to air flowing over a stationary body. “As it is to move the object against the motionless air,” he wrote around 1505, “so it is to move the air against the motionless object.”³⁵ Today, this is known as the principle of the wind tunnel, the most important experimental tool of modern aerodynamics.

Leonardo also realized that the relativity of motion implies that aerodynamic lift is generated by the same forces in both flapping and soaring flight. The passage in the Codex Atlanticus continues: “Therefore, the bird beating its heavy wings on the thin air causes it to compress and resist the bird’s descent. And if air moves against the motionless wings, that air sustains the heaviness of the bird in the air.” In the last part of this passage, Leonardo examines the three types of soaring flight—hovering in the wind, ascending, and descending:

When the power of the motion of the air is equal to the power of the descent of the bird, that bird will stay in the air motionless. And if the motion of the air is more powerful, it will win and will raise the bird up. And if the power of the motion of the air is less than the weight of the bird, that bird will come down.

On another folio in the Codex Atlanticus, Leonardo reiterates the identity of the aerodynamic principles underlying soaring and flapping flight:

When the bird finds itself within the wind, it can sustain itself on it without flapping its wings, because the function the wing performs against the air, when the air is motionless, is the same as that of the moving air against the wings when they are motionless.³⁶

From that time on, Leonardo’s notes on flight always treated flapping and soaring flight as equivalent. Both in his first set of notes in Manuscript K and in his more extensive records in the Codex on the Flight of Birds, the motion of birds is analyzed from these two perspectives.

In most of his analyses of flight, Leonardo reiterated his idea that birds are sustained in the air by the compression of air under their wings and the resulting upward rebound. This explanation is partially correct, but it is not the whole story. Today we know that aerodynamic lift is a consequence not just of the air pressure under the wing but also of the pressure difference between the air above and below the wing, and that the low pressure area above the wing actually generates most of the lift.

Late in his life, however, Leonardo realized the importance of the thin air above the wing. In fact, Manuscript E, composed around 1513–15 when Leonardo was over sixty, contains an exact description of the air densities around the body of a flying bird:

The air surrounding birds is above thinner than the usual thinness of the other air, as below it is thicker than the same; and it is thinner behind … and thicker in front of the bird.³⁷

If we replace the terms “thin air” and “thick air” in this passage by “low pressure” and “high pressure,” remembering that the concept of pressure was clearly defined only in the seventeenth century (see p. 169), the resulting description is very similar to how the pressure distribution around an airfoil is pictured in modern aerodynamics (see fig. 7-10b).

The flow of air around a bird’s wings produces not only the upward lift but also a drag on the bird’s forward motion due to air resistance in front of and turbulence behind the wings (see p. 261). When Leonardo examined these forces, he had already struggled for many years to understand the inertia and the dissipation of energy of bodies in motion, which he analyzed in terms of the medieval theory of impetus. By the time he composed the Codex on the Flight of Birds, he had convinced himself that drag was caused by the resistance of compressed air in front of the moving object, as well as by turbulence behind it (see p. 181).

Leonardo investigated the effects of drag for motion in both in the air and water. A sketch in Manuscript G shows three ships of different shapes, as well as two kinds of fish (fig. 7-12). Both causes of drag—the resistance of the water in front of the ship and the turbulence on the side and in the back—are clearly visible in the sketch. Leonardo concludes that the drag on the ship shown on top will be the smallest because of its streamlined shape (as we would say today), and notes that “it resembles the shape of birds and fishes such as the mullet.”³⁸

FIG. 7-12. Streamlined shapes of ships and fish. Ms. G, folio 50v (detail).

Leonardo also tried to quantify the resistance encountered by a body moving through air.³⁹ He postulated correctly that it is proportional to the surface area of the body, and also to the body’s velocity, which is incorrect (the air resistance is proportional to the square of the velocity). It is likely that the latter postulate was simply based on Leonardo’s belief in the privileged role of linear relationships in nature, which was also held by Galileo in his early work (see pp. 175–76).

