The desire to know is natural to good men.

—Leonardo da Vinci

f an abiding message is to be gleaned from an examination of Leonardo’s scientific and artistic legacy, it is the insatiable curiosity, the persistent questioning that defined his life—from resolving everyday problems to exploring grand-scale issues pertaining to the workings of nature. He observed and pondered; the commonplace became wondrous, the wondrous commonplace. “Nature is the best teacher,” he wrote. “Learn from nature, not from each other.” His curiosity encompassed diverse intellectual worlds: technical and nontechnical, scientific and artistic; and indeed it is by the conjoining of these intellectual worlds that he was able to produce works of such dazzling quality and diversity.

The typical liberal arts curriculum offered in the American undergraduate education system gives students exposure to different intellectual cultures. But then the need to develop specialized skills in one field or another tends to discourage further forays across the cultural divide. The higher education systems in most other countries call for this specialization to begin even earlier. Beyond formal education, normal maturation or aging itself is sadly accompanied by the monotonic dimming of one’s curiosity. Once the specialization process begins, those individuals who are more technically oriented have greater facility communicating with others who are technical, and those who are artistic or less technical similarly feel comfortable in the company of individuals with skills and interests akin to their own. In short, birds of a feather most clearly flock together, communicating with each other in their own languages.

In 1940 engineers from Indiana’s Department of Highways asked the state legislators to purchase an electric calculator for their office. One of the legislators challenged the engineers: “Why would you need a calculator? You are not mathematicians.” An engineer responded, “In order to build safe roads it is necessary to make precise calculations of curves and banking angles. We must factor π into our calculations; and π is an irrational number—a constant with a value of 3.141 593.…” The legislator was confused by the mathematical riposte. Shaking his head, he walked into his legislative meeting, taking with him the engineer’s explanation for the “justification” for the expenditure. After a while the legislator emerged with an answer: “We don’t have money for your calculator. But, we decided to change the value of π for you to 4.” Lack of communication between technical and the nontechnical cultures pervades all levels of society. Sixty years later the faculty senate in a selective liberal arts college in the United States voted to allow a certain course taught in the college’s geography department to satisfy the college’s natural science requirement, all the nonscientists in the senate voting, “aye,” all the scientists, outnumbered, voting “nay.” By committee vote, geography had been deemed a natural science—further evidence of the continuing malady of a cultural divide.

I opened the prologue of this book with C. P. Snow’s famous pronouncement on the two disparate intellectual cultures and on the lack of communication between them. What even Snow did not realize is that the seeds of another intellectual revolution had already been planted that would complicate the picture of the two cultures. Two decades earlier the gifted British mathematician Alan Turing, then at Princeton University, had published a general theory of computing.¹ Beginning in the late 1940s engineers, mathematicians, and physicists in the United States began to construct digital computers—those early mainframe machines, ENIAC and UNIVAC—and to create computer languages such as FORTRAN, COBOL, and ALGOL.²

In the late 1960s I was in a Ph.D. program in theoretical physics that still had a vestigial sense of a balanced liberal arts education, requiring doctoral candidates to demonstrate facility in two foreign languages. I had to write my own programs in one of the prevailing computer languages and, on the side, prepare for the language competency tests in French and German, besides the main task of mastering all the necessary physics. A year or two after I finished my degree the foreign language requirement was dropped. Ph.D. candidates would now have the option to substitute computer languages for foreign languages. Conversely, for the bachelor’s degree in computer science, the natural sciences are no longer a requirement, although there may be an institutional requirement to take such courses under the category of “general education.” By the late 1970s a new species had begun to propagate—“computer scientists”—with skills to write programs and create sophisticated general software packages that ran more efficiently than any program I could have conceived. And at approximately the same time there was the birth of the personal computer—desktops, followed by laptops—not replacing the mainframes, but joining them and beginning to communicate with each other, and all spreading at pandemic speed. The first personal computer was created in 1975. By July 2002, when the world population was estimated to be 6.2 billion, the sales of personal computers reached one billion.

At the beginning of the twenty-first century we find ourselves not only with a deeper chasm dividing C. P. Snow’s two cultures but also with a third intellectual culture formed by members of the seceding computer-literate “techies”—neither natural scientists nor humanists. Members of the other two intellectual cultures benefit from application programs such as word processing, mathematical computing, and cyberspace technology, developed by computer scientists. The details of the actual programs remain as incomprehensible, mystifying, and irrelevant to these two cultures as their own material is to each other. The cultural divide describing the dissociation of the first two cultures now demands a reorganization, with a “digital divide” now describing the further dissociation of a third culture.

