He who is fixed to a star does not change his mind.
—Leonardo da Vinci
t one level, the statement quoted above could be regarded as Leonardo’s derogation of astrology as an indicator of one’s fate; certainly it reflects his sentiment that one has to keep an open mind. Like other anatomists of his time, Leonardo had begun his studies from a Galenic frame of reference, but unlike the others, he was far more open to modifying his views when his observations appeared to contradict the prevailing understanding in the field. In astronomy, he started out with the earth-centered Ptolemaic view, but where intellectual honesty required revision of the accepted theory, he sought evidence presented by nature. In physics there existed Aristotelian formalism, but again Leonardo demanded proof before he could accept the tenets of that formalism. Coleridge’s words, “A man convinced against his will is of the same opinion still,” could not have rung with more truth for anyone else.
In the fourth century B.C. Aristotle, the most influential of all philosophers, taught that heavy bodies fall faster than light ones.1 In one passage Leonardo echoes this observation verbatim. But with systematic experimentation and observation, and with a quintessentially open mind, Leonardo established for himself that acceleration for a falling body was constant and independent of its weight. It was that extraordinary personal code of intellectual honesty, the skills to ask just the right questions of nature, that gave him the astonishing ability to extricate her secrets.
Regarding the Ptolemaic picture of the solar system, in which the sun orbits the earth, there is no doubt that Leonardo was engaged in a recurrent struggle to search out the most logical picture. In one geometric construction he depicts the earth orbited by the moon, and at a much greater distance, also by the sun—the geocentric picture. But in another instance he confronts Ptolemy head on, saying the sun “stands still.” He asserts the heliocentric picture: “The earth is not the center of the circuit of the sun, nor the center of the universe.” Unhappily, he left no other reference to this issue, but what other treasures must have been hidden in the original mass of manuscripts? Perhaps further ruminations regarding a sun-centered solar system? Or thoughts about what keeps the moon locked in an eternal orbit? We can only speculate.
The irrecoverable loss of two-thirds of Leonardo’s manuscripts leaves one with the same sorrow and frustration as the loss of the works stored in the library of Alexandria in the great fire of the late fourth century A.D. Among the works destroyed then were hundreds of thousands of scrolls from classical antiquity—including the original teachings of Aristarchus, Archimedes, and Eratosthenes, and of Herophilus, an Alexandrian physiologist who had firmly established the brain in distinction to the heart as the seat of reason. There were also the works of Barasus, the Babylonian historian whose accounts referred to a primeval flood, as well as events as far back as a hundred thousand years earlier. So many abstractions and discoveries—the idea of a spherical earth, the sun-centered universe, atoms, elements, democracy, and the recognition of the brain as the organ for reason—had had their origins in antiquity and had to be rediscovered in modernity. Leonardo’s manuscripts represent a similar treasure. So many of the insights and inventions of the notebooks prefigure the developments and discoveries of the following three hundred years. Had they been available to others in Leonardo’s time, the progress of science and technology would have been accelerated dramatically.
It would be more than a century after Leonardo that the law of falling bodies would be codified, expressed mathematically by Galileo.2 Almost another full century would pass before the physics of heaven and earth would be unified by Isaac Newton; and two centuries later still that in the hands of Albert Einstein a comprehensive theory of the mechanics of the cosmos would be formulated as the general theory of relativity. Perhaps because of their connection to human flight, Leonardo performed experiments related to free-fall. His conclusions turned out to be only partially correct. But in introducing a particularly resourceful approach, he was on the same track that led to the ultimate resolution of the problem. This draws him into the pantheon of physicists with Galileo, Newton, and Einstein. We shall examine here the history of this quest, and present the analysis in mostly qualitative terms. To eliminate all vestiges of what are mathematical theories, however, would undermine their essence as physical laws, and produce only dogma. A few very simple equations will be introduced, but relegated to the endnotes. To reiterate a frequently resonating message of this book: the universe is mathematical—a feature to be savored. In order to gain a deeper insight, however, one would need familiarity with that magnificent mathematical invention of differential calculus—formulated in the second half of the seventeenth century by Newton (and independently by Leibniz), but lying outside the scope of this book.
To reiterate a second message of this book: there is beauty in nature, beauty in the universe. For the scientist the prospect of uncovering a deep mystery of nature holds almost hypnotic allure. Mathematical models that help us to visualize physical phenomena are created and tested. Theories are formulated and are proved and accepted, disproved and abandoned, or (more often) modified to conform more closely with observations. Theories that rise to the level of the transformative are rare. George Bernard Shaw, in introducing Einstein at a banquet, had this refinement process in mind when he quipped, “Newton created a universe that lasted two hundred years. Einstein created a universe, and we don’t know how long [his] will last.” But before we can begin to see how we came to our present understanding of the underlying logic behind the universe and the significance of that understanding, we must first appreciate the dizzying scale of space and time.
The earth is a spherical ball 12,800 kilometers (8,000 mi.) in diameter, slightly more oblate (fatter at the equator) than strictly spherical. At the beginning of the second century B.C. the circumference and radius of the earth were determined with surprising accuracy by the astronomer and mathematician Eratosthenes of Cyrene. Head of the great library of Alexandria, Eratosthenes carried out a series of measurements. He knew from his predecessors that at noon on the longest day of the year (the day of the summer solstice) the sun would be directly overhead in Syene (today Aswan). A vertical post erected in Syene would have no shadow at that moment, whereas a post in Alexandria 800 kilometers (500 mi.) north of Syene would have a measurable shadow. It is this shadow that Eratosthenes measured with exquisite precision on that day. He determined the difference in the angles between the axes of the two posts to be 7 degrees. These axes, he reasoned, if extrapolated downward, would meet at the center of a spherical earth. That 7 degrees happens to be approximately one-fiftieth of a circle. Multiplying the 800-kilometer (500-mi.) distance between the two posts by 50, Eratosthenes obtained the circumference of the earth as 40,000 kilometers (25,000 mi.). Finally, dividing the circumference by π, he calculated the diameter of the earth to be 12,800 kilometers. If Columbus had known Eratosthenes’s value in 1492, he would not have been deluded into believing that he had reached India or Asia, and the islands of the Caribbean would not have been dubbed the “Indies.” He had no idea that the much more expansive Pacific lay to the west of the continent he had discovered.
