Early Modern Period

14. The Copernican Cosmos (1543)

The task of ordering the visible universe into an intelligible whole or cosmos has long challenged astronomers. According to Aristotle (384–322 BCE), the heavenly bodies—Moon, Sun, wandering stars or planets, and fixed stars—are embedded in rigidly rotating, Earth-centered, transparent spheres. In this way each “star” moves uniformly in a circle around a stationary earth. Ptolemy (90–168 CE), who was a great observer of the heavens, embellished Aristotle’s basic structure in order to better account for what he actually saw: planets whose brightness varied and planets that sped up, slowed down, and sometimes reversed direction.

Figure 23 illustrates how Ptolemy’s embellishments apply to a single planet. Accordingly, the planet moves in a relatively small circle or epicycle whose center, in turn, moves along the primary circular orbit or deferent. Thus, two concentric spheres border the epicycle’s orbit and determine its size. The epicycle itself accounts for the planet’s occasional backward or retrograde motion. The center of the epicycle’s bordering spheres can, in turn, be shifted from the center of the universe where the earth resides—a feature not illustrated here. Ptolemy’s task was to derive numbers that characterize these circles, motions, and shifts from known sequences of observed positions in order to produce an empirically accurate geocentric model of the cosmos. Such was his success that astronomers, astrologers, and calendar makers found his model useful for more than 1,400 years.

Other ancient astronomers, notably Aristarchus (310–230 BCE), placed the sun at the center of the universe, but because such solar systems implausibly require a moving earth they never gained many adherents. Not until 1543 when Nicolaus Copernicus (1473–1543) published his On the Revolutions of the Celestial Spheres did the heliocentrism of Aristarchus begin to replace the geocentrism of Aristotle and Ptolemy. Yet one wonders. What advantage did the cosmos of Aristarchus and Copernicus have over the cosmos of Aristotle and Ptolemy? After all, because Copernicus allowed only circular orbits, he had to employ even more epicycles than did Ptolemy in order to achieve an equivalent accuracy. The answer is that Copernicus’s cosmos requires retrograde motion, while Ptolemy’s merely allows for it. Copernicus’s system had a logical coherence that Ptolemy’s lacked.

Figure 24 illustrates this logical coherence. Four straight lines of sight from the earth to an outer planet and on to the “fixed” stars in its background are shown. Because Copernicus arranged the planets so that the more quickly moving ones are closer to the sun, the connected pairs of small open circles representing contemporaneous positions are further apart on the earth’s orbit than they are on the outer planet’s orbit. Therefore, as the earth moves from point A, near opposition, to point B the outer planet appears to move backward or to retrogress relative to the background of fixed stars. At C this retrogression diminishes and at D the outer planet’s forward motion resumes.

Retrograde motion is observed every time an outer planet nears opposition, that is, every time Sun, Earth, and outer planet line up in that order. Of course, Ptolemy’s geocentric cosmos also accounts for retrograde motion but only with just the right, individually chosen epicycles, non-concentric spheres, and planetary speeds.

Copernicus must have had the connection between the real motion of the earth and the apparent retrograde motion of the outer planets in mind in writing his introduction to On the Revolutions:

I finally discovered ... that if the movements of the other wandering stars are correlated with the circular movement of the Earth, and if the movements are computed in accordance with the revolution of each planet, not only do all their phenomena follow from that but also this correlation binds together so closely the order and magnitudes of all the planets and of their spheres or orbital circles and the heavens themselves that nothing can be shifted around in any part of them without disrupting the remaining parts and the universe as a whole.

After studying in Krakow, Bologna, Rome, and Padua, Copernicus performed the duties of a trustee, physician, translator, and diplomat for the diocese of Varmia in so-called Royal Prussia. Copernicus’s native language was probably German even though this region was then a part of the kingdom of Poland and today, after many vicissitudes, is again part of Poland, now a republic. He saw the birth of the Protestant movement launched by Martin Luther (1483–1546) in 1517 and the ravaging of his home by Teutonic knights. Against the wishes of his bishop, Copernicus extended hospitality to the Lutheran mathematician, Joachim Rheticus. Rheticus successfully urged Copernicus to publish On the Revolutions.

Copernicus held the first printed edition of On the Revolutions in his hands shortly before his death in 1543. We do not know whether Copernicus was aware of the anonymous preface, inserted by another Lutheran mathematician Andreas Osiander. If so, he would have been dismayed. For Osiander’s preface characterized heliocentricism as a mere calculational device that allows one to make accurate predictions without pretending to describe reality. Yet it is clear from the introduction to On the Revolutions that Copernicus believed in the reality of the heliocentric universe. Copernicus believed that he had discovered the “machinery of the world,” which had been constructed by the “Most Orderly Workman of all.”

15. The Impossibility of Perpetual Motion (1586)

A chain of fourteen identical, spherical beads is draped over a triangular support whose lower edge is parallel to the ground (figure 25). According to Simon Stevin (1548–1620), its Flemish originator and contemporary of William Shakespeare (1564–1616), this clootcrans or “wreath of spheres,” as it has been variously called, must remain stationary even if one assumes the beads can slip without friction on their supporting surface. Suppose, Stevin reasoned, that the wreath were to slip clockwise. Each sphere would soon take up a position previously held by an adjacent sphere. Then the wreath would recover its original aspect, and then slip again, and so on, ad infinitum. Since perpetual motion is clearly absurd, the wreath of spheres must remain in its original position.

The Nobel laureate Richard Feynman (1918–1988) referred to Stevin’s wreath, in his Lectures on Physics, in order to highlight the impossibility of perpetual motion. Feynman went on to say of Stevin’s wreath, “If you can get an epitaph like that on your gravestone, you are doing fine.” The impossibility of perpetual motion is, indeed, an important physical concept. Sadi Carnot appealed to it in 1824 in order to motivate his statement of the second law of thermodynamics. But we know neither where Simon Stevin is buried nor the location of his gravestone. Feynman must have meant, not Stevin’s gravestone, but rather the memorial statue of Stevin designed by Eugène Simonis and erected in Bruges in 1846 at a place now called Simon Stevin Plaza. The statue shows Stevin holding a scroll upon which the wreath of spheres is engraved. Simonis, no doubt, took the wreath of spheres from the title page of Stevin’s text The Principles of the Art of Weighing.

In his text, Stevin argues that since that part of the chain hanging below the supporting triangle is symmetrically arranged around a vertical line passing through its center, one can remove that part of the chain without disturbing the equilibrium of the wreath’s remaining parts—as shown in the left panel of figure 26. One can also replace the several beads on each inclined plane with a single bead of the same total weight without disturbing that equilibrium—as illustrated in the right panel. (I have, without modifying anything essential, added a frictionless pulley to the arrangement.) These transformations prove a theorem according to which: Two weights on inclined planes balance each other if the magnitude of each weight is proportional to the length of the inclined plane upon which it rests. This theorem had been discovered much earlier by one Jordanus Nemorarius, a figure about whom little is known—only that he wrote in Latin and flourished somewhere between 1050 and 1350. But it is Stevin’s method of proof, starting as it does from the impossibility of perpetual motion, that interests us.

As a young engineer, Stevin helped design and build wind-driven mills that drained the swamps of his native Flanders, now northern Belgium and the adjacent parts of Holland. Eventually Stevin was drawn into the struggle against the Spanish domination of the Low Countries—a struggle in which Stevin served Prince Maurice, the son of William of Orange, as tutor and military advisor. As Maurice’s military advisor, Stevin brought rational principles to laying sieges, building fortifications, and supplying armies. As Maurice’s tutor, Stevin compiled textbooks on dialectic, arithmetic, geometry, algebra, mechanics, astronomy, and music—textbooks that contain not only what was then known but also Stevin’s contributions to each subject. So much did Prince Maurice value Stevin’s textbooks he carried them with him on campaign.

Stevin is also famous for enthusiastically, if not always convincingly, promoting the use of his native Dutch. According to Stevin, the Dutch language is particularly suited to the scientific enterprise. For Stevin supposed that an efficiently constructed language should represent each single thing by a single word composed of a single syllable. And, according to his investigations, Dutch has more monosyllabic words available for this purpose than either Greek or Latin or, presumably, any of their derivative languages. Stevin even imagined Dutch to be the language of an ancient and enlightened Golden Age in which all people lived in peace and prosperity. Such were his fancies, but Stevin did materially contribute to the evolution of the Dutch language by coining new Dutch words for recently developed technical concepts. Stevin’s efforts to promote Dutch were a part of a larger movement toward employing the vernacular in scientific writing—a movement that attracted a new class of readers to the literature of science.

16. Snell’s Law (1621)

One of the most familiar manifestations of refraction is the broken appearance of a straight object resting in and projecting out of a glass of water. While the geometry of this particular phenomenon (involving as it does the light reflected from the object seen, its propagation through water and air, and its reception at the eye) is quite complicated, the essence of refraction is simple.

In a homogeneous medium composed of, for instance, water or air, light travels along straight lines. But when a beam of light or light ray leaves one medium and enters another, that part of the light not reflected at the boundary refracts or bends toward or away from the line normal (or perpendicular in a two-dimensional view) to the boundary between the two media. In particular, a ray inclines toward the normal when leaving air and entering water and inclines away from the normal when leaving water and entering air—as illustrated in figure 27 and in figure 19.

Ptolemy (90–168 CE), Alhazen (985–1040), and Kepler (1571–1630) all sought and failed to find an accurate mathematical description of refraction. Not until around 1621 did such a description—attributed in correspondence of the time to the Dutchman Willebrord Snell (1580–1626)—emerge. Snell’s law asserts that the angles, and , between the ray and the normal to the boundary in each medium are related to each other by the equation where the indices of refraction, and , characterize the two media. Snell and his contemporaries could, by measuring the two angles and , determine the ratio of one index to the other. When medium 2 is water and medium 1 is air, the ratio is about 4/3.