Under the rubric “Leonardo’s model” this book presented the mathematical and scientific basis underlying both science and art, and the tenets of the laws of nature, albeit all of it in an essentially qualitative manner. The interdisciplinary approach, it was seen, invited a discussion of the human side of doing science and art. Among the most creative artists and scientists—especially those who launched revolutionary transformations in the prevailing order in their fields—there abound tortured souls, baffling personalities very different from those in everyday life. There are also the historical and social conditions that made certain times ripe for such unusually creative individuals to spring up and leave their marks.

Beethoven on his deathbed reportedly raised his fist in defiance at a thunderstorm raging outside and cried, “I know I am an artist!” In the last hours before his death, Leonardo, with an air of forlorn resignation, remarked to an assistant, “Tell me, did anything get done?” For us, these are puzzling and poignant pleas for reassurance by two of the greatest creative individuals in history. In Leonardo’s case there also resonates the tone of frustration at unfulfilled ideas and unfinished work. One cannot deny that the reasoning out of solutions, especially in art, and the creating of mental inventions, especially theories in science and technology, were more important for Leonardo than bringing them to actuality. Thus his chronic problem of darting from one beckoning source of curiosity to the next was real. And as his reputation for failing to complete projects and commissions was well deserved. Through it all, however, he was always breaking significant new ground. From the threshold of the twenty-first century we can view the dizzying array and quality of his accomplishments, and we can wonder: “Could there have existed a dozen different and supremely gifted individuals all operating with the same name—Leonardo da Vinci?”

A retrospective that includes the comprehensive abundance of Leonardo’s achievements must be viewed against the specter of his inventory of demons. Leonardo displayed boundless variation in his passions, unparalleled originality, a superhuman drive, and an unusual personality. And these were all steeped in paradoxes, contradictions, and ironies. But the questions “What is a cause, what is an effect?” and “What is catalytic, what is coincidental?” must all be regarded as speculative. Nothing with Leonardo is easy. Some of the issues have the ring of central significance. Others certainly do not seem as important today, but we are five centuries removed from a time when they were regarded as serious demons and adversities with which Leonardo had to grapple.

Leonardo was an illegitimate son in a society that afforded little opportunity to such an individual. He was a vegetarian who found detestable the idea of becoming “a cage for dead animals,” and an animal rights activist who would purchase birds in the marketplace just to release them into the air. He was left-handed in a time when left-handedness was regarded as sinister. He may have been a homosexual at a time when society, guided largely by the Church, regarded this as sinful. He seems to have had no adult involvement with women, and yet painted the most enigmatic, beguiling and timeless portrait of a woman in all of Western art. In a time when dissecting human bodies was proscribed by law he spent endless hours studying cadavers. Under the intolerable conditions of fetid chambers he dissected human bodies in various states of decay, and created reams of anatomical drawings. He was a pacifist with an aversion to warfare who was employed as a military engineer, designing devices to fortify castles and machinery to breach them, shields to protect the bearer while striking fear into the enemy, and weapons as deadly as any conceived up to that time. As for that vexing curse—the reputation for failing to finish projects—was his mercurial focus and frenetic pace somehow connected to a higher form of attention deficiency hyperactive disorder? Or was it a case of just too many ideas for a finite lifetime? Labels tend to oversimplify phenomena; in Leonardo’s case, they are beyond the absurd.

Early in the book I had expressed, with some trepidation, the faith that presenting science through art and art through science would lead to an understanding of Leonardo’s mind. Who could be presumptuous enough to claim to fathom the mind of someone as complex as Leonardo? Complex he certainly was, but certain components of his psyche have indeed emerged that are surprisingly simple.

In the surviving paintings and notebooks are strewn a jumble of clues. Leonardo’s life can be viewed as a sort of kaleidoscope—creating art of sublime beauty and awesome power, abruptly darting to thoughts on turbulent flow in white water, the geometric patterns of polyhedra, the anatomical studies of man and beast metamorphosing into each other, human flight, spring-driven vehicles, then returning to create one more miraculous work of art. But amidst it all, there are the scattered unfinished projects, endlessly frustrating the most loyal patron.

Leonardo’s unusual style, dizzying in pace and diversity, defined his true modus operandi. The depiction of turbulent water in the daily notebooks is reflected in the golden curls of his subjects. The gently twisted helix, a product of his mathematical musings, is suspiciously evocative of the poses struck by the subjects of his portraits. Regarding painting, he wrote, “It embraces all the forms that are and are not found in nature.” The Platonic solids informed his architectural drawings, were seen as rough sketches in the codices, and are formalized in De divina proportione. There is the golden pyramid, which he appears to have discovered independently of past civilizations, organizing the grouping of the characters in the Virgin of the Rocks (and most likely also in the unfinished lost mural, The Battle of Anghiari). In studying the flight of birds he is at once a naturalist, an aerodynamic engineer, and an artist capturing the motion of wings in serial animated drawings; but his thoughts ultimately speculated on human flight. The helical spiral used earlier to give dynamism to the subjects in his paintings returned as the aerial screw, giving lift to the Leonardo helicopter. Totally preoccupied with manned flight, he examined free-fall and gravitation, issues of fundamental significance for physics. The experiments he performed were magnificently framed, leading to conclusions of constant acceleration, independent of the weight of the falling body. In Leonardo’s entire process there was total symbiosis, total synergy. His personal growth can be seen in all areas of his creativity, but also seen is the evidence of the deliberate and systematic effort, the mental exercises he performed to attain this growth. An “unlettered man,” he left behind lists of daily words to be mastered, in the manner of the autodidact striving for self-improvement.

