Revisiting Leonardo on Muscles: Intimations of Mathematical Biology and Biomechanics

Leonardo da Vinci’s extensive drawings and notes devoted to anatomy do not arise in a medical context. He does not engage with surgery or “physic.” Rather, his aim is to reveal what he understood to be the divine engineering of God’s greatest creation. His earliest anatomical drawings map the conduits for the “spirits” at a deep level not practiced by other artists interested in the human body. The first set of drawings he produced in 1489 describes skulls with brilliant draftsmanship. His notes for these drawings are predominantly concerned with the location of the “common sense” (sensus communis) at the geometrical center of the cranium. The next great set of drawings completed in ca. 1510 expounds the form and functions of the bones and muscles in meticulous mechanical detail, probably in collaboration with the young Paduan doctor, Marcantonio della Torre. Leonardo’s greatest realization of mathematics in anatomy comes when he looks at the valves of the heart; in studying the heart he also conceived a casting technique that he adapted to determine the form of the ventricles in the brain, radically revising the traditional arrangement of these structures. In a range of portrayals from diagrammatic to pictorial and from static to dynamic, Leonardo’s anatomical research is unrivalled in revealing the mechanics of the human body. His work in this context is an underappreciated precursor of the modern science of biomechanics. (Larger-sized versions of the Leonardo drawings presented here are available as supplementary material in the online version of this article.)

The book of the elements of machines with all its practice should precede the demonstration of movement and force in man and animals; and by means of these you will be able to prove all your propositions.

Leonardo da Vinci (MS A 10r)
Leonardo da Vinci's ( Fig. 1) anatomical drawings comprise the most remarkable vision of the functioning of the human body from any period. He was working in a context outside of medical practice. There is hardly a mention of the physician's or surgeon's practices. Nor does he concentrate on anatomy for the artist. From his earliest anatomical studies, he was delving into the deepest operations of the soul and body, the mind, the senses, birth and death, rather than limiting himself to the skin, muscles, and bones that Renaissance artists needed to create realistic figures in expressive contexts. His anatomies are best seen as his particular form of natural philosophy, devoted to expounding the wonders of God's greatest creation as a fit habitation for the soul during its temporary residence on earth.
Writing on the remarkable ensemble of muscles in the tongue, he invites us to consider the miracles of natural design: Although human ingenuity by various inventions with different instruments obeys the same end, it will never design an invention more beautiful, simpler or more direct than Nature, because in her inventions nothing is lacking and nothing is superfluous. She does not resort The Leonardo references adopt the standard format used in printed and electronic sources. The inventory numbers of the drawings in the Royal Collection at Windsor Castle are listed as RL 19000, and so on (recto and verso also noted). The latest numbering within the collection is RCIN 91900, and so on. The K&P citation refers to Keele andPedretti (1979-1980). to counterweights when she makes organs suitable for motion in the bodies of animals, but places the soul there. (RL 19115r) The basis of his discussions of the forms and functions of the whole and components of the body was an intense teleology in which all the components (what he called elementi in his machine designs) operated with no excess and no deficiency. He repeatedly prefaces an observation by saying, "nature does this…," or "nature has placed…," or "nature intends…," or "nature gives…," or "nature decrees…." Sometimes he cites "necessity," the principle that ensures automatic obedience to the laws of nature. God is directly adduced as governing the microcosm of the human bodywhat Leonardo calls the "lesser world." His earliest anatomical drawings map the conduits for the "spirits" at a deep level not practiced by other artists interested in the human body. His first set of drawings in 1489 describes skulls with brilliant draftsmanship. His notes are predominantly concerned with the location of the "common sense" (sensus communis) at the geometrical center of the cranium. The next great set of drawings in ca. 1510 expounds the form and functions of the bones and muscles in meticulous mechanical detail, probably in collaboration with the young Paduan doctor, Marcantonio della Torre. He is in effect conducting an early kind of biomechanics. One of his later sets of drawings deals with the pregnant uterus, setting a beautifully characterized fetus in a womb derived from a pregnant cow. Amongst other concerns, Leonardo is asking how the fetus is governed by two souls. His greatest realization of mathematics in anatomy comes when he looks at the valves of the heart, in particular the tricuspid valve. He subjected the heart to mathematical and mechanical analysis, conceiving the idea of a glass model of the neck of the aorta to study the flow of blood within the valve. This model was to be preceded by the casting of a valve in wax to determine its shape. Leonardo used a similar casting technique to determine the form of the ventricles in the brain, radically revising the traditional arrangement. In a range of portrayals from diagrammatic to pictorial and from static to dynamic, his anatomical research is unrivalled in revealing the mechanics of the human body.