Just as important to us as the accuracy of Snell’s law is what its form says about the nature of light. René Descartes (1596–1650) demonstrated that Snell’s law follows from the hypothesis that light is composed of tiny particles that upon crossing the boundary between two different media either speed up or slow down in the direction of the boundary normal. When, for instance, leaving air and entering water the particles of light speed up—at least according to Descartes. Given this hypothesis and that , light must be faster in water than in air by a factor of 4/3.

Descartes’s interpretation of Snell’s law is clever, but is neither compelling nor unique. Pierre de Fermat (1601–1665) reasonably objected that, on the contrary, light should travel more slowly in water than in air, since water, being denser than air, must offer more resistance to the particles of light. Elevating this supposition to a postulate, Fermat found that Snell’s law follows from an ingenious principle of his own invention: light travels between two points along the quickest route—a principle now known as Fermat’s principle or the principle of least time.

Compare, for instance, the path taken by light as it travels (from air to water) across the air-water interface, as shown in figure 27, to the path taken by a lifeguard as she runs along the beach, plunges into the water, and swims to a person in distress. Since the lifeguard can run faster on the beach than she can swim in water, she minimizes her travel time by covering more distance on the beach than in the water. Consequently, when entering the water, she bends her direction of travel toward the person in distress and, thus, toward the perpendicular to the beach-water interface.

Of course one could decide between Descartes’s and Fermat’s interpretations of the phenomenon by measuring the speed of light in water and comparing it to the speed of light in air. If light is faster in water than in air, Descartes is right; if light is slower in water than in air, Fermat is right. But technical difficulties delayed such measurements until 1850 at which time it was shown that light travels more slowly in water than in air. Fermat was right.

In the meantime the followers of Descartes sharply criticized Fermat’s principle as unphysical. They asked, “Is light supposed to try out all possible paths, compare their transit times, and then choose the quickest path?” It is not so surprising that lifeguards can discern the quickest path and also that ants can and do search out and occupy the quickest route between their nest and a supply of food, as illustrated in figure 28, for in both cases theirs is a learned, goal-oriented behavior. To the followers of Descartes and most other seventeenth-century natural philosophers, as scientists were then called, light propagation had to be a purely mechanical process whose explanation necessarily excludes such behavior. It was not until the wave theory of light triumphed in the early nineteenth century that the conflict between a mathematically sufficient description (based on Fermat’s principle of least time) and the expectations of a mechanistic physics was resolved.

17. The Mountains on the Moon (1610)

By the first decade of the seventeenth century the time had come for the telescope to be invented. In Holland several Dutch lens grinders and spectacle makers hit upon the same idea at the same time: a tube that aligned two lenses, one concave eyepiece and one convex light-gathering objective. In 1608 one of these Dutchmen, Hans Lippershey, attempted to patent a spyglass, as it was then called, that enabled one to see “things far away as if they were nearby.” Lippershey’s spyglass only magnified linear dimensions by a factor of three. Even so, because it had obvious military applications, the news of its discovery spread quickly across Europe.

Galileo heard of the spyglass in May 1609, discovered the principle of its construction, and that summer began making his own improved spyglasses. Eventually he achieved a magnifying power of 30. This number is important for, as Galileo later reported, one needs a magnifying power of at least 20 to see the astonishing sights he saw when he first pointed his spyglass toward the heavens: the moon’s surface not smooth, as had been supposed, but rough with mountains and craters; the Milky Way resolved into numerous individual stars; and, most amazing of all, four new wandering “stars” orbiting Jupiter.

Galileo understood the importance of these discoveries. Not only were they novelties, intrinsically interesting, and easily comprehended, they also had far-reaching consequences for our understanding of the cosmos. And because Galileo wanted to quickly communicate these discoveries to the scholars of Europe, he wrote his short report, Sidereus Nuncius or Starry Messenger, uncharacteristically in the Latin of his day rather in the vernacular Italian. Even so Starry Messenger created an immediate sensation in Italy and was discussed on the streets of Padua, Venice, Florence, and Rome.

Figure 29a is by Galileo’s own hand. It shows the surface of the moon as he saw it, roughened with mountains and craters. Figure 29b illustrates the method he used to determine the relative height of these mountains. Its key is that Galileo recognized in the moon phenomena similar to phenomena on Earth. In particular, he recognized that the white spots just to the left of the line dividing the moon’s shaded and illumined parts in figure 29a are mountaintops ablaze in the rising or setting sun. Figure 29b shows how the Pythagorean theorem, , relates the radius of the moon , the height of a mountain , and the distance from an illuminated mountaintop to the line dividing the shaded and lighted lunar hemispheres . Because Galileo had a value for the moon’s radius and could estimate the ratio from his drawings, he was able to use this equation to find values for the height of various lunar mountains. He found the highest of these mountains to be about four miles high—not far from modern determinations.

The very roughness of the surface of the moon undermines the distinction, so important in Aristotelian cosmology, between the imperfect sublunar realm of the earth with its atmosphere and the perfect realm of the heavens, the moon included. At the time of his telescopic discoveries, Galileo was a secret Copernican. Within a few years he would become a public advocate of the Copernican cosmos. It seems likely that these discoveries, especially of the rough surface of the moon and of the four orbiting satellites of Jupiter, confirmed and energized an advocacy that, eventually, led to Galileo’s conflict with the church.

Galileo’s Starry Messenger is itself a jewel of composition. Historians of science value it greatly. Scientists and science writers should also. Galileo’s later writing, found for instance in the dialogue Two Chief World Systems, is famous for its sarcastic polemics that destroyed his opponents’ positions. But in Starry Messenger we find no sarcasm and no polemics. Rather we find apt metaphors, rounded sentences, and efficient summaries that clearly express complex ideas with excitement. Starry Messenger is a delight for readers and a model for writers.

18. The Moons of Jupiter (1610)

Among the marvels Galileo saw when he turned his newly constructed telescope toward the heavens in the winter of 1609–1610 were the four brightest moons of Jupiter. He dubbed these the Medicean planets in order to flatter the man whose patronage he sought and to whom he dedicated the seventy-page booklet he published in March 1610 that reported on these telescopic discoveries. This was Sidereus Nuncius, that is, Starry Messenger, dedicated to “The Most Serene Cosimo II de Medici, Fourth Grand Duke of Tuscany.”

In this booklet, Galileo described the way in which he constructed a telescope capable of magnifying linear dimensions by a factor of 30 and the things he saw with his telescope: mountains on the moon, newly visible stars into which he resolved the Milky Way, the phases of Venus, and the finite-sized, disklike appearance of the planets. Then he announced, “There remains the matter which in my opinion deserves to be considered the most important of all—the disclosure of four PLANETS never seen from the creation of the world up to our own time.” On the night of January 7, 1610, Galileo observed that Jupiter appeared close to what he initially took to be three small stars in its background. But he also noticed, and later recalled, that these stars and Jupiter unaccountably lined up along the ecliptic, that is, along the band in which the planets move through the background of “fixed” stars. That night two of these stars were to the east of Jupiter and one to the west—as illustrated in figure 30. The next night, January 8, all three were to the west of Jupiter and again lined up along the ecliptic. Although these appearances interested Galileo, he did not yet understand that these “stars” were Jupiter’s moons.

Because the sky was overcast on January 9, Galileo made no observations. Then on January 10–11 only two stars appeared, on both occasions to the east of Jupiter. When Galileo began observing on January 12 he again saw only two stars: the brighter one to the east of Jupiter and the less bright one to the west. But as he was observing, another star emerged from the east side of Jupiter. On January 13 Galileo saw four stars—all lined up with Jupiter along the ecliptic.

Galileo kept observing every clear night for two months, long enough to conclude that there were four such stars, moons, or planets, as he variously called them, illumined by the sun and that they revolved, in unequal circles, around Jupiter but not long enough to determine their periods of revolution. Galileo thought that their varying degree of brightness, here and in Galileo’s original drawings crudely represented by size, is caused by different refractions of their images through Jupiter’s atmosphere. We now know that these moons spin on their axes and in this way bring into view different parts of their surface—parts that reflect sunlight in different degrees.

Galileo also noted, almost in passing, that these moons accompany Jupiter in its twelve-year orbit around the sun. It is at this point that Galileo, long a secret Copernican, became a public one. Important to this transition and to his subsequent evolution to public defender of Copernicus’s cosmology is Galileo’s observation “that the revolutions are swifter in those planets which describe smaller circles around Jupiter.” Jupiter and its moons validated, by reproducing in miniature, the Copernican cosmos in which those planets closer to the Sun move more swiftly. Furthermore:

Here we have a fine and elegant argument for quieting the doubts of those who, while accepting with tranquil mind the revolutions of the planets about the Sun in the Copernican system, are mightily disturbed to have the Moon alone revolve about the Earth and accompany it in an annual rotation about the Sun. Some have believed that this structure of the universe should be rejected as impossible. But now we have not just one planet rotating about another while both run through a great orbit around the Sun; our own eyes show us four stars which wander around Jupiter as does the Moon around the Earth, while all together trace out a grand revolution about the Sun in the space of twelve years.

Today, the four brightest moons of Jupiter are named after mythological figures: Ganymede, Callisto, Io, and Europa—all conquests of the god Jupiter. Sometimes we, quite appropriately, refer to this group as the Galilean satellites.

Altogether the orbits of sixty-seven moons of Jupiter have now been confirmed. Most of these were captured by Jupiter after its formation and, as a consequence, have highly elliptical orbits, highly inclined to the ecliptic. Had Galileo been able to observe some of these smaller satellites with irregular orbits as well as the four brightest ones, he might not have seen in Jupiter and its moons a miniature Copernican solar system.

