The Laws of Motion and Universal Gravitation
From Galileo's inclined planes to Newton's Principia: how a unified physics of motion and gravitation replaced the separate physics of heaven and Earth.
The Laws of Motion and Universal Gravitation
In 1687, Isaac Newton published a book that did for physics what no single work had done before: it gave a single set of mathematical laws that explained, with quantitative accuracy, both the fall of an apple and the orbit of the Moon. The Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) is the central document of the Scientific Revolution, and the laws it contains are still, with modifications from relativity and quantum theory, the working physics of most everyday situations. This section examines that synthesis. It is part of the Major Discoveries. It links upward to the Isaac Newton for biographical and historical context, and downward to two articles: What Are Newton’s Three Laws of Motion?, which examines the three laws in detail, and What Is Universal Gravitation?, which examines the inverse-square law and its consequences. For the unification of terrestrial and celestial physics in the broader history of the revolution, see Heliocentrism and the New Astronomy. For the optical and alchemical work that Newton also did, see Newton and Optics and Newton and Gravity.
The Problem the Principia Solved
At the start of the seventeenth century, the physics of motion was in a curious state. The Aristotelian tradition, which had dominated European universities for four centuries, taught that the natural state of a terrestrial object was rest, and that motion had to be continuously maintained by an external cause. A stone fell to the ground because the earth-element in it sought its natural place at the centre of the universe. An arrow flew through the air because the air behind it pushed it forward. A horse pulled a cart because the horse’s effort was transmitted through the harness to the cart. The celestial physics, by contrast, was the physics of perfect circular motion at constant speed, sustained by intelligences or angels and unchanging in the ideal substance of the aether. The two physics — terrestrial and celestial — operated by different rules, on different substances, and with different mathematical descriptions. The boundary between them lay at the orbit of the Moon. What Newton did was replace both of these with a single, unified physics in which the same three laws applied to every body in the universe, and in which the same law of universal gravitation explained both a falling apple and a falling Moon. The change was so total that the very word “physics” changed its meaning: from a branch of philosophy concerned with the nature of change, it became a mathematical science concerned with the quantitative description of motion and force.
Galileo’s Preliminary Work
The groundwork for Newton’s synthesis was laid in large part by Galileo Galilei. In his Two New Sciences (1638), published in the Netherlands while he was under house arrest in Italy, Galileo laid out the foundations of what we now call kinematics — the mathematical description of motion — and laid the ground for dynamics — the explanation of motion in terms of force. Galileo’s central experimental and theoretical achievements were several. He established, by careful experiments with balls rolling down inclined planes, that the distance travelled by a uniformly accelerating body starting from rest is proportional to the square of the elapsed time. He argued that a body in motion, in the absence of resistance, would continue in straight-line motion at constant speed — the principle that Newton state as the first law. He showed, by a famous thought experiment, that bodies of different masses fall at the same rate in a vacuum. He articulated a version of the principle of relativity: in a closed room below the deck of a steadily moving ship, one cannot tell from the behaviour of falling drops, flying birds, or swimming fish whether the ship is moving or at rest. Galileo’s programme was limited in two important ways. He did not develop a general mathematical theory of motion comparable to Newton’s. And he did not connect terrestrial motion to celestial motion — he left the Moon, planets, and stars to the astronomers. The unification of the two physics was Newton’s work.