An older contemporary of Eratosthenes, Aristarchus of Samos (c. 320–250 B.C.), had theorized just a generation earlier that the shape of the earth was spherical, offering as evidence a three-part argument: the field of stars changes with the latitude of the observer; the mast of a ship comes into view before its hull as the ship approaches the shore from a distance; and the shadow of the earth cast on the moon during a lunar eclipse is always round. Aristarchus, however, is not known to have attempted to measure the size of the earth.
In another project still, Aristarchus tried to compare the distances from the earth to the moon and the sun. When the moon is illuminated by the sun as a half-disk, he reasoned, the angle made by the line connecting the earth and moon and the line connecting the moon and sun would be exactly 90 degrees. Accordingly, he constructed a pair of similar triangles, and found the relative distances for those lines to have the ratio of 1:19. His method was correct, but his instruments were much too crude. The correct ratio is 1:395.
That value, 1:395, is also virtually the same ratio as the diameter of the moon to that of the sun, a happenstance that explains why, in viewing a total solar eclipse, we see the disk of the moon to fit almost precisely over the disk of the sun. The size and distance proportions of the sun and moon appear to have been known to Leonardo, who in a drawing in the Codex Leceister demonstrated that geometry. In a vertical cone representing light, the sun is a sphere at the opening of the cone, the moon is within the cone, and the vertex of the cone falls on a point on the surface of the earth. Indeed, this juxtaposition creates similar triangles in geometry.
But even more dramatic than Aristarchus’s assertion of the spherical shape of the earth and his attempts to measure the relative distances from the earth to a pair of heavenly bodies was his heliocentric (sun-centered) theory of the solar system. A full eighteen hundred years before Copernicus, he advanced a model of the solar system with the earth as a planet orbiting the sun along with the other planets. A few centuries later much of the erudition handed down by Aristarchus and Eratosthenes fell into disfavor in the eyes of the early Church. A geocentric (earth-centered) picture, placing mankind at the center of the universe, became the model of choice. Indeed, in time the teaching of the alternate theory would be deemed heresy punishable by death. It would not be until Copernicus in 1543,3 and Galileo and Kepler in the early seventeenth century, that the heliocentric picture would return. But only late in the seventeenth century, when Newton published the Principia, would it gain final acceptance.
Astronomical distances are expressed in a variety of units, each selected for its convenience of purpose. The distance between the earth and the sun is approximately 150 million kilometers (93 million miles); this distance also defines an astronomical unit (AU). For the first several planets the distances from the sun are 0.4 AU (Mercury), 0.7 AU (Venus), 1.0 AU (Earth), 1.6 AU (Mars), 5.2 AU (Jupiter) and 10 AU (Saturn).4 A more convenient unit, especially for longer distances than those encountered within the solar system, is “light-time.” It takes sunlight about 500 seconds, or about 8 minutes, to reach the earth. This distance is called “8 light-minutes.” On this scale, the distance to Saturn is 80 light-minutes, and the distance to the outermost planet, Pluto, is 5.3 light-hours. The closest neighboring star, Alpha Centauri, is 4.3 light-years away.
Spots in the night sky that appeared as nebular or diffuse stars were speculated to be “island universes” by the philosopher Immanuel Kant in the eighteenth century and first catalogued in Kant’s time by the French astronomer Messier. The catalogue of galaxies now honors Messier, in, for example, the designation M51 (or Messier 51), otherwise known as the celebrated Whirlpool galaxy, or M100, a magnificent spiral galaxy visible from the southern hemisphere (Plate 4, bottom right). Our galaxy, the Milky Way, with a population of approximately 400 billion stars, measures close to 120,000 light-years in diameter, and, as a spiral galaxy, it resembles the galaxy M100. The sun, with its array of planets, is approximately half way out from the center—28,000 light-years—on one of the spiral arms of the galaxy.
The entire galaxy rotates, not in the manner of a rigid pinwheel, but rather in differential irrotational motion, the inner parts turning much faster than the outer. At the radial distance from the center of the Milky Way, where our solar system is located, it takes about 200 million years to make one turn. Since the solar system (accompanied by the earth) has been around for about 4.5 billion years, it has made this giant galactic orbit approximately twenty-two or twenty-three times. The last time the solar system was in its present location, the dinosaurs were just emerging. And since they became extinct sixty-five million years ago (at the end of the Cretaceous Period), the entire era of the dinosaurs spanned two-thirds of one galactic orbit.
The Milky Way belongs to a small cluster of galaxies held together by their mutual gravitational field, all orbiting a common center of mass. One of these neighboring galaxies is the giant Andromeda galaxy (M31), approximately 2.2 million light-years away, with over a trillion stars. Although the universe is expanding, with most galaxies, and certainly clusters of galaxies, all moving away from each other, there are some galaxies that are drawing toward each other. For example, in about ten billion years the Milky Way and Andromeda will collide and merge, creating an even larger galaxy. The universe contains tens of billions of galaxies, each containing hundreds of billions of stars, producing a ballpark figure of 1023 stars for the entire universe. The size of the universe is around 14 billion light-years, and its age is 14 billion years. There is, however, no coincidence in these numbers: In the prevailing big bang theory of the universe, space and time have a simultaneous origin, with the universe growing at the speed of light for the past 14 billion years.