As it was, Galileo became a vocal champion of Copernicus, and as such unwisely responded to detractors who pitted Scripture against Copernican cosmology. In particular, these detractors pointed to Joshua’s command (Joshua 10:12–13) that the sun stand still and to the several Biblical references to the stability of the earth. Galileo, ever the faithful Catholic, did not doubt that Joshua had miraculously lengthened the day “until the nation took revenge on their enemies,” but contextualized the biblical account as written for readers who believed that the Sun moved around a stationary Earth. Such a defense did not impress church officials who reserved the office of interpreting Scripture to themselves.

19. Kepler’s Laws of Planetary Motion (1620)

Johannes Kepler (1571–1630) delighted in uncovering the hidden order of the universe. As a young man, he embraced the order in Copernicus’s heliocentric arrangement of the heavenly bodies. Copernicus’s universe was not significantly more accurate than Ptolemy’s geocentric one, but it was more ordered with each of its features logically entailing others.

Crucial to his search for order was Kepler’s encounter with Tycho Brahe (1546–1601) in February 1600. Tycho’s personal resources and connections had allowed him to construct the best pre-telescopic, astronomical observatories of his time: first the Uraniborg observatory on the island of Hven for the Danish king Frederick II and then, after falling out with Frederick’s heir, an observatory near Prague financed by the Holy Roman emperor, Rudolph II. Tycho, more so than any of his predecessors, saw the value of observing the same planet as it progressed through a complete orbit and of determining the uncertainty associated with each observation.

However, Tycho was not a Copernican. Rather, he promoted his own peculiar cosmology in which the five visible planets (Mercury, Venus, Mars, Jupiter, and Saturn) moved in concentric circles around the sun while the sun itself circled a stationary earth at the center of the Universe. Tycho wanted Kepler to use his data to verify this system, but because Tycho was uncertain of Kepler’s loyalties he shared his data with Kepler with great ambivalence, allowing, for instance, Kepler to view the data but not to copy it for later use.

On Tycho’s death in October 1601, Kepler inherited both Tycho’s position as imperial mathematician and Tycho’s data on condition that he complete the work of reforming theoretical astronomy on the basis of Tycho’s cosmology. Ironically, the high quality of Tycho’s data made this task impossible. For none of the established systems, Ptolemaic, Copernican, or Tychonic, could incorporate Tycho’s observations with the required precision. Eventually Kepler had to abandon these cosmologies, based as they were on combinations of circular motions. Kepler then tried to fit the orbit of Mars to an ellipse with the stationary Sun at one of its two foci—as illustrated in the right panel of figure 31—and found that this scheme worked perfectly.

The left panel shows how an ellipse can be constructed with a string, two pushpins, and a pencil. The ends of the string are attached to the pushpins and these, in turn, are stuck in a plane surface. The pencil marks out a circuit in the plane as it slides along and holds the string taut. Hence an ellipse is the set of points lying in a plane whose distances to each of two points in the plane sum to a constant. The two points are called the foci of the ellipse. If the two foci happen to coincide, the ellipse reduces to a circle. A line drawn through the two foci from one end of the ellipse to the other (2a in the diagram) is twice the semi-major axis a.

That each of the planets, Earth included, moves in an ellipse with the sun at one focus is known as Kepler’s first law of planetary motion. The right panel illustrates Kepler’s second law: A line extending from the sun to the planet sweeps out equal areas (e.g., the shaded areas) in equal intervals of time. Consequently, the closer a planet approaches the Sun, the faster it moves. While Kepler’s first two laws relate the different parts of a single planetary orbit to one another, his third law of planetary motion describes a relationship among the orbits of different planets. In particular, the square of the time required for a planet to complete a single revolution around the sun is proportional to the cube of the planet’s semi-major radius . In other words, the ratio is the same for all of the planets. Kepler suspected that these relationships were the result of the Sun’s push and pull on the planet but never discovered the form of that push or pull. He presented evidence for the first two of his three laws of planetary motion in New Astronomy (1609) and for the third in Harmonies of the World (1619).

Kepler was a generous spirit who sought to conciliate jealous rivals and a faithful Lutheran who twice uprooted his family in order to prevent their forced conversion to Catholicism. He was an unfortunate man who suffered the death of his first wife and eight of his twelve children, the need to defend his mother against a charge of witchcraft, and the outbreak of a war, the Thirty Years’ War, that devastated central Europe. Kepler was also deeply pious and deeply grateful for having discovered that for which he had long sought: new evidence of a universal harmony. At the close of his text Harmonies of the World, he offered to “Thee, O Lord Creator, who by the light of nature arouse in us a longing for the light of grace” the following prayer: “If I have been drawn into rashness by the beauty of thy works, or if I have pursued my own glory among men while engaged in a work intended for Thy Glory, be merciful, be compassionate, and pardon me; and finally deign graciously to effect that these demonstrations give way to Thy Glory and the salvation of souls and nowhere be an obstacle to them.”

20. Galileo on Free Fall (1638)

Galileo’s fame as a scientist rests on his ability to abstract the essential physics from complicated phenomena, to describe that physics in eloquent words and simple mathematics, and to verify that description with cleverly designed experiments. But Galileo had multiple talents. In fact, Galileo did so many things well his twentieth-century biographer, Stillman Drake, claimed that it is “hard to say whether the qualities of the man of the Renaissance were dominant, or those of our own scientific age.” He was an excellent prose stylist, an accomplished visual artist, an ardent gardener, a proficient lute player, and a vigorous debater.

One of Galileo’s tactics was to construct “thought experiments” that helped him explore the consequences of a hypothesis—including any absurdities that hypothesis might entail. Figure 32 illustrates a thought experiment that Galileo used in Two New Sciences (1638). Formally, Two New Sciences records a four-day-long conversation among three friends: Salviati, speaking for Galileo; Sagredo, questioning, intelligent, and open-minded; and Simplicio, naively representing what he (Simplicio) understood to be Aristotle’s position.

Aristotle had advanced plausible, if superficial, explanations of everyday phenomena. For instance, because objects in motion invariably slow down and come to a stop, continuous motion requires a continuously acting mover. And because heavier objects descend more quickly through water than do lighter ones, heavier objects descend more quickly through all media, including air, in direct proportion to their heaviness or weight and inversely proportional to the resistance of the medium—at least according to Aristotle.

Galileo’s persona, Salviati, contests these ideas by arguing as follows. Suppose, in accordance with Aristotle’s analysis, a one-pound stone falls with a speed of one cubit per second and a four-pound stone falls with a speed of four cubits per second. If tied together, the lighter stone should retard the speed of the heavier one and the heavier one increase the speed of the lighter one and, in this way, result in a speed intermediate between one and four cubits per second. On the other hand, the five-pound package of two stones, considered as a whole, should fall at a speed of five cubits per second. This contradiction can be avoided only if all objects fall from rest at the same speed.

However, Simplicio, the naïve Aristotelian, remained puzzled: “I am still at sea, he says, because it appears to me that the smaller stone when added to the larger increases its weight and by adding weight I do not see how it can fail to increase its speed or, at least, not to diminish it.” Salviati’s response, that is, Galileo’s response, to Simplicio surprises us:

It will not be beyond you when I have once shown you the mistake under which you are laboring. ... One always feels the pressure upon his shoulders when he prevents the motion of a load resting upon him; but if one descends just as rapidly as the load would fall how can it gravitate or press upon him? Do you not see that this would be the same as trying to strike a man with a lance when he is running away from you with a speed which is equal to, or even greater than, that with which you are following him? You must therefore conclude that, during free and natural fall, the small stone does not press upon the larger and consequently does not increase its weight as it does when at rest.

Evidently, when in free fall, neither stone presses upon the other, that is, neither has weight relative to the other—an idea that Albert Einstein (1879–1955) exploited to great effect almost three centuries later in constructing his theory of general relativity.

Galileo also appealed to actual experiments. While experiments that could support his case were hard to perform, he knew of at least two that challenged Aristotle’s conclusions. Simon Stevin, in 1586, and much earlier John Philoponus (490–570 CE) had dropped objects of very different weight from great heights and found that weight alone makes no significant difference in the rate of fall. (Galileo could also have dropped cannon balls from the Leaning Tower of Pisa for this purpose, but he never claimed to have done so.) It may be that Salviati refers to these earlier experiments in the following passage from their first day of conversation in Two New Sciences:

Aristotle says, “an iron ball of one hundred pounds falling from a height of one hundred cubits reaches the ground before a one-pound ball has fallen a single cubit.” I [Salviati] say that they arrive at the same time. You find on making the experiment, that the larger outstrips the smaller by two finger-breaths, that is, when the larger has reached the ground, the other is short of it by two finger breaths; now you would not hide behind these two fingers the ninety-nine cubits of Aristotle, nor would you mention my small error and at the same time pass over in silence his very large one.

After undermining Aristotle’s explanation of falling objects, Galileo proposed that in the absence of a restraining medium all objects accelerate downward at the same rate, in particular, by incrementing their speed equal amounts in equal intervals of time, that is, by about 32 feet (or 9.8 meters) per second every second. Galileo developed the mathematical consequences of this hypothesis and then devised an experiment to look for them. Because objects fall much too quickly for convenient measurement, he rolled spheres down an inclined plane in order to slow down the natural downward acceleration of free fall. This whole procedure (hypothesis, deduction, and experimental verification) worked brilliantly for Galileo and since his time has become standard practice for modern physics.

21. Galileo on Projectile Motion (1638)

In 1616 the office of Inquisitor General of the Roman Catholic Church warned Galileo to “relinquish altogether the said opinion that the Sun is the center of the world and immovable and that the Earth moves.” Consequently, Galileo promised not “to hold, teach, or defend in any way whatsoever, verbally or in writing” the said opinion—a promise that, by publishing Dialogue Concerning Two Chief World Systems in 1632, he broke in the most dramatic way. The conceit of the dialogue that placed powerful arguments in favor of a heliocentric universe and weak objections to it in the mouths of fictitious interlocutors fooled no one.