The Codex on the Flight of Birds

In the Codex on the Flight of Birds (Codice sul volo degli uccelli, or Codex Sul Volo for short), Leonardo summarized the observations and analyses of bird flight he made in Florence during a period of two years between 1503 and 1505. The elegant small Notebook is full of charming drawings of birds in flight (for example, plate 1), detailed descriptions of their turning maneuvers, their ability to maintain equilibrium in the wind, and various subtle features of active flight, as well as sketches of complex mechanisms he designed to mimic the birds’ precise movements.

When Leonardo recorded his occasional observations of birds in flight during his travels in central Italy, he already had a treatise on this subject in mind. In Manuscript K, one of the Notebooks he carried with him during that time, he outlined a clear plan for such a work:

Divide the treatise on birds into four books: the first on the flight maintained by beating the wings; the second on flight without beating wings, maintained by the wind; the third about flight common to birds, bats, fish, animals, insects; the last about instrumental motion.⁴⁰

In the Codex Sul Volo, Leonardo more or less followed this plan. For some reason, he composed the eighteen folios of this Notebook in reverse order, so that the conceptual sequence runs from back to front.⁴¹ The first part (folios with high numbers) deals mainly with flapping flight, while the second part (folios with low numbers) contains notes on how birds glide, soar, and maneuver in the wind. In both parts, the notes on bird flight are followed by sketches of mechanisms designed to imitate these natural movements with a mechanical “bird.”

On the opening folio of the Codex Sul Volo (folio 18r in the reverse order), Leonardo summarized the main points of his analysis of flapping flight in six short paragraphs. A striking feature of these notes is that he quite naturally identifies the bones and joints of the bird’s wing as the “elbow,” “hand,” “fingers,” and so on. It is an eloquent testimony to the maturity of his studies in comparative anatomy. Even today, avian anatomists speak of the wing’s elbow and wrist joints, of the hand (technically known as the carpometacarpus), and of its thumb (alula) and two fingers.⁴² In an anatomical study of a bird’s wing in the Windsor Collection (fig. 7-13), produced a few years later, these bones and joints, together with their tendons and ligaments, are pictured with great accuracy. The correspondences with Leonardo’s famous study of the bones of the human arm and hand (plate 3) are quite evident.

FIG. 7-13. Anatomy of a bird’s wing, c. 1513. Windsor Collection, Anatomical Studies, folio 187v.

Before evaluating Leonardo’s description of flapping flight, I shall summarize once more the main characteristics of the bird’s wing motion that have been identified by modern ornithologists (see p. 264). The wingbeat is tilted from the vertical, moving forward on the downstroke and backward on the upstroke. In addition, it is curved so that the wing tip describes an oval or more complex curve. The angle of attack is increased on the downstroke and decreased on the upstroke. The primary feathers are fully spread on the downstroke and are closed like a fan on the upstroke when the bird pulls the wings closer to the body. As in gliding flight, the pressure difference between the top and the bottom of the airfoil results in a net force on the wing, and because of the wingbeat’s curved path this force is tilted forward, generating both lift and thrust. In this complex motion, most of the lift is produced along the inner part of the wing and most of the thrust near its tip.

In view of the fact that the subtle details of the dynamics of flapping flight have been revealed only recently with the help of high-speed photography and slow-motion filming, it is truly astonishing how many of its basic characteristics Leonardo identified correctly with his sharp eye and his great capacity of visualization. In his description of the wingbeat, he notes the wing’s down-and-forward motion and accurately describes how its angle of attack is raised on the downstroke (by lowering the elbows):

The lowering of the elbows at the time the bird is putting its wings forward, somewhat edgewise on the wind, guided by the impetus already acquired, is the reason why the wind strikes under that elbow and becomes a wedge on which the bird, with the aforesaid impetus and without beating its wings, comes to rise.⁴³

The lift is attributed correctly to the relative airflow created by the thrust (“impetus”) when the angle of attack is raised (“somewhat edgewise on the wind”) but as in other passages, Leonardo does not recognize the important role of the low pressure above the wing, attributing the entire lift to the high pressure (“wedge”) below it.

On the same folio, Leonardo also notes that during the wingbeat the path of the wing tips is not straight but describes an oval. “The course of the finger tips is not the same in going as in returning,” he observed, “but is by a higher line, and the return is below it; and the figure made by the upper and lower lines is a long and narrow oval.”