Shortly before his death in 1543 Copernicus published his treatise De revolutionibus orbeum coelestium, reviving the long-dormant heliocentric picture of the universe, and effectively repudiating the Ptolemaic (geocentric) picture upheld by the Church. Though this conjecture might remove mankind from the center of the universe, he was not interested in challenging the authority of the Church. He was just hoping to set astronomical matters straight. However, his system, which seems obvious and irrefutable to us now, is not entirely correct. As it will be seen, Copernicus’s orderly picture of a sun-centered solar system with orbits describing perfect circles does not predict future positions of planets even as accurately as those given by the Alphonsine tables, based on a Ptolemaic picture.5 Accordingly, Copernicus’s conjecture could have simply been judged as poor science, originating in the scientific backwater of northern Europe. Also for Copernicus, living in Poland, considerably removed from the epicenter of the Inquisition sweeping Italy and Spain, one would have thought him to be somewhat safer. Moreover, an introduction to the book written by an anonymous author explained that Copernicus’s system represented an alternative method for determining positions of planets, rather than a fundamental shift in the paradigm representing physical reality. Finally, as a priest working within the Church, one also would have thought he might have been more sheltered than the secular scholars who were disseminating thoughts on the heavens. But this was not the case. The Church by then had already been shaken to its core by Martin Luther, the far more threatening heretical priest in Germany. But just to be safe from persecution, Copernicus took a pair of precautionary steps: he dedicated his book to the pope, and he waited until he was on his deathbed before publishing.
Neither the brilliance nor the foresight of Copernicus characterized another priest, Giordano Bruno, a Dominican friar born near Naples in 1547. Although Bruno would have deserved not even a footnote for his contributions to science, as a catalyst for bringing scientific progress in Italy to a halt, his negative significance cannot be overstated. His first and most important work, La cena de le ceneri, published in 1584, was actually about the Last Supper and the Eucharist, the Christian ceremony of Holy Communion. Bruno was a Copernican, but operating with a hidden agenda. He was a disagreeable man, regularly turning off hosts and patrons alike with his comportment. Moreover, he never quite understood the mathematics of Copernicus, but invoked scientific jargon and analogy as tools for his arguments, and was actually taking aim at the authority of the Church. He believed in demoting the Earth to a planet, thereby removing it and its human inhabitants from their hallowed position at the central eye of the universe, and then bestowing souls on them all—humans, planets, stars—drawing no distinction among them. He was a proponent of an “ancient true philosophy,” Hermeticism.6 And within the tenets of his own dubious version of this philosophy, he could equate Catholicism and Protestantism. But most flagrant and egregious of all of his beliefs was his denial of the divinity of Christ.
For his poor science the Church might have simply dismissed him, but for his heresy he could not be tolerated. He had formed some of his ideas at Oxford and London, and it was there he first published them. But when he unwisely visited Venice in 1592, he was arrested on a trumped-up charge and imprisoned for a year. The following year he was moved to Rome, where the Inquisitors tried him for heresy. At his trial and during his seven-year imprisonment he still could have saved himself simply by recanting his heretical notions. But he remained intransigent to the end. Given one last chance to disavow this heresy, he again renounced the authority of the Church. It is said that he made the pronouncement that the other planets were also inhabited, and “the inhabitants are looking down and laughing at us.” Unfortunately the Church had the last laugh, as Bruno was burned at the stake as an example to other heretics. Thus he became an unfortunate example of publish and perish! Ironically, at the Piazza Campo dei Fioro in Rome, indeed at the spot where Bruno met his death in February of 1600, there now stands a statue of the hapless Dominican monk in full habit.
Meanwhile on the island of Hven (today Ven) in the Baltic Sea, the Danish astronomer Tycho Brahe was gathering data on the movements of the planets, hoping to confirm the geocentric picture of the universe. The king of Denmark had bestowed on him the deed to the island, along with authority over the inhabitants—the local peasants—to do with as he wished. The peasants became his servants, helped him to build his observatory, and assisted him in his observations. In this pretelescope age his principal observational tools were a number of large astrolabes along with massive brass disks with holes in them—holes through which he could peer and track the movements of the planets. The servants were routinely assigned the task of lifting him up bodily, chair and all, and moving him from disk to disk.
A colorful nobleman, in his youth Tycho had lost his nose in a duel over a mathematics problem. He wore a golden nose secured by ties fastened behind his ears, similar to the manner in which Venetian masks are worn at Carnevale. Fortunately, his skills as an astronomer were far superior to his skills as a swordsman. Over almost four decades Tycho compiled extensive data on the heavens. Though he toiled assiduously, he was never able to reconcile his observations with the geocentric picture, and as an honest scientist he never fudged his data. His numbers awaited a far more clever and open-minded mathematician to determine the proper order behind planetary motion.
Johannes Kepler (1571–1630), a German-born mathematician, astronomer, and astrologer to royalty, plying his various trades in Prague, turned out to be that mathematician. In an effort to ascertain the laws of planetary motion, among his mathematical musings had been one in which the five regular polyhedra were regarded as spacers between the orbits of the six planets (see Figure 5.2). The scheme bore little fruit. By then it was general knowledge that Tycho possessed the most accurate data extant, but when Tycho realized that Kepler was a superior mathematician, he became excessively possessive of the planetary data he had collected, resolving to keep it out of the younger man’s hands at all cost.
Fortunately for science and unfortunately for Tycho and the Brahe family, calamity struck at a banquet given for the visiting Danish king when Tycho sustained a burst bladder and died. For a period afterward Tycho’s family felt a moral obligation to continue to keep his priceless data out of Kepler’s hands. Frustrated, Kepler simply stole the data and immediately started the task of trying to sort it all out. His attentions were focused on Mars, which with its pronounced retrograde motion turned out to be a propitious choice. As he analyzed the data he found that the orbits of the planets, including that of the earth, were not circular, as asserted by Copernicus, but elliptical, with the sun located at one of the foci, or focal points, of the ellipse. This statement, Kepler’s first law of planetary motion, resolved once and for all the controversy surrounding the geocentric versus the heliocentric picture.
A second law formulated by Kepler from Tycho’s data was that the area swept out on the plane of a planet’s orbit by the radial vector of the planet is the same for any equal period of time. As a consequence, during its elliptical motion around the sun a planet travels faster when it is close to the sun than when it is farther away. Mathematically speaking, the rate of change of area is constant. Kepler published his first two laws in his New Astronomy in 1608.