The inquisitors convicted Galileo of “vehement suspicion of heresy” and sentenced him to life in prison, eventually commuting his sentence to confinement to his villa in Arcetri. This time Galileo kept the promise extracted from him and no longer spoke or wrote, at least publically, on the structure of the world. Galileo was humiliated, but we have benefited, for in turning away from cosmology Galileo focused his remaining years on the subject of his greatest achievement: the description of motion. While Galileo’s telescopic discoveries of 1610 had demonstrated the unity of earthly and heavenly realms, he nevertheless clung to circular planetary orbits and, to Kepler’s dismay, ignored the latter’s evidence for elliptical ones. At Arcetri Galileo uncovered the foundations of a new science of motion that to this day remain current—a task for which by 1633 he had been preparing for many years.

The phenomenon of projectile motion had, up to Galileo’s time, resisted analysis. Today we might deploy a high-speed, digital video camera and special curve-fitting software, but Galileo had to measure intervals in space with a cord marked off in standard units and time intervals with the outflow of a water clock or, yet more imprecisely, the beating of his pulse.

Galileo first proposed a formal description of projectile motion and then tested the consequences of his proposal against painstakingly obtained experimental evidence. He had been aware since his school days of the distinction, originating in the fourteenth century, between uniform motion, in which a body traverses equal distances in equal intervals of time, and uniformly accelerated motion, in which a body increases its speed by equal amounts in equal intervals of time. Galileo proposed that the motion of a projectile was a combination of these two kinds of motion: uniform motion in the horizontal direction and uniformly accelerated motion in the downward direction. Galileo was aware that the distance traversed by a body moving with uniform speed increases as the first power of the time elapsed, while the distance traversed by a uniformly accelerating body increases as the square of the time elapsed.

Figure 33 shows snapshots, equally spaced in time, of a sphere that has been launched over the edge of a horizontal surface and, as a consequence, continues its uniform motion in the horizontal direction. The distance the object falls in each interval of time, indicated on the right by a sequence of odd integers 1, 3, 5, ... , ensures that the downward acceleration is uniform. For if during the first interval the sphere falls 1 unit of distance and in the second interval 3 units, in the third 5, and so on, then after 1 interval the sphere has fallen a total of units of distance, after 2 intervals the sphere has fallen a total of units, while after 3 intervals units, and so on. In general, after the n^th interval of time the sphere has fallen units of distance. In other words, the sphere falls a distance proportional to the square of the elapsed time—as is characteristic of uniformly accelerating motion. The result of these two motions is a parabolic trajectory, outlined in the diagram, in which the distance fallen is proportional to the square of the horizontal distance traversed.

Such was Galileo’s proposal, but how did he confirm it? Heavy objects, whether projected horizontally or simply falling straight down, increase their speed in the downward direction at a rate of 32 feet (9.8 meters) per second every second—accelerating too rapidly for careful observation. So in place of allowing an object to fall freely, Galileo let a bronze sphere roll down an inclined plane and, in this way, slowed down what he called the “natural acceleration” of gravity. He minimized the effect of friction by using smoothly polished, hard wood for his inclined plane. He gathered data by making different measurements on repeated trials of the same experimental arrangement.

Galileo’s study of projectile motion, presented on the “fourth day” of conversation in Two New Sciences (1638), broke new ground in ways that prefigure the modern practice of science. He abandoned the Aristotelian idea that continuous motion requires a continuously acting cause, and, indeed, strategically postponed the difficult search for the cause of projectile motion in order to focus on its description. In describing projectile motion, Galileo turned away from a concern with the whole universe and toward a phenomenon that could be isolated from its environment and from which he could abstract its essential elements. He searched for simple mathematical relationships that described this physics and tested these relationships with experiments that reproduced the idealized situation as closely as possible.

Galileo’s analysis of projectile motion is the basis of a simple device often used by physics teachers. Two identical steel balls are mounted on a wooden block. A spring-loaded lever launches one of the balls in the horizontal direction while simultaneously releasing the other and allowing it to drop from rest. The two steel balls strike the floor at the same time with a satisfyingly single “plunk.” Thus, projectiles and freely falling objects accelerate downward at the same rate: 32 feet or 9.8 meters per second every second.

22. Scaling and Similitude (1638)

Figure 34, in Galileo’s own hand, appears in his Two New Sciences (1638). Its exquisite shading tells us that these are not mere shapes for which outlines would have sufficed. Galileo wants us to see these figures as bones—bones with which he can illustrate the concepts of scaling and similitude and take the first steps toward a theory of appropriate size.

Mathematical objects with the same shape, such as two triangles that differ only in scale, are said to be similar. Three-dimensional objects, for example, pyramids, with parts in the same proportions yet of different size, are also similar. However, most natural objects and animals with similar shapes occupy a more or less limited range of sizes within which they can be large or small versions of themselves.

Sometimes our imaginations run away with the idea of scaling. In Jonathan Swift’s Gulliver’s Travels, a storm washes Lemuel Gulliver onto the island of Lilliput. Lilliput’s inhabitants are twelve times smaller than Gulliver. Later Gulliver is marooned on the island of Brobdingnag whose inhabitants are twelve times larger. Otherwise these relatively small and large people are, in virtue, in vice, in wisdom, and in folly, much like the humans of Swift’s time.

As entertaining or as instructive as such tales may be, Galileo would have recognized the flaw in Swift’s descriptions—for it is one thing to imagine indefinitely large or small mathematical objects and quite another to imagine such objects clothed with physical attributes within natural environments. According to Galileo, a giant twelve times as large as a normal human being would collapse under his own weight.

Galileo reasonably supposed that the cross-section of a limb determines the limb’s strength. After all, an animal’s muscles push or pull across a cross-sectional area. Furthermore, a bone is like a wooden beam. When a beam breaks, it does not break everywhere at once but breaks through in a single jagged cross-section. Yet the weight that a bone or beam must support is directly proportional to the mass of the whole structure to which it belongs. In general, the strength of an animal, or of any structure, is directly proportional to the area of its cross-section while the weight it supports is directly proportional to its total volume.

Since cross-sectional area increases with the square of a scale factor L, that is, with , and volume increases with the cube of this scale factor, that is, with , the ratio of an object’s strength to the weight it supports varies as , that is, as 1/L. This dependence has become known as the square-cube law. By reason of the square-cube law, the larger an animal or structure, the less able it is to support its own weight.

For the same reason, smaller creatures of the same general shape have a relative advantage in strength. “Thus,” says Galileo, “a small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size.” He could also have noted that an ant can carry more than ten times its own weight.

The larger of the two bones in Galileo’s drawing is about 10 times wider than the smaller one but only about 3 times as long. Therefore, these bones are not geometrically similar. Instead Galileo chose these proportions in order to better match their strength-to-weight ratio. Since the cross-sectional area of the larger bone is 10² or 100 times larger than that of the smaller bone, the larger bone can carry or support 100 times more weight than that carried or supported by the smaller bone. And if the animal to which the larger bone belongs is relatively thick-limbed and squat, that is, as they say “overbuilt” as shown here, it may manage to weigh only about 300 times more than the animal to which the smaller bone belongs. Thus, the larger animal would be relatively weaker than the smaller animal but still within the realm of possibility. If, on the other hand, the larger animal were geometrically similar and 10 times larger than the smaller one, it would weigh 1,000 times more while being only 100 times as strong—probably not within the realm of possibility. Apparently, nature departs from geometric similitude in order to preserve relative strength.

The science of determining what ratios are important in what contexts is called dimensional analysis. Comparative zoologists as well as engineers who build and test scale models, say, in wind tunnels and towing tanks, seek out, with the help of dimensional analysis, these ratios. In most cases the structure’s interaction with its environment is crucial to their task. Small water-borne insects, for instance, must cope not so much with gravity but rather with the surface tension and viscosity of water. And small mammals develop special strategies for maintaining their body temperature. Preserving the ratio of relevant forces (dynamical similitude), of relevant velocities (kinetic similitude), and of relevant thermal quantities (thermal similitude) is usually more important than preserving geometric similarity when scaling up or down.

23. The Weight of Air (1644)

The element mercury is relatively rare, but it does not blend easily with other elements in the earth’s crust, and, for this reason, is often found isolated in mineral deposits. These deposits have been mined for millennia for, in spite of mercury’s toxicity, our ancestors valued it, a shiny liquid metal, as a medicine, as an ornament, and for its high density.

Evangelista Torricelli (1608–1647) put mercury’s density, roughly fourteen times that of water, and its liquidity to good use in devising the first barometer as shown in figure 35. He prepared a narrow glass tube sealed at one end, filled it with mercury, stopped the open end with his finger, inverted it, and inserted it into a basin of mercury. Upon removing his finger the mercury in the tube fell and left a column about 1.3 cubits (2½ feet, 29 inches, or 74 centimeters) high. In this way Torricelli created something at the top of the tube that had long been thought impossible: a region containing nothing, that is, a vacuum, a Torricellian vacuum.

But is it not true that “Nature abhors a vacuum?” A few years earlier Galileo (1564–1642) had been drawn to reflect on this abhorrence by the testimony of a workman who had been called upon to repair a suction pump that drew water from a well (illustrated in figure 36). According to Galileo’s text Two New Sciences (on the “first day” of conversation) the workman claimed that “the defect was not in the pump but in the water which had fallen too low to be raised through such a height; and he [the workman] added that it was not possible, either by a pump or by any other machine working on the principle of attraction to lift water a hair’s breath above 18 cubits; whether the pump be large or small this is the extreme limit of the lift.” Galileo imagined that the vacuum created at the top of the pump suspended the column of water and that if the column got too long the water would break under its own weight just as a rod of wood or iron suspended at its top would, if sufficiently long, also break.

