Life, Scientific Methods, and Anatomical Works

Leonardo da Vinci was the illegimate son of Piero da Vinci, a respected Florentine notary, and a peasant girl named Caterina. The year that Leonardo was born, his father was rapidly married off to a girl of good family, Albiera di Giovanni Amadori. Genetically it is of some interest that Piero’s youngest legitimate son, Bartolommeo, an enthusiastic admirer of his half-brother Leonardo, deliberately repeated his father’s “experiment” by marrying a woman of Vinci and, as Vasari relates, “prayed God to grant him another Leoanrdo.” In fact he produced Pierino da Vinci, a sculptor of sufficient genius to win himself the name of “Il Vinci” long before he died at the age of twenty-two.

Young Leonardo’s education at Vinci was conventionally limited to reading and writing. His early manifested gifts for music and art induced his father to apprentice him, about 1467, to Andrea del Verrocchio, in whose workshop he studied painting, sculpture, and mechanics. During this period in Florence, Leonardo’s activities appear to have been directed predominantly toward painting and sculpture; the earliest of his pictures still extant, the “Baptism of Christ,” was painted in collaboration with Verrocchio in 1473. The “Adoration of the Magi” was still incomplete when he left for Milan in 1482, to enter the employ of Ludovico Sforza (Ludovico il Moro), duke of Milan. In his application to Ludovico, Leonardo revealed that a great deal of his attention had already been devoted to military engineering; only in concluding did he mention his achievements in architecture, painting, and sculpture, which could “well bear comparison with anyone else.”

Leonardo lived at Milan in the duke’s service until 1499. During these years his interest in the problems of mechanics and the physics of light grew steadily while his artistic output reached a peak in the fresco “The Last Supper” (1497) and in the clay model of his great equestrian statue of Francesco Sforza (1493). The notebooks of this period show the increase of his interest in mathematics, the physics of light, the physiology of vision, and numerous mechanical problems, including those of flight. Four projects for books—separate treatises on painting, architecture, mechanics, and the human figure—appear in his notes. These studies were to occupy him for the rest of his life. During his last years in Milan he collaborated with the mathematician Luca Pacioli on his Divina proportione; Leonardo drew the figures for the first book.

After the capture of the duke of Milan by the French, Leonardo left for Venice, eventually returning to Florence. He then entered the service of Cesare Borgia and was employed for about a year as chief inspector of fortifications and military engineer in the Romagna. Following this he was responsible for an unsuccessful attempt to divert the Arno near Pisa. While in Florence he began the portrait “Mona Lisa” and also the ill-fated fresco “Battle of Anghiari.” During these years in Florence, from 1500 to 1506, Leonardo began his systematic researches into human anatomy. Mathematics and the mechanical sciences, too, increasingly occupied his time; and he began to couple his study of the problem of human flight with intensive research on bird flight and meteorology. His studies on the movement of water, a lifelong preoccupation, were later compiled into the Treatise on the Movement and Measurement of Water.

In June 1506 Leonardo returned to Milan, where Charles d’Amboise, the French governor, showed him the keenest appreciation he had yet experienced. During this period he produced most of his brilliant anatomical drawings, perhaps stimulated for a short while by the young Pavian anatomist Marcantonio della Torre. Leonardo had now come to feel that mathematics held the key to the “powers” of nature; and his work in hydrology, geology, meteorology, biology, and human physiology was increasingly devoted to a search for the geometrical “rules” of those powers through visual experience, experiment, and reason.

After the French were expelled from Milan in 1513, Leonardo left for Rome, hoping that the Medici Pope Leo X and his brother Giuliano would provide him with encouragement and a good working environment. Nothing came of this hope; and in 1516 he resumed the French liaison, this time with Francis I, with whom he traveled to Amboise in 1516. He died there after a stroke.

Problems of Evaluating Leonardo’s Scientific Thought. There are many different opinions concerning Leonardo’s stature as a scientist. The essential reason that this is so lies in the grossly abbreviated form in which his work has come down to posterity. Leonardo himself noted that “abbreviators of works do harm to knowledge… for certainty springs from a complete knowledge of all the parts which united compose the whole” (Windsor Collection, fol. 19084r, in I. A . Richter, Selections …, p. 3). Unhappily, his extant notes are both abbreviated and confused. It is therefore necessary to assess these defects before evaluating Leonardo’s work in the history of science.

The Losses of Leonardo’s Manuscripts. Although Leonardo clearly intended to write treatises on painting, architecture, mechanics, and anatomy, he brought none to publication. Two known works are published under his name, the Treatise on Painting and Treatise on the Movement and Measurement of Water; both were compiled after his death from his notes. These remain of great value, although their respective compilers, Francesco Melzi and Luigi Arconati, constructed them according to their own ideas, rather than Leonardo’s. The great mass of surviving data consists of some 6,000 sheets of Leonardo’s manuscript notes. It is difficult to estimate what proportion of the whole these represent. Reti has calculated that 75 percent of the material used by Melzi for his compilation of the Treatise on Painting has since been lost. If only a corresponding proportion of Leonardo’s scientific notes are extant, this is indeed a severe truncation. The qualitative loss is probably even greater, since there is no trace of a number of “books” on which Leonardo frequently referred in his notes as “completed,” and which he used as references, among them an “Elements of Mechanics,” and a “Book on Water”—not to mention fifteen “small books” of anatomical drawing.

Sources of Confusion in the Notes. The notes that remain are in great confusion, in part because Leonardo himself made no effort to integrate them, and in part because after his death they underwent almost every conceivable kind of disarrangement and mutilation—as is illustrated by the great scrapbook of sheets composing the Codex Atlanticus. Thus the thousands of pages that do survive resemble the pieces of a grossly incomplete jigsaw puzzle.