Finally, Kepler gleaned the third of his planetary laws approximately a decade after he had discovered the first two. He found that the period for each planet (that is, its year, or the time for it to revolve around the sun) was indeed proportional to its distance from the sun, but that this was not a linear relationship. Rather, he found that the square of the period for a planet varied directly with the cube of its mean distance from the sun, thus T2 
R3, or T2/R3 = C, a constant.
As a simple example of the application of the third law, we can compute the period for Jupiter. The mean radius of Jupiter’s orbit is approximately 5.2 AU, compared to the earth’s 1.0 AU. With the earth’s period of T = 1 year, Kepler’s third law yields for Jupiter a period of 12 years.7 Saturn, revolving around the sun at a distance of 10 AU, takes approximately √(103), or about 31.6 years. In retrospect, the circumstances of Kepler’s theft of Tycho’s papers could be regarded as an example of how crime sometimes pays. But perhaps in this instance it is not an egregious crime. Tycho’s final pronouncement, so scientific lore goes, was to Kepler: “Don’t let my life seem to have been in vain.”
The scientific achievements of Copernicus, Tycho, and Kepler were all concerned with heavenly bodies and their motion. In southern Europe Galileo, Kepler’s contemporary and a confirmed Copernican, performed science on two separate fronts—terrestrial experimentation and celestial observation. Traditions regarding Galileo abound in his hometown of Pisa, for example there is a story that a seminal experiment with the pendulum was inspired by an oscillating chandelier in the cathedral following an earthquake.
A far more fundamental law that Galileo formulated just a short time later—by studying balls rolling down inclined boards—was the law of translational equilibrium: “a body placed in motion will continue in that state of motion forever unless a force is applied to stop it.” He rolled balls down one incline that was smoothly connected to a second, adjustable incline. No matter what angle the second incline made relative to the first, the balls would ascend until they reached exactly the same height from which they had been released. The line connecting the point of release and the point of final ascent was always a horizontal line, parallel to the ground. Galileo concluded that if the second incline were lowered to a horizontal position, the balls would have to keep rolling forever in order to reach the level at which they had been released; in geometry “parallel lines meet at infinity.” (The ball rolling down and increasing in speed, then rolling up and losing speed, can be effectively explained in terms of the constancy or “conservation” of mechanical energy: the kinetic plus the potential energy at every point in its motion remains the same.) Galileo’s law of motion contradicted the Aristotelian “law” proclaiming that bodies placed in motion would stop of their own accord unless a force were continually applied to them. Aristotle’s law certainly appears closer to everyday experience until one realizes that it is the force of friction that serves as the applied force slowing down traveling bodies. Although Galileo was the first to promulgate the law formally, it is now generally called Newton’s first law. Newton’s second law, that “force equals mass times acceleration,” is far more general. It accommodates the first law as a special case when the net force on a body is zero.
The use of a gently inclined board for measuring acceleration, however, was a particularly ingenious idea, and one that even Leonardo had not conceived, and explains why Galileo was able to collect data that was superior to Leonardo’s for accelerating bodies. The motion of a body could be slowed down to the point where precise measurements could be made while always maintaining the same relationship between acceleration and velocity and between velocity and displacement.
In the last chapter we also encountered “curvilinear motion” in the context of the trajectories of projectiles in two-dimensional space. Here we shall start out by examining “rectilinear motion,” or motion in a straight line. In the late sixteenth century natural philosophers still believed that the acceleration of heavy falling bodies would be greater than that of light ones. Even before Galileo, scholars had attempted to explain this acceleration, but no one had been able to properly quantify the phenomenon. A hundred years before Galileo, Leonardo da Vinci made his own study. Rather than ask how fast the body was descending, Leonardo sought to answer how far the body would descend in successive intervals of time. His conclusion was that the distances could be expressed as sequential integers: 1 unit of distance in the first interval, 2 units in the second, 3 units in the third, and so on. For example, after ten intervals the total distance the body would have dropped, according to Leonardo’s theory, would be given by 1 + 2 + 3 + 4 + … + 10 (55 units of distance. The sum of the values Leonardo obtained in his experiment of falling bodies, after time t, 1 + 2 + 3 + … t, is the average value ½ (1 + t), multiplied by the number of terms t, or s(t) = 1/x+1/x2 units of distance.8
A century later, when Galileo began to grapple with the problem of free-fall, he used precisely the same method Leonardo had used, namely to measure the distance a body falls in successive intervals of time. The young Galileo, according to Pisan tradition, dropped objects from the top of the campanile, or bell tower, which provided a convenient shape for his free-fall experiments (Figure 11.1). There is no historical basis for this tradition, and most likely Galileo made his discovery while serving as a professor at the University of Padua (1592–1610). We shall use the same tower, however, to demonstrate the sequence of distances for free-fall: the distances according to Leonardo (column L) and those according to Galileo (column G). In both columns the time intervals between the horizontal lines are the same. Unlike Leonardo, who had concluded that the distance covered in successive intervals were given by the sequence of integers (1, 2, 3, 4, …), Galileo determined these distances to be the sequence of odd integers only: 1, 3, 5, 7, …, his celebrated odd numbers rule. Thus after the first interval of time the distance would be 1 unit; after the first two intervals the total distance covered would be 1 + 3 = 4 units (or 22); after the first three intervals, 1 + 3 + 5 = 9 units (or 32) units, and after 4 intervals 16 units (or 42). The conclusion that Galileo reached was that the total distance a free-falling body would drop in t seconds would be given by s(t) = t2 units of distance. This equation differs from Leonardo’s expression describing free-fall.
Figure 11.1. Pisa. (left) The chandelier in the cathedral of Pisa that inspired Galileo to investigate the behavior of the pendulum. (right) Next to the cathedral is Pisa’s famous Leaning Tower, its campanile, or bell tower. The tower is used here as a backdrop to illustrate the data obtained by Leonardo and Galileo, who both investigated the law of falling bodies.
