by Mario Livio
For reasons that are not entirely clear, Newton essentially abandoned any serious research on the topics of gravitation and planetary motion until 1679. Then two letters from his archrival Robert Hooke renewed his interest in dynamics in general and in planetary motion in particular. The results of this revived curiosity were quite dramatic—using his previously formulated laws of mechanics, Newton proved Kepler’s second law of planetary motion. Specifically, he showed that as the planet moves in its elliptical orbit about the Sun, the line joining the planet to the Sun sweeps equal areas in equal time intervals (figure 28). He also proved that for “a body revolving in an ellipse…the law of attraction directed to a focus of the ellipse…is inversely as the square of the distance.” These were important milestones on the road to Principia.
Figure 28
Principia
Halley came to visit Newton in Cambridge in the spring or summer of 1684. For some time Halley had been discussing Kepler’s laws of planetary motion with Hooke and with the renowned architect Christopher Wren (1632–1723). At these coffeehouse conversations, both Hooke and Wren claimed to have deduced the inverse-square law of gravity some years earlier, but both were also unable to construct a complete mathematical theory out of this deduction. Halley decided to ask Newton the crucial question: Did he know what would be the shape of the orbit of a planet acted upon by an attractive force varying as an inverse-square law? To his astonishment, Newton answered that he had proved some years earlier that the orbit would be an ellipse. The mathematician Abraham de Moivre (1667–1754) tells the story in a memorandum (from which a page is shown in figure 29):
In 1684 Dr Halley came to visit him [Newton] at Cambridge, after they had been some time together, the Dr asked him what he thought the curve would be that would be described by the planets supposing the force of attraction towards the sun to be reciprocal to the square of their distance from it. Sr Isaac replied immediately that it would be an Ellipsis [ellipse], the Doctor struck with joy and amazement asked him how he knew it, why saith he [Newton] I have calculated it, whereupon Dr Halley asked him for his calculation without any farther delay, Sr Isaac looked among his papers but could not find it, but he promised him to renew it and send it.
Halley indeed came to visit Newton again in November 1684. Between the two visits Newton worked frantically. De Moivre gives us a brief description:
Figure 29
Sr Isaac in order to make good his promise fell to work again but he could not come to that conclusion wch he thought he had before examined with care, however he attempted a new way which thou longer than the first, brought him again to his former conclusion, then he examined carefully what might be the reason why the calculation he had undertaken before did not prove right, &…he made both his calculations agree together.
This dry summary does not even begin to tell us what Newton had actually accomplished in the few months between Halley’s two visits. He wrote an entire treatise, De Motu Corporum in Gyrum (The Motion of Revolving Bodies), in which he proved most aspects of bodies moving in circular or elliptical orbits, proved all of Kepler’s laws, and even solved for the motion of a particle moving in a resisting medium (such as air). Halley was overwhelmed. To his satisfaction, he at least managed to persuade Newton to publish all of these staggering discoveries—Principia was finally about to happen.
At first, Newton had thought of the book as being nothing but a somewhat expanded and more detailed version of his treatise De Motu. As he started working, however, he realized that some topics required further thought. Two points in particular continued to disturb Newton. One was the following: Newton originally formulated his law of gravitational attraction as if the Sun, Earth, and planets were mathematical point masses, without any dimensions. He of course knew this not to be true, and therefore he regarded his results as only approximate when applied to the solar system. Some even speculate that he abandoned again his pursuit of the topic of gravity after 1679 because of his dissatisfaction with this state of affairs. The situation was even worse with respect to the force on the apple. There, clearly the parts of the Earth that are directly underneath the apple are at a much shorter distance to it than the parts that are on the other side of the Earth. How was one to calculate the net attraction? The astronomer Herbert Hall Turner (1861–1930) described Newton’s mental struggle in an article that appeared in the London Times on March 19, 1927:
At that time the general idea of an attraction varying as the inverse square of the distance occurred to him, but he saw grave difficulties in its complete application of which lesser minds were unconscious. The most important of these he did not overcome until 1685…It was that of linking up the attraction of the earth on a body so far away as the moon with its attraction on the apple close to its surface. In the former case the various particles composing the earth (to which individually Newton hoped to extend his law, thus making it universal) are at distances from the moon not greatly different either in magnitude or direction; but their distances from the apple differ conspicuously in both size and direction. How are the separate attractions in the latter case to be added together or combined into a single resultant? And in what “centre of gravity,” if any, may they be concentrated?
The breakthrough finally came in the spring of 1685. Newton managed to prove an essential theorem: For two spherical bodies, “the whole force with which one of these spheres attracts the other will be inversely proportional to the square of the distance of the centres.” That is, spherical bodies gravitationally act as if they were point masses concentrated at their centers. The importance of this beautiful proof was emphasized by the mathematician James Whitbread Lee Glaisher (1848–1928). In his address at the bicentenary celebration (in 1887) of Newton’s Principia, Glaisher said:
No sooner had Newton proved this superb theorem—and we know from his own words that he had no expectation of so beautiful a result till it emerged from his mathematical investigation—than all the mechanism of the universe at once lay spread before him…How different must these propositions have seemed to Newton’s eyes when he realised that these results, which he had believed to be only approximately true when applied to the solar system, were really exact!…We can imagine the effect of this sudden transition from approximation to exactitude in stimulating Newton’s mind to still greater efforts. It was now in his power to apply mathematical analysis with absolute precision to the actual problem of astronomy.