Interviewed on the occasion of the exhibition Leonardo da Vinci: Master Draftsman at the Metropolitan Museum of Art, Carmen Bambach explained, “As an architect, he begins to think of buildings in terms of plans, sections, elevations, three-dimensional perspectival views of form. Now that sounds to us very normal, but that’s basically Leonardo’s legacy! He is doing this in architecture by 1490, precisely the moment when he undertakes his studies for the human skull, and guess what—he applies the same technique of the section, the dimensional view, and the elevation to create a consistent vocabulary for anatomical description.”³

In this one man’s works there is more cross-semination of ideas from different intellectual worlds than among the works of generations of specialists in any number of disciplines. Herein lies a timeless lesson. Dramatic progress in any field is most effectively catalyzed by cross-fertilization with others. It is certainly not the only way to achieve progress, but it is an extraordinarily effective way. Leonardo is always making connections and driving it all is the freakish trait of never outgrowing an exceedingly fertile childlike curiosity. This is a trait that was also to characterize the scientists Newton and Einstein and the composer Mozart, each at the pinnacle of his respective field. Einstein’s lifelong curiosity is seen in the pronouncement: “The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”

We are all impressed by the rate of learning of young children. In the parlance of psychology children exhibit steep learning curves that allow short doubling and tripling times in their knowledge, but obviously this ability attenuates with age. Along with the insatiably inquisitive intellect, Leonardo retained throughout his life that childlike, virtually vertical learning curve and an endless passion for making connections. In his late fifties he was collaborating with a distinguished mathematician, and separately with a world-class anatomist, to learn all they knew about their respective fields and to raise the quality of the research to much higher levels.

Among his faculties there is the suggestion of a preternatural vision, an ability to take mental snapshots and, in his mind’s eye, to virtually freeze motion. This ability is manifested in his renditions of parabolic trajectories of cannonballs, the flapping wings of birds, and the swirling vortices in water. Although one has always to guard against the temptation to impute supernatural ability and even omniscience to Leonardo, an endowment with such acute vision is not to be dismissed out of hand. Ted Williams, the legendary baseball player renowned for his extraordinary ability to hit a ball, use to claim that he could see the seams on a pitched baseball.

A century ago Sigmund Freud put forth a conjecture that it was the sublimated sexual energy of a homosexual Leonardo that was the source of his superhuman drive, an explanation that subsequent psychoanalysts chose to discredit. Sherwin Nuland makes a compelling case that Freud’s explanation should not be dismissed out of hand. If Freud and Nuland are correct, then the hypothesis of the repressed sexual energy, coupled with unsurpassed intelligence, could also go a long way toward explaining Isaac Newton’s unrivaled productivity in mathematics and physics. But all this is largely speculative; there is little rigorous science in it. So, how much have we really penetrated Leonardo’s mind?

Through the ages both the arts (including the visual arts and literature) and the sciences (the natural sciences and mathematics) have generally evolved gradually, incrementally. But there have also taken place, however infrequently, fundamental changes in the prevailing order—changes of kind rather than degree, akin to a phase transition between vapor and liquid, or liquid and solid. Initiating these “revolutions” have been a finite number of artistic and intellectual revolutionaries—better characterized as transformative geniuses or “magicians.”⁴ Their reasoning processes do not appear to be constrained by the familiar topography of logic, a topography that includes the slopes and valleys. Rather, reaching conclusions for them is reminiscent of leaping from mountaintop to mountaintop. But their works represent no less than the periodic syntheses of the accumulated knowledge in a field; they succeed by defining human limits and elevating the human spirit. Leonardo, Michelangelo, Shakespeare, Newton, Beethoven, and no more than three or four others⁵—all magicians, all towering intellects, all consumed by a fire raging within—also appear to share one common quality: they rarely allow any profound penetration into their psyche; their creative processes are forever steeped in mystery. One thing is clear: the more time one devotes to understanding the mind of the magician, the deeper he recedes into the mist.⁶ This is a curious phenomenon that distinguishes the magician from others, including the ordinary genius.