In 1604 Galileo observed in the night sky a new star, invisible one moment and shining bright the next (which we now realize was a supernova, or exploding star). The observation of a supernova is a rare event for the unaided eye, but far more common when telescopes are used. Indeed, since 1604 another such event has been seen only once—in 1987—from the southern hemisphere. From the perceived parallax effect Galileo knew that this was a member of the region of immutable stars, far beyond the sublunar region where phenomena were believed to be susceptible to change, according to Aristotelian astronomy. He gave a set of special lectures in Padua about the new star, pointing out the alarming evidence of Aristotle’s mistake.
In 1609 Galileo received the portentous news of the German-Dutch lens maker Hans Lippershey’s invention of the telescope just a few months earlier. Galileo immediately undertook the construction of a more powerful instrument for his own use. In making his own lenses he tried both Florentine glass and Murano glass from Venice; he found the former superior for its optical properties. Taking his telescope to the top of the campanile in Venice, he offered the rulers of the city a tool that would enable them to see approaching enemy warships and the merchants a means of obtaining advanced notice of arriving goods. When he turned his instrument to the heavens he observed unexpected wonders: the night sky teeming with hitherto unseen stars, sunspots (blemishes on a celestial body beyond the moon), and further evidence against Aristotelian astronomy. He saw mountains and valleys on the moon and a set of four moons around Jupiter. Galileo published his observations in a 1610 treatise, Sidereal Messenger. The book was dedicated to the Medicis, his new patrons in Florence, and indeed, he named the four satellites of Jupiter the Medicean stars (Figure 11.2).
In 1613 Galileo observed the phases of Venus—evidence that Venus was orbiting the sun—and immediately published his findings. In 1615 the inquisitors summoned him to Rome. There he pleaded, “Please … look through my telescope.” “It is not necessary,” suggested the Church, “you have faulty vision! Only the Earth can have a moon.” As for those mountains he claimed to have seen, “As a heavenly body, the moon has to be a perfectly smooth sphere!” Galileo was ordered “to abjure, curse and detest … and to recant the heretical teachings!” Remembering well the tragic lesson of Giordano Bruno, he swore, in effect, that he was mistaken, and that the sun revolved around a stationary earth. With the mediation of powerful friends, especially the Medicis, he was released with a stinging reprimand, but without incarceration or physical torture.
In 1623, when his old friend and patron Cardinal Mafeo Barberini was made Pope Urban VIII, Galileo might have hoped for better days. In 1624 he even had a series of six audiences with Urban, receiving tacit permission to write a book about the plausibility of the Copernican system, but only “if he would treat it as a hypothetical and mathematical possibility … not one to endorse over the accepted Aristotelian view.” But in 1632, twenty years after his first brush with the Inquisition, Galileo’s controversial book Dialogue on the Two Chief World Systems—Ptolemaic and Copernican, was published. The superiority of the Copernican view was more than evident in Galileo’s treatment. Moreover, depicted in the frontispiece in his own skillful hand were the three characters: Simplicio, the Aristotelian simpleton; Sagredo, the sagacious and reflective thinker; and Salviati, the intelligent layman serving as the interlocutor. Galileo, a talented artist, in creating the cartoon, had evidently drawn Simplicio in the likeness of the pope—or so the pope was led to believe by his advisors. In reality, the pope may never have seen the book.
Figure 11. 2. (top) Detail of title page of Galileo’s Sidereal Messenger (1610). This rare presentation copy of the book is inscribed “To the most illustrious Sig. Gabriel Chiabrera,” and signed “Galileo Galilei.” (bottom) Excerpt showing Galileo’s own annotation in his personal copy of the Dialogue on the Two Chief World Systems. (Courtesy History of Science Collections of the University of Oklahoma, Norman)
Again Galileo was invited to Rome for an audience with the inquisitors. Although he recanted readily this time, no amount of apology and penance could save him. After a lengthy trial, Galileo, then sixty-nine years old, was convicted and condemned—although spared the fate that had befallen Bruno. He would thereafter live under house arrest. He was never to write another book, and no publisher was ever to print any of his heretical works. Meanwhile, the pope displayed neither the patience nor the political courage to come to his aid, being mired in his own troubles with the Thirty Years’ War. The year turned out to be Galileo’s annus horribilis.
In 1638 while he was under house arrest and suffering from a number of debilitating maladies, his most important book was published. Two New Sciences distilled Galileo’s years of experimentation and analyses on problems of physics and strength of materials. The manuscript had been smuggled out to a friend, the publisher Louis Elzevir of Leyden, Holland. As with his other books, the language was Italian rather than the usual scholarly Latin, and just as with the Sidereal Messenger and the Dialogue on the Two Chief World Systems, there were three characters engaged in a philosophical dialogue. In one typical exchange, Simplicio declares, “When two objects of different weight are dropped simultaneously, the heavier object hits the ground before the lighter—displaying greater acceleration.” Sagredo’s riposte, “If you tie the two bodies together, creating a heavier body still, will the aggregate fall faster still, or will the lighter component impede the motion of the heavier one?” completely confounds Simplicio. The Interlocutor’s facetious comment, devoid of a decision on who had offered the more compelling argument, is “Oh, yes, I see.”
Galileo’s physics had gotten him into a great deal of difficulty with the Church. But ultimately it was not the physics of free-fall, nor any of the other aspects of his classical mechanics that contradicted Aristotelian mechanics, rather his endorsement of the Copernican view of the heliocentric universe that was manifestly unacceptable to the Church.
A recent book by Dava Sobel offers an account of the extraordinarily warm relationship that existed between Galileo and his eldest daughter, presented against the backdrop of his less than congenial relationship with the Church.9 It is evident that Galileo had a warm human side while dealing with his daughter, but there is little doubt that he did not suffer fools easily. The attitudes of humanists and theologians who were guided by old teachings and dogma, rather than by observation and reasoning, bedeviled him. He wrote, “I do not feel obliged to believe that the same god who has endowed us with sense, reason, and intellect has intended us to forgo their use.” The Church, however, saw him as a reincarnate Giordano Bruno, although never quite the equal of the reviled and defrocked priest. Nonetheless, he had become a thorn in their side, gaining a reputation for being ambitious, quarrelsome, and arrogant, and demonstrating an endless appetite for antagonizing former patrons and friends.
