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setembro 9, 2019

   

The feelings of love, peace, bliss are tuning measures of the spiritual and physical realms resulting from our actions. There is where perfect life lies within.

    

 

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(0) This text presents some sensible quotes extracted from the book Isaac Newton, written by James Gleick, First Edition, Pantheon Books, New York, 2003. This Isaac Newton’s biography I first read in the middle of 2004, and now, in the middle of 2018, I proposed myself a second reading of such invaluable adventure, which is the Newton’s Life! I hope You Enjoy this Summary of Quotes!

    

(1) His name betokens a system of the world. But for Newton himself there was no completeness, only a questioning – dynamic, protean, an unfinished. He never fully detached matter and space from God. He never purged occult, hidden, mystical qualities from his vision of nature. He sought order and believed in order but never averted his eyes from the chaos. He of all people was no Newtonian. Information flowed faintly and perishably then, through the still human species, but he created a method and a language that triumphed in his lifetime and gained ascendancy with each passing century. He pushed open a door that led to a new universe: set in absolute time and space, at once measureless and measurable, furnished with science and machines, ruled by industry and natural law. Geometry and motion, motion and geometry: Newton joined them as one.

   

(2) He did not know what he wanted to be or do, but was not tend sheep or follow the plow and the dung cart. He spent more time gathering herbs and lying with a book among the asphodel and moonwort, out of the household’s sight. He built waterwheels in the stream while his sheep trampled the neighbors’ barley. He watched the flow of water, over wood and around rocks, noting the whorls and eddies and waves, gaining a sense of fluid motion.

   

(3) He felt learning as a form of obsession, a worthy pursuit, in God’s service, but potentially prideful as well. He taught himself a shorthand of esoteric symbols – this served both to save paper and encrypt his writing – and he used it, at a moment of spiritual crisis, to record a catalogue of his sins. Among them were ‘neglecting to pray’, ‘negligence at the chapel’, and variations on the theme of falling short in piety and devotion.

   

(4) He read Aristotle through a mist of changing languages, along with a body of commentary and disputation. The words crossed and overlapped Aristotle’s was a word of substances. A substance possesses qualities and properties, which taken together amount to a form, depending ultimately on its essence. Properties can change; we call this motion. Motion is action, change, and life. It is an indispensable partner of time; the one could not exist without the other. If we understood the cause of motion, we would understand the cause of the world.

   

(5) By the 1660s – new news every day – readers of esoterica knew well enough that the earth was a planet and that the planets orbited the sun. Newton’s notes began to include measurements of the apparent magnitude of stars. Descartes proposed a geometrical and mechanical philosophy. He imagined a universe filled throughout with invisible substance, forming great vortices that sweep the planets and stars forward.

   

(6) He set authority aside. Later he came back to this page and inscribed an epigraph borrowed from Aristotle’s justification for dissenting from his teacher. Aristotle had said, ‘Plato is my friend, but truth my greater friend.’ Newton inserted Aristotle’s name in sequence: ‘Amicus Plato amicus Aristoteles magis amica veritas.’ He made a new beginning.

   

(7) Cambridge in 1664. At end of that year, just before the winter solstice, a comet appeared low in the sky, its mysterious tail blazing toward the west. Newton stayed outdoors night after night, noting a path against the background of fixed stars, watching till it vanished in the light of each dawn, and only then returned to his room, sleepless and disordered. A comet was a frightening portent, a mutable and irregular traveler through the firmament. Nor was that all: rumors were reaching England of a new pestilence in Holland – perhaps from Italy or the Levant, perhaps from Crete or Cyprus. Hard behind the rumors came the epidemic. Three men in London succumbed in a single house; by January the plague, this disease of population density, was spreading from parish to parish, hundreds dying each week, then thousands. Before the outbreak ran its course, in little more than a year, it killed one of every six Londoners.

   

(8) Newton returned home. He built bookshelves and made a small study for himself. He opened the nearly blank thousand-page commonplace book he had inherited from his stepfather and named it his Waste Book. He began filling it with reading notes. These mutated seamlessly into original research. He set himself problems; considered them obsessively; calculated answers, and asked new questions. He pushed past the frontier of knowledge (though he did not know this). The plague year was his transfiguration. Solitary and almost incommunicado, he became the world’s paramount mathematician. Most of the numerical truths and methods that people had discovered, they had forgotten and rediscovered, again and again, in cultures far removed from one another. Mathematics was evergreen. One scion of Homo sapiens could still comprehend virtually all that species knew collectively.

   

(9) Newton was inspired by the leap of Descarte’s Géométrie, a small and rambling text, the third and last appendix to his Discours de la Méthode. This forever joined two great realms of thought, geometry and algebra. Algebra (a ‘barbarous’ art, Descartes said, but it was his subject nonetheless) manipulated unknown quantities as if they were known, by assigning them symbols. Symbols recorded information, spared the memory, just as the printed book did. Indeed, before texts could spread by printing, the development of symbolism had little point. With symbols came equations: relations between quantities, and changeable relations at that. This was new territory, and Descartes exploited it.

   

(10) He taught himself to find real and complex roots of equations and to factor expressions of many terms – polynomials. When the infinite number of points in a curve correspond to the infinite solutions of its equation, then all the solutions can be seen at once, as a unity. Then equations have not just solutions but other properties: maxima and minima, tangent and areas. These were visualized, and they were named.

   

(11) No one understands the mental faculty we call mathematical intuition; much less, genius. People’s brains do not differ much, from one to the next, but numerical facility seems rarer, more special, than other talents. It has a threshold quality. In no other intellectual realm does the genius find so much common ground with the idiot savant. A mind turning inward from the world can see numbers as lustrous creatures; can find order in them, and magic; can know numbers as if personally. A mathematician, too, is a polyglot. A powerful source of creativity is a facility in translating, seeing how the same thing can be said in seemingly different ways. If one formulation doesn’t work, try another. Newton’s patience was limitless. Truth, he said much latter, was “the offspring of silence and meditation”. And he said: “I keep the subject constantly before me and wait ‘till the first dawnings open slowly, by little and little, into full and clear light”.

   

(12) Newton seemed now to possess a limitless ability to generalize, to move from one or a few particular cases to the universe of all cases. Yet it turns out that the human mind, though bounded in a nutshell, can discern the infinite and take its measure.

   

(13) Far away across the country multitudes were dying in fire and plague. Numerologists had warned that 1666 would be the Year of the Beast. Most of London lay in black ruins: fire had begun in a bakery, spread in the dry wind across thatch-roofed houses, and blazed out of control for four days and four nights. The new king, Charles II – having survived his father’s beheading and his own fugitive years, and having outlasted the Lord Protector, Cromwell – fled London with his court. Here at Woolsthorpe the night was strewn with stars, the moon cast its light through the apple trees, and day’s sun and shadows carved their familiar pathways across the wall. Newton understood now: the projection of curves onto flat planes; the angles in three dimensions, changing slightly each day. He saw an orderly landscape. Its inhabitants were not static objects; they were patterns, process and change. What he wrote, he wrote for himself alone. He had no reason to tell anyone. He was twenty-four and he had made tools.

   

(14) We call the Scientific Revolution an epidemic, spreading across the continent of Europe during two centuries: ‘It would come to rest in England, in the person of Isaac Newton,’ said the physicist David Goodstein. ‘On the way north, however, it stopped briefly in France…’ Or a relay race, run by a team of heroes who passed the baton from one to the next: COPERNICUS to KEPLER to GALILEO to NEWTON. Or the overthrow and destruction of the Aristotelian cosmology: a worldview that staggered under the assaults of Galileo and Descartes and finally expired in 1687, when Newton published a book. For so long the earth had seemed the center of all things. The constellations turned round in their regular procession. Just a few bright objects caused a puzzle – the planets, wanderers, like gods or messengers, moving irregularly against the fixed backdrop of stars. In 1543, just before his death, Nicolaus Copernicus, Polish astronomer, astrologer and mathematician, published the great book De Revolutionibus Orbium Celestium (‘On the Revolutions of the Heavenly Spheres’). In it he gave order to the planets’ paths, resolving them into perfect circles; he set the earth in motion and placed an immobile sun at the center of the universe. Johannes Kepler, looking for more order in a growing thicket of data, thousands of painstakingly recorded observations, declared that the planets could not be moving in circles. He suspected the special curves known to the ancients as ellipses. Having thus overthrown one kind of celestial perfection, he sought new kinds, believing fervently in a universe built on geometrical harmony. He found an elegant link between geometry and motion by asserting that an imaginary line from a planet to the sun sweeps across equal areas in equal times. Galileo Galilei took spy-glasses – made by inserting spectacle makers’ lenses into hollow tube – and pointed them upward toward the night sky. What he saw both inspired and disturbed him: moons orbiting Jupiter; spots marring the sun’s flawless face; stars that had never been seen – ‘in numbers ten times exceeding the old and familiar stars.’ He learned, ‘with all the certainty of sense evidence,’ that the moon ‘is not robed in a smooth polished surface but is in fact rough and uneven.’ It has mountains, valleys, and chasms.

   

(15) He read them in a new book from London, titled Micrographia: ‘The Science of Nature has been already too ling made only a work of the Brain and the Fancy. It is now high time that it should return to the plainness and soundness of Observations on material and obvious things.’ The author was Robert Hooke, a brilliant and ambitious man seven years Newton’s senior, who wielded the microscope just as Galileo had the telescope. These were the instruments that penetrated the barrier of scale and opened a view into countries of the very large and the very small. Wonders were revealed there.

   

(16) Newton’s status at Trinity improved. In October 1667 the college elected fellows for the first time in three years. He bought a set of old books on alchemy, along with glasses, a tin furnace, and chemicals: aqua fortis, sublimate, vinegar, white lead, salt and tartar. With these he embarked on a program of research more secret than ever.

   

(17) Barrow showed him a new book from London, Logarithmotechnia, by Nicholas Mercator, a mathematics tutor and member of the Royal Society. It presented a method of calculating logarithms from infinite series and thus gave Newton a shock: his own discoveries, rediscovered. Mercator had constructed an entire book – a useful book, at that – from few infinite series. For Newton these were merely special cases of powerful approach to infinite series he had worked out at Woolsthorpe. Provoked, he revealed Barrow a bit more of what he knew. He drafted a paper in Latin, ‘On Analysis by Infinite Series.’ He also let Barrow post this to another Royal Society colleague, a mathematician, John Collins, but he insisted on anonymity. Only after Collins responded enthusiastically did he let Barrow identify him: ‘I am glad my friends paper giveth you so much satisfaction. His name is Mr Newton; a fellow of our College, & very young… but of an extraordinary genius and proficiency in these things.’ It was the first transmission of Newton’s name south of Cambridge.