Even more remarkably, still on folio 18r, he sketches the changes in the angle of attack along this oval in a small drawing (fig. 7-14) together with the note:

The palm of the hand goes from a to b [downstroke] always at about the same angle, pressing down the air, and at b it immediately turns edgewise and turns back, rising along the line cd [upstroke], and arriving at d it immediately turns full face and sinks along the line ab, and in turning it always turns around the center of its breadth.

As ornithologists do today, Leonardo compared this circular motion to that made by a swimmer’s hand. In Manuscript F, composed a few years after the Codex Sul Volo, he noted:

As the hand of the swimmer acts when it strikes and presses against the water and makes his body glide forward in a contrary motion, so acts the wing of the bird in the air.⁴⁴

And finally, Leonardo did not fail to notice that the bird produces thrust and lift with two different parts of its wing. The thrust (“impetus”) is produced by the outer wing (the “hand”), he explains, and the lift by the raised angle of attack of the inner wing (the “elbow”):

The hand of the bird is what causes the impetus, and then its elbow … assumes a slanting position, and the air on which it rests becomes slanting, as if in the form of a wedge on which the wing raises itself up.⁴⁵

Apparently, Leonardo realized the different functions of the inner and outer wing even before his methodical observations of bird flight in Florence. During his travels a couple of years earlier, he jotted down a quick note in his pocket book: “In beating the wings to remain up high and to go forward, from the hand back causes to stay up, and the hand causes it go forward.”⁴⁶

FIG. 7-14. Changes of angle of attack along oval wingbeat. Codex Sul Volo, folio 18r (detail), and reconstruction.

In Leonardo’s notes on birds in flight, there are also some contradictions and erroneous statements, which is not surprising given the great complexity of the subject. However, the Codex on the Flight of Birds leaves no doubt that he fully understood the essential features of both soaring and flapping flight. Thus aeronautical engineer John Anderson concludes in his History of Aerodynamics: “It is clear that Leonardo was the first person to understand the mechanics of bird flight.”⁴⁷

In the Codex Sul Volo, Leonardo’s summary of his understanding of flapping flight is followed immediately by two folios with designs of mechanical wings. The first drawing (fig. 7-15) shows the left wing of the machine from the front.

The wing is connected to two foot pedals via a system of cords and pulleys, which allows the pilot to raise and lower it with his feet. In addition, the wing can be rotated to change its angle of attack by means of a handlebar, to be operated by the pilot’s hands. It is evident that Leonardo attempted here to imitate, as closely as possible, the complex motions of flapping flight he described on the preceding folio.

While he studied the motion of birds in flight and designed mechanical wings to imitate it, Leonardo turned once more to the problem of measuring human muscle power, which had occupied him so intensely ten years earlier (see p. 254). But this time he approached the problem from the perspective of comparative anatomy. “You will make the anatomy of the wings of a bird together with the muscles of the breast, the movers of those wings,” he wrote in a note to himself, “and you will do the same for man in order to show that there is the possibility in man to sustain himself in the air by the flapping of wings.”⁴⁸

FIG. 7-15. Design for mechanical wing imitating the flapping flight of birds. Codex Sul Volo, folio 17r.

Leonardo carried out these comparative anatomical studies with various birds and recorded them in the Codex Atlanticus on several folios.⁴⁹ The results convinced him that flying by beating mechanical wings might not be possible because of the limitations of our anatomy. “The sinews and muscles of a bird [are] incomparably more powerful than those of man,” he explained in the Codex Sul Volo, “because all the fleshiness of the big muscles and fleshy parts of the breast goes to facilitate and increase the power of the wings’ motion, and with that bone of the breast of one piece, which provides the bird’s very great power.”⁵⁰ This power is so strong, he observed, that it enables big raptors to sustain themselves in the air even while carrying heavy loads:

All this strength is provided to enable [the bird], over and above the ordinary sustaining action of the wings, to double and triple its motion in order to escape from its predator or to pursue its prey. For this purpose, it has to double or triple its strength and, moreover, carry in its claws as much weight through the air as it weighs itself. Thus we see a falcon carry off a duck and an eagle a hare.⁵¹

However, Leonardo noted in the same passage that birds “need little power to keep themselves up in the air and balance on their wings and flap them on the currents of air and steer along their paths, a little movement of the wings sufficing; and the larger the bird, the slower [the wing motion].” He concluded from this observation that, even though human muscle power was too weak to lift a flying machine off the ground by flapping mechanical wings, a machine for soaring and gliding flight might be feasible, since this would require much less force.