A key aspect of Leonardo's science of the human body is that his analyses aspire to high truths of mathematics. In the many studies of Leonardo's more than 200 pages of anatomical drawings and notes in the Royal Collection at Windsor Castle, the more mathematical aspects of his analyses of the workings of the human body have not been pulled together for specific study, in spite of his rather severe injunction at the head of this essay. The main reason for this appears to be that the more geometrical drawings are generally small and diagrammatic, often squeezed in at a late stage in the filling of a folio. They are also comprehensively overwhelmed by the compelling visual qualities of Leonardo's "portraits" of body parts. My strategy here is to select some exemplary episodes of his application of mathematical principles to the physical operations of the body. 1 The main digital resource for the Windsor anatomies is https:// www. rct. uk/ colle ction/ searc h#/ page/1. See also the app: https:// apps. apple. com/ gb/ app/ leona rdo-da-vinci-anato my/ id520 564038.
For online digital images and transcriptions of the manuscripts as a whole, see https:// www. leona rdodi gitale. com. It includes all the manuscripts to which reference has been made in the text: MSS A-M owned by the Institut de France and on deposit in the Bibliothèque Nationale, Paris; the Codex Arundel in the British Library; and the Codex Leicester owned by Bill Gates.

Muscular Mechanics
The peak of Leonardo's studies of the muscles and bones in dynamic action comes in a series of intensely observant and communicative drawings from about 1511, when he declared his intention "in this winter …to finish all this anatomy" (RL 19016r). The series comprises more than twenty folios and was undertaken when he was back in Milan and spending time in the country villa owned by the family of his aristocratic pupil, Francesco Melzi.
His studies of the pronation and supination of the arm convey the quality that runs throughout the series (RL 19000v, Fig. 2; see Online Resource 1 for larger-sized versions of Leonardo's drawings). Not only are the bony structures beautifully characterized, but Leonardo also shows their implications for the rotation of the hand and arm. He explains that "[the radius] makes exactly half a turn" as a Fig. 2 Bones of the arm with schematic indications of muscles. The notes describe how the supination and pronation of the arm are driven by the biceps. Leonardo stresses that drawings should be made from multiple viewpoints to give "true knowledge of the shape of any form." A small and obscure geometrical demonstration in the right margin shows how the reorientation of the bones in the forearm from straight to oblique should make the forearm shorter. ( 1 3 result of the action of the radius crossing the ulna. He also recognizes that pronation and supination are driven somewhat unexpectedly by the biceps with the pronator muscle. The reduction of the mass of each muscle to a "cord" along its line of force allows its mechanical action to be clearly understood. For good measure, he proposes to make "this demonstration" from four viewpoints with and without the clavicle, with the bones abutting normally and then separated by a short distance-showing as well each bone "sawn through." Here, as so often with Leonardo, his prolific desire to leave no visual stone unturned works against the practical realization of his projected book on "the human figure." In the accompanying notes he is much concerned to map how the longitudinal proportions of the arm and hand are affected by this rotation. Leonardo was the first to consider how proportions of the body were affected by the bending of joints. He brings dynamics and aesthetics into the same field of enquiry. It is often in the little diagrams added late to a page that he undertakes his most original analysis of the engineering of bodily components. The analysis results in large part from the act of drawing. In a series of five scrappy diagrams on the reverse of the previous sheet (RL 19000r, Fig. 3 detail), he argues that, Nature has placed the glandular bone [the sesamoid] under the joint of the great toe because if the tendon to which this bone is attached were to be without this glandular bone it would receive great damage from the friction made by such a great weight as that of a man walking, when at each step he raises himself on the ankles of his feet. This is followed by a detailed mechanical analysis of how the components of the big toe operate to best effect. The little diagrams correspond to what he called conclusioni (conclusions) in his written analyses.