   

(18) Like no institution before it, the Royal Society was born dedicated to information flow. It exalted communication and condemned secrecy. ‘So far are the narrow conceptions of a few private Writers, in a dark Age, from being equal to so vast a design,’ its founders declared. Science did not exist – not as an institution, not as an activity – but they conceived it as a public enterprise. They imagined a global network, an ‘Empire in Learning.´ Those striving to grasp the whole fabric of nature ‘ought to have their eyes in all parts, and to receive information from every quarter of the earth, they ought to have a constant universal intelligence: all discoveries should be brought to them: the Treasuries of all former times should be laid open before them.’

   

(19) Far away in Cambridge Newton inhaled all this philosophical news. He took fervid notes. Rumors of lunar influence: ‘Oysters & Crabs are fat at the new moone & leane at the full.’ Then in 1671 he heard directly from the voice of the Royal Society. ‘Sr’, Oldenburg wrote, ‘Your Ingenuity is the occasion of this addresse by a hand unknowne to you…’ He said he wished to publish an account of Newton’s reflecting telescope. He urged Newton to take public credit. This peculiar historical moment – the manners of scientific publication just being born – was alert to the possibilities of plagiarism.

   

(20) It led him (or so he reported) to the Experimentum Crucis – the signpost at a crossroads, the piece of experience that shows which path to trust. Years before, in his earliest speculation, he had asked himself, ‘Try if two Prismas the one casting blue upon the other’s red doe not produce a white.’ They did not. Blue light stayed blue and red stayed red. Unlike white (Newton deduced) those colors were pure. ‘And so the true cause of the length of that Image was detected,’ Newton declared triumphantly – ‘that Light consists of Rays differently refrangible.’ Some colors are refracted more, and not by any quality of the glass but their own predisposition. Color is not a modification of light but an original, fundamental property. Above all: white light is a heterogeneous mixture. ‘But the most surprising, and wonderful composition was that of Whiteness. There is no one sort of Rays which alone can exhibit this. ‘Tis ever compounded, and to its composition are requisite all the aforesaid primary Colours, mixed in due proportion. I have often in Admiration beheld, that all the Colours of the Prisme being made to converge, and thereby to be again mixed,… reproduced light, intirely and perfectly white.’

   

(21) In 1675 Newton journeyed to London and finally appeared at the Royal Society. He met in person these men who had till then been friends and antagonists twice removed, their spirits channeled through Oldenburg’s mail.

   

(22) This sheaf of papers posted to Oldenburg blended calculation and faith. It was a work of the imagination. It sought to reveal nothing less than the microstructure of matter. For generations it reached no further than the few man who heard it read and then raptly debated it through all the meetings of the Royal Society from December 1675 to the next February. Newton had peered deeper into the core of the matter than could be justified by the power of the microscopes. Through a series of experiments and associations he seemed to feel nature’s fundamental particles just beyond the edge of the vision. Indeed, he predicted that instruments magnifying three or four thousand times might bring atoms into view.

   

(23) Irregular motions, he emphasized – and he saw no way to explain them mechanically, purely in terms of matter pressing on matter. It was no static world, no orderly world he sought to understand now. Too much to explain at once: the world in flux; a world of change and even chaos. He gave out poetry: ‘For nature is a perpetuall circulatory worker, generating fluids out of solids, and solids out of fluids, fixed things out of volatile, & volatile out of fixed, subtile out of gross, and gross out of subtile, Some things to ascend & make the upper terrestriall juices, Rivers and the Atmosphere; and by consequence others to descend…’ The ancients had often supposed the existence of ether, a substance beyond the elements, purer than air or fire. Newton offered the ether as a hypothesis now, describing it as a ‘Medium much of the same constitution with the air, but far rarer, subtiler & more strongly Elastic.’ As sound is a vibration of the air, perhaps there are vibrations of the ether – these would be swifter and finer. He estimated the scale of sound waves at a foot or half-foot, vibrations of ether at less than a hundred thousandth of an inch.

   

(24) Oldenburg – no friend to Hooke – chose to surprise him with a public reading of Newton’s rejoinder at the next Royal Society meeting. Finally, after years of jousting by proxy, Hooke decided to take pen in hand and address his adversary personally. He adopted a meek and philosophical tone. Newton’s famous reply came a fortnight later. He called Hooke a ‘true Philosophical spirit.’ And then, for the matter of their dispute, he put on record a finely calibrated piece of faint praise and lofty sentiment: ‘What Des-Cartes did was a good step. You have added much several ways, & especially in taking the colours of thin plates into philosophical consideration. If I have seen further it is by standing on the sholders of Giants.’ The private philosophical dialogue between Newton and Hooke never took place. Almost two years passed before they communicated again at all. By then Oldenburg had died, Hooke had succeeded him as Secretary of the Royal Society, and Newton had withdrawn ever more deeply into the seclusion of his Trinity chambers.

   

(25) His devotion to philosophical matters grew nonetheless. He built a special chimney to carry away the smoke and the fumes. No one could understand till centuries latter – after his papers, long hidden and scattered, began finally to be reassembled – that he had been not only a secret alchemist but, in the breadth of his knowledge and his experimentation, the peerless alchemist of Europe. Much later, when the age of reason grew mature, a fork was seen to have divided the road to the knowledge of substances. On one path, chemistry: a science that analyzed the elements of matter with logic and rigor. Left behind, alchemy: a science and an art, embracing the relation of the human to the cosmos; invoking transmutation and fermentation and procreation. Alchemists lived in a realm of exuberant, animated forces. In the Newtonian world of formal, institutionalized science, it became disreputable. But Newton belonged to the pre-Newtonian world. Alchemy was in its heyday. Newton was a mechanist and a mathematician to his core, but he could not believe in a nature without spirit.

   

(26) To alchemists nature was alive with process. Matter was active, not passive; vital, not inert. Many processes began in the fire: melting, distilling subliming, and calcining. Newton studied them and practiced them, in his furnaces of tin and bricks and firestones. In sublimation vapors rose from the ashes of burned earths and condensed again upon cooling. In calcination fire converted solids to dust; ‘be you not weary of calcination,’ the alchemical fathers had advised; ‘calcination is the treasure of a thing.’ When a crimson-tinged earth, cinnabar, passed through the fire, a coveted substance emerged: ‘silvery water’ or ‘chaotic water’ – quicksilver. It was liquid and a metal at once, lustrous white, eager to form globules. Some though a wheel rimmed with quicksilver could turn unaided – perpetual motion. Alchemists knew quicksilver as Mercury (as iron was Mars, copper Venus, and gold the Sun); in their clandestine writings they employed the planet’s ancient symbol. Or they alluded to quicksilver as ‘the serpents.’

   

(27) Rather than turn away from what he could not explain, he plunged in more deeply. Dry powders refuse to cohere. Flies walked on water. Heat radiated through a vacuum. Metallic particles impregnated mercury. Mere though caused muscles to contract and dilate. There were forces in nature that he would not be able to understand mechanically, in terms of colliding billiard balls or swirling vortices. They were vital, vegetable, sexual forces – invisible forces of spirit and attraction. Later, it had been Newton, more than any philosopher, who effectively purged science of the need to resort to such mystical qualities. For now, he needed them. When he was not stoking his furnaces and stirring his crucibles, he was scrutinizing his growing hoard of alchemistical literature. By the century’s end, he had created a private Index chemicus, a manuscript of more than a hundred pages, comprising more than five thousand individual references to writings on alchemy spanning centuries. This, along with his own alchemical writing, remained hidden long after his death.

   

(28) He had seldom returned home to Lincolnshire since the sojourn of the plague years, but in the spring of 1679 his mother succumbed to a fever. He left Cambridge and kept vigil with her over days and nights, till she died. He, the first-born son, not his half-brothers or sisters, was her heir and executor, and he buried her in the Colsterworth churchyard next to the grave of his father.

   

(29) In the next year a comet came. In England it arose faint in the early morning sky for a few weeks in November till it approached the sun and faded in the dawn. Few saw it. A more dramatic spectacle appeared in the nights of December. Newton saw it with naked eye on December 12: a comet whose great tail, broader than the moon, stretched over the full length of King’s College Chapel. He tracked it almost nightly through the first months of 1681. A young astronomer travelling to France, Edmond Halley, a new Fellow of the Royal Society, was amazed at its brilliance. Robert Hooke observed it several times in London. Across the Atlantic Ocean, where a handful of colonists were struggling to survive on a newfound continent, Increase Mather delivered a sermon, ‘Heaven’s Alarm to the World,’ to warn Puritans of God’s displeasure.

 

(30) Hooke and Newton had both jettisoned the Cartesian notion of vortices. They were explaining the planet’s motion with no resort to ethereal pressure (or, far that matter, resistance). They had both come to believe in a body’s inherent force – its tendency to remain at rest or in motion – a concept for which they had no name. They were dancing around a pair of questions, one the mirror of the other: What curve will be traced by a body orbiting another in an inverse-square gravitational field? (An ellipse). What gravitational force law can be inferred from a body orbiting another in a perfect ellipse? (An inverse-square law). Hooke finally did put this to Newton: ‘My supposition is that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall’ – that is, inversely as the square of distance. Hooke had finally formulated the problem exactly. He acknowledged Newton’s superior powers. He set forth a procedure: find the mathematical curve, suggest a physical reason. But he never received a reply.

   

(31) Four years latter Edmond Halley made a pilgrimage to Cambridge. Halley had been discussing planetary motion in coffee-houses with Hooke and the architect Cristopher Wren. Some boasting ensued. Halley himself had worked out (as Newton had in 1666) a connection between an inverse-square law and Kepler’s rule of periods – that the cube of the planet’s distance from the sun varies as the square of its orbital year. Wren claimed that he himself had guessed at the inverse-square law years before Hooke, but could not quite work out the mathematics. Hooke asserted that he could show how to base all celestial motion on the inverse square law and that he was keeping the details secret for now, until more people had tried and failed; only then would they appreciate his work. Halley doubted that Hooke knew as much as he claimed. Halley put the question to Newton directly in August 1684: supposing an inverse square law of attraction toward the sun, what sort of curve would a planet make? Newton told him: an ellipse. He said he had calculated this long before. He would not give Halley the proof – he said he could not lay his hands on it – but promised to redo it and send it along.

   

(32) He could not, or would not, give Halley a simple answer. First he sent a treatise of nine pages, ‘On the Motions of Bodies in Orbit.’ It firmly tied a centripetal force, inverse proportional to the square of distance, not only to specific geometry of the ellipse but to all Kepler’s observations of orbital motion. Halley rushed back to Cambridge. His one copy had become an object of desire in London. Flamsteed complained: ‘I believe I shall not get a sight of [it] till our common friend Mr Hooke & the rest of the towne have been first satisfied.’ Halley begged to publish the treatise, and he begged for more pages, but Newton was not finished.

