e talk about space so readily and easily: “There's no space for an office in this apartment” or “Give me some space, man.” The concept of space seems self-apparent to the casual observer. But on deeper reflection its true nature remains elusive. “As a rule, people differentiate between matter, space, and time. Matter is what exists in space and endures through time. But this does not tell us what space is. . . . It is matter which we see, touch, and hear, which causes sensations to arise within us. . . ,” the British philosopher Ian Hinckfuss once noted. Space is perceived yet not felt, observed, or heard. So what, after all, is space?

Perceiving the limits and bounds of space was probably one of Homo sapiens' earliest accomplishments. Space was first and foremost an orientation among familiar objects, discerning the effort needed to reach a nearby river, rock, or tree. A sense of space may even have preceded a perception of time, since we describe moments of time as “short” or “long,” words that usually describe spatial categories. And

with the emergence of agriculture came the need to measure space exactly for practical considerations, such as planting a field or digging an irrigation ditch.

From these humble beginnings arose a more esoteric contemplation of space. In ancient Greece philosophers developed the concept of the void—the pneuma apeiron—a vacancy that allowed for a separation of things. Democritus, father of the atomic theory, required such a void—a nothingness bereft of matter—for his idea to work. Space was the empty extension that allowed his bits of matter—his atoms—to move about. Such discussions soon extended to thinking about space in the abstract. A Greek philosopher named Archytas asked what would happen if you journeyed to the end of the world and stretched out your hand. Would your hand be stopped by the boundaries of space? Lucretius, Democritus's pupil, answered no, and he had an interesting proof for space being unbounded. He asked: Suppose a man runs forward to the very edge of the world's borders and throws a winged javelin. Unable to conceive that anything could get in the javelin's way, Lucretius concluded that the universe must stretch on and on without end. Aristotle, on the other hand, took the opposite stand. He stated that it was “clear that there is neither place nor void nor time beyond the heaven.” For him the universe was finite. If a stone falls to the Earth to find its natural place at the center of the universe, he argued, then fire moving upward in the opposite direction must also face a limit. In Aristotle's physics, upward and downward motions had to be balanced. Moreover, any objects in the farthest reaches of an infinite universe, forced to rotate about a motionless Earth, would end up traveling at infinite velocities, a situation that Aristotle considered patently absurd.

Space was the subject of fierce intellectual debates into the Renaissance. In medieval times theological concerns often prejudiced the debate. To think of an immovable void, as outlined by the Greek atomists, meant God created something He could not budge. Such a situation challenged His omnipotence. Thus, this idea was deemed heretical and was avoided. But as natural philosophers, starting as early as the fourteenth century, began to consider kinematics, the motion of objects, it became necessary to contemplate some kind of fixed space.

They needed to imagine a special motionless container in order to understand such physical concepts as velocity and acceleration. One man in particular would change the landscape of science in making this assumption as he searched for the mathematical rules by which motions could be predicted. That man was Sir Isaac Newton.

The dreaded Black Death appeared in London in 1665. With the plague spreading northward to the university town of Cambridge, Newton fled that summer to his childhood manor home, Woolsthorpe, in Lincolnshire. In that rural setting he worked intensely for two years. Still in his early twenties, he was laying down the mathematical and physical foundations of his most important ideas, which had been germinating throughout his college years: the theory of color, the construction of the calculus, and, most importantly, the laws of gravitation. He returned to Cambridge in 1667, at the age of 24, and within two years became Lucasian Professor of Mathematics, a high honor at the university. Secretive, obsessive, and fearful of exposing his work to criticism—a man chockful of neuroses—Newton let many of his revolutionary thoughts go unpublished. It was not until 1684, sparked by the questions and persistent prodding of Edmond Halley (of comet fame), that Newton was at last convinced to write his masterpiece, Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy). He abandoned his work on alchemy, his most recent fascination, and applied his legendary power of concentration completely on the Principia (as it is most familiarly known) for nearly two years.