The Concept of Inertia
The most important conceptual step from Aristotle to Newton was the principle of inertia. In Aristotle’s physics, a body in motion came to rest when the force pushing it stopped acting. The intuition is everyday: a cart stops rolling when the horse stops pulling, a stone falls to the ground, an arrow comes to rest in the air. Galileo saw that this intuition is misleading when friction and air resistance are involved. A ball rolling down a smooth inclined plane speeds up, but a ball rolling up a smooth inclined plane slows down, and the two effects are symmetrical: the same surface smoothness produces the same magnitude of change in speed in the same time. Galileo inferred that on a perfectly smooth horizontal surface, with no friction, the ball would neither speed up nor slow down: it would roll forever at constant speed. The same reasoning, applied to a body falling through a medium with no resistance, would yield a uniform fall; applied to a body projected horizontally, would yield a parabolic trajectory. Newton generalised this into the first of his three laws. Every body persists in its state of being at rest, or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed. This is not a statement about how things actually behave in the everyday world, where friction and air resistance are pervasive; it is a statement about the limiting behaviour when those disturbances are removed. Modern physics, with relativity, modifies it: a body free of forces moves on a geodesic of spacetime, which in the flat limit of special relativity becomes a straight line at constant speed.
Newton’s Three Laws
The three laws of motion, as Newton stated them in the Principia, are deceptively simple. The first law is the law of inertia, just described. The second law states that the change of motion of a body is proportional to the impressed force and occurs in the direction of the straight line along which that force is impressed. In modern notation, this is the equation F = ma, where F is the net force on a body, m is its mass, and a is the resulting acceleration. The third law states that for every action there is an equal and opposite reaction: if body A exerts a force on body B, body B exerts an equal and opposite force on body A. The three laws are unpacked in detail in the article What Are Newton’s Three Laws of Motion?. Together, they are the axioms from which the whole of classical mechanics is derived, and on which most of the engineering of the modern world — from bridges to rockets — has been built.
Universal Gravitation
The second of Newton’s two great achievements in the Principia was the law of universal gravitation. Every particle of matter in the universe attracts every other particle with a force directed along the line joining them, with magnitude proportional to the product of their masses and inversely proportional to the square of the distance between them. This single statement, in modern notation F = G m₁ m₂ / r², unifies the falling apple and the orbiting Moon. The same force that pulls a dropped stone to the ground pulls the Moon toward the Earth, and pulls the planets toward the Sun, and keeps the stars in their courses. The trajectory of a comet, the rise and fall of the tides, the precession of the equinoxes, and the slight wobble of the Earth on its axis are all consequences of the same equation. The law is examined in detail in the article What Is Universal Gravitation?. For the chain of evidence that confirmed the law, see Newton and Gravity, which covers the apple-to-Moon leap, the perturbation of the Moon’s orbit, and the prediction of the return of Halley’s Comet.
From Gravity to the Heavens
Once the law of universal gravitation was in hand, the work of showing that it explained the observed motions of the heavens began in earnest. Newton showed, in the Principia, that a body moving in an inverse-square central force directed toward a fixed point travels on a conic section. For a closed orbit, that conic is an ellipse with the central body at one focus — which is exactly Kepler’s first law. The conservation of angular momentum gives the equal-areas law — Kepler’s second. The harmonic law, Kepler’s third, follows from the inverse-square law for circular orbits (and approximately holds for nearly circular ones). The derivations are worked out in Kepler’s Three Laws and in the Isaac Newton. Newton went further. He accounted for the mutual perturbations of the planets; the slow precession of the equinoxes due to the equatorial bulge of the Earth; the variation of the apparent weight of an object with latitude; the behaviour of pendulums; the shape of rotating fluid bodies; the motion of the Moon; and the trajectories of comets. Halley’s Comet, observed in 1682, showedn by Edmund Halley, using Newton’s theory, to be the same object that had appeared in 1531, 1607, and 1682, and to be expected to return in 1758. It did, on Christmas night of that year, vindicating both the law of gravitation and the method by which it had been derived.
The Unification of Physics
The deepest significance of the Principia was philosophical as much as physical. By showing that the same mathematical laws applied to the Earth and to the heavens, it dissolved the Aristotelian distinction between the terrestrial and celestial realms. The Moon, the planets, the Sun, and the stars were made of the same kind of matter as the Earth and obeyed the same laws. The universe was unified, even if it was not yet known to extend beyond the visible stars. This unification also changed the meaning of “explanation” in science. To explain a phenomenon was no longer to fit it into a scheme of natural places and qualitative tendencies; it was to derive it from a small number of general laws, by mathematical argument, in such a way that its quantitative features fell out of the calculation. The success of the Principia established this new standard for what counted as a scientific explanation, and it is the standard that physics has worked to ever since.