   

(33) The alchemical furnaces went cold; the theological manuscripts were shelved. A fever possessed him, like none since the plague years. He ate mainly in his room, a few bites standing up. He wrote standing at his desk. When he did venture outside, he would seem lost, walk erratically, turn and stop for no apparent reason, and disappear inside once again. Though he had dropped alchemy for now, Newton had learned from it. He embraced invisible forces. He knew he was going to have to allow planets to influence one another from distance. He was writing the principles of philosophy. But not just that: the mathematical principles of natural philosophy. ‘For the whole difficult of philosophy’, he wrote, ‘seems to be to discover the forces of nature from the phenomena of motions and then demonstrate the other phenomena from these forces’. The planets, the comets, the moon, the sea. He promised a mechanical program – no occult qualities. He promised proof. Yet there was mystery in forces still.

   

(34) Our eyes perceived only relative motion: a sailor’s progress along his ship, or the ship’s progress on the earth. But the earth, too, moves, in reference to space – itself immovable because it is purely mathematical, abstracted from our senses. Of time and space he made a frame for the universe and a credo for a new age.

   

(35) ‘It was ordered, that a letter of thanks be written to Mr Newton’, recorded Halley, as clerk of the Royal Society, on April 28, 1686, ‘…and that in the meantime the book be put into the hands of Mr Halley’. Only Halley knew what was in ‘the book’ – a first sheaf of manuscript pages, copied in Cambridge by Newton’s amanuensis and dispatched to London with the grand title Philosophiae Naturalis Principia Mathematica. Halley had been forewarning the Royal Society: ‘a mathematical demonstration of the Copernican hypothesis’; ‘makes out all the phenomena of the celestial motions by the only supposition of a gravitation towards the centre of the sun decreasing as the squares of the distances therefrom reciprocally’. Hooke heard him.

   

(36) Without further ado, having defined his terms, Newton announced the laws of motion. Law 1. ‘Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed’. A cannonball would fly in a straight line forever, were it not for air resistance and the downward force of gravity. The first law stated, without naming, the principle of inertia, Galileo’s principle, refined. Two states – being at rest and moving uniformly – are to be treated as the same. If a flying cannonball embodies a force, so does the cannonball at rest. Law 2 ‘A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed’. Force generates motion, and these are quantities, to be added and multiplied according to mathematical rules. Law 3 ‘To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction’. If a finger presses a stone, the stone presses against the finger. If a horse pulls a stone, the stone pulls the horse. Actions are interactions – no preference of vantage point to be assigned. If the earth tugs at the moon, the moon tugs back. He presented these axioms, to serve as the foundation for an edifice of reasoning and proof. ‘Law’ – lex – was a strong and peculiar choice of words. Bacon had spoken of laws, fundamental and universal. It was no coincidence that Descartes, in his own book called Principles of Philosophy, had attempted a set of three laws, regulae quaedam sive leges naturae, specifically concerning motion, including a law of inertia. For Newton, the laws formed the bedrock on which a whole system would lie. A law is not a cause, yet it is more than a description. A law is a rule of conduct: here God’s law, for every piece of creation. A law is to be obeyed, by inanimate particles as well as sentient creatures. Newton chose to speak not so much of God as of nature. ‘Nature is exceedingly simple and conformable to herself. Whatever reasoning holds for great motions, should hold for lesser ones as well’.

   

(37) Book III gave The System of the World. It gathered together the phenomena of the cosmos. It did this flaunting an exactitude unlike anything in the history of philosophy, Phenomenon 1: the four known satellites of Jupiter. Newton had four set of observations to combine. He produced some numbers: their orbital periods in days, hours, minutes, and seconds, and their greatest distance from the planet, to the nearest thousandth of Jupiter’s radius. He did the same for the five planets, Mercury, Venus, Mars, Jupiter, and Saturn. And for the moon.

   

(38) He said he had tested gold, silver, lead, glass, sand, salt, wood, water, and wheat – suspending them in a pair of identical pendulums so precisely that he could detect a difference of one part in a thousand. Furthermore, he proposed, the heavenly bodies must perturb one another: Jupiter influencing Saturn’s motion, the sun influencing the earth, and the sun and moon both perturbing the sea. ‘All the planets are heavy toward one another’. He pronounced: ‘It is now established that this force is gravity, and therefore we shall call it gravity from now on’.

   

(39) These elements meshed and turned together like the parts of a machine, the work of a perfect mechanic, like an intricate clock, a metaphor that occurred to many as news of the Principia spread. Yet Newton himself never succumbed to this fantasy of pure order and perfect determinism. Continuing to calculate where calculation was impossible, he saw ahead to the chaos that could emerge in the interactions of many bodies, rather than just two or three. The center of the planetary system, he saw, is not exactly the sun, but rather than just two or three. The center of the planetary system, he saw, is not exactly the sun, but rather the oscillating common center of gravity. Planetary orbits were not exact ellipses after all, and certainly not the same ellipse repeated. ‘Each time a planet revolves it traces a fresh orbit, as happens also with the motion of the Moon, and each orbit is dependent upon the combined motions of all planets, not to mention their actions upon each other’, he wrote.

   

(40) Yet he solved another messy, bewildering phenomenon, the tides. He had assembled data, crude and scattered though they were. Samuel Sturmy had recorded observations from the mouth of the River Avon, three miles below Bristol. Samuel Colepress had measured the ebb and flow in Plymouth Harbor. Newton considered the Pacific Ocean and the Ethiopic Sea, bays in Normandy and at Pegu in the East Indies. Halley himself had analyzed observations by sailors in Batsha Harbor in the port of Tunking in China. None of these lent themselves to a rigorous chain of calculation, but the pattern of two high tides per twenty-five hours was clear and global. Newton marshaled the data and made his theoretical claim. The moon and sun both pull the seas; their combined gravity creates the tides by raising a symmetrical pair of bulges on opposite sides of the earth.

   

(41) He had declared at the outset that his mission was to discover the forces of nature. He deduced forces from celestial bodies’ motion, as observed and recorded. He made a great claim – the System of the World – and yet declared his program incomplete. In fact, incompleteness was its greatest virtue. He bequeathed to science, that institution is its throes of birth, a research program, practical and open-ended. There was work to do, predictions to be computed and verified.

   

(42) Besides mathematics Newton has returned to the most tortuous unfinished problem in the Principia: a full theory of the moon’s motion. This was no mere academic exercise; given a precise recipe for predicting the moon’s place in the sky, sailors with handheld astrolabes should finally be able to calculate their longitude at sea. A lunar theory should follow from Newton’s theory of gravity: the ellipse of the lunar orbit crosses the earth’s own orbital plane at a slant angle; the sun’s attraction twists the lunar orbit, apogee and perigee revolving over a period of roughly nine years. But the force of solar gravity itself varies as the earth and moon, in their irregular dance, approach and recede from the sun. With a revised edition of the Principia in mind, he needed more data, and this meant calling upon the Astronomer Royal. Late in the summer of 1694 he boarded a small boat to journey down the river Thames and visit, for the first time, Flamsteed in Greenwich. He pried loose fifty lunar observations and a promise of one hundred more.

   

(43) He did ultimately produce a practical formula for calculating the moon’s motion: a hybrid sequence of equations and measurements that appeared first in 1702, as five Latin pages inside David Gregory’s grand Astronomiae Elementa. Gregory called it Newton’s theory, but in the end Newton had omitted any mention of gravitation and buried his general picture under a mass of details. Halley quickly reprinted Newton’s text as a booklet in English, saying, ‘I though it would be a good service to our Nation… For as Dr. Gregory’s Astronomy is a large and scarce Book, it is neither everyone’s Money that can purchase it’. Halley hailed the theory’s exactness and hoped to encourage people to use it, but ‘the Famous Mr. Isaac Newton’s Theory of the Moon’ was little noted and quickly forgotten. Newton abandoned his Cambridge cloister for good in 1696. His smoldering ambition for royal preferment was fulfilled. Trinity had been his home for thirty-five years, but he departed quickly and left no friends behind. As he emphatically told Flamsteed, he was now occupied by the King’s business. He had taken charge of the nation’s coin.

   

(X) Reader’ Personal Note :: 1696 was a turning point in Newton’s life. He left not only Trinity, but the Natural Philosophy principles. From Good to evil, he became obsessively engaged in politics and power struggles. Newton embraced the dark side of himself from 1696 on. He was fifty-three.

 

(44) The new Chancellor of the Exchequer, Charles Montague, set a radical program in motion: a complete recoinage – all old coins to be withdrawn from circulation. Montague had known Newton at Cambridge and with this support the king named him Warden of the Mint in April 1696, just as the recoinage began. Newton supervised an urgent industrial project, charcoal fires burned around the clock, teams of horses and men crowding in upon one another, garrisoned soldiers standing watch. It was tumultuous time at the Tower and in London: the terms of the recoinage had strangled the supply of money essential to daily commerce and, not incidentally, effected a transfer of national wealth from the poor to the rich. Newton grew rich himself, as Warden and then, from 1770 onward, Master. (From his first months he complained to the Treasury about his remuneration, but as Master he received not only a salary of 500 pounds but also a percentage of every pound coined, and these sums were far greater). He found a house in Jermyn Street, bought luxurious, mainly crimson furniture, engaged servants, and invited his twenty-year-old niece, Catherine Barton, the daughter of his half-sister, to live with him as housekeeper. She became renowned in London society for beauty and charm. Jonathan Swift was an admirer and frequent visitor. Within a half-decade she became the lover of Newton’s patron Montague, by now the Earl of Halifax.

   

(45) A portent of future trouble came from Leibnitz, by second hand: ‘to Mr. Newton, that man of great mind, my most devoted greeting’ – and ‘another matter, not only did I recognize that the most profound Newton’s Method of Fluxions was like my differential method, but I said so… and I also informed others’. In passing this on, the elderly mathematician John Wallis begged Newton to let some of his treasure out from darkness. His (Newton’s) return to the Royal Society ha waited, all these years, for Hooke’s exit. Hooke died in March 1703; within months Newton was chosen president. Past presidents had often been honorary, political figures. Newton seized power now and exercised it authoritatively.

   

(46) With Hooke dead, he also finally took Wallis’s advice and released for publication his second great work – in English, rather than Latin, and, more important, in prose rather than mathematics. This time he needed no editor. He had three ‘books’ based on his work from thirty years earlier on the nature of light and color: the geometry of reflection and refraction; how lenses form images; and the workings of the eye and the telescope. The origin of whiteness; prisms; the rainbow. He added much more, in the form of ‘Queries’: queries on heat; queries on the ether; occult qualities, action at a distance, inertia. For good measure he included a pair of mathematical papers, the first he ever published. He titled the book Opticks – or, a Treatise on the Reflexions, Refractions, Inflexions and Colours of Light. He presented it to the Royal Society with the ‘Advertisement’ in which he explained why he had suppressed this work since 1675. The reason: ‘To avoid being engaged in Disputes’. Not only had Hooke died but the world had changed. Newton’s style, integrating theories with mathematical experimentation, had become familiar to philosophers, and they accepted readily the same propositions that had stirred skepticism and scorn in the 1670s.