Torricelli respected Galileo to the point that he waited upon the man, blind and under house arrest, during the last three months of his life. However, Torricelli dismissed Galileo’s hypothesis that the vacuum itself suspends the weight and held that, on the contrary, because “we live submerged at the bottom of an ocean of air,” the surrounding air, by pushing down on the surface of the barometer basin and the well water, pushes up the columns of mercury and water. Because mercury is roughly fourteen times more denser than water, air will support a column of water roughly fourteen times higher, that is, about 18 cubits (34 feet or 10 meters) high. Torricelli intended for his barometer to measure weather-related changes in the weight of air, but this effect could not, in his time, be easily separated from temperature-induced changes in the volumes of its glass and mercury parts.

Torricelli’s explanation survived while Galileo’s did not. For if the barometer’s column of mercury is pushed up by the weight of the surrounding air rather than held up by the abhorrence of the vacuum, the mercury column should fall as the barometer is carried to a higher elevation where the air was known to be thinner and less weighty. The philosopher, mathematician, and physicist Blaise Pascal (1623–1662) tested this idea by arranging for his sister’s husband, the judge Florin Périer, to transport a barometer up a mountain, the so-called Puy de Dôme that rises 900 meters (3,000 feet) above the nearby municipality of Clermont-Ferrand in France.

Pascal’s 1648 report The Great Experiment on the Weight of the Mass of the Air includes his careful instructions to his brother-in-law and Périer’s exciting description of the actual experiment. Its result: a complete vindication of the idea that air has weight. Périer constructed two mercury barometers and left one attended at the foot of the Puy de Dôme while he and his companions carried the other to its top. There he found that the mercury column had fallen 8 centimeters (3⁵/₃₂ inches) during the course of their ascent. Périer repeated this test on a smaller scale by ascending the 37-meter (120-foot) high tower of the Notre Dame de Clermont as did Pascal himself by ascending a 46-meter (150-foot) high tower in Paris. In each case the mercury column fell a distance proportional to the ascent of the barometer. The only credible explanation was Torricelli’s that “we live submerged at the bottom of an ocean of air.” Pascal concluded his report with these words:

Does nature abhor a vacuum more in the highlands than in the lowlands? ... Is not its abhorrence the same on a steeple, in an attic, and in the yard? ... let them [the Aristotelians] learn that experiment is the true master that one must follow in Physics; that the experiment made on mountains has overthrown the universal belief in nature’s abhorrence of a vacuum, and given the world the knowledge, never more to be lost, that nature has no abhorrence of a vacuum, nor does anything to avoid it; and that the weight of the mass of the air is the cause of all the effects hitherto ascribed to that imaginary cause.

24. Boyle’s Law (1662)

Robert Boyle’s first trial of the experiment depicted here ended in disaster. He had a long glass tube bent in the shape of a “U” with its two unequal legs parallel to one another. The longer leg was more than six feet in length and the shorter one was sealed at its end. Then he poured mercury into the open end of the long leg of the tube. His object was to record paired values of the distances marked and in the third panel of figure 37, indicating how much higher the mercury is in the longer leg than in the shorter one and indicating the length of the column of air trapped in the shorter leg. But before he could gather data, he accidently broke the unwieldy tube and, presumably, spilled the expensive mercury.

As the well-educated son of the fabulously wealthy first Earl of Cork, Boyle (1629–1691) had both the know-how and the means to do this experiment—and to redo it as necessary. And because Boyle was acquainted with Torricelli’s barometer (1643) and with Pascal’s demonstration of the weight of air (1648), he was able to interpret his results as a demonstration of what is known, in England and in the United States, as Boyle’s law, the first enunciation of which Boyle published in an appendix (1662) to an earlier work, A Defense of the Doctrine Touching the Spring and Weight of Air (1660).

Figure 37 illustrates the idea behind Boyle’s law. In the first panel the air in the shorter leg is just barely connected to the air in the longer leg that, in turn, is open to the atmosphere. Given the behavior of Torrecilli’s barometer, we know that the pressure of the air that surrounds us is enough to hold up a column of mercury a little more than 29 inches high or, alternatively, a column of water 34 feet high or a column of air extending to the top of the atmosphere. As the mercury continues to flow into the longer leg, the air in the shorter leg is cut off from the atmosphere. And as the mercury in both legs rises—more quickly in the longer leg than in the shorter one—the air in the shorter column is compressed. Boyle kept a record of the coordinate values of the three quantities , , and to the nearest sixteenth of an inch. The numbers in the table are Boyle’s values listed to the nearest inch.

H	H+29	h
0	29	12
1	30	12
3	32	11
4	33	11
6	35	10
8	37	10
10	39	9
12	42	9
15	44	8
18	47	8
21	50	7
25	54	7
29	58	6
35	64	6
41	70	5
49	78	5
58	88	4
71	100	4
88	117	3

When the mercury in the longer leg is 29 inches higher than in the shorter one, the air pressure in the shorter leg must be enough to hold up the 29-inch-high column of mercury and a column of air extending to the top of the atmosphere (equal in weight to another 29-inch-high column of mercury). At this point the column of air in the shorter leg is compressed from its initial value of 12 inches to 6 inches in length. As more mercury is poured into the longer leg the quantity grows and the quantity h shrinks in inverse relation so that . Thus, when and, therefore, , the original value of h is diminished by a factor of 3. Since is directly proportional to the pressure P exerted on and by the column of trapped air and h is proportional to the trapped air’s volume V, these data demonstrate Boyle’s law, which in algebraic form is as illustrated in figure 38.

Boyle may have known that obtains only when the temperature of the gas is constant—as it was in his experiment—but never mentioned this limiting condition. It was Edme Marriote (1620–1684) who, having independently discovered Boyle’s law in 1667, made this condition explicit. For this reason, Europeans often refer to Marriote’s law or to the Marriote-Boyle law.

Boyle, like Francis Bacon before him, was a champion of empirical study—such as that which led to his eponymous law. Also, like Bacon, he was suspicious of overarching theories. Speculative hypotheses and mathematical expression of physical laws were helpful only when motivated by data gathered from close observation and careful experimentation.

Like many from wealthy Anglo-Irish families Boyle was educated partly at home with private tutors and partly in an English “public” school, in this case, Eton College near Windsor. He left England for study on the Continent with a tutor as a twelve-year-old in 1639. While Boyle was in Europe, his father died and left him a handsome inheritance. When he returned in 1644, the Irish were rebelling against English rule and England was in the midst of a civil war. His brothers and sisters were on both sides of this latter conflict, but Boyle became a partisan of neither. As he remarked in a letter to his former tutor, he felt exposed “to the injuries of both parties, and the protection of neither.” Throughout his life, Boyle observed “a very great caution” in all matters political and religious.

Boyle is sometimes called the “father of chemistry”—probably for rejecting both the Aristotelian doctrine of four elements (earth, air, fire, and water) and the Paracelsian doctrine of three principles (salt, sulfur, and mercury) and in their place emphasizing that all chemical phenomena should be understood in terms of the mechanics of particles in motion. During the late 1640s in London, he began meeting weekly with a group of like-minded natural philosophers to witness and discuss physical demonstrations. He called this group, which later evolved into the Royal Society of London, “the invisible college.”

Boyle had wide interests and wrote prolifically on medicine, theology, and language as well as on physical science. Nearly half of his literary output was devoted to theological topics, especially to the relation of theology to the new philosophy of experimental science. Boyle’s last will and testament established a series of lectures on Christian apologetics that were revived in 2004 with the express purpose of exploring the relationship between Christianity and science.

25. Newton’s Theory of Color (1666)

Isaac Newton (1642–1727) entered Trinity College, Cambridge University, in 1661, the year after the Restoration of the English monarchy that, in turn, followed the beheading of Charles I and the decade-long dictatorship of Oliver Cromwell. At that time the university had entered a long period of decline from which it did not emerge until after Newton’s death. Although nominally dedicated to educating young men, chiefly for the clergy, the fellows of Cambridge were not obliged to tutor, lecture, publish, or even remain in residence. Many, in fact, chose to absent themselves for months and years at a time. Even so they drew their stipends. Only three offenses were cause for dismissing a fellow: committing voluntary manslaughter, becoming a heretic, and getting married.

Yet in many ways Cambridge was a perfect place for Newton. He was self-motivated and independent-minded and would not have followed the guidance of a good teacher were one available. All he needed were some books and tools (he installed a lathe in his student lodgings), and to be left alone. He learned by continually thinking about a subject of his own choosing. His youthful obsessions were mathematics, mechanics, and optics.

Newton received his master of arts and became a fellow at Trinity College, and, at the age of thirty, was elected Lucasian Professor of Mathematics, one of the most lucrative professorships in the kingdom. So steady was his advancement that Richard Westfall, his twentieth-century biographer, believes that, in an age when the church or the court dictated most academic appointments, the young Newton must have had a powerful patron—now unknown to us. Upon assuming his professorship, Newton was obliged to give a series of inaugural lectures. The phenomenon of color was the subject of these lectures.

Newton’s theory of color is so thoroughly our own, we struggle to imagine another. But for two thousand years people believed that sunlight was pure and simple. Color was somehow added to originally colorless sunlight during its reflection from or refraction (or bending) through different transparent materials. Newton, for instance, tried to imagine light as a collection of identical globules that, like tennis balls, could acquire a spin during reflection and refraction. Different rates of spin would correspond to different colors. But these ideas fell by the wayside once Newton bought some prisms and started experimenting on his own.