Galileo died in 1642, a broken man, blinded by the ravages of an eye disease, likely caused by damage to the fovea (part of the retina) that he had sustained in repeated observations of the sun through his telescope. A poignant meeting in his last years was with a visitor and ardent admirer, John Milton, who would himself go blind a few years later. In his great epic poem Milton would refer to the wonders revealed by Galileo’s telescope:
his ponderous shield
Etherel temper, massy, large and round,
Behind him cast; the broad circumference
Hung on his shoulders like the Moon, whose Orb
Through Optic Glass the Tuscan Artist views
At Ev’ning from the top of Fesole,
Or in Valderno, to describe new Lands,
Rivers or Mountains in her spotty Globe
—Paradise Lost, 1667, I:284–91
“The Tuscan artist!” What reverence, what veneration for the great scientist from the great poet, accepting him as a fellow artist.
Some of the lenses that Galileo personally ground, as well as a number of his own telescopes, are on display in the Museum of the History of Science in Florence. When his body was moved from the chapel of Saint Cosmas and Damian to Santa Croce for reburial in 1737, an admirer removed the middle finger of his right hand—now a relic, mounted in a glass case in the same room as the one housing the telescopes. And those four satellites he had discovered orbiting Jupiter are now known to astronomers as the Galilean moons.
The last great creator of the Italian Renaissance had been a scientist who established an enduring modus operandi for the modern scientist. And in reality Galileo’s technique of scientific inquiry had been little different than Leonardo’s, except in one critical respect. Unlike Leonardo, Galileo saw an urgent need to disseminate news of his discoveries, the effect of which would be to immediately influence the course of future research. Ironically, the publication of his results, especially those refuting misapprehensions of the Church, brought terrible personal consequences for him. And indeed, with the persecution and death of Galileo serious scientific inquiry in Italy and most of the rest of Catholic Europe came to an end, as scientific inquiry’s center of gravity shifted to the north.
Galileo’s Dialogue was placed on the list of banned books, the Index librorum prohibitorum, in 1633, where it remained until 1821. Finally, in 1992 Pope John Paul II courageously convened a commission to revisit the case of Galileo’s persecution by the Church. What was at stake was the issue of papal infallibility—a possible misjudgment by a predecessor who had sanctioned the trial some 360 years earlier. When the commission returned with its findings, there was no apology on behalf of the Church, no judgment that Galileo had been wronged. However, the final pronouncement that was published—“Galileo was a great man!”—was a tacit vindication of Galileo Galilei and a final closure to loose ends that had prevailed for all those years. The statement read, “Galileo sensed in his scientific research the presence of the Creator who, stirring the depths of his spirit, stimulated him, anticipating and assisting his intuitions.”
Just as 1564 had been a transitional year, with the death of Michelangelo and the birth of Galileo and Shakespeare, so too was 1642. Galileo died on January 8, and Isaac Newton was born on Christmas Day in Woolsthorpe, a village in the county of Linconshire in England. Only a numerologist might assign significance to the coincidence of those years, but certainly the coincidence of the dates makes them easier to remember. Nature operates with conservation laws—conservation of momentum, conservation of angular momentum, conservation of energy/mass, conservation of charge, and so on—but there is no conservation law for genius. When an individual of towering genius dies, talent of similar magnitude does not have to be born. There is always the possibility, however, that the prevailing intellectual climate is especially conducive for individuals with inherent gifts to step forward. Galileo spent his lifetime in combat against the prejudices of the past—the dogma and the intellectual tradition that had taken two millennia to become entrenched. Newton could ignore all that, taking dead aim at the future.
Demonstrating early on in his life that he would make a less-than-effective farmer, in 1661 Isaac Newton was sent off to Cambridge University. There is no evidence that he knew any more mathematics in 1661 than anyone else, but there is plenty of evidence of his possessing transcendent gifts for mathematics and analytic reasoning just four years later. In his final year as an undergraduate at Cambridge in 1665 he had already developed the binomial theorem, a mathematical expression to raise the sum of a pair of terms to a constant power—a power that could be positive or negative, integer or fraction. Just about the time he completed his studies at Cambridge, the bubonic plague that had been ravaging the Continent crossed the English Channel and began to decimate the population. The universities were closed, and the scholars sent home.
Sixteen months later, when the plague subsided the scholars were called back to the universities, and academic activities resumed. Isaac Newton, on his return to Cambridge, reported to his mentor Isaac Barrow with a list of his “mental inventions” (theories). The list included (1) applying the binomial theorem to determine the slopes of continuous curves, (2) the formulation of the “method of fluxions,” (3) experiments in optics and the invention of the reflecting telescope, (4) the formulation of the “inverse method of fluxions,” (5) the formulation of the three laws of motion (now known as Newton’s laws), and as a crowning glory, (6) the formulation of the universal law of gravitation. In that sixteen-month period he laid the foundations for optics and classical mechanics and for the unification of classical mechanics and astronomy.
The invention in mathematics of “fluxions” and “inverse fluxions,” as differential and integral calculus, respectively, earned for him the mantle of the greatest mathematician in history. It was Einstein’s opinion that Newton’s greatest single contribution was in developing the differential law, or showing that the laws of the universe could be expressed as differential equations. The period from 1665 to 1666 is referred to as Newton’s annus mirabilis—his miracle year.
Two years later Newton’s mentor, Isaac Barrow, holder of the first Lucasian Chair of Mathematics at Cambridge, resigned his professorship to take a position as a professor of theology; he recommended Newton as his successor. The episode is evocative of Leonardo’s mentor Verrocchio, putting down his paintbrush permanently upon seeing Leonardo’s work on the angel in the Baptism of Christ—Barrow and Verrocchio, each a distinguished master, recognized staggering, inscrutable genius.