   

(47) By now he and Newton were in open conflict. Leibniz, four years Newton’s junior, had seen far more of the world – a stoop-shouldered, tireless man of affairs, lawyer and diplomat, cosmopolitan traveler, courtier to the House of Hanover. The two men had exchanged their first letters – probing and guarded – in the late 1670s. In the realm of mathematics, it was paradoxically difficult to stake effective claims to knowledge without disclosure.

   

(48) Now, decades later, Newton had a purpose in publishing his pair of mathematical papers with the Opticks, and he made his purpose plain. In particular, ‘On the Quadrature of Curves’ laid out for the first time his method of fluxions. In effect, despite the utterly different notation, this was Leibniz’s differential calculus. Where Leibniz worked with successive differences, Newton spoke of rates of flow changing throughout successive moments of time. Leibniz was chunklets – discrete bits. Newton was the continuum. A deep understanding of the calculus ultimately came to demand a mental bridge from one to the other, a translation and reconciliation of two seemingly incompatible symbolic systems. Newton declared not only that he had made his discoveries by 1666 but also that he had described them to Leibniz. He released the correspondence, anagrams and all. Soon an anonymous counterattack appeared in Acta Eruditorum suggesting that Newton had employed Leibniz’s methods, though calling them ‘fluxions’ instead of ‘Leibnizian differences’. This anonymous reviewer was Leibniz. Newton’s disciples fired back in the Philosophical Transactions, suggesting that it was Leibniz who, having read Newton’s description of his methods, then published ‘the same Arithmetic under a different name and using a different notation’. Between each of these thrusts and parties, years passed. But a duel was under way. Partisans joined both sides, encouraged by tribal loyalties more than any real knowledge of the documentary history. Scant public record existed on either side. The principals joined the fray openly in 1711. A furious letter from Leibniz arrived at the Royal Society, where it was read aloud and ‘deliver’d to the President to consider the contents thereof’. The society named a committee to investigate ‘old letters and papers’. Newton provided these. Early correspondence with John Collins came to light; Leibniz had seen some of it, all those years before. The committee produced a document without precedent: a detailed, analytical history of mathematical discovery. No clearer account of the calculus existed, but exposition was not the point; the report was meant as a polemic, to condemn Leibniz, accusing him of a whole congeries of plagiarisms. It judged Newton’s method to be not only the first – ‘by many years’ – but also more elegant, more natural, more geometrical, more useful, and more certain. It vindicated Newton eloquence and passion, and no wonder: Newton was its secret author. The Royal Society published it rapidly. It also published a long assessment of the report, in the Philosophical Transactions – a diatribe, in fact. This, too, was secretly composed by Newton.

   

(49) Newton understood the truth full well: that he and Leibniz had created the calculus independently. Leibniz had not been altogether candid about what he had learned from Newton – in fragments, and through proxies – but the essence of invention was his. Newton has made his discoveries first, and he had discovered more, but Leibniz had done what Newton had not: published his work for the world to use and to judge. It was secrecy that spawned competition and envy. The plagiarism controversy drew its heat from the gaps in the dissemination of the knowledge. In a young and suddenly fertile field like the mathematics of the seventeenth century, discoveries had lain waiting to be found again and again by different people in different places.

   

(50) The obsessions of Newton’s later years disappointed modernity in some way. Just when science began to coalesce as an English Institution, Newton made himself its autocrat. He purged the Royal Society of all remnants of Hooke. He gained authority over the Observatory and wrested from Flamsteed the astronomer’s own life’s work, a comprehensive catalogue of the stars.

    

(51) He had concealed so much, till the very end. As his health declined, he kept writing. His niece’s new husband, John Conduitt, saw him in his last days working in near darkness on an obsessional history of the world – he wrote at least a dozen drafts – The Chronology of Ancient Kingdoms Amended. In his chambers, after a painful fit of gout, he sat with Conduitt before a wood fire and talked about comets. In his deathbed he refused the sacrament of the church. Nor could a pair of doctors ease his pain. He died early Sunday morning, March 19, 1727. On Thursday the Royal Society recorded in its Journal Book, ‘The Chair being Vacant by the death of Sir Isaac Newton there was no Meeting this Day’.

   

(52) In eight-four years he had amassed a fortune: household furniture, much of it upholstered in crimson; crimson curtains, a crimson mohair bed, and crimson cushions; a clock; a parcel of mathematical instruments and chemical glasses; several bottles of wine and cider; thirty-nine silver medals and copies in plaster of Paris; a vast library with nearly two thousand books and his many secret manuscripts; gold bars and coins – the whole state valued at 31,821 pounds, a considerable legacy. Yet he left no will.

   

(53) The relativity of Einstein appeared as a revolutionary assault on absolute space and time. Motion distorts the flow of time and the geometry of space, he found. Gravity is not just a force, ineffable, but also a curvature of space-time itself. Mass, too, had to be redefined; it became interchangeable with energy. Einstein did shake space-time loose from pins which Newton had bound it, but he lived in Newton’s space-time nonetheless: absolute in its geometrical rigor and its independence of the world we see and feel. He happily brandished the tools Newton had forged. Einstein’s is no everyday or psychological relativity. ‘Let no one suppose’, he said in 1919, ‘that the mighty work of Newton can really be superseded by this or any other theory. His great and lucid ideas will retain their unique significance for all time as the foundation of our whole modern conceptual structure in the sphere of natural philosophy’.

   

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(23r0) Simple excerpts from Siddhartha, an Indian tale, by Hermann Hesse.

 

(0n3) Tenderly, he looked into the rushing water, into the transparent green, into the crystal lines of its drawing, so rich in secrets. Bright pearls he saw rising from the deep, quiet bubbles of air floating on the reflecting surface, the blue of the sky being depicted in it. With a thousand eyes, the river looked at him, with green ones, with white ones, with crystal ones, with sky blue ones. How did he love this water, how did it delight him, how grateful was he to it! In his heart he heard the voice talking, which was newly awaking, and it told him: Love this water! Stay near it! Learn from it! Oh yes, he wanted to learn from it, he wanted to listen to it. He who would understand this water and its secrets, so it seemed to him, would also understand many other things, many secrets, all secrets.

 

But out of all secrets of the river, he today only saw one, this one touched his soul. He saw: this water ran and ran, incessantly it ran, and was nevertheless always there, was always at all times the same and yet new in every moment! Great be he who would grasp this, understand this! He understood and grasped it not, only felt some idea of it stirring, a distant memory, divine voices.

 

(7w0) In a friendly manner, he lived side by side with Vasuveda, and occasionally they exchanged some words. Vasuveda was no friend of words; rarely, Siddhartha succeeded in persuading him to speak. “Did you,” so he asked him one time, “did you too learn that secret from the river: that there is no time?”

 

Vasuveda’s face was filled with a bright smile. “Yes, Siddhartha,” he spoke. “It is this what you mean, isn’t it: that the river is everywhere at once, at the source and at the mouth, at the waterfalls, at the ferry, at the rapids, in the sea, in the mountains, everywhere at once, and that there is only the present time for it, not the shadow of the past, not the shadow of the future?

 

“This is it,” said Siddhartha. “And when I had learned it, I looked at my life, and it was also a river, and the boy Siddhartha was only separated from the man Siddhartha and from the old man Siddhartha by a shadow, not by something real. Also, Siddhartha’s previous births were no past, and his death and his return to Brahma was no future. Nothing was, nothing will be; everything is, everything has existence and is present.”

 

(7hr33) Quoth Siddhartha, smiling from his old eyes: “Do you call yourself a searcher, oh venerable one, though you are already of an old in years and are wearing the robe of Gotama’s monks?” “It’s true, I’m old,” spoke Govinda, “but I haven’t stopped searching. Never I’ll stop searching, this seems to me my destiny. You too, so it seems to me, have been searching. Would you like to tell me something, oh honourable one?”

 

Quoth Siddhartha: What should I possibly have to tell you, oh venerable one? Perhaps that you’re searching far too much? That in all that searching, you don’t find the time for finding?” “How come?” asked Govinda.

 

“When someone is searching,” said Siddhartha, “then it might easily happen that the only thing his eyes still see is that what he searches for, that he is unable to find anything, to let anything enter his mind, because he always thinks of nothing but the object of his search, because he has a goal, because he is obsessed by the goal. Searching means: having a goal. You, oh venerable one, are perhaps indeed a searcher, because, striving for your goal, there are many things you don’t see, which are directly in front of your eyes.”

 

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This article presents some quotes carefully extracted from the book ‘The science of Leonardo: inside the mind of the great genius of the Renaissance’, Fritjof Capra, Anchor Books, 1st Edition, December 2008.

          

1) …in the collection of his notes on painting, known as Trattato della pittura (Treatise on Painting), he writes: The science of painting extends to all the colors of the surface of bodies, and to shapes of the bodies enclosed by those surfaces… [Painting] with philosophic and subtle speculation considers all the qualities of forms… Truly this is science, the legitimate daughter of nature, because painting is born of nature.

          

2) Nature as a whole was alive for Leonardo. He saw the patterns and processes in the microcosm as being similar to those in the macrocosm. He frequently drew analogies between human anatomy and the structure of the Earth, as in the following beautiful passage from Codex Leiscester: We may say that the Earth has vital force of growth, and that its flesh is the soil; its bones are the successive strata of the rocks which form the mountains; its cartilage is the porous rock, its blood the veins of the waters. The lake of blood that lies around the heart is the ocean. Its breathing is the increase and decrease of the blood in the pulses, just as the in the Earth it is the ebb and flow of the sea.

            

3) As a painter, Leonardo felt that he should use gestures to portray the frames of mind and emotions that provoked them. He asserted that, in the painting of a human figure, the most important task was to “express in gesture the passion of its soul.” Indeed, to portray the body’s expression of the human spirit was the artist’s highest aspiration, in Leonardo’s view. And it was one in which he himself excelled, as the paintings of his mature period attest. As art historian Irma Richter explains in the introductory comments to her classical selections from the Notebooks, for Leonardo, “the human body was an outward and visible expression of the soul; it was shaped by its spirit.” We shall see that this vision of soul and spirit, unmarred by the mind-body split that René Descartes would introduce in the seventeenth century, is perfectly consistent with the conception of the “embodied mind” in today’s cognitive science.

         

4) Leonardo did not pursue science and engineering to dominate nature, as Francis Bacon would advocate a century later. He had a deep respect for life, a special compassion for animals, and great awe and reverence for nature’s complexity and abundance. While a brilliant inventor and designer himself, he always thought that nature’s ingenuity was vastly superior to human design. He felt that he would be wise to respect nature and learn from her. It is an attitude that has reemerged today in the practice of ecological design. Leonardo’s synthesis of art and science is infused with a deep awareness of ecology and systems thinking. It is not surprising that he spoke with great disdain of the so-called “abbreviators”, the reductionists of his time: The abbreviators of works do injury to knowledge and to love… Of what value is he who, in order to abbreviate the parts of those things of which he professes to give complete knowledge, leaves out the greater part of the things of which the whole is composed?… Oh human stupidity!… You don’t see that you are falling into the same error as one who strips a tree of its adornment of branches full of leaves, intermingled with fragrant flowers and fruit, in order to demonstrate that the tree is good for making planks.