Newton’s experiments suggested a radically different theory of color. He arranged for sunlight to enter his chamber through a small circular hole in a closed shutter and to fall upon a triangular glass prism as shown on the left side of the figure 39. That such prisms produce colored light was already so well-known as to be celebrated, to use Newton’s word. René Descartes, Robert Boyle, and Robert Hooke (1635–1705) had all reported on the phenomenon, but none had projected a refracted beam on a surface more than a few feet distant from the prism that caused the refraction. As a result, the size and shape of their refracted beams were not noticeably changed. Newton, on the other hand, projected his beam onto the opposing wall of his chamber twenty-two feet distant and found that the originally circular image now extended five times more in one direction than in the other and displayed a series of colors in the extended direction. Moreover, the size of the oblong image increased in direct proportion to the distance from the prism. Newton’s concluded that sunlight is a composition of different colors, and the different colors refract or bend in different amounts, blue more than yellow and yellow more than red.

Figure 39 also shows an extension of this experiment that Newton used to confirm these ideas—what he called his experimentium crucis. As before, a beam of sunlight refracts through a triangular glass prism and the oblong image projects on an opaque surface. This time the light of a single color (here yellow) passes through a hole in that surface and falls on a second prism. If a prism creates colors by modifying the beam rather than by separating the beam into its constituent colors, the second prism should also modify the color of the yellow beam. But the beam remained yellow on passing through the second prism. Newton still believed that light was composed of particles, but he was now also convinced that (1) sunlight is a heterogeneous mixture of colors, (2) which refract in different amounts, and (3) upon refraction separate into a continuous spectrum of colors. Furthermore, (4) opaque objects appear differently colored because they preferentially reflect one color.

Newton’s theory of color had an immediate consequence. He abandoned his effort to produce lenses, for telescopes, that perfectly focus starlight. For since lenses focus light by refracting, that is, by bending their rays, and different-colored rays bend in different amounts and starlight, like sunlight, is a mixture of different colors, starlight will never focus to a single point in a refracting telescope as is required for perfect image formation. This realization may have motivated Newton in 1668 to build the first reflecting telescope, which avoids refraction altogether.

Richard Westfall confesses, in the preface to his nine-hundred-page biography of Newton, Never at Rest (1983), that the more he learned of the man, the more alien he seemed. Newton never married and, apparently, had few friends. Yet his Cambridge colleagues viewed him with respect if not awe. Newton would sometimes draw figures, such as in figure 39, in the newly prepared gravel walks of Trinity College, and they, for a time, would carefully walk around these drawings in order to preserve them for the Lucasian Professor’s use.

26. Free-Body Diagrams (1687)

At a certain stage in their study, a physics student is called upon to master an important diagrammatic tool: the free-body diagram. Free-body diagrams help us analyze a situation in terms of forces. A general Newtonian principle is that only bodies exert forces on one another. What are the forces and what are the bodies that exert and that experience these forces? A free-body diagram helps us keep track of our answers.

Consider, for example, the seemingly simple situation of a book resting on a table that in turn rests on the ground. We know that both the book and the table are heavy, that is, both are attracted downward toward the center of the earth. This gravitational force—after the Latin gravis for “heavy”—is an example of an action-at-a-distance force because the bulk of the earth need not touch either the book or the table in order to pull them down. In contrast, an object that must touch another object in order to exert a force on it is called a contact force.

Figure 40 shows the book and the table and a free-body diagram of the book. Free-body diagrams represent objects as dots or circles and the forces on the object as directed line segments or arrows whose tails are attached to the object and whose heads point in the direction of the force. The length of the arrow is proportional to the magnitude of the force applied to the object. A longer arrow represents a larger magnitude force and equal-length arrows represent equal-magnitude forces. We also attach superscripts to each force symbol indicating the nature of the force, action-at-a-distance or contact, and subscripts indicating the force’s source and the object to which it is applied. Thus, stands for the action-at-a-distance force exerted by the earth on the book. We usually call this particular force the book’s weight.

If the force of gravity were the only force on the book, it would, according to Newton’s second law, accelerate downward. However, our initial description indicates that the book is at rest. Therefore, its acceleration must be zero. Consequently, the net force on the book must vanish and a force other than the earth’s gravity must be applied to the book in order for it to remain at rest.

We deduce the source of this additional force on the book by a process of elimination. A force must be either an action-at-a-distance force or a contact force. The gravitational pull of the earth is the only action-at-a-distance force on the book. (We ignore the relatively small gravitational attraction between book and table.) And, clearly, since the table is the only object touching the book, the table is the only object capable of exerting a contact force on the book. Therefore, because, and only because, we know the book is at rest, we know that the contact force the table exerts on the book points in the opposite direction and is of equal magnitude to the action-at-a-distance force the earth exerts on the book .

But how do contact forces work? How, in particular, does the table exert a force on the book? Interestingly, the surfaces of the table and the book behave approximately as if they were composed of a multitude of tiny springs that resist compression. When the book is placed on the table, a book-shaped array of “springs” in the table is compressed until the total upward force of the table on the book is equal and opposite to the downward force of gravity on the book. The depression in the table is typically so slight as to be invisible to the naked eye. If the table were not strong enough to generate and maintain this contact force, the book would crash through the table.

A similar kind of analysis applies to the table with the interesting difference that now two objects, the ground on which the table rests and the book, touch and push on the table. For this reason the corresponding free-body diagram of the table in figure 41 shows three forces on the table: the downward gravitational force of the earth on the table , the downward contact force of the book on the table , and the upward contact force of the ground on the table . These forces must sum to zero because we know the table is at rest and, therefore, not accelerating.

Free-body diagrams can also be drawn of bodies that are not at rest. Consider, for instance, a pear that has fallen from its tree but has not landed on the ground. Figure 42 illustrates the falling pear and its free-body diagram. Since nothing touches the pear, the gravitational force of the earth on the pear is the only force on the pear. Thus, the net force on the pear is its weight. According to Newton’s second law, the apple will accelerate toward the center of the earth at a rate equal to where m is the mass of the pear.

27. Newton’s Cradle (1687)

Many of us have played with a “Newton’s cradle,” composed of several steel balls suspended as shown in figure 43. A simplified version of this toy composed of only two steel balls best illustrates our concern (see figure 44). The left panel of the diagram shows the black ball hanging at rest while the elevated white ball is released and allowed to swing down and strike the black ball. The right panel shows the two balls shortly after their collision. The white ball is now at rest, and the black ball has now completely captured the motion originally in the white ball. Eventually the black ball will swing up to a level close to that originally occupied by the white ball. Sometimes we hear this device referred to as a “double pendulum,” so called because two pendula mimic the motion of one.

The collision of these two steel balls is approximately elastic. In a perfectly elastic, head-to-head collision of two identical objects, one originally at rest and the other moving, the one originally at rest completely captures the motion of the originally moving object. The collision of two billiard balls is approximately elastic. However, collisions are not always perfectly or even approximately elastic. One could, for instance, easily arrange for the two swinging balls to stick together on impact and so reproduce an example of a perfectly inelastic collision. After a perfectly inelastic collision, identical balls move off together at half the speed of the moving ball just before collision.

Could there be one principle behind both—indeed behind all—kinds of collisions? Newton found such a principle. It was his effort to understand collisions that set him on a path leading to his three laws of motion that encapsulate the foundations of classical mechanics.

Observe that in both kinds of collision the ball originally at rest speeds up by the same amount the ball originally in motion slows down. The technical word for speeding up and slowing down is acceleration; speeding up is positive acceleration and slowing down is negative acceleration. Therefore, both when the two identical balls stick together and when they do not, the ball originally at rest accelerates by the same amount the ball originally in motion decelerates. Given that, according to Newton’s second law, forces cause acceleration, the force on one of the balls must be equal and oppositely directed to the force on the other ball.

The general principle behind this last statement is Newton’s third law. Newton’s third law is probably the least understood of Newton’s three laws of motion—possibly because its original Latin was first rendered into English by the seemingly meaningless phrase For every action there is always an equal and opposite reaction. This widely reproduced statement is more a mnemonic for than an accurate description of the law. When used correctly, this mnemonic should remind us that all forces occur in action-reaction pairs of equal magnitude and opposite direction. Thus whenever object A exerts a force on object B, object B exerts a force of equal magnitude and opposite direction on object A.

The left panel of figure 45 again shows colliding balls, while the right panel consists of two free-body diagrams that label the forces on each ball. These free-body diagrams clearly show that the left ball exerts a contact force on the right ball equal in magnitude and opposite in direction to the contact force the right ball exerts on the left ball . These diagrams are typical in another way. They show that each of the two forces in an action-reaction pair applies to a different object.

We take advantage of Newton’s third law every day. When we begin to walk, having been in a resting position, our body accelerates in the forward direction, and, according to Newton’s second law, whenever a body accelerates there is a net force on that body in the direction of that acceleration. What is this force that causes us to begin walking? The force of gravity is in the wrong direction. It could pull us through the floor, but it cannot accelerate us along the floor. We cleverly solve this physics problem by pushing backward on the floor with one foot. Consequently, the floor must, according to Newton’s third law, exert a force on this foot that accelerates us in the forward direction. Of course, this happens whether or not we are aware of Newton’s third law. Our bodies understand Newton’s third law and exploit it every day.

28. Newtonian Trajectories (1687)

Galileo’s telescopic observations of the mountains and valleys on the moon suggest that the earthly and heavenly realms are similarly composed. Newton (1643–1727) confirmed this suggestion by generalizing to cosmic scale another of Galileo’s discoveries: the parabolic trajectory. Pitch a baseball horizontally and imagine its trajectory unfolding in space and time. The baseball covers ground in direct proportion to the first power t of the time elapsed and falls downward in direct proportion to the second power t² of the time elapsed. The result is a parabolic trajectory.