It was not until 1687 that Newton, at the behest of his friend Edmund Halley (whose namesake is the famous comet), published his theories on classical mechanics in a book entitled Philosophia naturalis principia mathematica (mathematical principles of natural philosophy), or simply, the Principia. In synthesizing classical mechanics and astronomy the book unified the physics of terrestrial and celestial phenomena. The work is unrivaled in its significance in the history of science. As for Newton’s reasoning of the universal law of gravitation, an integral topic of the book, the great twentieth-century theoretical physicist Richard Feynman described it as “the greatest achievement of the human mind.”10
Newton asked the question, “What keeps the moon in its orbit?” The prevailing explanation of monks in monasteries had been “angels flapping their wings with great vigor.” Most popular renditions of Newton’s discovery of the universal law of gravitation include an apple: “Newton saw an apple fall,” “Newton was struck by an apple.” A century after Newton’s death, Lord Byron in Don Juan alluded to the connection with the apple, when he wrote: “And this is the sole mortal who could grapple / Since Adam, with a fall or with an apple.” There may or may not have been an incident with an apple, but in Newton’s explanation an apple does figure prominently, if just as an example of a mass. Newton sought to compare the distance an apple fell in the first second with how far the moon “fell” in that second. The notion of the moon falling simply refers to the distance its trajectory has deviated from a straight line. The information he had available was (1) the distance between the earth and the moon, measured center-to-center, is about 240,000 miles; (2) the radius of the earth, 4,000 miles; (3) the period of the moon, about 28 days; and (4) the distance an apple falls in the first second after it has been released, 16 feet. In the hands of Newton this data sufficed. From his second law of motion, Newton knew that the acceleration of a body was proportional to the force applied on the body, and that it was the mass of the body that represented the constant of proportionality. But since the distance a body travels depends on its acceleration (and time), Newton’s question reduced to comparing the distance the moon fell toward the earth (or deviated from a straight line) with how far the apple fell in the same time.
With the use of similar triangles Newton showed that in the first second, while the apple falls 16 feet, the moon falls 1/19 of an inch. The ratio of the two distances is close to 3,600. Meanwhile, the ratio of the distance between the apple and the earth (measured center-to-center) and the distance between the moon and the earth is 1/60. In correlating these, Newton arrived at his result: the dependence of the gravitational force on distance must be 1/602 (= 1/3,600). Finally, the general relationship—the universal law of gravitation, describing the gravitational attraction between two masses—is obtained as an inverse-square law. Specifically, the law says the force is the product of the two masses, m and m’, divided by the distance of their separation squared (a proportionality constant G, the universal gravitational constant, appears as a multiplier of the expression).11 The value of G was determined experimentally at the end of the eighteenth century by the Cambridge physicist Henry Cavendish, who published a paper entitled “Weighing the Earth.”
Newton’s third law states succinctly, “For every action (force), there is an equal and opposite reaction (force),” these forces acting on separate or contrary bodies. Applied to the apple and earth, the force with which the earth attracts the apple (action) exactly equals in magnitude the force with which the apple attracts the earth (reaction). Since the magnitudes of these forces are the same, we can then invoke Newton’s second law: the mass (of the apple) multiplied by the acceleration (g) of the apple downward equals the mass of the earth multiplied by the acceleration (a) of the earth upward. Accordingly, if the apple is held out at arms length and released, it descends with the normal acceleration g = 9.8 m/sec2, and simultaneously the earth ascends with an acceleration of 9.8 × 10–25m/sec2—approximately a millionth of a millionth of a millionth of a millionth of a meter/sec/sec—in order to meet the apple.
Applying this result to celestial bodies, we can specify the force in Newton’s second law as that of gravitation, and the acceleration as centripetal (center-seeking). The velocity of a body in orbit can then be computed. In the frontispiece of the Principia a cannonball is fired horizontally from a mountaintop at v, its range depending on the velocity (Figure 11.3). Fired at 30,000 km/hr (18,000 mi/hr or 5 mi/sec), the cannonball achieves a low-earth orbit. A velocity of approximately 3,700 km/hr (2,300 mi/hr) is sufficient to keep the moon in its orbit. The trajectories are all elliptical, and indeed the parabolic trajectories—illustrated by Leonardo and described mathematically by Galileo—are actually small-scale approximations of Newton’s and Kepler’s parabolic trajectories.12 The Newton formulation also shows that, fired at 42,000 km/hr (25,000 mi/hr or 7 mi/sec), the cannonball possesses “escape velocity”; it can reach the moon. Two hundred and seventy years before a Sputnik became the first artificial satellite, Newton had already computed the velocity needed to achieve orbit or to visit the moon and outer planets. It was the technology that had to catch up with the science.
Newton’s explanation of gravitation was much more powerful than its application to just the apple, the moon, and the earth. It applied to every particle of mass in the universe, thus the expression “universal law of gravitation.” Now if the expression for orbital velocity is set equal to the distance the satellite travels divided by its period, also a measure of its speed, what emerges is Kepler’s third law—demonstrating that the square of the period of a satellite is related directly to the cube of the orbit’s radius.
Figure 11.3. Detail of the frontispiece of Newton’s Principia, showing elliptical trajectories of cannonballs fired from the top of a mountain at V (Courtesy History of Science Collections of the University of Oklahoma, Norman)
The Principia synthesized astronomy and classical mechanics, in effect unifying the physics of the heavens and the earth. It gave a consistent mechanical system for the operation of the universe with mathematical precision. The calculus, as the mathematics of change, initially developed to solve problems in physics, proved to be the most powerful tool in every quantifiable field of intellectual endeavor. By 1687, however, Newton was essentially finished with his scientific queries and concentrated on two other fields at loggerheads with science—alchemy and religion, both of which he pursued with greater tenacity than he had science or mathematics. In 1693 he had a debilitating nervous breakdown, taking several years to recover before reverting to the reclusive, taciturn, and irascible man he had always been. The cause of the breakdown has been a puzzle for generations of psychologists, historians, and physicists. What was the reason for his personal crisis? The clues in his notebooks from that period suggest that the cause may have been his poor laboratory practices. In the early 1980s locks of his hair from that period—maintained in the Royal Society Museum—were tested with neutron activation (a sample of hair or fingernails are irradiated in a reactor, then analyzed for their gamma ray spectrum). The culprit was revealed. Newton had indeed damaged his brain by ingesting heavy elements, especially mercury, during his alchemical experiments. A pair of ironies presents itself in this connection: one of the most brilliant brains in history had destroyed itself; and the science he spawned more effectively than anybody else had explained how he had done it. But the revelation was three centuries too late to help him.