             

5) Leonardo’s physical beauty in his youth and middle aged years must have been exceptional, as it is mentioned by all his contemporary commentators, even though this was not customary at the time. An anonymous writer called the Anonimo Gaddiano exclaimed, “He was so unusual and many-sided that nature seemed to have produced a miracle in him, not only in the beauty of his person, but in many gifts with which she endowed him and which he fully mastered.” Others marveled at the unique combination of physical strength and grace seemed to embody. Many authors, including Vasari, referred to him with the ultimate epithet – il divino.

          

6) Throughout his life, Leonardo displayed an air of serene self-confidence, which helped him to overcome professional setbacks and disappointments with equanimity and allowed him to calmly pursue his research even during times of great political turbulence. He was aware of his unique genius and skill, yet he never boasted about them. Nowhere in his Notebooks does he vaunt the originality of his inventions and discoveries, nor does he flaunt the superiority of his ideas, even as he explains how they differ from traditional beliefs. This lack of arrogance and ego was remarkable indeed. Another quality that distinguished him was his passion for life and for all living things.

            

7) The artist’s fascination with the grotesque forms also led him to devise the most extravagant, and often quite macabre, practical jokes, which delighted the courtiers in Milan and Rome. At the papal court in Rome, Vasari tells us that Leonardo obtained a large lizard to which he attached “with a mixture of quicksilver some wings, made from the scales stripped from other lizards, which quivered as it walked along. Then, after he had given it eyes, horns, and a beard he tamed the creature, and keeping it in a box he used to show it to his friends and frightened the life out of them.”

           

8) During Leonardo’s time, the term “genius” did not have our modern meaning of a person endowed with extraordinary intellectual and creative powers. The latin word genius originated in Roman religion, where it donate the spirit of the gens, the family. It was understood as a guardian spirit, first associated with individuals and then also with people and places. The extraordinary achievements of artists and scientists were attributed to their genius, or attendant spirit. This meaning of genius was prevalent throughout the Middle Ages and the Renaissance. In the eighteen century, the meaning of the word changed to its familiar modern meaning to denote these individuals themselves, as in the phrase “Newton is a genius”.

            

9) The first is an intense curiosity and great enthusiasm for discovery and understanding. This was indeed an outstanding quality of Leonardo, whom Kenneth Clark called “the most relentlessly curious man in history.” Another striking sign of genius is an extraordinary capacity of intense concentration over long periods of time. Isaac Newton apparently was able to hold a mathematical problem in his mind for weeks until it surrendered to his mental powers. When asked how he made his remarkable discoveries, Newton is reported to have replied, “I keep the subject constantly before me and wait until the first dawnings open little by little into the full light.” Leonardo seems to have worked in a very similar way, and most of the time not only on one but on several problems simultaneously.

           

10) Indeed the Italian humanists were so bold as to compare artistic creations to the creations of God. This comparison was first applied to the creativity of poets, and was then extended, especially by Leonardo, to the painter’s creative power: If the painter wants to see beauties that make him fall in love, he is the lord who can generate them, and if he wants to see monstrous things that frighten, or funny things that make him laugh, or things that truly arouse compassion, he is their lord and God… In fact, whatever there is in the universe, by essence, presence, or imagination, he has it first in his mind and than in his hands.

           

11) He wished to achieve relief through the scientific use of the light and shade. According to Leonardo, such an achievement is “the soul of painting”. Leonardo’s technique of using light and shade to give his figures “great vigor and relief,” as Vasari put it, culminated in his celebrated creation of sfumato, the subtle melting of shades that eventually became the unifying principle of the paintings. “Leonardo’s sfumato was the power behind the poetry of his paintings,” Arasse claimed, “and the mystery that seems to emanate from them.”

          

12) Leonardo could have not developed his mastery of chiaroscuro, nor his characteristic sfumato style, without a major advance in Renaissance paint – the use of oil-based paints. Oil painting makes it possible to put layers of paint on top of each other without blurring the colors (provided the layers are allowed to dry individually), to go back over work again, and to mix paints at ease, all of which were essential for Leonardo to achieve his special effects of relief and sfumato.

           

13) Over the years, Leonardo achieved a sublime mastery in applying the finest layers of paint to create the luminous color tones that give his paintings their special magic. As Serge Bramly describes it, “The light passes through his paintings as if through stained glass, straight on to the printed surface beneath, which reflects it back, thus creating the impression that it emanates from the figures themselves.

          

14) On the other hand, Leonardo’s completed masterpieces always involved radical innovations at several levels – artistic, philosophical, and scientific. For example, the Virgin of the Rocks was not only revolutionary in its rendering of light and dark. It also represented a complex and controversial meditation on the destiny of Christ, expressed through the gestures and relative positions of the four protagonists, as well as in the intricate symbolism of the surrounding rocks and vegetation.

          

15) In a similar vein, Vasari refers to Leonardo as “Florentine painter and sculptor” in the title of his biography. And yet, we have no known sculpture from Leonardo’s hand. His reputation rests on a single piece of work: a monumental bronze horse that was never cast, but which occupied Leonardo intensely for over ten years.

          

16) Good designers have the ability to think systematically and to synthesize. They excel at visualizing things, at organizing known elements into new configurations, at creating new relationships; and they are skillful in conveying these mental processes in the form of drawings almost as rapidly as they occur. Leonardo, off course, possessed all these abilities to a very high degree. In addition, he had an uncanny knack of perceiving and solving technical problems – another key characteristic of a good designer – so much so, in fact, that it was almost second nature to him.

          

17) What made Leonardo unique as a designer and engineer, however, was that many of the novel designs he presented in his Notebooks involved technological advances that would not be realized until several centuries latter. And second, he was the only man among the famous Renaissance engineers who made the transition from engineering to science. Like painting, engineering became a “mental discourse” for him. To know how something worked was not enough for Leonardo, he also needed to know why. Thus an inevitable process was set in motion, which led him from technology and engineering to pure science. As art historian Kenneth Clark notes, we can see the process at work in Leonardo’s manuscripts: First, there are questions about the construction of certain machines, then… questions about the first principles of dynamics; finally questions which had never been asked before about winds, clouds, the age of the earth, generation, the human heart. Mere curiosity has become profound scientific research, independent of the technical interests which had preceded it.

          

18) In other words, the problems Leonardo addresses are theoretical problems of architectural design. The questions he asks are the same questions he explores throughout his science of organic forms – questions about patterns, spatial organization, rhythm, and flow. The notes accompanying his drawings (written in his customary mirror writing, and hence intended for himself) can be seen as fragments of a treatise on architecture that Leonardo, according to Heydenreich, may have intended to compose.

           

19) In view of Leonardo’s central focus on understanding nature’s forms, both in the macro- and the microcosm, it is not surprising that he emphasized similarities between architectural structures and structures in nature, especially in human anatomy. In fact, this linking of architecture and anatomy goes back to antiquity and was common among Renaissance architects, who recognized the analogy between a good architect and a good doctor. As Leonardo explained, “Doctors, teachers, and those who nurse the sick should understand what man is, what is life, what is health, and in what manner a parity and concordance of the elements maintains it… The same is also needed for the ailing cathedral, that is, a doctor-architect who understands well what buildings is and from what rules the correct way of building derives.”

          

20) Leonardo’s science, by contrast, cannot be reduced to a single foundation, as we have seen. Its strength does not derive from a single trunk, but from the complex interconnectedness of the branches of many trees. For Leonardo, recognizing the numerous patterns of relationships in nature was the hallmark of a universal science. Today, we, too, sense a greater need for such universal, or systemic, knowledge, which is one of the reasons why Leonardo’s unified vision of the world is so relevant to our time.

          

21) Leonardo showed greatly artistic talent early in his youth; his synthesis of art and science was also foreshadowed early on. This is vividly illustrated in a story related by Vasari. When Piero da Vinci was asked by a peasant to have a “buckler” (a small wooden shield) decorated with a painting in Florence, he did not give the shield to a Florentine artist but instead asked his son to paint something on it. Leonardo decided to paint a terrifying monster. “To do what he wanted,” writes Vasari, “Leonardo carried into a room of his own, which no one else entered except himself, a number of lizards, crickets, serpents, butterflies, locusts, bats, and various strange creatures of this nature. From all these he took and assembled different parts to create a fearsome and horrible monster… He depicted the creature emerging from a dark cleft of a rock, belching forth venom from its open throat, fire from its yes, and smoke from its nostrils in so macabre a fashion that the effect was altogether monstrous and horrible. Leonardo took so long over the work that the stench of dead animals in his room became unbearable, although he himself failed to notice because of his great love of painting.” When Ser Piero came to see the finished painting, Leonardo went back into the room, put the buckler on an easel in the light, and shaded the window. Then he asked Piero to come and see it. When his eyes fell on it, Piero was completely taken by surprise and gave a sudden start, not realizing that he was looking at the buckler and that the form he saw was, in fact, painted on it. As he backed away, Leonardo stopped him and said: ‘This work certainly serves its purpose. It has produced the right reaction, so now you can take it away.’”

          

22) Other inventions he created from that time involved fire and a hot air. In addition to the self-regulating spit mentioned earlier, Leonardo invented a method of creating a vacuum to raise water by means of a fire burning in a closed bucket, based on the observation that a burning flame consumes air. During these early years he also developed his first versions of a diving apparatus. During a visit to Vinci he designed an olive press with more efficient leverage than the presses used at the time. While he was engaged in these multiple projects of invention, design, and engineering, Leonardo also painted his Annunciation, two Madonnas, and the portrait of Ginevra de’ Benci.

          

23) He drew [a] long series of diagrams showing the effect of light falling on spheres and cylinders, crossing, reflecting, intersecting with endlessly variety… The calculations are so complex and abstruse that we feel in them, almost for the first time, Leonardo’s tendency to pursue research for its own sake, rather than as an aid to his art.

          

24) He was asked by Ludovico to paint a portrait of the Moor’s mistress, the young and lovely Cecilia Gallerani. Leonardo painted her holding an ermine, a symbol of purity and moderation which, because of its Greek name, gale, was also a veiled allusion to her name, Gallerani. Lady with an Ermine, as it is called today, was a highly original portrait in which Leonardo invented a new pose, with the model looking over her shoulder with an air of surprise and subdued delight, caused, perhaps, by the unexpected arrival of her lover. Her gesture is graceful and elegant, and is echoed in the animal’s twisting movement.

          

25) For Leonardo himself, the 1940s were a period of intense creative activity. With two major projects – the equestrian statue and The Last Supper – his artistic career was at its peak, he was consulted repeatedly as an expert on architectural design, and he embarked on extensive and systematic research in mathematics, optics, mechanics and the theory of human fly.