The Principia deals with both gravity and the mechanics of motion. Forces in nature, declared Newton, are not needed to keep things moving (as Aristotle argued); rather, forces change motion and in predictable ways. Newton clarified what Galileo had begun to infer from experimental tests: an object in motion does not naturally come to a stop; instead, it will remain in motion unless altered by an outside force, such as friction. The effect of a force is to get an object moving, to stop it, or to change its direction. And when it comes to gravity, Newton revealed that the strength of the gravitational attraction between two objects depends on two things: the total amount of matter in each object and the distance between them. The greater the mass of each ob-

ject, the stronger the pull; conversely, the larger the separation between the two masses, the weaker the attraction. Or as Newton put it, two objects exert a gravitational force on one another that is in direct proportion to their masses and in inverse proportion to the square of their distance. More importantly, he realized that what draws an apple to the ground (an event he presumably witnessed as a young man at Woolsthorpe during his early ruminations on gravity) also keeps the Moon in orbit about the Earth. Moreover, he deduced the exact equations for determining those motions. Newton had discovered that nature uses a mathematical treatise as its playbook. As if with one monumental stroke of the pen, he established that motions everywhere, in the heavens and on Earth, are described by the same physical laws. Before this insight, philosophers generally believed that the heavens were distinctly different from the domain of man. Earthly things were mortal—subject to change and transition—while the stars and planets were eternal and incorruptible. But with Newton's new laws the cosmos and terra firma were blissfully wedded. An all-encompassing set of mathematical rules could now explain events in both domains: the ocean tides, the motion of comets and planets, as well as the projectile paths of cannon balls. All these phenomena could be tracked with the same clocklike precision. So great was this achievement that Newton was the first person in England to be knighted for his scientific work.

Skydivers and bungee jumpers, plummeting toward the ground, have great respect for the force of gravity pulling them downward. Gravity is also the ruling force in determining the universe's evolution and its grand structure. Yet gravity is the weakest force in the cosmos. It seems paradoxical. A toy magnet can easily pick up a paper clip against the gravitational pull of the entire Earth. Or take two protons sitting next to each other. The gravitational force between them is a trillion trillion trillion times weaker than the electrostatic force pushing on them. Gravity gains collective strength only when masses accumulate and exert their effect over larger and larger distances. In this way gravity comes to control the motions of planets, stars, and galaxies.

Space and time loom large in Newton's laws, for laws need a framework. Take Newton's first law of motion. A body either remains at rest or in continuous uniform motion (traveling in a straight line at a con-

stant speed), unless an outside force causes that state to change. But at rest in relation to what? Or in motion to or away from what location? As soon as one talks of “motion,” one must establish a home base. Think of a child reading a book in a moving car. To someone on the curb watching the car whiz by, the book is moving fast. To the child inside, it is perfectly still. Newton's critical choice was to establish a reference frame in the universe at large. Space itself became his motionless laboratory: flat, penetrable, yet forever the same. He was not the first to think this way—Galileo, for one, posited a continuous three-dimensional void—but Newton made it an integral component of the Western canon. “Absolute space in its own nature, without relation to anything external, remains always similar and immovable,” he authoritatively stated. Space was at rest, and everything else in the universe moved with respect to that. To Newton, space was an empty vessel. You were either at rest or in motion with regard to this container. Positions, distances, and velocities were all measured in regard to this fixed space. Only by establishing this framework—this unchanging cosmic landscape—could his equations work.

Measuring motions in this absolute space also required a universal clock, which ticked off the seconds for all the inhabitants of the cosmos. Events everywhere, from one end of the universe to the other, were in step with the ticks of this grand cosmic timepiece, no matter what their speed or position. A clock sitting at the edge of the universe or zipping about the cosmos at high speed would register the same passage of time, identical minutes and identical seconds, as an earthbound clock. This meant that two cosmic observers, perched on opposite sides of the universe, could synchronize their watches instantaneously. Moreover, “the flowing of absolute time is not liable to any chance,” said Newton in the Principia. His clock was never affected by the events going on around it. Like some cosmic Big Ben, time stood aloof, as galaxies collided, solar systems formed, and moons orbited planets in this vast universe of ours.