The other point that was apparently still troubling Newton when he wrote the early draft of De Motu was the fact that he neglected the influence of the forces by which the planets attracted the Sun. In other words, in his original formulation, he reduced the Sun to a mere unmovable center of force of the type that “hardly exists,” in Newton’s words, in the real world. This scheme contradicted Newton’s own third law of motion, according to which “the actions of attracting and attracted bodies are always mutual and equal.” Each planet attracts the Sun precisely with the same force that the Sun attracts the planet. Consequently, he added, “if there are two bodies [such as the Earth and the Sun], neither the attracting nor the attracted body can be at rest.” This seemingly minor realization was actually an important stepping-stone toward the concept of a universal gravity. We can attempt to guess Newton’s line of thought: If the Sun pulls the Earth, then the Earth must also pull the Sun, with equal strength. That is, the Earth doesn’t simply orbit the Sun, but rather they both revolve around their mutual center of gravity. But this is not all. All the other planets also attract the Sun, and indeed each planet feels the attraction not just of the Sun, but also of all other planets. The same type of logic could be applied to Jupiter and its satellites, to the Earth and the Moon, and even to an apple and the Earth. The conclusion is astounding in it simplicity—there is only one gravitational force, and it acts between any two masses, anywhere in the universe. This was all that Newton needed. The Principia—510 dense Latin pages—was published in July of 1687.
Newton took obser
vations and experiments that were accurate to only about 4 percent and established from those a mathematical law of gravity that turned out to be accurate to better than one part in a million. He united for the first time explanations of natural phenomena with the power of prediction of the results of observations. Physics and mathematics became forever intertwined, while the divorce of science from philosophy became inevitable.
The second edition of the Principia, edited extensively by Newton and in particular by the mathematician Roger Cotes (1682–1716), appeared in 1713 (figure 30 shows the frontispiece). Newton, who was never known for his warmth, did not even bother to thank Cotes in the preface to the book for his fabulous work. Still, when Cotes died from violent fever at age thirty-three, Newton did show some appreciation: “If he had lived we would have known something.”
Curiously, some of Newton’s most memorable remarks about God appeared only as afterthoughts in the second edition. In a letter to Cotes on March 28, 1713, less than three months before the completion of Principia’s second edition, Newton included the sentence: “It surely does belong to natural philosophy to discourse of God from the phenomena [of nature].” Indeed, Newton expressed his ideas of an “eternal and infinite, omnipotent and omniscient” God in the “General Scholium”—the section he regarded as putting the final touch on the Principia.
Figure 30
But did God’s role remain unchanged in this increasingly mathematical universe? Or was God perceived more and more as a mathematician? After all, until the formulation of the law of gravitation, the motions of the planets had been regarded as one of the unmistakable works of God. How did Newton and Descartes see this shift in emphasis toward scientific explanations of nature?
The Mathematician God of Newton and Descartes
As were most people of their time, both Newton and Descartes were religious men. The French writer known by the pen name of Voltaire (1694–1778), who wrote quite extensively about Newton, famously said that “if God did not exist, it would be necessary for us to invent Him.”
For Newton, the world’s very existence and the mathematical regularity of the observed cosmos were evidence for God’s presence. This type of causal reasoning was first used by the theologian Thomas Aquinas (ca. 1225–1274), and the arguments fall under the general philosophical labels of a cosmological argument and a teleological argument. Put simply, the cosmological argument claims that since the physical world had to come into existence somehow, there must be a First Cause, namely, a creator God. The teleological argument, or argument from design, attempts to furnish evidence for God’s existence from the apparent design of the world. Here are Newton’s thoughts, as expressed in Principia: “This most beautiful system of the sun, planets and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centers of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One.” The validity of the cosmological, teleological, and similar arguments as proof for God’s existence has been the subject of debate among philosophers for centuries. My personal impression has always been that theists don’t need these arguments to be convinced, and atheists are not persuaded by them.
Newton added yet another twist, based on the universality of his laws. He regarded the fact that the entire cosmos is governed by the same laws and appears to be stable as further evidence for God’s guiding hand, “especially since the light of the fixed stars is of the same nature [emphasis added] with the light of the sun, and from every system light passes into all the other systems: and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed these systems at immense distances one from another.”