          

26) Leonardo’s research in statics and dynamics was concerned not only with the workings of machines but also, and even more important, with understanding the human body and its movements. For example, he investigated the body’s ability to generate various amount of forces in several positions. One of the key aims was to find out how a human pilot might generate enough force to lift a flying machine off the ground by flapping its mechanical wings. In his studies of machines during that period, Leonardo began to separate individual mechanisms – levers, gears, bearings, couplings, etc. – from the machines in which they were embedded. This conceptual separation did not arise again in engineering until the eighteenth century. In fact, Leonardo planned (and may even have written) a treatise on Elements of Machines, perhaps influenced by his discussions with Fazio Cardano of Euclid’s celebrated Elements of Geometry in Pavia.

          

27) Leonardo’s Last Supper, generally considered the first painting of the High Renaissance (the period of art between, approximately 1495 and 1520), is dramatically different from earlier representations of the subject. Indeed, it became famous throughout Europe immediately after his completion and was copied innumerable times. The firstly highly imaginative feature one notices is the way Leonardo integrated the fresco into the architecture of the refectory. Demonstrating his mastery of geometry, Leonardo contrived a series of visual paradoxes to create an elaborate illusion – a complex perspective that made the room of the Last Supper look like a refectory itself, in which the monks ate their meals. One consequence of this complex perspective is that from every viewing position in the room, the spectator is drawn into the drama of the picture’s narrative with equal force. And dramatic it is. Whereas traditionally the Last Supper was pictured at the moment of communion, a moment of calm, individual meditation for each apostle, Leonardo chose the ominous moment when Jesus says, “One of you will betray me.” The words of Christ have stirred up the solemn company, creating powerful waves of emotion. However, the effect is far from chaotic. The apostles are clearly organized into four groups of three figures, with Judas forming one of the groups together with Peter and John. This is another striking compositional innovation. Traditionally, Judas was pictured sitting on the other side of the table, facing the apostles, with his back to the spectator. Leonardo had no need to identify the traitor by isolating him in this way. By given the apostles carefully expressive gestures, which together cover a wide range of emotions, the artist made sure that we immediately recognize Judas, as he shrinks back into the dark of John’s shadow, nervously clutching his bag of silver. The depiction of the apostles as embodiments of individual emotional states and the integration of Judas into the dramatic narrative were so revolutionary that after Leonardo, no self-respecting artist could go back to the previous static configuration.

          

28) Soon after they began their study sessions, Leonardo and Fra Luca decided to collaborate on a book, titled De divina proportione, to be written by Pacioli and illustrated by Leonardo. The book, presented to Ludovico as a lavish manuscript and eventually published in Venice, contains an extensive review of the role of proportion in architecture and anatomy – and in particular of the golden section, or “divine proportion” – as well as detailed discussions of the five regular polyhedra known as the Platonic solids. It features over sixty illustrations by Leonardo, including superb drawings of the Platonic solids in both solid and skeletal forms, testimony to his exceptional ability to visualize abstract geometric forms. What further distinguishes this work is that it is the only collection of drawings by Leonardo published during his lifetime.

          

29) In the Madonna and Child with Saint Anne, as the paint is called today, Leonardo had again broken new ground with both his composition and the theological interpretation of a traditional religious theme. Rather than presenting Mary and her mother, Saint Anne, in static configuration – seated next to each other with Jesus in Mary’s arms between them, or with Saint Anne seated higher in a majestic, hierarchical composition – Leonardo upset tradition by adding a lamb as a fourth figure. Jesus, having slipped to the ground, reaches for the lamb as Mary tries to restrain him, and Saint Anne seems to hold her back. The theological message embodied in Leonardo’s highly original composition can be seen as a continuation of his long meditation on the destiny of Christ, which he had begun with the Virgin of the Rocks. Mary, in an anxious gesture, attempts to pull her soon away from the lamb, the symbol of Passion, while Saint Anne, representing Mother Church, knows that Mary’s gesture is futile – the Passion is Christ’s destiny and cannot be avoid.

          

30) When he had built flight machines in Milan and tested them in his workshop in Corte Vecchia, Leonardo’s main concern had been to find out how human pilot could flap mechanical wings with enough force and velocity to compress the air underneath and be lifted up. For these tests he had designed various types of wings modeled after those of birds, bats, and flying fish. Now, ten years later, he embarked on careful and methodical observations of the flight of birds. He spent hours in the hills surrounding Florence, near Fiesole, observing the behavior of birds in flight, and filled several Notebooks with drawings and comments that analyzed the birds’ turning maneuvers, their ability to maintain their equilibrium in the wind, and the detailed mechanisms of active flight. His aim was to design a flying machine that would be able, like a bird, to maneuver with agility, keep its balance in the wind, and move its wings with enough force to allow it to fly.

          

31) In his Anatomical Studies, Leonardo gives a vivid description of the dreadful conditions under which he had to work. As there were no chemicals to preserve the cadavers, they would begin to decompose before he had time to examine and draw them properly. To avoid accusations of heresy, he worked at night, lighting his dissection room by candles, which must have made the experience even more macabre. “You will perhaps be impeded by the fear of living through the night hours in the company of these corpses, quartered and flayed and frightening to behold.”

          

32) One will see darkly gloomy air beaten by the rush of different and convoluting winds, which are mingled with the weight of continuous rain, and which are carrying helter-skelter an infinite number of branches torn from the trees, entangled with countless autumn leaves. The ancient trees will be seen uprooted and thorn to pieces by the fury of the winds… Oh how many will you see closing their ears with their hands to shut out the tremendous noises made in the darkened air by the raging of the winds… Others, with gestures of hopelessness, took their own lives, despairing of being able to endure such suffering; and of these, some flung themselves from high rocks, others strangled themselves from high rocks, others strangled themselves with their own hands.

          

33) The drawings that illustrate his apocalyptic narrative are dark, violent, menacing, and disturbing. Nonetheless, they are astonishingly accurate in their renderings of water and air turbulence. Throughout his life, Leonardo had carefully studied the forms of waves, eddies, waterfalls, vortices, and air currents. Here, in old age, he summed up his knowledge of turbulence. Beyond their expressive emotional power, the deluged drawings can be seen as sophisticated mathematical diagrams, presenting a visual catalog of turbulent flows that would not look out of place in a modern textbook on fluid dynamics.

          

34) In Leonardo’s mind, his science of living forms was certainly an integrated whole. At the end of his life, his problems were no longer conceptual; they were simply the limitations of time and energy. As he wrote several years before his death, “I have been impeded neither by avarice nor by negligence, but only by time.” And yet, Leonardo never gave up. In June 1518 he wrote what may have been the last entry in his Notebooks: “I shall go on.”

          

35) Nor was he perturbed by contemplating his approaching death. “Just well-spent day brings a happy sleep,” he had written thirty years earlier, “so a well-employed life brings a happy death.”

          

36) A few days after completing his will, on May 2, 1519, Leonardo da Vinci died in the manor of Cloux – according to legend, in the arms of the king of France.

          

37) To appreciate Leonardo’s science, it is important to understand the cultural and intellectual context in which he created it. Scientific ideas do not occur in a vacuum. They are always shaped by the technologies available at the time. The entire constellation of concepts, values, perceptions, and practices – the “scientific paradigm” in the terminology of science historian Thomas Kuhn – provides the context that is necessary for scientists to pose the great questions, organize their subjects, and define legitimate problems and solutions. All science is built upon such an intellectual and cultural foundation. Hence, when we recognize ancient or medieval ideas reflected in Leonardo’s scientific writings, this do not mean that he was less of a scientist, Leonardo consulted the traditional texts and used their conceptual framework as his starting point. He then tested the traditional ideas against his own scientific observations. And, in accordance with scientific method, he did not hesitate to modify the old theories when his experiments contradicted them.

          

38) The leading figure in the movement to weave the philosophy of Aristotle into Christian teachings was Saint Thomas Aquinas, one of the towering intellects of the Middle Ages. Aquinas taught that there could be no conflict between faith and reason, but the two books on which they were based – the Bible and the “book of nature” – were both authored by God. Aquinas produced a vast body of precise, detailed, and systematic philosophical writings in which he integrated Aristotle’s encyclopedic works and medieval Christian theology into a magnificent whole. The dark side of this seamless fusion of science and theology was that any contradiction by future scientists would necessarily have to be seen as heresy. In this way, Thomas Aquinas enshrined in his writings the potential for conflicts between science and religion – which indeed arose three centuries later in Leonardo’s anatomical research, reached a dramatic climax with the trial of Galileo, and have continued to the present day.

          

39) A few years later, at the height of his anatomical work in Milan, Leonardo added a technical note about the reproduction of his drawings to his famous assertion of the superiority of drawing over writing. He insisted that his anatomical drawings should be printed from copper plates, which would be more expensive than woodcuts but much more effective in rendering the fine details of his work. “I beg you who come after me”, he wrote on the sheet that contains his magnificent drawings of the vertebral column, “not let avarice constrain you to make the prints in [wood].”

          

40) The conception of the Renaissance worldview was the conceptions of the universe that had been developed in classical Greek science: that the world was a kosmos, an ordered and harmonious structure. From its beginnings in the sixth century B.C., Greek philosophy and the science understood the order of the cosmos to be that of a living organism rather than a mechanical system. This meant that all its parts had an innate purpose to contribute to the harmonious functioning of the whole, and that objects moved naturally toward their proper places in the universe. Such an explanation of natural phenomena in terms of their goals, or purposes, is known as teleology, from the Greek telos (purpose). It permeated virtually all of Greek philosophy and science. The view of the cosmos as an organism also implied for the Greeks that its general properties are reflected in each of its parts. This analogy between macrocosm and microcosm, in particular between the Earth and the human body, was articulated most eloquently by Plato in his Timaeus in the fourth century B.C., but it can also be found in the teachings of the Pythagoreans in other earlier schools. Over time, this idea acquired the authority of common knowledge, which continued throughout the Middle Ages and into Renaissance. In early Greek philosophy, the ultimate moving force and source of all life was indentified with the soul, and its metaphor was that of the breath of life. Indeed, the root meaning of both the Greek psyche and the Latin anima is “breath”. Closely associated with that moving force – the breath of life that leaves the body at death – was the idea of knowing. For the early Greek philosophers, the soul was both the source of movement and life, and that which perceives and knows. Because of the fundamental analogy between micro- and macrocosm, the individual soul was thought to be part of the force that moves the entire universe, and accordingly the knowing of an individual was seen as part of a universal process of knowing. Plato called it the anima mundi, the “world soul”.

          

41) The culmination of the early phase of Greek mathematics was reached around 300 B.C. with Euclid, who presented all of the geometry and other mathematics known in his days in a systematic, orderly sequence in his celebrated Elements. The thirteen volumes of this classical textbook were not only widely read during the Renaissance, but remained the foundation for the teaching of geometry until the end of the nineteenth century.