Newton's law of gravity, brilliant in its ability to predict the future paths of planets, did have an Achilles' heel. It provided no explanation for the mechanism underlying gravity. There was no medium or physical means to push and pull the planets and other objects around.

Newton's tendrils of gravitational force just appeared to act instantaneously over vast distances, as if by magic. This feat appeared more resonant with the occult. As one wry critic of the time noted, “Newton calculated everything and explained nothing.” For some, the lack of a cause was tantamount to bad science. Newton was aware of these difficulties and lamented that one body acting on another through a vacuum, “ without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophical matters a competent faculty of thinking, can ever fall into it.” But his decision to stand by his laws was a practical one. He chose the path that allowed successful predictions to be made. As Einstein said in an imaginary talk with Newton, “You found the only way which, in your age, was just about possible for a man of highest thought and creative power.” Newton's decision to define an absolute space and absolute time was flawed, but it became ingrained into the very fabric of physics simply because it came up with the right answers.

Newton's conception of an absolute space and absolute time would influence the course of physics for some 200 years, but it was not universally accepted. There were critics who raised their voices loudly. The most notable were the British philosopher George Berkeley and the German diplomat and mathematician Gottfried Wilhelm Leibniz, who was Newton's archrival for his claim to have invented the calculus first. To them space and time were not fixed entities at all. Leibniz declared that “space and time are orders of things, and not things. ” Space and time could only be defined with regard to their relation with objects of matter. For Muslim philosophers, who had developed similar theories, this avoided the question, “Where was God before the creation?” The answer was simply that there was no “place.” Space did not exist until the creation of matter. Toward the end of his life, Newton found solace for his worries about absolute space and time in a religious explanation: “[God] endures forever, and is everywhere present; and by existing always and everywhere, he constitutes duration and space.” Newton's perception would indeed endure and for one simple reason: his equations worked. Mathematics had largely been an aethestic experience for the Greek philosophers. Newton

changed that attitude by demonstrating, with his powerful law of gravitation, that mathematics offered a road to discovery. He transformed mathematical law into physical law, rules by which the operations of nature—planetary motions, the propagation of light, mechanics—could be predicted. Eventually belief in the infallibility of these laws grew so strong that when Newton's equations appeared to fail (as in the case of explaining the orbital motions of the planet Uranus) it was immediately assumed that the equations were still valid and that an unseen planet was lurking beyond Uranus to account for the discrepancy. In that particular case, such resolute faith paid off handsomely. Adherence to Newton led to the discovery of the planet Neptune in 1846. It was difficult for Newton's critics to fight such success.

In his mathematical choices Newton assumed that space was “Euclidean,” holding all the properties defined by the famous Greek geometer Euclid in the third century B.C. Although the rudiments of geometry had been fashioned in Egypt along the banks of the Nile River—knowledge gained by the Pharaoh's surveyors, the harpedonaptai or rope-stretchers—those rules turned into a mathematical discipline when they traveled to Greece. The Greek philosophers saw in geometry a pure set of truths, which could be arrived at through logic alone. Geometry was proof that knowledge of the physical world could be gleaned through pure reason. So revered was geometry that when Plato established his Academy it was said that the sign over the door announced, “Let no one enter here unless he knows geometry.” Euclid stood at the zenith of this movement. Around 300 B.C. he wrote The Elements, which expressed all geometrical knowledge then known as a concise set of axioms and postulates. It served as the basis for all mathematical thought for the next 2,000 years.

In this major work Euclid defined a space that was flat, which is exactly the world we perceive and measure around us when confined to the ground. He also made a list of all the geometric ideas we take for granted—for example, that a straight line can be drawn between any two points or that all right angles are equal to one another. These were self-evident truths. His fifth postulate considers a line and any point not on that line. According to the ancient Greek geometer, there is

one—and only one—line that can be drawn through that point that is parallel to the original line. The two lines, like two parallel railroad tracks, will never meet. To our senses there appears to be no other possible configuration. While it seems self-apparent that parallel lines will never converge, later mathematicians were caught up in studying this particular axiom in more depth. Rather than assuming it was a given, they wondered whether the rule could be derived directly from Euclid 's other axioms. They wanted to prove it explicitly, rather than just state it as true.