Leonardo’s mode of expressing himself often challenges interpretation. Apart from his “mirror” script, he habitually presented his thoughts visually, sometimes covering an astonishing range of phenomena with very few words, as in Windsor Collection, fol. 12283 (Figure 1). This page contains segments of circles, a geometrical study for the “rule of diminution” of a straight line when curved, a study of curly hair (with a note on its preparation), and drawings of grasses curling around an arum lily, an old man with curly hair, trees, billowing clouds, rippling waves in a pool, a prancing horse, and a screw press. All can be seen as studies of curves, viewed from different aspects, each of which is developed in detail in various pats of his notes. The sheet provides a good example of Leonardo’s visual approach to any problem—by integration—a mode which has until recently been frowned upon by orthodox science. Interpretation of the meaning of many of the drawings must necessarily be speculative; but it is always dangerous to call any drawing a “doodle,” since some of Leonardo’s most interesting scientific concepts appear as casual, small, inartistic figures.

A further source of confusion in Leonardo’s notes derives from his habit of periodically returning to the same subject. This often resulted in incompatible statements in different notes, which Leonardo himself recognized, referring in 1508 to his notes as “a collection without order taken from many papers … for which O reader blame me not because the subjects are many, and memory cannot retain them…because of the long intervals between one time of writing and another” (Codex Arundel, fol. lr).

Leonardo’s thinking was intensely progressive during these “long intervals” of time. A not made in 1490 may therefore differ markedly from one on the same subject from 1500 or 1510. With this understanding, contradictory statements can often be transformed into sources of comprehension if they can be dated—a means of tracing the progress of Leonardo’s thought that has only recently become available.

In 1936, with the facsimile production of the Forster codices, it was thought that publication of extant Leonardo notes was complete. Nevertheless, in 1967 two further codices, containing 340 folios, were found in Madrid. These still await publication; and one wonders how many more will be found.

Leonardo’s Scientific Method. Leonardo lived at a time when theology was the queen of the sciences. Theological thinking emphasizes the divine incomprehensibility and mystery of natural phenomena; even the Neoplatonic thought of Leonardo’s contemporary Marsilio Ficino focused on a direct, revealed link between the mind of man and God and was not concerned with the visual exploration of natural phenomena. Leonardo, on the other hand, declared that faith that is the foundation of all science: there is a logic in natural phenomena detectable by the senses and comprehensible to the mathematical logic of the human mind. Beyond this he postulated God as the creator of all.

Leonardo felt his way to his scientific outlook through his study of the theory of painting, which led him to analyze visual phenomena. As he declared,

Painters study such things as pertain to the true understanding of all the forms of nature’s works, and solicitously contrive to acquire an understanding of all these forms as far as possible. For this is the way to understand the Creator of so many admirable things, and this is the way to love such a great Inventor. In truth great love is born of great knowledge of the thing that is loved [Treatise on Painting, Para. 80.]

For Leonardo the world of nature was created by God, and the science or theory of painting “is a subtle invention which with philosophical and ingenious speculation takes as its theme all the various kinds of forms, airs and scenes, plants, grasses and flowers which are surrounded by light and shade. And this truly is a science and the true-born daughter of Nature” (MS Ashburnham 2038, fol. 20r, in E. MacCurdy, Notebooks, II , 229).

Science for Leonardo was basically visual: “The eye, the window of the soul, is the chief means whereby the understanding can most fully and abundantly appreciate the infinite works of Nature; and the ear is second” (MS Ashburnham 2038, fol. 20r, in E. MacCurdy, Notebooks,II , 227). He asserted his belief that the patterns of natural phenomena can conform with the patterns of form created in the human mind—“Painting compels the mind of the painter to transform itself into the very mind of Nature to become an interpreter between Nature end the art. It explains the causes of Nature’s manifestations as compelled by its laws…” (Treatise on Painting, para. 55). These are “laws of necessity the artificer of nature—the bridle, the law and the theme” (Codex Forster,III , fol. 43v); while “Necessity constrains all effects to the direct result of their causes” (Codex Atlanticus, fol. 345v-b). For Leonardo these causes, effects, and laws could be expressed visually in the form of a geometry that includes the movement of things as well as their resting forms.

Leonardo’s Mathematics of Science. Leonardo defined science as “that mental analysis which has its origin in ultimate principles beyond which nothing in nature can be found which is part of that science” (Treatise on Painting, Para. I). Geometry was such a science, in which “the point is that than which nothing can be smaller. Therefore the point is the first principle of geometry and nothing in nature or the human mind can be the origin of the of the point” (ibid.). He differentiated between the “Natural” point, which has the characteristics of an atom, and the “conceptual” point of geometry: “The smallest natural point is larger than all mathematical points, because a natural point is a continuous quantity and as such is infinitely divisible, while the mathematical point is indivisible, having no quantity” (quoted from J. P. Richter, Literary Works of Leonardo, 44). In this way he linked geometry with physics and denied the term “science” to any investigation that is not capable of “mathematical” demonstration, such as the so-called sciences that “begin and end in the mind” and are “without the test of experience.” He proclaimed the necessity for both geometry and experience, taking as his example astronomy, which was all “visual lines, which enclose all the different shapes of bodies created by nature, and without which the art of geometry is blind” (Treatise on Painting, para. 15). Thus, via geometry, human experience could interpret nature.

Leonardo divided his inquiry into three coincident parts: the geometry of vision, the geometry of nature, that is, physics or “natural philosophy,” and pure geometry. His investigation of vision was the first of his physiological researches; it pertained primarily to perception and derived largely from the traditional concept of a cone of vision. (Pure geometry—that is, Euclidean geometry—is discussed in the mathematics section, below.)