Leonardo habitually referred to his drawings of anatomy as dimostrazioni (demonstrations); that is to say, the components assume the graphic form that serves best to inform the viewer about the functioning of the body parts. Not infrequently visual properties of different kinds, ranging from relatively naturalistic to entirely diagrammatic, appear on the same sheet and even in the same illustration. His dense page of eight demonstrations of the complex system of muscles of the back and neck exemplifies his illustrative strategies (RL 19015r, Fig. 4.) Beginning at the top right he progressively explores deeper dissections as part of his program to define the mechanisms of respiration, together with those that act to stabilize the spine and to tilt the head from side to side. He notes that there are ten muscles, "5 on each side" of each vertebra, which act in concert in an opposite (or antagonistic) series. He knows that each muscle can only pull and not push. The portrayals of the muscles themselves transition from semi-naturalistic via "cord diagrams" to small line diagrams. They are "demonstrations" of form and function, not eyewitness portrayals. Indeed, we cannot confidently describe any of the surviving studies as drawings made directly from dissections. The characterization of a muscle as a "cord" that pulls along its line of action is partly graphic convention and partly "anatomical" in that the fibers that comprise each nerve were thought to merge into the fibers of each muscle. He characterizes a demonstration that uses cords as having high status as a "ragione scientificha" ("scientific explanation"). His diagrammatic reasoning is extended on two related folios that consist largely of notes. A schematic diagram at the bottom of the right margin of a page devoted to a lengthy discussion of the role of the dorsal muscles serves to promote the kind of analogy of which he was so fond (RL 19015v, Fig. 5). For Leonardo, demonstration of an analogy between various components in the natural world serves as a form of proof of their shared operation. In a similar way, a parallel between a component in the body and a human invention could be cited as a conclusive demonstration of how the body works. Thus he places his sketch of three vertebrae with their "cords" or ropes next to a diagram of the mast of a boat. This demonstrates that a wider angle of attachment results in a stronger support. (The word for mast is albero; that is to say, "tree.") He realizes that there is trade-off here between the greatest stability of the system and the realities of nature's design of the shoulders and neck.
A comparable "demonstration" of the mechanisms of the upper spine occurs in a characteristically mixed page devoted to the scapula, with a hanging biceps (called the "fish" by Leonardo because of its shape) and schematic blood vessels of the "old man" dissected by Leonardo early Fig. 4 Demonstrations of muscles of the neck, shoulders and upper spine, at progressively deeper levels, with two "cord diagrams." The drawings and added texts were undertaken over a period of time and develop away from naturalism. The chief concern is to demonstrate the antagonistic "cords" operating on the spine: "every tendon [corda] fastened to a vertebra has another tendon acting via the spur that sustains the vertebra. Every muscle of the neck that pulls in one direction has another that pulls in the contrary direction." (RCIN 919015r/RL 19015r/K&P 149r; courtesy Royal Collection Trust/ © His Majesty King Charles III 2023) 1 3 in 1507 (RL 19049r, Fig. 6). In the top left are diagrams of a round head with the lateral flexor muscles. As he explains, If nature had joined up the muscle ac in order to bend the head towards the shoulder it would have been necessary for the cervical spine to be bent in the same way that a bow is bent by its cord. Whence nature in order to escape from such a difficulty, made the muscle ab which draws down the side of the cranium a but with but little bending of the bone of the neck because the muscle ab draws the side of the cranium a towards b, the root of the cervical spine; and because the cranium is on a little pole on top of the surface of the neck, it is bent with great ease to right and left without excessive curvature of the bone of the neck etc.