          

42) Health, according to the Hippocratic writings, requires a state of balance among environmental influences, the way in which we live, and the various components of human nature. One of the most important volumes in the Hippocratic Corpus, the book on Airs, Waters and Places, represents what we might now call a treatise on human ecology. It shows in greater detail how the well-being of individuals is influenced by environmental factors – the quality of air, water, and food, the topography of the land, and general living habits. During the last two decades of the fifteenth century, this and several other volumes from the Hippocratic Corpus were available to scholars in Latin, most of them derived from Arabic translations.

          

43) Leonardo da Vinci shared with his fellow humanists their great confidence in the capabilities of the human individual, their passion for voyages and exploration, and their excitement about the rediscovery of the classical texts of antiquity. But he differed dramatically from most of them by refusing to blindly accept the teachings of the classical authorities. He studied them carefully, but then he tested them by subjecting them to rigorous comparisons with his own experiments and his direct observations of nature. In doing so, I would argue, Leonardo single-handedly developed a new approach to knowledge, known today as scientific method.

          

44) All scientific models and theories are limited and approximate. This realization has become crucial to the contemporary understanding of science. Twentieth-century science has shown repeatedly that all natural phenomena are ultimately interconnected, and that their essential properties, in fact, derive from their relationships to other things. Hence, in order to explain any one of them completely, we have to understand all the others, which is obviously impossible. This insight has forced us to abandon the Cartesian belief in the certainty of scientific knowledge and to realize that science, to put into bluntly, we never deal with truth, in the sense of a precise correspondence between our descriptions and the described phenomena. We always deal with limited and approximate knowledge. This may sound frustrating, but for many scientists the fact that we can formulate approximate models and theories to describe an endless web of interconnected phenomena, and that we are able to systematically improve our models and approximations over time, is a source of confidence and strength. As the great biochemist Louis Pasteur put it, “Science advances through tentative answers to a series of more and more subtle questions which reach deeper and deeper into the essence of all natural phenomena.”

          

45) “All our knowledge has its origins in the senses,” he noted in his first Notebook, the Codex Trivulzianos. “Wisdom is the daughter of experience,” we read in the Codex Forster, and in his Treatise on Painting, Leonardo asserted: “To me it seems that those sciences are vain and full of errors that are not born of experience, mother of all certainty… that is to say, which do not at their beginning middle, or end pass through any of the five senses.” Such an approach to the study of nature was unheard-of in Leonardo’s day, and would fully emerge again only in the seventeenth century, the era of the Scientific Revolution.

          

46) He recognized that learning from skilled masters was important in the arts, but he also observed that such masters were rare. “The surer way,” he suggested, “is to go to the objects of nature, rather than those that are imitated with great deterioration, and so acquire sad habits; for he who can go to the well does not go to the water jar.”

          

47) He was deeply aware of the fundamental interconnectedness of all phenomena and of the interdependence and mutual generation of all parts of an organic whole, which Immanuel Kant in the eighteenth century would define as “self-organization.” In the Codex Atlanticus, Leonardo eloquently summarized his profound understanding of life’s basic processes by paraphrasing a statement by the Ionian philosopher Anaxagoras: “Everything comes from everything, and everything is made of everything, and everything turns into everything, because that which exists in the elements is made up of the elements.”

          

48) Only in the twentieth century did the limits of Newtonian science become fully apparent, and the mechanistic Cartesian worldview begin to give way to a holistic and ecological view not unlike that developed by Leonardo da Vinci. With the rise of systemic thinking and its emphasis on networks, complexity, and patterns of organization, we can know more fully appreciate the power of Leonardo’s science and its relevance for our modern era. Leonardo’s science is a science of qualities, of shapes and proportions, rather than absolute quantities. He preferred to depict the forms of nature in his drawings rather than describe their shapes, and he analyzed them in terms of their proportions rather than measured quantities. Proportion was seen by Renaissance artists as the essence of harmony and beauty. Leonardo filled many pages of his Notebooks with elaborate diagrams of proportions between the various parts of the human figure, and he drew corresponding diagrams to analyze the body of the horse.

          

49) Leonardo was always impressed by the great diversity and variety of living forms. “Nature is so delightful and abundant in its variations,” he wrote in a passage about how to paint trees, “that among trees of the same kind there would not be found one plant that resembles another nearby, and this is not only of the plant as a whole, but among the branches, the leaves, and the fruit, not one will be found that looks precisely like another.”

          

50) Leonardo was fascinated by water in all its manifestations. He recognized its fundamental role as life’s medium and vital fluid, as the matrix of all organic forms. “It is the expansion and the humor of all living bodies,” he wrote. Without it nothing retains its original form.” Throughout his life, he strove to understand the mysterious processes underlying the creation of nature’s forms by studying the movements of water through earth and air.

          

51) At the center of Leonardo’s investigations of turbulence lies the water vortex, or whirlpool. Throughout the Notebooks, there are countless drawings of eddies and whirlpools of all sizes and types – in the currents of rivers and lakes, behind piers and jetties, in the basin of waterfalls, and behind objects of various shapes immersed in flowing water. These often very beautiful drawings are testimony to Leonardo’s endless fascination with the ever-changing and yet stable nature of this fundamental type of turbulence. I believe that this fascination came from a deep intuition that the dynamics of vortices, combining scalability and change, embody an essential characteristic of the living forms.

          

52) To investigate the mechanics of muscles, tendons, and bones, Leonardo immersed himself in a long study of the “science of weights,” known today as statics, which is concerned with the analysis of loads and forces on physical systems in static equilibrium, such as balances, levers, and pulleys. In the Renaissance this knowledge was very important for architects and engineers, as it is today, and the medieval science of weights comprised a large collection of works compiled in the late thirteenth and fourteenth centuries.

          

53) Leonardo applied his knowledge of mechanics not only to his investigations of the movements of the human body, but also to his studies of machines. Indeed, the uniqueness of his genius lay in his synthesis of art, science, and design. In his lifetime, he was famous as an artist, and also as a brilliant mechanical engineer who invented and designed countless machines and mechanical devices, often involving innovations that were centuries ahead of his time.

          

54) Based on these designs, British engineers recently built a glider and tested it successfully in a flight from the chalk cliffs in southeast England know as the Sussex Downs. This maiden flight of “Leonardo’s glider,” reportedly, exceeded the first attempts by the Wright brothers in 1900. Although the machines with movable mechanical wings were not destined to fly, the models built from Leonardo’s designs are extraordinary testimonies to his genius as a scientist and engineer. In the words of art historian Martin Kemp: “Using mechanical systems, the wings flap with much of the sinuous and menacing grace of a gigantic bird prey… [Leonardo’s] designs retain their conceptual power as archetypal expressions of man’s desire to emulate the birds, and remain capable of inspiring a sense of wonder even in a modern audience, for whom the sight of tons of metal flying through the air has become a matter of routine.”

          

55) Leonardo’s careful and patient studies of the movements of the heart and the flow of blood, undertaken in old age, are the culmination of his anatomical work. He not only understood and pictured the heart like no one before him, but also observed subtleties in its actions and in the flow of blood that would elude medical researchers for centuries.

          

56) Leonardo’s success in cardiac anatomy [is] so great that there are aspects of the work which are not yet equaled by modern anatomical illustration… His consistent practice of illustration of the heart and its valves, both in systole and in diastole, with a comparison of the position of the parts, has rarely if ever been performed in any anatomical textbook.

          

57) Leonardo’s embryological drawings are graceful and touching revelations of the mysteries surrounding the origins of human life. In the words of physician Sherwin Nuland, “[His] depiction of a five-month fetus in the womb is a thing of beauty… It stands as a masterwork of art, and, considering the very little that was at the time understood of embryology, a masterwork of science perception as well.” Leonardo knew very well that, ultimately, the nature and origin of life would remain a mystery, no matter how brilliant his scientific mind was. “Nature is full of infinite causes that have never occurred in experience,” he declared in his late forties, and as he got older his sense of mystery deepened. Nearly all the figures in his last paintings have that smile that expresses the ineffable, often combined with a pointing finger. “Mystery to Leonardo”, wrote Kenneth Clark, “was a shadow, a smile and a finger pointing into darkness.”

          

58) Leonardo’s approach to mathematics was that of a scientist, not a mathematician. He wanted to use mathematical language to provide consistency and rigor to the descriptions of his scientific observations. However, in his time there was no mathematical language appropriate to express the kind of science he was pursuing – explorations of the forms of nature in their movements and transformations. And so Leonardo used his powers of visualization and his great intuition to experiment with new techniques that foreshadowed branches of mathematics that would not be developed until centuries later. These include the theory of functions and fields of integral calculus and topology.

          

59) The really important mathematics for him was geometry, which is evident from his praise of the eye as “the prince of mathematics.”

          

60) Like most mathematicians of his time, Leonardo frequently used geometrical figures to represent algebraic relationships. A simple but very ingenious example is his pervasive use of triangles and pyramids to illustrate arithmetic progressions and, more generally, what we now call linear functions. He was familiar with the use of pyramids to represent linear proportions from his studies of perspective, where he observed that “All the things transmit to the eye their image by means of a pyramid of lines. By ‘pyramid of lines’ I mean those lines which, starting from the edges of the surface of each object, converge from a distance and meet in a single point… placed in the eye.”

          

61) Leonardo realized very early on that the mathematics of his time was inappropriate for recording the most important results of his scientific research – the description of nature’s living forms in their ceaseless movements and transmutations. Instead of mathematics, he frequently used his exceptional drawing facility to graphically document his observations in pictures that are often strikingly beautiful while, at the same time, they take the place of mathematical diagrams. His celebrated drawing of “Water falling upon water”, for example, is not a realistic snapshot of a jet of water falling into a pond, but an elaborate diagram of Leonardo’s analysis of several types of turbulence caused by the impact of the jet.

          

62) Arasse makes an interesting point: Whenever Leonardo rendered objects in their sharp outlines, these pictures represented conceptual models rather than realistic images. And whenever he produced realistic images of objects, he blurred the outlines with his famous sfumato technique, in order to represent them as they actually appear to the human eye.

           

63) What Leonardo found especially attractive in geometry was its ability to deal with continuous variables. “The mathematical sciences… are only two,” he wrote in the Codex Madrid, “of which the first is arithmetic, the second is geometry. One encompasses the discontinuous quantities [i.e., variables] the other the continuous.” It was evident to Leonardo that a mathematic of continuous quantities would be needed to describe the incessant movements and transformations in nature. In the seventeenth century, mathematicians developed the theory of functions and the differential calculus for that very purpose. Instead of these sophisticated mathematical tools, Leonardo had only geometry at his disposal, but he expanded it and experimented with new interpretations and new forms of geometry that foreshadowed subsequent developments.