The Geometry of Vision. Without the test of visual experience there could be no science for Leonardo, for “the observer’s mind must enter into nature’s mind” (Treatise on Painting, para. 40). The problem of vision involved his concept of the spread of light from its source. This he conceived of by analogy to the transverse wave-spread set up by dropping a stone into a still pond. “Just as a stone flung into water becomes the centre and cause of many circles … so

any object placed in the luminous atmosphere diffuses itself in circles and fills the surrounding air with images of itself” (MS A, fol. 9r). These images spread out in circles from the surfaces of the object with steady diminution of power to the eye (Figure 2). As pond waves spread by the power of percussion, so the light wave varies in power with the force of percussion and inversely with its distance in the form of an infinite number of pyramids (like perspective), diminishing as they approach the eye (Figure 3). The

pyramidal form which Leonardo used to represent this decline of power has the characteristic that

… if you cut the pyramid at any stage of its height by a line equidistant from its base you will find that the proportion of the height of this section from its base to the total height of the pyramid will be the same as the proportion between the breadth of this section and the breadth of the whole base [MS M, fol. 44r].

Leonardo thus represented the power of percussion by the base or diameter of a cone or pyramid, and its decline by its breadth at any height of the pyramid. When the spreading light image reaches the pupil, it finds not a point but a circle, which in its turn contracts or expands with the power of light reaching it and forms the base of another cone or pyramid of light rays directed by refraction through the lens system of the eye to the optic nerve (Figure 4). The image then travels via the central foramen up the nerve to the “imprensiva” of the brain, wherein the percussion is impressed. Here it joins with the impressions made by the “percussions” from the nerves that serve the senses of hearing, smell, touch, and pain—all of which are activated by similar percussive mechanisms following similar pyramidal laws.

Thus in the imprensiva a spatiotemporal verisimilitude of the external environment is produced, while a truly mathematical, geometrically conditioned model of reality is experienced by the “senso commune” in

the third ventricle where all these sensations come together (Figure 5). “Experience therefore does not ever err, it is only your judgment that errs in promising itself results which are not caused by your experiments” (Codex Atlanticus, fol. 154r). Leonardo later located the imprensiva in the lateral ventricles of the brain when he discovered them. “Memory” and “judgment” he located fourth and third ventricles, respectively. Thus his distinction between sensory “experience” and “judgment” have physiological as well as psychological connotations.

Leonardo recognized that errors of sensory observation can occur. He attributed these to “impediments"—the resulting experience “partaking of this impediment in a greater or less degree in proportion as this impediment is more or less powerful.”

The Geometry of Nature. The pyramidal form that Leonardo imposed upon the propagation of visual images he further applied to the propagation of the four “powers” of nature: “movement, force, weight, and percussion.” He saw it as perspectival, i.e., simple, pyramidal proportion. It was in this form, therefore, that he quantified these powers, whether dealing with light and vision, as above, or with other forms of the powers of nature, as, for example, weights, levers, the movements of water, and power transmission in machines, including the human heart or limbs. For example, he described the force of a spring as “pyramidal because it begins at a point or instant and with each degree of movement and time it acquires size and speed” (MS G,fol. 30r); he further described the movement of compound pulleys as “Pyramidal since it proceeds to slowness with uniform diminution of uniformity down to the last rope” (ibid., fol. 87v); see mechanics section, below. Such phrasing is reminiscent of the terminology of the Merton School and reminds one that the names “Suisset” and “Tisber” occur in Leonardo’s notes.

Leonardo’s close relationship with Luca Pacioli in Milan strongly stimulated his interest in mathematics. At this time he made an intensive study of algebra and the “manipulation of roots,” describing algebra as “the demonstration of the quality of one thing with another” (MS K, II, fol. 27b). He does not, however, appear to have applied this concept of algebraic equations to his scientific work. On the other hand, he did apply square roots to the calculation of wingspan in relation to the weight of the human flying machine. His formula ran thus: “If a pelican of weight 25 pounds has a wingspan of 5 braccia, then a man of weight 400 pounds will require a wingspan of braccia” (Codex Atlanticus, fol. 320r-b). This calculation was made about the time of his final attempt at flight (1504). Leonardo never applied the inverse–square law to any “power” or “force.”

Leonardo’s Methods of Observation and Experiment.

Leonardo’s acquisition of “experience” was remarkable for three main methods: measurement, models, and markers. From about 1490, when he concluded that the “powers” of nature had geometrical relationships, Leonardo attempted to quantify his observations and experiments. Measurement entered into all his observations; measurements of weights, distances, and velocities. His notes from about this time contain descriptions of hodometers, anemometers, and hygrometers, as well as balances of various ingenious types.

In general one is struck, however, by the crudity of his measurements of both space and time. He rarely mentioned any spatial measurement other than the braccio (approximately twenty-four inches). For small intervals of time, such as the heart rate, he used musical tempi. Here he explained, “of which tempi an hour contains 1080” (Quaderni d’ anatomia, II , 11r). For timing he used sandglasses and water clocks. Although he certainly did not invent the balance, he did use balances in a highly original way—for example, by placing a man in one pan and observing whether the downward movement of a levered wing to which he was attached would raise him or not, and by weighing objects at different temperatures.

Models were Leonardo’s favorite form of experimental demonstration. He used them particularly for observations of the flow of water or blood currents. Perhaps his smallest model was a glass cast of an aorta with its valves (Figure 6); the largest was one of

the Mediterranean, which he built to demonstrate the effects of the rivers entering it.

Markers were another favorite method for visualizing the movements of water, both in Leonardo’s models and in actual rivers. His models for this purpose consisted of tanks with three glass sides through which he could view the seeds, bits of paper, or colored inks used a markers; he had a special set of floats, each designed to be suspended at a different depth. With such observations and experiments he accumulated an enormous mass of data regarding the directions and velocities of water movements, their angles of reflection, the effects of percussion, and the movements of sand and stones in water movements, their angles of reflection, the effects of percussion, and the movements of suspended solids. He traced the movements of sand and stones in water and their mode of deposition, as well as the action of water on surrounding surfaces—for instance, the erosion of river banks. He also applied these results to such problems as the movement of lock gates or aortic valves. He extended the marker method to the other “elements,” particularly to the movement of solid bodies in air, including projectiles, dust, and “birds” (models).