But the earth’s surface is not flat; neither is the strength of its gravitational attraction independent of the distance from its center. Things do not really fall down, but rather toward the center, and the magnitude of the resulting centripetal (to the center) acceleration diminishes with distance from the center. What is unimportant in the context of a pitched baseball becomes important for trajectories that are as large as the earth itself. This is the lesson of figure 46, which is taken from Newton’s major work Philosophia Naturalis Principia Mathematica (1687).

Newton’s Principia completes a line of thought Newton had begun in 1664–1666, during which time Cambridge University had closed and sent its students and fellows away in order that they might protect themselves from an epidemic of the plague then sweeping through England’s cities and towns. Newton returned to his boyhood home in the village of Woolsthorpe and to his twice-widowed mother Hannah Smith. He built bookshelves, read, and thought—and tried to do little else. He invented a scheme of calculation in which ratios of indefinitely small quantities and their infinite sums had finite limits—a scheme we now call calculus.

He also reflected on the effect of gravity reaching far beyond his mother’s apple-laden trees all the way up to the orbit of the moon. How, he asked, would the moon fall if it fell like an apple? Newton supposed the moon’s orbit was the result of a double tendency: one to continue in straight-line motion and the other to accelerate toward the center of the earth. He found that this combination produced, in the simplest case, a circular orbit. Newton also wondered how the magnitude of the acceleration caused by gravity diminished with distance d from the Earth’s center—as the first power of the inverse distance or as the second power ? Guided by Kepler’s third law—The period of the planetary orbits increases as the 3/2 power of their average distance from the Sun—Newton settled on the second power. His calculations were “pretty nearly” consistent with what he observed: a month-long revolution of the moon around the earth.

The seventeenth-century equivalent of a pitched baseball is a cannonball fired from a horizontally directed muzzle. Place the cannon on top of the highest mountain on earth, eliminate the earth’s atmosphere, and you have the situation illustrated in Newton’s drawing. At relatively low velocities, the cannonball follows what appears to be a parabolic trajectory. As the cannonball’s initial speed increases, its range increases until the cannonball orbits the earth in a perfect circle. The two outermost orbits are ellipses with one focus at the center of the earth. The diagram suggests that a parabolic trajectory is, in actual fact, a small part of an ellipse. The larger part of the elliptical trajectory of a baseball or a cannonball is never realized because the bulk of the earth intervenes.

The Principia demonstrates mathematically what Newton’s drawing can only suggest. The inverse square law of gravitation and Newton’s laws of motion result in the approximate parabolic trajectories of projectiles near the surface of the earth, in the circular and elliptical orbits of artificial satellites and the moon, and in the elliptical and hyperbolic orbits of planets and comets.

The Principia proposes a universally operating law of gravitation according to which each point of mass in the universe attracts every other point of mass with a force proportional to the product of the two masses and the inverse square of their separation. Newton carried his calculations so far as to develop an algorithm for predicting the size and timing of high tides by accounting for the oceans’ simultaneous threefold attraction to Earth, Moon, and Sun.

Universal gravitation was a success, but Newton knew that the idea was incomplete. After all, how could the sun exert a force that kept the earth in its orbit without the benefit of an intervening mechanism? For Newton and other seventeenth-century natural philosophers to explain a phenomenon meant to reveal the mechanism in terms of which objects, in direct contact, push or pull on one another. Newton abhorred the concept of a force that could project itself across empty space. Yet he cautiously refrained from offering a mechanical explanation of gravity, and, in the end, accepted the universal law of gravitation as a means of efficient and precise mathematical description.

The first edition of the Principia (1687) made Newton famous. During his remaining forty years, Newton revised and extended the Principia and, at long last, brought his optical and mathematical writings into print. In 1696 he left his professorship at Cambridge and accepted an appointment as warden of the Royal Mint and, in 1700, as its master. He represented Cambridge in Parliament (1701) and was elected president of the Royal Society (1703). Queen Anne knighted him in 1705.

Sir Isaac Newton died in 1727, but his intellectual legacy endures. In no sense has Newton’s physics been overthrown. Relativity and quantum mechanics embed Newtonian physics as a limiting case—the particularly important case that describes the world of human-sized objects. The Newtonian synthesis remains the core of physics education and provides the structure that gives meaning to new discoveries.

29. Huygens’s Principle (1690)

Constantijn Huygens intended that his son Christiaan (1629–1695) follow him into the Dutch diplomatic service and for this reason gave him a liberal education in languages, music, history, rhetoric, logic, mathematics, and natural philosophy and also training in fencing and riding. But when the House of Orange lost its power, Constantijn lost a patron, and Christiaan lost his opportunity in diplomacy. Fortunately for us, Christiaan’s true interests were in mathematics and natural philosophy. He developed the theory of pendulum motion and of colliding objects, invented an algorithm for computing the digits of the irrational number , and constructed a telescope with which he identified Saturn’s rings, its moon Titan, and the Orion nebula. But he is most remembered for his reflections on the nature of light.

The nature of light had long fascinated scientists. By the late seventeenth century, two theories were current: (1) light was a stream of high-speed particles, and (2) light was a disturbance that propagated through an invisible medium called the ether. Descartes and Newton, who agreed about little else, both supported the particle theory while Huygens promoted the disturbance or wave theory.

Popular opinion favored particles. After all, we can hear but not see around corners—facts that support the idea that sound is a wave disturbance that propagates through the air and around corners while light is composed of small particles that, except at reflecting and refracting boundaries, travel in straight lines. But in 1660 the Jesuit Francesco Grimaldi (1618–1693) observed that light does, indeed, show a slight tendency to diffuse or diffract around small objects such as pins and through and around narrow slits in an opaque barrier just as would be expected if light were a wave disturbance.

Huygens’s Treatise on Light (1690) is quite narrow in scope. Huygens does not mention that white light is composed of a spectrum of colors, or Grimaldi’s discovery of diffraction, or the properties we usually associate with waves such as periodicity and wavelength. Instead, Huygens asks: If light is a stream of particles, why don’t the particles scatter from one another as different streams cross and, in this way, make ordinary vision impossible? And he observes that it is the nature of sound waves to propagate through each other without distortion. Otherwise conversation in a noisy room would be impossible. Huygens concludes that light must be like sound: a disturbance or wave that propagates from source to receiver.

But a question remains. What is the medium through which light waves propagate? It cannot be air since light travels through a glass jar from which the air has been pumped while sound does not. Evidently, light has its own medium that Huygens calls the ether: an invisible material that penetrates transparent objects yet as a whole forms an elastic fluid through which disturbances propagate at high speed. The option of allowing light waves to propagate without a medium, the modern point of view, was not available to Huygens and other seventeenth-century scientists who were materialists and, as Huygens put it, dedicated to “the true philosophy, in which one conceives the causes of all natural effects in terms of mechanical motions.”

Huygens’s contribution to the discussion begins by observing that every point on a luminous body is a source of light. And if the speed of light is the same in all directions, a disturbance originating from one point will, if unhindered, soon occupy the surface of a finite sphere centered on that point. But how exactly does one spherical disturbance evolve into another larger, concentric spherical disturbance? Huygens’s principle, according to which every point in a disturbed medium is a new point source, provides the answer. Four points are identified on the inner circle of the left panel in figure 47 and each one of these, according to Huygens, sends out its own spherical disturbance that is sometimes called a secondary wave. The envelope of these secondary waves (here identified with a dashed circle) constructs a new surface of disturbed medium. On the right, light propagates from a point source so distant that its spheres of disturbance appear in figure 47 as parallel lines. In this case, the envelope of secondary waves also constructs a new line in the direction of forward propagation.

All the well-known properties of light: straight-line propagation in homogeneous media, equality of the angles of incidence and reflection, and Snell’s law of refraction, follow from Huygens’s principle. Consider, for instance, the wave surfaces depicted in figure 48 as they approach an air-water interface. Huygens supposed that the material through which light travels modifies the speed of light—the denser the material, the slower the light. Therefore, the distance between the waves should shrink as they enter the water. One could imagine the secondary waves produced at the interface: two concentric half-circles, the one in water smaller than the one in air. But one could also, equivalently, imagine the wave crests as rows of a marching band closing up as the band leaves a smooth walking surface and enters a rough piece of ground—assuming that the march tempo remains steady while the marchers’ pace shortens. The result: the lines of wave disturbance incline toward the normal of the interface in just the amount dictated by Snell’s law given that the speed of light in water is about three quarters that in air.

The particle and the wave interpretations of light coexisted throughout the eighteenth and early nineteenth centuries. In the mid-nineteenth century, the speed of light in water was found to be three quarters that in air. Only Huygens’s theory of light, which had by then evolved into a complete wave theory, was consistent with this result, even though direct evidence of the ether has never been found. Evidently light waves do not need a medium through which to propagate.

30. Bernoulli’s Principle (1733)

Daniel Bernoulli’s (1700–1782) good fortune (and also his misfortune) was to have his own father as a tutor. Daniel’s father, Johann Bernoulli (1667–1748), was a professor at the University of Basel in Switzerland and the foremost mathematician in Europe. Johann and his brother Jakob, Daniel’s uncle, were among the first mathematicians to master calculus after its invention by Newton and Gottfried Wilhelm Leibniz (1646–1716) in the second half of the seventeenth century. Johann’s son, Daniel, was probably the ablest of several generations of Bernoulli mathematicians.

Such was Daniel’s precocity and his broad talent that at twenty-one years of age he would have made the University of Basel a fine professor in any one of several fields: natural philosophy, mathematics, logic, and physiology. But his application for a faculty position was passed over on two occasions, not because his qualifications fell short, but because the university chose the successful candidate by lot from among those qualified. Daniel was simply unlucky.