With its use of analogy and application of engineering to anatomy, this is nicely representative of Leonardo's diagrammatic studies of bones and muscles in action.  5 Mechanical analyses of the muscles attached to the ribs, spine, and neck, with a diagram of a ship's mast. Leonardo added a note at the top that says, "one treats of man following the instrumental method and not vice versa." The two major topics are how the ribs move in respiration and how the heavy head is held upright and can move. The smallest of the diagrams demonstrate by reference to the mast of a boat that "those ropes more readily prevent the collapse of the mast [albero] to which they are joined at their end that converge on their junction with the mast at equal angles." (RCIN 919015v/RL 19015v/K&P 149v; courtesy Royal Collection Trust/ © His Majesty King Charles III 2023)

Levers and Laws
Leonardo was much concerned with the proportional laws that govern mechanical systems, especially balances and levers. Such proportional systems operate as a kind of music of machines. He brings the mathematics of levers to bear upon the action of the jaws and teeth. On one of the skull studies from 1489 Leonardo later made sketches of the four kinds of teeth (RL 19058v). He then explains that their shape corresponds to their position on the levers of the jaws. A little diagram is accompanied by an explanatory text (RL 19041r, Fig. 7): That tooth has less power that is more distant from the center of its movement. Thus, if the center of the movement of the teeth is at a, so much less is their power of biting. Thus, dv is less powerful in its bite than the teeth at bc. From this follows the corollary which states that the tooth is more powerful the nearer it is to the center of its movement… That is, the bite of the teeth bc is more powerful than the teeth de. Nature makes those teeth less capable of penetrating food, and with blunter points, which are of greater power… The teeth bc will be more obtuse than the teeth de in the proportion that they Fig. 6 Demonstrations of the mechanics of the neck with the bending of the upper spine, the scapula, the muscle of the biceps, and pelvic vessels. This page, twice labeled "on the old man," is earlier than the preceding illustrations, and is his earliest mechanical analysis of the spine and head. In this case, Leonardo states that the system is not like the bending of a bow by its bowstring because of the small fulcrum at the base of the cranium and the limited flexion of each of the cervical vertebrae. (RCIN 919049r/ RL 19049r/K&P 58r;courtesy Royal Collection Trust/ © His Majesty King Charles III 2023) are nearer to the fulcrum or axis… Nature has made the chewers [molars] with large prominences for masticating the food and not for penetrating or cutting. And in front she has made the cutting and penetrating teeth which are not adapted to chewing the food…And she has made canine teeth between the molars and the incisors.
The system of dental proportions is perfectly designed to crush, crunch, and cut; as Leonardo writes, "in animals 2 teeth close on the prey, 4 cut it up, and six grind it up." Corresponding laws of leverage are applied to the action of the biceps in a monkey and a man. Quick sketches of two arms show, in an exaggerated manner, that the angle subtended by the biceps and the bones of the forearm accords proportionately less power to the human arm than that of the monkey (RL 19026v, Fig. 8). "The nearer the tendon cd which flexes the bone op is to the hand, so much greater is the weight that the hand can lift. And this makes the monkey more powerful in its arms than man according to this proportion." Leonardo perpetually sought a set of fundamental laws that governed the behavior of every aspect of nature and was delighted that such a manifold variety of effects could result from such a compact set of physical causes. It was on this basis that the skilled engineer could be called "a second nature" in the world (Arundel 151v).

Hearts and Hydraulics
His greatest set of anatomical studies (and maybe his last) are those devoted to the operation of the heart in two contexts: the laws of the motion of fluids; and the Fig. 8 Comparison of the attachment of the biceps in a monkey and a man, plus a mechanical analysis. The drawing on the left is labeled "monkey," on the right "man." (RCIN 919026v/RL 19026r/K&P 112r; courtesy Royal Collection Trust/ © His Majesty King Charles III 2023) implicit geometry of bodily components. His probing researches were undertaken on an ox heart. A large double folio (RL19073v -4v, Fig. 9) majestically portrays the exterior of the heart and its major vessels, including the coronary artery, the silting-up of which he adduced as the cause of the death of the "old man." Amidst the grand organic array, he severs the arteries in such a way that the underlying geometry of their tricuspid valves becomes apparent. Overwhelmed by the eyewitness portrait is a small diagram of two pulleys around which runs a rope from which hang two lateral weights and a central weight (Fig. 10 detail). The structure is in equilibrium.
At first sight, this looks like the familiar case of unrelated things appearing together on the same page of Leonardo's manuscripts. However, this is not the case. He is diagrammatically thinking about the action of the muscular forces on the leaflets and cords that govern the aperture of the atrioventricular valves. The pull of a papillary muscle is represented by the central weight, which on relaxation permits the closure of the valve. Again, a small "demonstration" literally carries of a lot of weight.