Leonardo frequently advocated the repetition of experiments, “for the experiment might be false whether it deceived the investigator or not” (MS M, fol. 57r), If a confirmed experiment did not agree with a suggested mathematical “rule,” he discarded the rule. By this means, and clearly to his surprise, he had to reject the Aristotelian “rule” that “if a power moves a mobile object with a certain speed, it will move half this mobile object twice as swiftly.” By experiment Leonardo tested the rule, comparing the movements of “atoms” of dust in air with those of large objects fired from a mortar; and from his observation he draw a moral—mistrust those who have used nothing but their imagination and have not verified their statements by experiment. For Leonardo mathematical rules had to give way to experimental verification or refutation.

Leonardo applied his experimental or mathematical results only to visual space. Unlike such medieval antecedents as Oresme, he refused to quantify such abstract qualities as beauty or glory; he specifically stated that geometry and arithmetic “are not concerned with quality, the beauty of nature’s creations, or the harmony of the worked” (Treatise on Painting, para. 15). He honored the precept that these are fields for the creative arts rather than for the creative sciences.

Leonardo’s “Rules.” Not only did Leonardo advocate repetition of experiments, but he performed experiments in series, each repeated with one slight variation to resemble continuous change. For example, on investigating flight with model birds, he wrote: “Suppose a suspended body resembling a bird and that the tail is twisted to different angles. By means of this you will be able to derive a general rule as to the various movements of birds occasioned by bending their tails” (MS L, fol.61v). Demonstrating the distribution of weight of a beam suspended by two cords, he wrote: “I make my figures so that you shall know all the cases the are placed under one simple rule” (Codex Atlanticus, fol.274r-b). The statement is carried out in the following three pages of patient depiction of all the variable distributions of weight (ibid., fols.274r-b, r-a,v-b).

Thus by repeated observation, repeated experiment, and calculatedly varying his experiments, Leonardo built up quantitative data into limited generalized “rules” “These rules”, he wrote, “are the causes of making you know the true from the false” (ibid., fol. 119v-a). Since the rule is mathematical in form, he averred, “These is no certainty where one cannot apply any of the mathematical science” (MS G,fol. 96v); and clearly it follows that “Now man who is not a mathematician should read the elements of my work” (Windsor Collection, fol. 19118v, in I. A. Richter, Selections…,p.7).

The Four Powers. Leonardo’s notes contain long debates on the nature of movement, weight, force, and percussion, all in the context of the Aristotelian elements of earth, air, fire, and water. From this debate emerged the statement: “Weight, force, together with percussion, are to be spoken of as the producers of movement as well as being produced by it” (Codex Arundel, fol. 184v). He finally saw weight as an accidental power produced by a displaced own element “desiring” to return to its natural place in its own element. Thus “gravity” and “levity” were two aspects of the same drive. Force was caused by violent movement stored within bodies. Sometimes Leonardo called force “accidental weight,” using the Aristotelian meaning of “accidental.” Movement of an object from one point to another could be natural, accidental, or participating. Impetus was derived movement arising from primary movement when the movable thing was joined to the mover. He defined percussion as “an end of movement created in an indivisible period of time, because it is caused at the point which is the end of the line of movement” (Codex Froster, III, fol. 32r. in I. A. Richter, Selections …,p.77).

This remarkable simplification of his experiences of “nature” was applied by Leonardo to all fields of investigation and practical creation, that is, to science and its derived art of invention. Clearly these four powers were both derived from, and most applicable to, mechanics, “Mechanics is the pharadise of mathematical science because here one comes to the fruits of mathematics” (MS E,fol. 8v). This statement was generalized when Leonard asserted: “Proportion is not only found in numbers and measurements but also in sounds, weight, times, spaces, and whatsoever powers there be” (MS K, 497, Ravaisson-Mollien). From these two statements his essentially integrative approach to all problems can be glimpsed. In each field of investigation he built up, from quantitative observation and experiment defined mathematical “rules” which he applied to the particular problem occupying him. For example, with regard to the flight of the bird, when discussing the problem of its being overturned, he set out the forces concerned and then stated: “This is proved by the first section of gravity send down their opposite sides …” (Codex on Flight …, fol. 8r). He continued, “And in this way it will return to a position of equlilibrium. This is proved by the fourth of the third, is acted on by the grater forces; also by the fifth of the third, according to which a resistance is weaker the farther it is from its fixed point” (ibid., fol.9r). In this way Leonard’s “rules” were built into the structure of his thought about all the powers of nature, imparting a remarkable consistency in all fields, whether mechanics, light, sound, architecture, botany, or human physiology. Unfortunately, the books to which he refers in this way, showing how he reached these generalizations, have been lost.

Leonardo’s Achievements in Science. Leonardo’s achievements in art and science depended primarily on his remarkable acuity of vision, and his particular sensitivity to the geometrical consequences of that vision. These gifts he combined with the creative technology of the engineer and the artist. These creative aspects emerge in his achievements in scientific technology.

Mathematics. Leonardo’s approach to mathematics was predominantly physical. Even when appearing abstract, as in his “Book on Transmutation” (of forms), his practical object was revealed by his reference to the metalworker or sculptor. Typical of his contribution to mathematics were proportional and parabolic compasses. His explorations of the properties of the pyramid were, as is to be expected, thorough. As a result of an experimental investigation of the center of gravity of a tetrahedron, he discovered that the center of gravity is at “a quarter of the axis of that pyramid” (MS F, fol. 51r). (For his development of pure mathematics, see the section on mathematics.)