Limits and Successors
The Newtonian system was not the end of the story. The first major limitation to emerge was the failure of Newtonian gravitation to account fully for the observed precession of Mercury’s perihelion, a discrepancy of about forty-three arcseconds per century that was eventually explained by Einstein’s general theory of relativity in 1915. The second was the failure of Newtonian mechanics at very small scales, which motivated the growth of quantum mechanics in the early twentieth century. The third was the failure of Newtonian absolute space and time to account for the constancy of the speed of light, which motivated special relativity in 1905. These limitations, however, did not diminish the achievement of the Principia. They extended it. Newton’s laws remain the working physics of almost every engineering discipline, of celestial mechanics, of ballistics, of spaceflight, and of the analysis of structures. The first man to walk on the Moon did so using a flight plan computed with Newton’s equations.
The Continental Response: The Leibniz-Clarke Debate
The Newtonian system was accepted in Britain with relatively little resistance, but on the Continent it generated a sustained philosophical debate that engaged some of the leading minds of Europe. The principal controversy, known as the Leibniz-Clarke debate, took place in 1715–1716 between Samuel Clarke, defending Newton’s views, and Gottfried Wilhelm Leibniz, attacking them in correspondence with Caroline of Ansbach, the Princess of Wales. Leibniz raised four principal objections. First, Newton’s absolute space and absolute time — in which the universe is situated and events are dated against a fixed background — were, in Leibniz’s view, philosophically incoherent: space was nothing but the order of co-existence of bodies, and time was nothing but the order of succession of events, and to postulate an absolute space and time over and above these orders was to multiply entities without necessity. Second, Newton’s claim that God occasionally intervened in the solar system to correct the irregularities produced by mutual planetary perturbations (a position Newton flirted with in the Opticks) was theologically and physically inadequate. A perfect creator, Leibniz argued, would have made a self-sustaining machine that did not need such interventions. Third, Newton’s action-at-a-distance gravity, in which the Sun pulls the Earth without any intervening medium, was occult and unintelligible. Fourth, the conservation of “vis viva” (what we would now call kinetic energy) suggested that the universe contained a fixed amount of motion, which would gradually dissipate without continued divine intervention. The debate was less a contest of science than a contest of metaphysical commitments, and the two sides talked past each other to some degree. Clarke’s replies to Leibniz were widely regarded as evasive, and Newton’s system was forced, in the century that followed, to defend itself against the formidable Cartesian-Leibnizian tradition of the Continent. The eventual triumph of Newton’s physics, especially in France after the work of Voltaire and Émilie du Châtelet in the 1730s and 1740s, owed as much to the explanatory power of the theory as to its adoption in British engineering. By the late eighteenth century, Newtonian mechanics was the working physics of the whole of Europe.
The Mathematisation of Nature
The Principia established a new standard for what counted as a scientific explanation. To explain a phenomenon was to derive it, by mathematical argument, from a small set of general laws. The success of this standard in mechanics, astronomy, and optics led to its extension, in the eighteenth and nineteenth centuries, to heat (thermodynamics), electricity and magnetism (Maxwell’s equations), and chemistry (the periodic table and chemical bonding). The full programme of the Principia — that nature, when properly interrogated, would yield to mathematical description — has been, on the whole, vindicated. It is, however, important to recognise the limitations of the Newtonian programme even within its own period. The phenomena that resisted mathematisation — chemistry, biology, geology, meteorology — turned out to require new mathematical tools, new instruments, and a different kind of theorising. The biology of the cell, the chemistry of the molecule, the geology of the deep time, and the meteorology of the atmosphere all required the growth of new mathematical physics in the nineteenth and twentieth centuries, and they remain active areas of mathematical research today.