For Leonardo, this was a momentous insight. After ten years of observations, studies, and design projects, he now saw a concrete way in which his unwavering belief in the possibility of human flight could be turned into reality. During the same year in which he compiled the Codex on the Flight of Birds, he summed up his new insight in a long and carefully worded passage in the Codex Atlanticus:

The [mechanical] bird is an instrument working according to mathematical law, an instrument which it is within the power of man to make with all its motions, but not with such power [as the natural bird], its power extending only to the balancing movements. We may say therefore that such an instrument designed by man lacks only the soul of the bird, which must be counterfeited with the soul of the man.⁵²

Leonardo was well aware that, even if he succeeded in designing a machine that could fly like a bird, he would never be able to completely match the bird’s instinctive capacity to maneuver in the wind, keeping its equilibrium by responding to changes in air currents with subtle movements of its wings and tail. “The soul of the bird,” he continued his summary, “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.”* However, he concluded that imitating the clearly perceptible balancing movements of birds should be sufficient to prevent the flying machine from crashing:

The varieties of perceptible motions that we observe in birds … can be learned by man, and … he will to a great extent be able to prevent the destruction of the instrument for which he is the soul and driver.

The enthusiasm Leonardo must have felt when he reached these conclusions is reflected in his famous prophecy:

The large bird will take its first flight from the back of the great Swan [Monte Ceceri, near Florence], filling the universe with amazement, filling all writings with its fame and [bringing] eternal glory to the nest where it was born.⁵³

Leonardo placed this exuberant declaration on the inside back cover (that is, at the beginning) of the Codex on the Flight of Birds, and it seems that he intended it as an epigraph for his treatise. Apparently not quite satisfied with the wording, he composed a shorter version on the subsequent page:

From the mountain that takes its name from the great bird, the famous bird will take its flight, which will fill the world with its great fame.⁵⁴

Having convinced himself that a flying machine could be designed to imitate the soaring and gliding flights of birds, Leonardo used the last part of the Codex Sul Volo (pages with low numbers) to study how a gliding bird keeps its balance in the wind. For example, he devotes an entire folio to the analysis of how the bird, when pushed by a lateral gust of wind, balances itself by spreading or folding one or the other wing to various degrees, because “the forces of the wind striking the two wings will be of the same proportion as their extensions.”⁵⁵ On the subsequent folios, Leonardo describes in detail how the bird’s tail supports the balancing actions of the wings in steering and controlling the flight.⁵⁶

These copious notes on the balancing actions of birds are followed, once again, by design sketches that attempt to imitate the birds’ natural movements. This time, however, Leonardo’s focus is no longer on the wingbeat but rather on the flexions and extensions of the wings that are critical to the balancing maneuvers in gliding flight (see fig. 7-8).

In Manuscript G, composed a few years later, Leonardo discusses a variety of techniques used by birds for takeoff, among them one that is especially relevant to his new design ideas for flying machines:

The second method employed by birds at the beginning of their flight is when they descend from a height. They merely throw themselves forward and at the same time spread out their wings upward and forward, and in the course of their leap they lower their wings downward and backward, and thus, rowing, they proceed on their slanting descent.⁵⁷

It is evident that this comes very close to describing the takeoff maneuvers of a modern hang glider.

Throughout the entire Codex Sul Volo, Leonardo shows supreme confidence in the feasibility of human flight. At times, he sounds as if mechanical flight had already become so routine for him that he could sprinkle his treatise with pieces of practical advice for would-be pilots. Thus he recommends, without any apparent sense of irony:

The movement of the [mechanical] bird should always be above the clouds so that the wing does not get wet, and to survey more country, and to avoid the dangers of swirls of winds within the mountain passes, which are always full of gusts and eddies of wind.⁵⁸

The Science of the Winds

In the years after the completion of the Codex Sul Volo, during his second period in Milan, Leonardo only rarely recorded observations on the flight of birds, and no further designs of flying machines have come down to us. It was not until eight years later that he returned to the subject of flight. He was then living in Rome, over sixty and rather lonely and depressed, his reputation as a painter having been eclipsed by younger rivals like Michelangelo and Raphael.⁵⁹ In spite of his somber state of mind, however, he continued his studies with great diligence.