Mechanics. (Here brief mention is made only of the main areas studied by Leonardo. For fuller development of these, see the section on mechanics.) Leonardo’s point of departure for statics was Archimedes’ work on the lever, of the principles of which he had some understanding. His appreciation that equilibrium of a balance depends arithmetically upon the weights and their distances from the fulcrum gave him many examples of pyramidal (arithmetical) proportion.

Leonardo never quite reached the concepts of mass or inertia, as opposed to weight. He still used the “halfway” concept of impetus when he said, “All moved bodies continue to move as long as the impression of the force of their motors remains in them” (Codex on Flight…, fol. 12r). This is as near as he came to Newton’s first law. And although Leonardo disproved the Aristotelian relationship between force, weight, and velocity, he did not reach the concept of acceleration resulting from action of the force on a moving body. Newton’s third law, however, Leonardo did state clearly in his concrete way and apply persistently. For example, “An object offers as much resistance to the air as the air does to the object” (Codex Atlanticus, fol. 38lv-a). And after a repetition of this statement he added: “And it is the same with water, which a similar circumstance has shown me acts in the same way as air” (Codex Atlanticus, fol. 395r). One can see here the process by which he built up to one of his “rules”—Newton’s third law.

Gravity for Leonardo was the force of weight which is exerted “along a central line with with straightness is imagined from the thing to the center of the world” (MS I, fol. 22v). In his investigations of it Leonardo dropped weights from high towers. For example, he dropped two balls of similar weight together and observed at the bottom whether they still touched (MS M, fol. 57r). From such experiments he note: “In air of uniform density the heavy body which falls at each degree of time acquires a degree of movement more than the degree of preceding time… the aforesaid powers are all pyramidal, seeing that they commence in nothing and proceed to increase in degrees of arithmetical proportion” (ibid., fol. 44r). he concluded that the velocity is proportional to the time of fall but, incorrectly, that the distance of fall is also proportional directly to the time, not to its square

Leonardo appreciated and used the concepts of resolving and compounding of forces. He described the parallelogram of forces, although he did not draw it, in analyzing the flight of the bird (Codex on Flight, fol. 5r); and he resolved the movement of a weight down an inclined plane into two components (MS G, fol. 75r). Such movement led him to study friction, a force which, like so many others, he divided into simple and compound. Leonardo appreciated the importance of the pressure or weight factor and the nature of the surface and their independence of area. hus, for “a polished smooth surface” he found a coefficient of friction of 0.25. He was aware that such friction produces heat.

Hydrodynamics. All the “rules” described above were carried into Leonardo’s studies of the movement of water. Here two phenomena received particular attention: wave formation and eddies. He found the velocity of flow of water to be inversely proportional to the dimension of the passage, whether rivers or blood vessels are being discussed. Currents of water, as observed by his marker experiments, percussed so that the angles of incidence and reflection were equal, similar to those made by bouncing balls. The formation of the transverse wave from the percussion of a stone in water was similarly explained. The cohesion of a drop of water against the force of gravity led Leonardo to postulate a force “like that of a magnet” which was also responsible for capillary attraction. The extent of his studies of water is reflected in the Codex Leicester, where he draws “732 conclusions as to water” (Codex Leicester, fol. 26v).

Cosmology. Water, more than the other elements, led Leonardo to compare “the greater and the lesser world.” The macrocosm–microcosm analogy was very real to him , since he believed that similar laws governed both the cosmos and the body of man. He considered at one time, for example, that the movement of the waters of the earth, particularly to the tops of mountains, was analogous to the movement of blood to the head, both being produced by heat. At first he postulated that water was drawn up by the heat of the sun; later he suggested that the heat arose from subterranean fires, from volcanos in the earth. He posited such subterranean fires as an analogy to the human heart, which he saw not only as the percussor of blood to the periphery but also as a source of heat to the body through the friction of the blood in its chambers (Figure 7).

Leonardo saw the land of the world as emerging from its surrounding element of water by a process of growth. Mountains, the skeleton of the world, were formed by destructive rain, frost, and snow washing the soft earth down into the rivers. Where large landslides occurred, inland seas were formed, from which the waters eventually broke through gorges to reach the sea, their “natural” level. No better summary of Leonardo’s neptunist concept of these geological changes can be found than the background of his “Mona Lisa.”

Biology. From about 1489 to 1500 Leonardo investigated the physiology of vision and the external powers of the human body. During this period he applied the principles of the “four powers”—movement, force, weight, and percussion—to every conceivable human activity. Many of the drawings of men in action found in the Treatise on Painting date from this period, during which he also completed a book, now lost, on the human figure. He drew men sitting, standing, running, digging, pushing, pulling, and so on, all with such points of reference as the center of gravity or the leverage of the trunk and limbs. These simple movements he elaborated into studies of men at work and so to the transformation of human power into machines at work, the field of many of his technological inventions (Figure 8). (For Leonardo’s development of this field, see the section on technology.)

Anatomy . Leonardo later investigated the internal powers of the human body. He described his approach to human anatomy and physiology in a passage headed “On Machines” :

Why Nature cannot give movement to animals without mechanical instruments is shown by me in this book, On the Works of Motion Made by Nature in Animals. For this I have set out the rules on the 4 powers of nature without which nothing can through her give local motion to these animals. Therefore we shall first describe this local motion, and how it produces and is produced by each of the other three powers….

He then briefly defined these powers (Quaderni d’ anatomia, I, fol. 1r) and added an admonition: “Arrange it so that the book of the elements of mechanics with examples shall precede the demonstration of the movement and force of man and other animals, and by means of these you will be able to prove all your propositions” (Anatomical Folio A, fol. 10r).

As usual, Leonardo gave intense thought to his methods of approaching the problem of man the machine. “I shall describe the function of the parts from every side, laying before your eyes the knowledge of the whole healthy figure of man in so far as it has local motion by means of its parts” (Quaderni d’ anatomia, 1, fol. 2r). He applied his gift of visual artistry and his concepts of the four powers to the human body with results that remain unique. The mechanics of the musculoskeletal system were displayed and explained, such complex movements as pronation and supination being correctly analyzed and demonstrated.