In the bottom right of the sheet, he begins to examine the geometry of a three-cusp valve, which features on no less than ten of the pages, traversing a graphic range from Fig. 9 A large double sheet, folded once. Studies of the exterior of the heart of an ox, showing the major vessels truncated, the tricuspid valve, and the blood supply to the heart muscles. Having earlier neglected the auricles, Leonardo now crucially states that "the blood is made more subtle where it is more beaten, and this percussion is made by the flux and reflux of the blood generated from the integral ventricles of the heart into the two external ventricles, called auricles… and then they contract returning the blood to the integral ventricle." The auricles serve to cushion the "percussion" in a way comparable to bales of wool that are used to reduce the impact of bombardments on naval boats. The drawing of the exterior of the heart at the top left of the right-hand page illustrates the vessel M, that "arises from the atrial vessel, and it nourishes the substance of the heart." (RCIN 919073v/RL 19074v-19073v/K&P 166vA;courtesy Royal Collection Trust/ © His Majesty King Charles III 2023) geometrical diagram to fleshy portrait. Most remarkable is a higgledy-piggledy folio that covers a range of topics: the potential collapse of the leaflets, the stereometric geometry of the three-cusp valves, the papillary muscles and chords, and cross sections of the neck of an aorta (RL19028r, Fig. 11). He is not adducing the flow of the blood within the neck of the aorta on the basis of direct observation of blood in the heart, which would have been impossible to see. Rather he is using his knowledge of water in motion to model the spiraling turbulence, which assumes as geometrical a formation as the capital of an ionic column. His portrayal relies upon his observations of water as an engineer of canals and locks. He also invented an experimental tank with glass slides for "laboratory" studies in his workshop. Leonardo three times sketches the tank in the Codex Leicester (9v and 29v). The result was spectacular demonstrations of turbulent motion (RL 12660v).
The dynamics at work were those of medieval impetus theory. An object set in motion by a force acquires a certain amount of impetus, proportional to that force. The rule of motion decrees the inexorable draining of impetus from an object that has been set in motion. Successive impacts of impetuous water in water result in revolving motions within which the impetus is progressively and proportionately expended.
The most consequential of the drawings on the page is in fact the rather graceless diagram at the upper right. He tells us what he proposes to do in the note written inside the line drawing: "a mold of plaster within which is blown thin glass, and then break it at a n, at the top and bottom. But first pour wax into the gate of a bull's heart so that you may see the true shape of this gate." Looking at his characterization of the vortices across a number of drawings, and comparing them with modern imaging, I think it is likely that the glass model was actually made, using fine grass seeds to render the motions visible in the way that he suggested. In any event, the proposal to construct an experimental model of a natural process is conceptually remarkable.
It was exceptional that an anatomist should require a knowledge of mathematics to allow the reader to understand what is being demonstrated in studies of anatomy. We now take it for granted that mechanical analyses can be brought to bear on the forms and forces of bodily mechanisms. In the history of anatomy, we are familiar with the march towards a full understanding of bodily structures and mechanisms, not least in the wake of Andreas Vesalius's De humani corporis fabrica libri septem (On the fabric of the human body in seven books) in 1543. Leonardo yields nothing to Vesalius in the cogency of his demonstrations. However, he does not really belong to the standard historical progression of anatomical science. There were no comprehensive publications of his anatomical studies until the late 19th century. As we noticed at the outset, his drawings and notes are not really an integral part of the history of medicine. In his thousands of pages of drawings and texts, there is hardly a hint of involvement in the practice of medicine by surgeons and physicians. His vision of the human body remains as one of the greatest visual achievements of any era, standing alongside the Mona Lisa or Michelangelo's David as a supreme masterpiece of visual culture.
Today, in a variety of sciences of the human body mathematics and computer modeling play conspicuous roles, not least in the mapping of the motion of blood within the heart. Leonardo's vision of what might be possible as ways of understanding and depicting the forms and functions of the human body transcends the more obvious and common question of whether he was right or wrong in particular aspects of his demonstrations. His anatomies can best be regarded as magnificent and prophetic visual achievements in their own right, standing outside the conventional history of anatomy and presaging theoretical biology.