In his book Opticks, Newton made it clear that he did not believe that the laws of nature by themselves were sufficient to explain the universe’s existence—God was the creator and sustainer of all the atoms that make up the cosmic matter: “For it became him [God] who created them [the atoms] to set them in order. And if he did so, it’s unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chaos by the mere Laws of Nature.” In other words, to Newton, God was a mathematician (among other things), not just as a figure of speech, but almost literally—the Creator God brought into existence a physical world that was governed by mathematical laws.
Being much more philosophically inclined than Newton, Descartes had been extremely preoccupied with proving God’s existence. To him, the road from the certainty in our own existence (“I am thinking, therefore I exist”) to our ability to construct a tapestry of objective science had to pass through an unassailable proof for the existence of a supremely perfect God. This God, he argued, was the ultimate source of all truth and the only guarantor of the reliability of human reasoning. This suspiciously circular argument (known as the Cartesian circle) was already criticized during Descartes’ time, especially by the French philosopher, theologian, and mathematician Antoine Arnauld (1612–94). Arnauld posed a question that was devastating in its simplicity: If we need to prove God’s existence in order to guarantee the validity of the human thought process, how can we trust the proof, which is in itself a product of the human mind? While Descartes did make some desperate attempts to escape from this vicious reasoning circle, many of the philosophers who followed him did not find his efforts particularly convincing. Descartes’ “supplemental proof” for the existence of God was equally questionable. It falls under the general philosophical label of an ontological argument. The philosophical theologian St. Anselm of Canterbury (1033–1109) first formulated this type of reasoning in 1078, and it has since resurfaced in many incarnations. The logical construct goes something like this: God, by definition, is so perfect that he is the greatest conceivable being. But if God did not exist, then it would be possible to conceive of a greater being yet—one that in addition to being blessed with all of God’s perfections also exists. This would contradict God’s definition as the greatest conceivable being—therefore God has to exist. In Descartes’ words: “Existence can no more be separated from the essence of God than the fact that its angles equal two right angles can be separated from the essence of a triangle.”
This type of logical maneuvering does not convince many philosophers, and they argue that to establish the existence of anything that is consequential in the physical world, and in particular something as grand as God, logic alone does not suffice.
Oddly enough, Descartes was accused of fostering atheism, and his works were put on the Catholic Church’s Index of Forbidden Books in 1667. This was a bizarre charge in light of Descartes’ insistence on God as the ultimate guarantor of truth.
Leaving the purely philosophical questions aside, for our present purposes the most interesting point is Descartes’ view that God created all the “eternal truths.” In particular, he declared that “the mathematical truths which you call eternal have been laid down by God and depend on Him entirely no less than the rest of his creatures.” So the Cartesian God was more than a mathematician, in the sense of being the creator of both mathematics and a physical world that is entirely based on mathematics. According to this worldview, which was becoming prevalent at the end of the seventeenth century, humans clearly only discover mathematics and do not invent it.
More significantly, the works of Galileo, Descartes, and Newton have changed the relationship between mathematics and the sciences in a profound way. First, the explosive developments in science became strong motivators for mathematical investigations. Second, through Newton’s laws, even more abstract mathematical fields, such as calculus, became the essence of physical explanations. Finally, and perhaps most importantly, the boundary between mathematics and the sciences was blurred beyond recognition, almost to the point of a complete fusion between mathematical insights and large swaths of exploration. All of these developments created a level of enthusiasm for mathematics perhaps not experienced since the time of the an
cient Greeks. Mathematicians felt that the world was theirs to conquer, and that it offered unlimited potential for discovery.
CHAPTER 5
STATISTICIANS AND PROBABILISTS: THE SCIENCE OF UNCERTAINTY
The world doesn’t stand still. Most things around us are either in motion or continuously changing. Even the seemingly firm Earth underneath our feet is in fact spinning around its axis, revolving around the Sun, and traveling (together with the Sun) around the center of our Milky Way galaxy. The air we breathe is composed of trillions of molecules that move ceaselessly and randomly. At the same time, plants grow, radioactive materials decay, the atmospheric temperature rises and falls both daily and with the seasons, and the human life expectancy keeps increasing. This cosmic restlessness in itself, however, did not stump mathematics. The branch of mathematics called calculus was introduced by Newton and Leibniz precisely to permit a rigorous analysis and an accurate modeling of both motion and change. By now, this incredible tool has become so potent and all encompassing that it can be used to examine problems as diverse as the motion of the space shuttle or the spreading of an infectious disease. Just as a movie can capture motion by breaking it up into a frame-by-frame sequence, calculus can measure change on such a fine grid that it allows for the determination of quantities that have only a fleeting existence, such as instantaneous speed, acceleration, or rate of change.
Continuing in Newton’s and Leibniz’s giant footsteps, mathematicians of the Age of Reason (the late seventeenth and eighteenth centuries) extended calculus to the even more powerful and widely applicable branch of differential equations. Armed with this new weapon, scientists were now able to present detailed mathematical theories of phenomena ranging from the music produced by a violin string to the transport of heat, from the motion of a spinning top to the flow of liquids and gases. For a while, differential equations became the tool of choice for making progress in physics.