In his anatomical writings Leonardo, always fastidious, noted his intense dislike of “passing the night hours in the company of corpses, quartered and flayed, and horrible to behold.”

Human Anatomy. On several occasions Leonardo laid out comprehensive plans for demonstrating the anatomy of the human body. In all of them he put great stress on the necessity for presenting the parts of the body from all sides. He stated, moreover, that each part must be dissected specifically to demonstrate vessels, nerves, or muscles:

…you will need three [dissections] in order to have a complete knowledge of the veins and arteries, destroying all the rest with very great care; three others for a knowledge of the membranes, three for the nerves, muscles and ligaments, three for the bones and cartilages…. Three must also be devoted to the female body, and in this there is a great mystery by reason of the womb and its fetus.

Leonardo thus hoped to reveal “in fifteen entire figures… the whole figure and capacity of man in so far as it has local movement by means of its parts” (Quaderni d’ anatomia, 1, fol. 2r). His extant anatomical drawings show that he was in fact able to carry out a large part of this extensive program.

In addition to conventional methods of dissection, Leonardo brought to his study of anatomy his own particular skill in modelmaking. He discovered the true shape and size of the cerebral ventricles through making wax casts of them, and he came to appreciate the actions of the ventricles of the heart and the aortic valves by making casts of the cardiac atria and ventricles and glass models of the aorta.

Since his approach was primarily mechanical, Leonardo regarded the bones of the skeleton as levers and their attached muscles as the lines of force acting upon them. A firm knowledge of these actions could not be reached without a detailed demonstration of the shapes and dimensions of both bones and muscles. About 1510, Leonardo executed for this purpose a series of drawings, which constitute a large part of the collection Anatomical Folio A. Here he systematically illustrated all the main bones and muscles of the body, often accompanying the drawings with mechanical diagrams to show their mode of action. In these drawings Leonardo frequently adopted the technique of representing muscles by narrow bands, corresponding to what he called their “lines of force” (Figure 9), thus facilitating demonstration of their mechanical powers. Since an understanding of the mode of articulation of joints is clearly necessary to an appreciation of their movements in leverage, Leonardo illustrated these structures by an exploded view, whereby the joint surfaces are separated and the surrounding tendons of muscles are severed to show their exact lines of action in moving the joints. Such illustrations permitted Leonardo to demonstrate, for example, the subtleties of the movements of the upper cervical spine and the shoulder joint.

Possibly because the visual sense was so overwhelmingly important to him, Leonardo appears to have made his earliest anatomical studies on the optic nerves, cranial nerves, spinal cord, and peripheral nervous system. In the course of these studies he traced back the optic nerves, beautifully illustrating the optic chiasma and the optic tracts. In his attempts to elucidate the central distribution of sensory tracts in the brain, he injected the cerebral ventricles with wax, thereby ascertaining and demonstrating their approximate shape, size, and situation On this discovery he based his own mechanical theory of sensation through percussion.

Leonardo conceived the eye itself to be a camera obscura in which the inverted image, formed by light penetrating the pupil, is reinverted by the action of the lens. By constructing a glass model of the eye and lens, into which he inserted his own head with his own eye in the position corresponding to that of the optic nerve, he came to the conclusion that the optic nerve was the sensitive visual receptor organ. Here his own experimental brilliance deceived him.

On pithing a frog he noted, “The frog instantly dies when the spinal cord is pierced, and previous to this it lived without head, heart, bowels, or skin. Here therefore it would seem lies the foundation of movement and life” (Quaderni d’ anatomia, V, fol. 21v). He here contradicted both Aristotelian and Galenic concepts, and opened the way for his own mechanical theory of sensation and movement.

Leonardo’s interest in the alimentary tract arose with his dissection of the “old man” in the Hospital of Santa Maria Nuova in Florence. From this dissection he made drawings of the esophagus, stomach, liver, and gallbladder, and drew a first approximation of the distribution of the coils of small and large intestines in the abdomen. Perhaps most notable of all was a detailed sketch of the appendix (its first known representation) accompanied by the hypothesis that it served to take up superfluous wind from the bowel. The heat necessary for digestive coction of the food, Leonardo held, was derived from the heat of the heart—not from the liver, as the Galenists stated. The movement of food and digested products down the bowel was, in Leonardo’s view, brought about by the descent of the diaphragm and the pressure produced by contraction of the transverse muscles of the abdomen. Since he did not vivisect, he did not observe peristalsis. Always aware of the necessity of feedback mechanisms to preserve the equilibrium of life, Leonardo was greatly concerned with the fate of superfluous blood, since blood was thought to be continuously manufactured by the liver. Superfluous blood, Leonardo thought, was dispersed through the bowel, contributing largely to the formation of the feces.

Leonardo understood in principle the mechanism of respiration, clearly described by Galen. Describing the movements of the ribs, he wrote, “… since there is no vacuum in nature the lung which touches the ribs from within must necessarily follow their expansion; and the lung therefore opening like a pair of bellows draws in the air” (Anatomical Folio A, fol. 15v). He considered, however, that the most powerful muscle of inspiration is the diaphragm, the “motor of food and air,” as he called it, since by its descent it draws in air and presses food down the gut. After a long debate Leonardo decided that expiration is mostly passive, induced by the diaphragm rising with contraction of the abdominal muscles, and subsidence of the ribs.