One tried-and-true mathematical trick to test a postulate is to assume it is false and see what happens. That's precisely what a Jesuit priest named Girolamo Saccheri did in 1733. He assumed that the parallel axiom was false and then showed it would lead only to absurdities —hence, the name of this technique, reductio ad absurdum. Saccheri discovered that he could get more than one line through a point to be parallel to a given line. Figuring this was clearly ridiculous, he concluded that he had proven what he had set out to do—show that the axiom was clearly true as stated so elegantly by Euclid. Saccheri failed to see that he had accidentally stumbled onto a whole new geometry.

By 1816, after years of thinking about the problem, another mathematician arrived at the same insight and backed off as well but this time out of fear of ridicule. The great German polymath Carl Friedrich Gauss uncovered the same absurdities that Saccheri did. He did not reject them outright, though; he just knew that such challenges to the great Euclid would be considered heresy. As a result, Gauss never officially reported his findings during his lifetime (although he did discuss the new geometry he developed in private correspondence with colleagues). A reclusive man unwilling to start a public dispute that would disrupt his peace of mind, Gauss carefully guarded his secret, fearing as he put it, “the clamor and cry of the blockheads ” over questioning mathematics' sacred gospel. Euclid's framework, sturdy for centuries, served as the very foundation of mathematics. Gauss was also a perfectionist, who kept much of his work to himself. He was terribly reluctant to publish any idea until he had polished its proof to a fine

sheen. It's not surprising that his personal seal, a tree with sparse fruit, bore the motto Pauca sed matura (“few but ripe”).

Realizing that non-Euclidean geometries were a possibility (at least on paper), Gauss gradually began to wonder whether a non-Euclidean geometry might describe true physical space. Perhaps space wasn' t flat, as Newton assumed, but rather curved. His musings were amplified by a practical concern. He was commissioned by his government in the 1820s to conduct a geodetic survey of the region around the city of Göttingen in Hannover. This endeavor enhanced the thoughts he was already having about curved space. Curvature, he realized, need not be restricted to two dimensions, such as the rounded surface of a planet. In a 1824 letter to Ferdinand Karl Schweikart, a professor of law and a geometer as well, Gauss bravely mentioned that space itself, in all its three dimensions, might be curved or “anti-Euclidean ” as he called it. “Indeed,” he wrote, “I have . . . from time to time in jest expressed the desire that Euclidean geometry would not be correct.” He may even have tested this hypothesis during his geodetic work. Using light rays shining from peak to peak in the Harz Mountains, Gauss had surveyed a triangle of pure space formed by three mountains, the Hohenhagen, the Brocken, and the Inselberg. By his metric figuring, the sides of the triangle measured 69, 85, and 107 kilometers. No deviation from flatness, however, was detected.

Others were more open in exploring this new geometric terrain. Between 1829 and 1832, while Gauss kept mum publicly from his faculty post at the University of Göttingen, two mathematicians independently published papers stating that it was possible to have geometries that disobeyed Euclid. One proof was done by the Russian mathematician Nikolai Lobachevsky, the other by the Austro-Hungarian János Bolyai, allegedly the best swordsman and dancer in the Austrian Imperial Army in his day. Like Saccheri a century earlier, Lobachevsky and Bolyai had asked what would happen if the fifth postulate were wrong. If it were in error, what type of mathematics arises? What if it is assumed that an infinite number of lines can be drawn through a point near a given line without any of the lines intersecting? In this way the two mathematicians came to describe a space of negative curvature.

“From nothing I have created another entirely new world,” wrote Bolyai to his father, who had struggled with the fifth postulate himself when he was a student friend of Gauss. Bolyai's new world can be visualized by imagining a triangle drawn on the surface of a saddle. The triangle, with its sides curved inward, would appear a bit shrunken. So the sum of its angles is not 180 degrees, as authoritatively stated in high school textbooks as the standard Euclidean answer. Instead, it is less than that. Being concave the saddle's surface also allows many lines, which never meet, to be drawn through a point near a given line. Lobachevsky called this new system his “imaginary geometry.”