A decade later Daniel had forged an international reputation chiefly for his work at the Imperial Academy of Sciences in St. Petersburg. He excelled at finding problems ripe for solution with the new methods. But he hated the harsh climate of St. Petersburg and, in 1732, returned to Basel in order to accept the position finally offered: a professorship of anatomy and botany—two subjects in which, by that time, he had little interest. Only toward the end of his productive career did Bernoulli assume professorships that reflected his abiding interests: in physiology in 1743 and in natural philosophy in 1760.

Figure 49 illustrates one of Daniel Bernoulli’s discoveries. The thick dark lines outline a section of tubing through which an incompressible fluid (for instance, water) flows from left to right. As the tube’s diameter shrinks in the direction of fluid flow, the fluid’s speed increases in order that the fluid entering the tube from the left might leave at the same rate on the right. The thin lines are streamlines—reproducible trajectories of a lightweight object immersed in the fluid. Where the streamlines crowd together, the fluid more flows quickly.

Leonardo da Vinci (1452–1619) had, much earlier, understood this behavior. Evidently, Leonardo spent many pleasant afternoons dropping seeds into flowing brooks and watching their trajectories unfold in space and time—the seed speeding up and slowing down as the brook narrowed and widened. This simple inverse relation between a fluid’s speed and the cross-sectional area of its channel, sometimes called the principle of continuity, expresses conservation of mass.

Bernoulli, who was well aware of the principle of continuity, searched for a second principle linking a fluid’s speed to the pressure it exerts. Because Bernoulli was a former medical student, he knew that measuring the pressure of a moving fluid, such as the pressure of arterial blood, presented a problem. Physicians in his day simply cut open a patient’s artery and observed how high the blood spurted. Bernoulli sought a less wasteful and less dangerous method. He experimented with water flowing at various speeds through pipes of various diameters. He punched holes in the pipes and fitted these holes with vertical glass tubes open at both ends. When water flowed through the pipe, water ascended the tube. The pressure of the moving water equals the pressure of the ambient air plus an amount proportional to the height of the water supported in the open glass tube.

Bernoulli’s technique quickly became standard medical practice. For the next 170 years, physicians inserted the sharpened end of an open glass tube into a patient’s artery and observed how high the blood ascended in the tube—the higher its ascent, the higher the patient’s blood pressure. This method was better than the old one, but still painful and dangerous. Not until 1896 was the current noninvasive way of measuring blood pressure devised.

More importantly, Bernoulli discovered that in each experimental arrangement the sum of the pressure exerted by the flowing fluid and the energy density of its bulk flow, where is the fluid pressure, is its speed, and is its mass per unit volume, remains constant along a streamline—a relationship we now call Bernoulli’s principle. Evidently an incompressible fluid loses some of its pressure as it speeds up upon entering a narrower section of tube.

Today we use the principle of continuity and Bernoulli’s principle to explain how airplane wings produce lift. Figure 50 shows the cross-section of an airplane wing and the streamlines of the air that flows around it. Straight, evenly spaced streamlines far above and immediately below the wing indicate undisturbed air. Immediately above the wing section the streamlines necessarily crowd together. According to the principle of continuity, the air immediately above the wing must flow more quickly than the air below it, while, according to Bernoulli’s principle, the more quickly flowing air above the wing exerts less downward pressure on the wing than the more slowly moving air below the wing exerts upward pressure. The result is a net upward force on the wing.

Most fathers would be proud of a son with Daniel’s accomplishments, but not Johann. Instead he saw Daniel as a competitor. When in 1734 the two tied for first place in a competition sponsored by the Paris Academy of Sciences to which they had submitted independent solutions to a problem in celestial mechanics, Johann angrily denounced his son—and also the prize committee for not recognizing his superior achievement. Then in 1743 Daniel discovered that his father had reproduced or, as he suspected, plagiarized and published in Hydraulica (1743) much of what Daniel had published ten years earlier on moving fluids in his similarly titled Hydrodynamica (1733). Worse yet, Johann had asked the printer to backdate the publication of Hydraulica to 1732 in order to establish priority over his son. Daniel never forgave his father and the two remained unreconciled at the elder Bernoulli’s death in 1748.

31. Electrostatics (1785)

Figure 51 illustrates a well-known interaction: Like charges repel and unlike charges attract. The two kinds of charges are here indicated by a plus and by a minus sign. Insulating strings suspend the charged balls. (An insulator is made of material in which charges are not free to move, and a conductor is of material in which charges are free to move.)

In the eighteenth century these balls were composed of pith (a spongy organic material) and the strings were of silk—both good insulators. The natural philosophers of that time generated positive charge on a glass rod, for instance, by rubbing the rod with their hands, and transferring the charge to the pith balls by stroking the latter with the glass rod.

Today we can illustrate the same interactions more easily. Simply detach about 18 inches (46 cm) of Scotch tape from its spool. Fold one end of the tape over on itself to make a nonsticky handle and press the remaining sticky side to a smooth table surface. Prepare another tape in exactly the same way, and pull both tapes up from the surface at once—one in each hand. The identically prepared tapes are identically charged and will repel each other as shown in the right panel of figure 52. In order to create oppositely charged tapes press one to the table and stick another on top of the first—sticky side to nonsticky side with the two handles at the same end. Pull the tapes up while stuck together. Then, carefully peel them apart. They will be oppositely charged and attract each other as shown in the left panel.

Any two dissimilar materials in close contact, for instance, tape and table surface, will result in charge moving from one material to the other. The traditional method of charging by rubbing is simply one way of making close, repeated contact between two dissimilar materials. Placing the sticky side of one tape on the nonsticky side of a second tape is another.

Benjamin Franklin (1706–1790) invented the names positive and negative to denominate the two kinds of charge, but they could have been given other names, and, indeed, were by Charles du Fay (1698–1739), who was the first to recognize the phenomenon of like charges repelling and unlike charges attracting. Du Fay called the charge created on glass by rubbing glass vitreous (after the Latin root for glass) and the charge created on amber by rubbing amber resinous (after the Latin for fossilized tree resin). In Franklin’s jargon, vitreous charge was positive and resinous charge negative.

The names positive and negative suggest Franklin’s theory according to which all materials contain a single electric fluid. When an object has an excess of this fluid it is charged positive, and when an object has a deficit it is charged negative. Other people explained the same phenomena by appealing to two different kinds of fluid, each with its own charge. Compelling evidence in favor of two different charges (and their two different particles) did not emerge until the discovery of the electron in the late nineteenth century and the proton in the early twentieth century.

If you try the experiment with charged Scotch tape, you may notice that your hand attracts the tape regardless of its charge. The explanation is simple. Under normal conditions, skin is a good conductor. As your hand approaches, for instance, a positively charged tape, negative charges within your hand move closer to the tape and positive charges move farther away. Your hand (actually your whole body) becomes polarized. Since the closer the charges the stronger their attraction, your hand attracts the tape independently of the kind of charge the tape contains. Figure 53 shows how, in similar fashion, an uncharged (horizontal) conducting rod becomes polarized and attracts a (vertical) positively charged tape.

The eighteenth century was the first age of electricity. It was also the age of enlightenment—a period when men and women believed that rational thinking, rather than adherence to tradition, would solve our problems and improve our lives. The natural philosophers of the eighteenth century not only discovered that like charges repel and unlike ones attract but invented means of mechanically generating large amounts of charge. Some went in for fantastic demonstrations that, for instance, polarized or charged small boys suspended from bundles of silk threads. Franklin not only drew electricity from storm clouds with his wet (and so conducting) kite string, but also managed to cook a turkey by passing electric charges through it. Stephen Gray (1666–1736) conducted an electrostatic signal some 800 feet (244 meters) along a metal wire—a precursor to the telegraph. Others hawked the therapeutic effect of electric shocks.

Charles Coulomb’s (1736–1806) contribution crowned the efforts of Gray, Du Fay, and Franklin. In order to accurately measure the strength of the earth’s magnetism Coulomb eliminated friction from a compass needle by suspending it from a fine thread. The more force exerted on the ends of the needle, the more it rotated and twisted the suspending thread. Coulomb used this technique to devise a sensitive device, now called a torsion balance, for measuring the force between two small, charged spheres. One sphere was attached to the end of a light, horizontal, counterbalanced rod suspended from a fine thread. The other stationary, charged sphere was placed near the first. The larger the force exerted on the charged sphere attached to the suspended rod, the more the rod rotated and turned its suspending thread—the rotation in direct proportion to the force applied. Coulomb found that the force F exerted between two small charges, and , is inversely proportional to the square of the distance between them and directly proportional to the product of the charges—a result, , now known as Coulomb’s law.

While this result, mimicking as it does Newton’s universal law of gravitation, had been suggested before, Coulomb was the first to demonstrate it with a simple, convincing experiment. He did not speculate on the cause of this electrostatic force, but, rather, like Newton before him, was satisfied with its precise mathematical description. Coulomb is one of the seventy-two notable French engineers, scientists, and mathematicians whose names are inscribed on the Eiffel Tower.

H	H+29	h
0	29	12
1	30	12
3	32	11
4	33	11
6	35	10
8	37	10
10	39	9
12	42	9
15	44	8
18	47	8
21	50	7
25	54	7
29	58	6
35	64	6
41	70	5
49	78	5
58	88	4
71	100	4
88	117	3

H	H+29	h
0	29	12
1	30	12
3	32	11
4	33	11
6	35	10
8	37	10
10	39	9
12	42	9
15	44	8
18	47	8
21	50	7
25	54	7
29	58	6
35	64	6
41	70	5
49	78	5
58	88	4
71	100	4
88	117	3

Notes

H	H+29	h
0	29	12
1	30	12
3	32	11
4	33	11
6	35	10
8	37	10
10	39	9
12	42	9
15	44	8
18	47	8
21	50	7
25	54	7
29	58	6
35	64	6
41	70	5
49	78	5
58	88	4
71	100	4
88	117	3