Leonardo’s studies of the heart and its action occupy more anatomical sheets than those of any other organ. By tying off the atria and ventricles and injecting them with air (later wax), he obtained an accurate idea of their shape and volume. He thus came to recognize that the atria are the contracting chambers of the heart which propel blood into the ventricles, a discovery to which he devoted pages of discussion and verification because it was entirely opposed to Galenic physiology. He applied the same technique to the root of the aorta to discover the aortic sinuses, subsequently named after the anatomist Antonio Maria Valsalva. Having established the shape and position of the valve cusps, Leonardo applied the same marker technique which he used so often in his studies of rivers and water currents. To a glass model of the aorta, containing the aortic valve ring and cusps of an ox, he attached the ventricle (or a bag representing it), which he squeezed so that water passed through the valve. The water contained the seeds of panic grass, which served to demonstrate the directional flow of the currents passing the valve cusps. From these experiments Leonardo showed that the aortic valve cusps close in vertical, not horizontal, apposition—that is, from the side, by pressure of eddying currents, not from above, by direct reflux. Although much modern evidence confirms this view, such action has not yet been directly visualized through angiocardiography.

Other features of Leonardo’s study of the anatomy of the heart include his detailed demonstration of the coronary vessels and of the moderator band which some have named after him. By observation of the movements of the “spillo,” the instrument that slaughterers used to pierce the heart of a pig, Leonardo deduced the relation of systole to the production of the pulse and apex beat. He considered that the force of cardiac percussion propelling the blood was exhausted by the time the blood reached the periphery of the body, however, and thus missed the Harveian concept of its circulation.

Although his drawings of the kidney, ureter, and bladder are relatively sophisticated, Leonardo’s mechanistic physiology did not take him far beyond Galen’s views on urology. He saw the kidney as acting as a kind of filter, as did Vesalius after him. His representation of the uterus as a single cavity marks a great advance, particularly in his drawing of a five-month fetus in utero, although even in his drawing Leonardo showed the cotyledons of the bovine uterus. His drawings of male and female genitalia that pertain to coitus show an austere emphasis on procreation that is perhaps reflected in his comment: “The act of procreation and the members employed therein are so repulsive that if it were not for the beauty of the faces and the adornments of the actors and the pent-up impulse, nature would lose the human species” (Anatomical Folio A, fol. 10r).

Comparative Anatomy. Although Leonardo concentrated his anatomical investigations on the human body, he by no means confined them to that field. From the artistic point of view, he was equally interested in the structural mechanics of the horse: and from the point of view of the ever-present problem of flying, the anatomy of the bird and bat took priority.

There are scattered through Leonardo’s extant notebooks illustrations and notes on the anatomy of horses, birds, bats, oxen, pigs, dogs, monkeys, lions, and frogs. Most of these were studies undertaken to solve particular “power” problems. His anatomy of the horse, however, forms an important exception, for Lomazzo, Vasari, and Rubens all refer to the existence of Leonardo’s “book on the anatomy of the horse,” which has since been lost. Possible remnants of it remain in some of the drawings of the proportions of the horse in the Royal Library, Windsor, the Codex Huygens, and the musculoskeletal sketches in MS K, folios 102r and 109v. But perhaps the best-known comparative study of the hind limbs of the horse and the legs of man has come down to us in his drawing on Quaderni d’ anatomia, V, fol. 22r, where he wrote: “Show a man on tiptoe so that you may compare man with other animals.”

In a number of cases Leonardo repeated Galen’s mistakes in substituting animal for human parts. These mistakes diminished as his increasing knowledge of anatomy revealed to him surprising variation of form in organs serving similar physiological functions. He then turned to such variations to gain knowledge of the function. Such was the basis of his numerous studies of the shapes of the pupils of the eye in men, cats, and different birds. And it was by dissection of the lion that Leonardo came to say:

I have found that the constitution of the human body among all the constitutions of animals is of more obtuse and blunt sensibilities, and so is formed of an instrument less ingenious and of parts less capable of receiving the power of the senses. I have seen in the leonine species how the sense of smell forming part of the substance of the brain descends into a very large receptacle to meet the sense of smell…. The eyes of the leonine species have a great part of the head as their receptacle, so that the optic nerves are in immediate conjunction with the brain. With man the contrary is seen to be the case, for the cavities of the eye occupy a small part of the head, and the optic nerves are thin, long and weak [Anatomical Folio B, fol. 13v].

That Leonardo appreciated the importance of these studies in comparative anatomy is evinced in his note “Write of the varieties of the intestines of the human species, apes, and such like; then of the differences found in the leonine species, then the bovine, and lastly the birds; and make the description in the form of a discourse” (Anatomical Folio B, fol. 37r). His awareness of homologous structures was most pronounced in relation to the limbs, the form and power of which in both man and animals were of outstanding artistic and scientific importance to him. “Anatomize the bat, study it carefully and on this construct your machine,” he enjoys on MS F, folio 41v. Comparing the arm of man and the wing of the bird, he pointed out: “The sinews and muscles of a bird are incomparably more powerful than those of man because the whole mass of so many muscles and of the fleshy parts of the breast go to aid and increase the movement of the wings, while the breastbone is all in one piece and consequently affords the bird very great power” (Codex on Flight…,fol. 16r). To elucidate this problem he embarked on a comparative mechanical anatomy of the proportions of the wings of the bat, the eagle, and the pelican, and the arm of man. He even reduced the forelimb to a three-jointed model of levers representing humerus, forearm, and hand, manipulating this artificial limb by pulley mechanisms. He was clearly using comparative anatomy as a means of solving the great problem of human flight.

Such comparisons led Leonardo to suggest repeatedly that he should represent the hands and feet of “the bear, monkey, and certain birds” in order to see how they differ from those of man. Examples of such completed work are scattered in the notes.

From such studies Leonardo came to realize the anatomicance significant of “the movements of animals with four feet, amongst which is man, who likewise in his infancy goes on four feet, and who moves his limbs crosswise, as do other four-footed animals, for example the horse in trotting” (MS E, fol. 16; Codex Atlanticus, fol. 297r).