Like Gauss, Lobachevsky also thought of the possibility that three-dimensional space might be curved but figured that distances far longer than the spans between alpine mountains would be needed to test such a radical idea. He suggested conducting certain parallax measurements on distant stars. When the measurements were carried out, no change from flatness was discovered. Hence, it was assumed that Euclid' s rules continued to reign supreme throughout the universe.

Meanwhile, Gauss's fascination with the new geometry was passed on to a brilliant student at the University of Göttingen, Bernhard Riemann, who developed another non-Euclidean geometry altogether. The new construct was revealed during a trial lecture the timid 27-year-old student gave while seeking appointment as a Privatdozent (lecturer) at the university in 1854. In the course of his lecture, prepared in only seven weeks and later described as a high point in the history of mathematics, Riemann introduced a geometry in which no parallel lines can be drawn through a point near a given line. He was dealing with a space of positive curvature, best typified by the surface of a sphere. Here, the shortest distance between two points is not a straight line; instead, the shortest path is an arc, a segment of a great circle that encompasses the entire sphere. Like the lines of longitude on Earth, each great circle eventually intersects with every other great circle at the poles of the sphere. Consider two adjacent lines at the equator aligned directly north to south. Locally, they seem as parallel as they can be. Yet extend these lines around the world and they eventually cross and meet. Consequently, there are no parallel lines in this special kind of geometry. A triangle on such a curved surface would look

Three different geometries: flat space (top); negatively curved space (middle); and positively curved space (bottom).

inflated; the sum of its angles would be more than 180 degrees. Like Gauss, Lobachevsky, and Bolyai before him, Riemann was discovering that a mathematician can imagine many different geometric worlds. Euclid did not corner the market after all.

Infamous for his stern and critical demeanor, Gauss displayed a rare enthusiasm at the end of Riemann's presentation. He was perhaps the only one in the audience that day who recognized that Riemann had surpassed his predecessors by extending non-Euclidean geometry much farther. Riemann generalized the geometry of curved spaces to higher dimensions, spaces involving four, five, and even more dimensions. While at the time these manipulations may have appeared to be no more than a mathematical game, this work would later prove invaluable when Einstein faced the awesome task of developing his general theory of relativity. Riemann was fashioning the tools that allowed Einstein to envision a completely different view of space and time. Riemann served as a vanguard for the Einsteinian revolution to

come. At one point he dared to suggest that the true nature of space would not be found in ancient manuscripts from Greece but rather from physical experience. He even imagined that the universe might close in on itself, forming a sort of four-dimensional ball. Such a curvature would be noticed only over great distances, hence our common experience of seeing our local universe as flat. Interestingly, Riemann went on to consider whether the structure of space was somehow molded by the presence of matter, creating what he called a metrical field akin to an electromagnetic field. It was a prescient vision, but he spoke too soon. Physics was not yet ready to give up on its pleasant Newtonian world, a world of absolute and rigid space, unvarying and unchanging. The idea that space might display a distinct geometry elicited fury in certain philosophers of that era. Space was still considered an empty vessel devoid of physical properties.

Riemann's life was tragically cut short. He died of tuberculosis at the age of 39 in the village of Selasca on Lake Maggiore. He had gone to Italy to attempt a cure. One of Riemann's greatest desires was to unify the laws of electricity, magnetism, light, and gravity. Such a project was premature, but his mathematics would still become the vital ingredient of the new physics to come. “Riemann left the real development of his ideas in the hands of some subsequent scientist whose genius as a physicist could rise to equal flights with his own as a mathematician,” said mathematician Hermann Weyl. After a lapse of 49 years, that mission would at last be fulfilled by Einstein. Had Riemann lived to a ripe old age, it is conceivable that Einstein would have thanked him in person.