          

64) In the course of his explorations of circles and squares, Leonardo tried his hand at the problem of squaring the circle, which had fascinated mathematicians since antiquity. In its classical form, the challenge is to construct a square with an area equal of that of a given circle, and to do so by using only ruler and compass. We know today that this is not possible, but countless professional and amateur mathematicians have tried. Leonardo worked on the problem repeatedly over a period of more than a dozen years. In one particular attempt, he worked by candlelight through the night, and by dawn he believed that he had finally found the solution. “On the night of St. Andrew,” he excitedly recorded in his Notebook, “I found the end of squaring the circle; and at the end of the light of the candle, of the night, and of the paper on which I was writing, it was completed; at the end of the hour.” However, as the day progressed, he came to the realization that this attempt, too, was futile.

          

65) When we look at Leonardo’s geometry from the point of view of present-day mathematics, and in particular from the perspective of complexity theory, we can see that he developed the beginnings of the branch of mathematics now known as topology. Like Leonardo’s geometry, topology is a geometry of continuous transformations, or mappings, in which certain properties of geometric figures are preserved. For example, a sphere can be transformed into a cube or a cylinder, all of which have similar continuous surfaces. A doughnut (torus), by contrast, is topologically different because of the hole in its center. The torus can be transformed, for example, into a coffee cup where the hole now appears in the handle. In the words of historian of mathematics Morris Kline: Topology is concerned with those properties of geometric figures that remain invariant when the figures are bent, stretched, shrunk, or deformed in any way that does not create new points or fuse existing points. The transformations presupposes, in other words, that there is a one-to-one correspondence between the points of the original figure and the points of the transformed figure, and that transformation carries nearby points into nearby points. This latter property is called continuity.

          

66) During the last twelve years of his life, Leonardo spent a great deal of time mapping and exploring the transformations of his “geometry done with motion.” Several times he wrote of his intention to present the results of these studies in one or more treatises. During the years he spent in Rome, and while he was summing up his knowledge of complex turbulent flows in his famous deluge drawings, Leonardo produced a magnificent compendium of topological transformations, titled De ludo geometrico (On the Game of Geometry), on large double folio in the Codex Atlanticus. He drew 176 diagrams displaying a bewildering variety of geometric forms, built from intersecting circles, triangles and squares – row after row of crescents, rosettes and other floral patterns, paired leaves, pinwheels, and curvilinear stars. Previous this endless interplay of geometric motifs was often interpreted as the playful doodling of an aging artist – “a mere intellectual pastime,” in the words of Kenneth Clark. Such assessments were made because art historians were generally not aware of the mathematical significance of Leonardo’s geometry of transformations. Close examination of the double folio shows that its geometric forms, regardless of how complex and fanciful, are all based upon strict topological principles.

          

67) Since Leonardo’s science was a science of qualities, of organic forms and their movements and transformations, the mathematical “necessity” he saw in nature was not one expressed in quantities and numerical relationships, but one of geometric shapes continually transforming themselves according to rigorous laws and principles. “Mathematical” for Leonardo referred above all to the logic, rigor, and coherence according to which nature has shaped, and is continually reshaping, her organic forms. This meaning of “mathematical” is quite different from the one understood by scientists during the Scientific Revolution and the subsequent three hundred years. However, it is not unlike the understanding of some of the leading mathematicians today. The recent development of complexity theory has generated a new mathematical language in which the dynamics of complex systems – including the turbulent flows and growth patterns of plants studied by Leonardo – are no longer represented by algebraic relationships, but instead by geometric shapes, like the computer-generated strange attractors or fractals, which are analyzed in terms of topological concepts. This new mathematics, naturally, is far more abstract and sophisticated than anything Leonardo could have imagined in the fifteenth and sixteenth centuries. But is used in the same spirit in which he developed his “geometry done with motion” – to show with mathematical rigor how complex natural phenomena are shaped and transformed by the “necessity” of physical forces. The mathematics of complexity has led to a new appreciation of geometry and to the broad realization that mathematics is much more than formulas and equations. Like Leonardo da Vinci five hundred years ago, modern mathematicians today are showing us that understanding of patterns, relationships, and transformations is crucial to understand the living world around us, and that all questions of pattern, order, and coherence are ultimately mathematical.

          

68) From perspective, he proceeded in two opposite directions – outward and inward, as it were. He explored the geometry of light rays, the interplay of light and shadow, and the very nature of light, and he also studied the anatomy of the eye, the physiology of vision, and the pathways of sensory impressions along the nerves to the “seat of the soul”. To a modern intellectual, used to exasperating fragmentation of academic disciplines, it is amazing to see how Leonardo moved swiftly from perspective and the effects of light and shade to the nature of light, the pathways of the optic nerves, and the actions of the soul. Unencumbered by the mind-body split that Descartes would introduce 150 years later, Leonardo did not separate epistemology (the theory of knowledge) from ontology (the theory of what exists in the world), nor indeed philosophy from science and art. His wide-ranging examinations of the entire process of perception led him to formulate highly original ideas about the relationship between physical reality and cognitive processes – the “actions of the soul”, in his language – which have reemerged only very recently with the development of a post-Cartesian science of cognition.

          

69) As architectural historian James Ackerman points out, the geometry of perspective developed by the Florentine artists was the first scientific conception of three-dimensional space: As a method of constructing an abstract space in which any body can be related mathematically to any other body, the perspective of the artists was a preamble to modern physics and astronomy. Perhaps, the influence was indirect and unconsciously transmitted, but the fact remains that artists were the first to conceive a generalized mathematical model of space and that it constituted an essential step in the evolution from medieval symbolism to the modern image of the universe.

          

70) Leonardo demonstrated his throughout understanding of linear perspective not only in his art, but also in his scientific drawings. While he was conducting his experiments on the geometry of perspective, he also investigated the anatomical connections between the eye and the brain. He documented his findings in a series of magnificent pictures of the human skull, in which the foreshortening of visual perspective is employed with great effect. Leonardo combined this technique with delicate renderings of light and shade to create a vivid sense of space within the skull, in which he exhibited anatomical structures that had never been seen before and located them with complete accuracy in three dimensions.

          

71) From the earliest years in Verrocchio’s workshop, Leonardo was familiar with the grinding of lenses and the use of concave mirrors to focus sunlight for welding. Throughout his life he tried to improve the design of these burning mirrors, and when he became seriously interested in the theory of optics, he undertook careful studies of their geometries. He was fascinated by the intricate intersections of the reflected rays, which he explored in a series of precise and beautiful diagrams, tracing their pathways from parallel beams of light through their reflections to the focal point (or points). He showed that in spherical mirrors, the rays are focused in an area along the central axis, whereas parabolic mirrors are true “mirrors of fire”, focusing all the rays in a single point. He also made several attempts to solve Alhazen’s problem, and late in his life, while experimenting with parabolic mirrors in Rome, found an ingenious solution by employing an instrument with hinged rods.

           

72) According to Leonardo, shadow is the central element in the science of painting. It allows the painter to effectively represent solid bodies in relief, emerging from the backgrounds of the painted surface. His poetic definition of shadow in Codex Atlanticus is clearly written from the artist’s point of view: Every opaque body is surrounded, and its whole surface is enveloped, in shadow and light…. Besides this, shadows have in themselves various degrees of darkness, because they are caused by the absence of a variable amount of the luminous rays…. They clothe the bodies to which they are applied.

          

73) As Kenneth Clark has remarked, “The calculations are so complex and abstruse that we feel in them, almost for the first time, Leonardo’s tendency to pursue research for its own sake, rather than as an aid to his art.”

          

74) He marveled at the swift velocity of light: “Look at the light of the candle and consider its beauty,” he wrote. “Blink your eye and look at it again. What you see of it was not there before, and what was there before is not anymore.” But he also realized that, however fast light moves, its velocity is not infinite. He asserted that the speed of sound is greater than that of elastic waves in earth, and that light moves faster than sound, but that the mind moves even faster than light. “The mind jumps in an instant from East to the West,” he noted, “and all the other immaterial things have velocities that are by a long way inferior.”

          

75) The structure of the eye and the process of vision were natural wonders for Leonardo that never ceased to amaze him. “What language can express this marvel?” he writes about the eyeball, before continue with rare expression of religious awe: “Certainly none. This is where human discourse turns directly to the contemplation of the divine.” In the Treatise on Painting, Leonardo waxes enthusiastic about the human eye: Don’t you see that the eye embraces the beauty of the whole world? It is the master of astronomy, it practices cosmography, it counsels and corrects all human arts; its transports man to different parts of the world. [The eye] is the prince of mathematics; its sciences are most certain. It has measured the heights and sizes of the stars; it has discovered the elements and their locations…. It has created architecture, perspective, and divine painting…. [The eye] is the window of the human body, through which [the soul] contemplates and enjoys the beauty of the world.

          

76) “The pupil of the eye,” he concluded, “changes to as many different sizes as there are differences in the degrees of brightness and darkness of the objects which present themselves before it…. Nature has equipped the visual faculty, when irritated by excessive light, with the contraction of the pupil… and here nature works like someone who, having too much light in his house, closes half of a window, and more or less according to necessity.” And then he added: “You can observe that in nocturnal animals such as cats, screech owls, tawny owls and others, which have the pupil small at midday and very large at night.”

          

77) During the last two decades of the twentieth century, however, a novel conception of the nature of mind and consciousness emerged in the life sciences, which finally overcame the Cartesian division between mind and body. The decisive advance has been to reject the view of mind as a thing; to realize that mind and consciousness are not entities but processes. In the past twenty-five years the study of mind from this new perspective has blossomed into a rich interdisciplinary field known as cognitive science, which transcends the traditional frameworks of biology, psychology, and epistemology. One of the central insights of cognitive science is the identification of cognition, the process of knowing, with the process of life. Accordingly, the interactions of a living organism – plant, animal, or human – with its environment are understood as cognitive interactions. Thus life and cognition become inseparably connected. Mind, – or, more accurately, mental activity – is immanent in matter at all levels of life. This new conception represents a radical expansion of the concept of cognition and, implicitly, the concept of mind. In the new view, cognition involves the entire process of life – including perception, emotion, and behavior – and does not even necessarily require a brain and a nervous system.

          

78) At the end of life, the reverse process takes place: “While I thought I was learning how to live, I had been learning how to die,” Leonardo wrote movingly late in his life.

          

79) Nature, as a whole was alive and animated for Leonardo, a world in continual flux and development, in the macrocosm of the Earth as in the microcosm of the human body.

          

80) Leonardo himself never boasted about his unique talents and skills, and in his thousands of pages of manuscripts he never vaunted the originality of so many of his ideas and discoveries. But he was well aware of his exceptional stature. In the Codex Madrid, in the midst of extensive discussions of the laws of mechanics, we find two lines that can stand as his own definitive epitaph: Read me, O reader, if in my words you find delight, for rarely in the world will one such as I be born again.

          

Synthesis Ratio :: 274 total of pages read / 23 typewritten pages :: 11.9

           

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