Newton was honored first by his countrymen, and then beginning in the Enlightenment of the eighteenth century, by all intellectuals—scientists, philosophers, and enlightened monarchs alike. William Wordsworth, ruminating on the view from his Cambridge room, wrote in his epic poem, the Prelude:
Near me hung Trinity’s loquacious clock,
Who never let the quarters, night or day
Slip by him unproclaimed, and told the hours
Twice over with a male and female voice.
Her pealing organ was my neighbor too;
And from my pillow looking forth by light
Of moon or favoring stars, I could behold
The antechapel where the statue stood
Of Newton with his prism and silent face,
The marble index of a mind for ever
Voyaging through the seas of Thought, alone.…”
—Prelude, 1850, III: 53–63
In molding Western civilization, Cambridge University casts a shadow rivaled by few other institutions. The ancient university boasts scholars who have left monumental impact in virtually every field of intellectual endeavor: Duns Scotus, Roger Bacon, Christopher Marlowe, John Milton, John Harvard, William Harvey, Oliver Cromwell, Henry Cavendish, William Wordsworth, Samuel Taylor Coleridge, Lord Byron, Charles Babbage, Charles Darwin, Lord Tennyson, Rupert Brooke, James Clerk Maxwell, Robert Walpole, Lord Kelvin, Srinavasa Ramanujan, Bertrand Russell, Ludwig Wittgenstein, John Keynes, Paul Dirac, C. S. Lewis, Alan Turing, Louis Leakey, James Watson, Francis Crick, Jane Goodall, Freeman Dyson, Steven Hawking, and a list that includes seventy-nine Nobel Prize winners (twenty-nine from the Cavendish Laboratory alone).13 During the 1930s one college at Cambridge claimed that it was “home to more Nobel Prize winners than France.” In the antechapel of Trinity College the two facing long walls are lined with some of the busts of the great sons of Cambridge—all turned toward the end of the hall, all gazing with veneration at a full size statue—Newton staring out the window. The inscription: Newton, qui genus humanum ingenio superavit (Newton, who surpassed human genius).
The French have never been quick to praise the English. But Voltaire, a man of uncommon wisdom and intellectual integrity, excoriated France’s own scientists, “You have confirmed in these tedious places what Newton found without leaving his room.” When Newton died in 1727 at the age of eighty-five, he became the first scientist to be buried in Westminster Abbey. And for the occasion a contest was held to choose a suitable epitaph to place over his tomb. The entry that won the contest is lengthy, and not particularly memorable. The one that is best remembered, however, is the couplet by Alexander Pope from his Essay on Man: “Nature and Nature’s Laws lay hid in night. / God said, ‘Let Newton be!’ and all was light.”
In 1987, when institutions of science throughout the world celebrated the three-hundredth anniversary of the publication of the Principia, I was in Washington, D.C. with some of my students to tour an exhibit of Newton memorabilia—manuscripts, spectacles, a lock of hair, and a priceless copy of the first edition of the Principia—all on display at the National Museum of American History. As I stood before a geometric proof posted on a wall, admiring its conciseness and precision, I glimpsed a guide sweep right through the room with her charges, a group of visitors. When they paused momentarily in front of a case in which some of Newton’s manuscripts lay, she announced, with the air of authority bestowed on her by her position, “This collection is to commemorate the work of Sir Isaac Newton, author of a famous book published exactly three hundred years ago.” She continued, “He was a famous English scientist. He may have been gay, but we are not quite sure.” Newton, who invented calculus, formulated the universal law of gravitation, unified the physics of terrestrial and celestial mechanics, and perhaps even had a hand in sowing the seeds of the Industrial Revolution, “… he may have been gay, but we are not sure!” Her remark, delivered in passing, seemed to echo endlessly in that room. This was all she could say about Isaac Newton. And right across from the National Museum of Natural History on the Mall, the grassy rectangular patch lined with great museums, stands the National Air and Space Museum—exhibiting satellites that orbit the earth, space capsules, replicas of the rocket ships that took astronauts to the moon, and indeed samples of rocks these astronauts brought back with them. That is the real monument honoring Isaac Newton.
Leonardo realized early that mathematics comprised the very firmament of science, the pillars on which science stood. “Oh, students,” he wrote, “study mathematics and do not build without foundations.” How prescient he was in that intuitive conviction. But it would be Newton who finally demonstrated the mathematical nature of the universe, inextricably melding mathematics and physics. In the eighteenth and nineteenth centuries physicists refined the formulation of Newton’s classical mechanics and went on to develop the sciences of heat, light, electricity, and magnetism. In 1864, while the Civil War was raging in North America, the great Scottish physicist James Clerk Maxwell, then a professor at Cambridge University, succeeded in synthesizing the laws of electricity and magnetism, reducing them to just four equations. Few physicists would argue that Maxwell was the finest physicist from the time of Newton to the time of Einstein, characterized by a notable contemporary as “incapable of making a mistake in solving a physics problem.”
By the end of the nineteenth century many physicists, with Lord Kelvin as the most vocal proponent, were convinced that most of the significant questions in physics had been answered and all that remained was to tweak things here and there, just refine some of the measurements. Ironically, physics was poised at the threshold of a pair of revolutions—one leading to relativity, the other to quantum mechanics.