From such steps Leonardo reached the generalization: “All terrestrial animals have a similarity of their parts, that is their muscles, nerves and bones, and these do not vary except in length and size, as will be demonstrated in the book of Anatomy…. Then there are the aquatic animals which are of many varieties, concerning which I shall not persuade the painter that there is any rule, since they are of almost infinite variety, as are insects” (MS G, fol. 5v). “Man differs from animals.” concludes Leonardo, “only in what is accidental, and in this he is divine….” (Anatomical Folio B,fol. 13v). After referring to skill in drawing, knowledge of the geometry of the four powers, and patience as being necessary for completing his anatomical researches, he finally wrote of his work: “Considering which things whether or no they have all been found in me, the hundred and twenty books which I have composed will give their verdict, yes or no. In these I have not been hindered either by avarice or negligence, but only by want of time. farewell” (Quaderni d’ anatomia, I, fol. 13r).

Botany. Leonardo’s botanical studies developed, as it were, in miniature along lines similar to those of his anatomy, with the important difference that in this field he had no predecessors like Galen and Mondino to aid (or impede) his personal observations. These studies commenced in his early years with representation of the external forms of flowering plants, and it was not long before evidence appeared of his interest in plant physiology. About 1489, on a page containing two of his most beautiful designs for cathedrals, he presented a series of four botanical diagrams, one being described by the words: “If you strip off a ring of bark from the tree, it will wither from the ring upwards and remain alive from there downwards” (MS B, fol. 17v). Experiments and observations on the movement of sap and growth of plants and trees continued. They reached a climax about 1513, when he was in Rome. Many of these are gathered together in Ms G and book VI of the Treatise on painting. For many years Leonardo saw movement of sap in the plant as analogous to the movement of blood in the animal and water in the sun and downward by its own “natural” weight:

Heart that is poured into animated bodies moves the humours which nourish them. The movement made by this humour is the conservation of itself. The same cause moves the water through the spreading veins of the earth as that which moves the blood in the human species…. In the same way so does the water that rises form the low roots of the vine to its lofty summit and falling through the severed branches upon the primal roots mounts anew [Codex Arundel, fols, 234, 235].

Incidentally, he postulated this same mechanism for the absorption of food from the human intestine by the portal veins.

Leonardo described an experiment with a gourd:

The sun gives spirit and life to plants and the earth nourishes them with moisture. In this connection I once made an experiment of leaving only one very small root on a ground and keeping it nourished with water and the gourd brought to perfection all the fruits that it could produce, which were about sixty. And I set myself diligently to consider the source of its life; and I perceived that it was the dew of the night which penetrated abundantly with its moisture through the joints of its great leaves to nourish this plant with its offspring, or rather seeds which have to produce its offspring…. The leaf serves as a nipple or breast to branch or fruit which grows in the following year [MS G, fol. 32v].

Thus Leonardo reached a concept of the necessity of Sunlight “giving spirit and life” through its leaves to a plant, while the moisture of its sap came from both root and leaf. He also perceived upward and downward movement of sap. He located the cambium when he stated: “The growth in thickness of trees is brought about by the sap, which in the month of April is created between the ’Camicia’[cambium] and the wood of the tree. At that time this cambium is converted into bark and the bark acquires new cracks” (Treatise on Painting,pt. VI, McMahon ed., p.893).

Appreciating this seasonal growth of wood from the cambium, Leonardo came to recognize its part in the annual ring formation in trees. He wrote: “The age of trees which have not been injured by men can be countered in years by their branching from the trunk. The trees have as many differences in age as they have principal branches…. The circles of branches of trees which have been cut show the number of their years, and also show which years were wetter or drier according to their greater or lesser thickness….” (ibid.,p.900). In these words Leonardo showed himself to be originator of dendrochronology.

Leonardo’s observation on the growth of the tree trunk and their patterns of its branches and leaves, combined with his views on plant nutrition, initiated his study of phyllotaxy:

The lower trunks of trees do not keep their roundness of size when they approach the origin of their branches or roots. And this arises because the higher and lower branches are the organs [membra] by which the plants (or trees) are nourished; that is to say, in summer they are nourished from above by the dew and rain through the leaves and in winter from below through the contact which the earth has with their roots…. Larger branches do not grow toward the middle of the tree. This arises because every branch naturally seeks the air and avoids shadow. In those branches which turn to the sky, the course of the water and dew descends… and keeps the lower part more humid than the upper, and for this reason the branches have more abundant nourishment there, and therefore grow more [ibid., p.885].

Thus the trunk of the tree, being most nourished, is the thickest part of the tree. Leonardo saw here the mechanical implications of the force of winds on trees: “the part of the tree which is farthest from the force which percusses because of its greater leverage. Thus nature has provided increased thickness in that part, and most in such trees as grow to great heights like pines” (Codex Arundel,fol. 277v). Leonardo asserted that the branch pattern of a plant or tree follows its leaf patter: “The growth of the branches of trees on their principal trunk is like that of the growth of their leaves, which develop in four ways, one above the other. the first and most general is that the sixth leaf above always grows above the sixth below; and the second id for the third pair of leaves to be above the third pair below; the third way is that for the third leaf to be above the third below” (Treatise on painting, McMohan ed., p.889). this whole pattern of trunk, branch, and leaf Leonardo saw as designed to catch maximum sun, rain, and air—the leaves and branches were arranged “to leave intervals which the air and sun can penetrate, and drops that fall on the first leaf can also fall on the fourth and sixth leaves, and so on” (ibid.).

“All the flowers that see the sun mature their seed and not the others, that is, those that see only the reflection of the sun” (MS G, fol. 37v). And of seeds he wrote, “All seeds have the umbilical cord which breaks when the seed is ripe; and in like manner they have matrix [uterus] and secundina [membranes], as is shown in herbs and seeds that grow in pods” (Quaderni d’anatomia, III, fol. 9v). These words, written on a page of drawings of the infant in the womb, vividly reveal Leonardo’s integrating mode of thought.

Kenneth D. Keele