During those years, 1513–15, Leonardo collected his scientific thoughts—many of a general, reflective nature—in a small Notebook now known as Manuscript E. This is the Notebook that contains the famous passage on his empirical method,⁶⁰ in addition to various notes on many of the grand themes he had pursued during his life: the “science of weights,” geometry, motion, the flows of water, and especially the science of flight.

In the opening passage of the section on flight in this Notebook, which I have already quoted in part, Leonardo defines the proper theoretical framework for such a science:

To give the true science of the movement of birds in the air, it is necessary first to give the science of the winds, and this we shall prove by means of the movements of water within itself. And this science, accessible to the senses, will serve as a ladder to arrive at the knowledge of things flying in the air and the wind.⁶¹

This passage is remarkable for several reasons. As I have discussed, Leonardo declares here that his science of flight is grounded in aerodynamics and, more generally, in fluid dynamics (to use modern scientific terms). To gain knowledge about the “science of the winds,” he explains, he will study turbulent flows of water (“the movements of water within itself”), knowing from his lifelong observations that the principles of flow are the same for water and air (see p. 33). But unlike the movements of air, those of water are visible (“accessible to the senses”) and hence can serve as a model (“a ladder”) to gain knowledge about aerodynamics and about flight. Once again we encounter here an aspect of Leonardo’s scientific thought that puts him centuries ahead of his time—the recognition of flow as a universal phenomenon of liquids and gases, and his use of the former as models of the latter.

Most of Leonardo’s notes on flight in Manuscript E are concerned with his theoretical studies of the “science of the winds.” Indeed, fluid dynamics was very much on his mind at that time, both in his science and in his art. Those were the years when he created his celebrated “deluge drawings”—violent and disturbing images that represent a visual catalogue of different types of turbulences in water and air (see p. 61).

Having defined the “science of the winds” as the proper framework for the study of the flight of birds, Leonardo then restates three of his most important discoveries in aerodynamics. The first is the fact that air, unlike water, is compressible. “Air can be compressed and rarefied almost infinitely,” he notes, and he adds that, because of the thin air at high altitudes,* only large birds with great wingspans are able to fly there.⁶²

Leonardo’s second important discovery in aerodynamics is the principle of the wind tunnel, that is, the relativity of motion between a solid object and the surrounding air. Its formulation in Manuscript E is virtually identical to the one given ten years earlier in the Codex Atlanticus: “As it is to move the air against the motionless thing, so it is to move the thing against the motionless air.”⁶³ His third discovery, finally, is that of the pressure distribution in the flow of air around a bird’s wing—higher pressure on the bottom surface and lower pressure on the top surface—which is described correctly for the first time in Manuscript E, as I have discussed (see p. 267).

When Leonardo recorded these notes, he apparently no longer had the energy to review his previous observations on the flight of birds in the light of his late insight into the density distribution around the wings. Still, his full understanding of the origin of aerodynamic lift, together with his concise formulation of the principle of the wind tunnel, establishes Leonardo da Vinci as one of the great pioneers of aerodynamics. Indeed, in the opinion of aeronautical engineer John Anderson, “[Leonardo’s] aerodynamic concepts were amazingly advanced and would have constituted a quantum jump in the state of the art of aerodynamics if they had been widely disseminated.”⁶⁴

Leonardo’s mechanical “birds” with flapping wings were not destined to fly, even though their designs were based on sound aerodynamic principles (see p. 258). Nevertheless, the models built from those designs in recent years are extraordinary testimonies to his genius as a scientist and engineer. In the words of art historian Martin Kemp:

Using mechanical systems, the wings flap with much of the sinuous and menacing grace of a gigantic bird of prey … [Leonardo’s] designs retain their conceptual power as archetypal expressions of man’s desire to emulate the birds, and remain capable of inspiring a sense of wonder even in a modern audience, for whom the sight of tons of metal flying through the air has become a matter of routine.⁶⁵