The expression “struggling with issues of creativity,” a valid topic for the psychologist, is entirely alien to the magician. He creates spontaneously without ever questioning the process personally, often unable to explain the process. There is inspiration; there is intuition. In the parlance of electronics, there may even be a special hard wiring of the brain. For example, Einstein’s brain, although entirely normal in size, was missing the left parietal cortex, and the inferior parietal lobe had grown approximately 15 percent larger than it normally would be, in order to fill the area of the missing operculum.⁷ The inferior parietal lobe is thought to be the site where analytical reasoning and mathematical modeling are carried out. Another noteworthy feature of Einstein’s brain was the existence of an unusually shallow Sylvian fissure, a groove slicing through the brain. This feature, the researchers suggest, allows the brain cells in the area to be packed closer together, permitting more interconnections, and functionally, allowing better cross-referencing of information and ideas. This is of critical importance for analytical processing.

Leonardo, mastering virtually every discipline known in his time and inventing fields that would have to be reinvented hundreds of years later (for example, geology, aeronautical engineering, and automotive engineering), nowhere bothers with the “creativity” question. Ironically, his subjects, including his portraits of the “three women”—Ginevra de’ Benci, Cecilia Gallerani, and the Mona Lisa—and the cast of thirteen characters in the Last Supper—are all unsurpassed psychological portraits. They speak with their facial expressions. The sense of imminent motion, which is achieved by Michelangelo with strained muscles, is achieved by Leonardo with psychology. Leonardo and Shakespeare may well represent the greatest applied psychologists ever, just as Leonardo may well represent the greatest anatomist ever, the latter a sentiment expressed by Walter Pater and echoed by Nuland.

Sherwin Nuland, in that unusually beautiful biography of Leonardo to which I have referred a number of times, wrote a particularly stirring passage:

If he is, as Sir Kenneth Clark so appropriately calls him, “the most relentlessly curious man in history,” he is also the historical figure about whom we are most relentlessly curious.… The dates, the facts, the known events are far fewer than we need, if we are to understand how such a being could have existed. The enigma of the Mona Lisa’s smile is no less than the enigma of her creator’s life force. Or perhaps the smile is in itself Leonardo’s ultimate message to the ages: There is even more to me than you can ever capture; though I have spoken so intimately to you in my notebooks even as I have spoken to myself, I have kept final counsel only with the depths of my spirit and the inscrutable source that has made me possible; seek as you may, I will commune with you only so far; the rest is withheld, for it was my destiny to know things you will never know.⁸

The lore associated with the magician is spawned by our own need to understand such an individual—extroverted and intimate with us at one level, and utterly introverted and inaccessible at another. Perhaps the best explanation is also the simplest, that genius at this scale is entirely isolated by its own genius.⁹ Leonardo da Vinci, paragon artist-scientist-engineer, a cynosure of the Italian Renaissance, more so than any other magician manifests that quality. Does it really matter that Leonardo will always remain this mysterious and mythical creature, who as a part-time artist produced the epitome in art? His works number so few that even the expression “part-time artist” overstates the case. But it is certainly not the quantity that counts. Goethe’s throwaway line, “In art, the best is good enough,” encapsulates the circumstance. Leonardo also developed scientific methodology a century before Galileo, and anticipated future technologies centuries before they were realized. What really matters is that he demonstrated a model of supreme efficacy, a lesson for the ages: seeking connections, conjoining different intellectual worlds, is indeed the source of profound understanding and appreciation in any one of these worlds or in all of them.

What exactly can account for a creature of such transcendent gifts will most likely never be known. It is too easy to say that he was a happy accident of nature and nurture—the concatenation of his mother’s and father’s genes, the effects of the prevailing social, political, and intellectual winds sweeping Renaissance Florence. In Leonardo’s case, there is so much self-nurture. We simply do not know much about his upbringing. He repeatedly admonished others—artists and scientists alike—to “learn from nature, not from each other.” In a timeless irony, we must first learn from him, then observe, and ponder. Avoid taking anything for granted, test it before accepting it. Don’t ever give up the aspiration for personal growth no matter what stage of your life you are in, read incessantly, read critically, even look up words you don’t know with a view toward increasing your vocabulary. Carry a small pad with you and draw sketches (even if you have convinced yourself that you cannot draw). Creating sketches will make you more observant.¹⁰ Observe in the manner of the scientist, savor in the manner of the artist. Record your observations. Experiment, knowing full well that some experiments will fail. But that is how to attain a deeper understanding. It is important to be curious, and important to explore different intellectual worlds, but it is essential to seek their connections. The model that worked magnificently for him will never make any of us another Leonardo, a man called by so many scholars “the greatest genius who ever lived.” But it cannot fail to make us each far more creative and more effective practitioners in the intellectual world that we inhabit.