Even without knowing the value of G, Newton could use his theory of gravitation to calculate the ratios of the masses of various bodies in the solar system. (See Technical Note 35.) For instance, knowing the ratios of the distances of Jupiter and Saturn from their moons and from the Sun, and knowing the ratios of the orbital periods of Jupiter and Saturn and their moons, he could calculate the ratios of the centripetal accelerations of the moons of Jupiter and Saturn toward their planets and the centripetal acceleration of these planets toward the Sun, and from this he could calculate the ratios of the masses of Jupiter, Saturn, and the Sun. Since the Earth also has a Moon, the same technique could in principle be used to calculate the ratio of the masses of the Earth and the Sun. Unfortunately, although the distance of the Moon from the Earth was well known from the Moon’s diurnal parallax, the Sun’s diurnal parallax was too small to measure, and so the ratio of the distances from the Earth to the Sun and to the Moon was not known. (As we saw in Chapter 7, the data used by Aristarchus and the distances he inferred from those data were hopelessly inaccurate.) Newton went ahead anyway, and calculated the ratio of masses, using a value for the distance of the Earth from the Sun that was no better than a lower limit on this distance, and actually about half the true value. Here are Newton’s results for ratios of masses, given as a corollary to Theorem VIII in Book III of the Principia, together with modern values:10
Ratio
Newton’s value
Modern value
m(Sun)/m(Jupiter)
1,067
1,048
m(Sun)/m(Saturn)
3,021
3,497
m(Sun)/m(Earth)
169,282
332,950
As can be seen from this table, Newton’s results were pretty good for Jupiter, not bad for Saturn, but way off for the Earth, because the distance of the Earth from the Sun was not known. Newton was quite aware of the problems posed by observational uncertainties, but like most scientists until the twentieth century, he was cavalier about giving the resulting range of uncertainty in his calculated results. Also, as we have seen with Aristarchus and al-Biruni, he quoted results of calculations to a much greater precision than was warranted by the accuracy of the data on which the calculations were based.
Incidentally, the first serious estimate of the size of the solar system was carried out in 1672 by Jean Richer and Giovanni Domenico Cassini. They measured the distance to Mars by observing the difference in the direction to Mars as seen from Paris and Cayenne; since the ratios of the distances of the planets from the Sun were already known from the Copernican theory, this also gave the distance of the Earth from the Sun. In modern units, their result for this distance was 140 million kilometers, reasonably close to the modern value of 149.5985 million kilometers for the mean distance. A more accurate measurement was made later by comparing observations at different locations on Earth of the transits of Venus across the face of the Sun in 1761 and 1769, which gave an Earth–Sun distance of 153 million kilometers.11
In 1797–1798 Henry Cavendish was at last able to measure the gravitational force between laboratory masses, from which a value of G could be inferred. But Cavendish did not refer to his measurement this way. Instead, using the well-known acceleration of 32 feet/second per second due to the Earth’s gravitational field at its surface, and the known volume of the Earth, Cavendish calculated that the average density of the Earth was 5.48 times the density of water.
This was in accord with a long-standing practice in physics: to report results as ratios or proportions, rather than as definite magnitudes. For instance, as we have seen, Galileo showed that the distance a body falls on the surface of the Earth is proportional to the square of the time, but he never said that the constant multiplying the square of the time that gives the distance fallen was half of 32 feet/second per second. This was due at least in part to the lack of any universally recognized unit of length. Galileo could have given the acceleration due to gravity as so many braccia/second per second, but what would this mean to Englishmen, or even to Italians outside Tuscany? The international standardization of units of length and mass12 began in 1742, when the Royal Society sent two rulers marked with standard English inches to the French Académie des Sciences; the French marked these with their own measures of length, and sent one back to London. But it was not until the gradual international adoption of the metric system, starting in 1799, that scientists had a universally understood system of units. Today we cite a value for G of 66.724 trillionths of a meter/second2 per kilogram: that is, a small body of mass 1 kilogram at a distance of 1 meter produces a gravitational acceleration of 66.724 trillionths of a meter/second per second.
After laying out Newton’s theories of motion and gravitation, the Principia goes on to work out some of their consequences. These go far beyond Kepler’s three laws. For instance, in Proposition 14 Newton explains the precession of planetary orbits measured (for the Earth) by al-Zarqali, though Newton does not attempt a quantitative calculation.
In Proposition 19 Newton notes that the planets must all be oblate, because their rotation produces centrifugal forces that are largest at the equator and vanish at the poles. For instance, the Earth’s rotation produces a centripetal acceleration at its equator equal to 0.11 feet/second per second, as compared with the acceleration 32 feet/second per second of falling bodies, so the centrifugal force produced by the Earth’s rotation is much less than its gravitational attraction, but not entirely negligible, and the Earth is therefore nearly spherical, but slightly oblate. Observations in the 1740s finally showed that the same pendulum will swing more slowly near the equator than at higher latitudes, just as would be expected if at the equator the pendulum is farther from the center of the Earth, because the Earth is oblate.
In Proposition 39 Newton shows that the effect of gravity on the oblate Earth causes a precession of its axis of rotation, the “precession of the equinoxes” first noted by Hipparchus. (Newton had an extracurricular interest in this precession; he used its values along with ancient observations of the stars in an attempt to date supposed historical events, such as the expedition of Jason and the Argonauts.)13 In the first edition of the Principia Newton calculates in effect that the annual precession due to the Sun is 6.82° (degrees of arc), and that the effect of the Moon is larger by a factor 6⅓, giving a total of 50.0" (seconds of arc) per year, in perfect agreement with the precession of 50" per year then measured, and close to the modern value of 50.375" per year. Very impressive, but Newton later realized that his result for the precession due to the Sun and hence for the total precession was 1.6 times too small. In the second edition he corrected his result for the effect of the Sun, and also corrected the ratio of the effects of the Moon and Sun, so that the total was again close to 50" per year, still in good agreement with what was observed.14 Newton had the correct qualitative explanation of the precession of the equinoxes, and his calculation gave the right order of magnitude for the effect, but to get an answer in precise agreement with observation he had to make many artful adjustments.
This is just one example of Newton fudging his calculations to get answers in close agreement with observation. Along with this example, R. S. Westfall15 has given others, including Newton’s calculation of the speed of sound, and his comparison of the centripetal acceleration of the Moon with the acceleration of falling bodies on the Earth’s surface mentioned earlier. Perhaps Newton felt that his real or imagined adversaries would never be convinced by anything but nearly perfect agreement with observation.
In Proposition 24, Newton presents his theory of the tides. Gram for gram, the Moon attracts the ocean beneath it more strongly than it attracts the solid Earth, whose center is farther away, and it attracts the solid Earth more strongly than it attracts the ocean on the side of the Earth away from the Moon. Thus there is a tidal bulge in the ocean both below the Moon, where the Moon’s gravity pulls water away from the Earth, and on the opposite side of the Earth, where the Moon’s g
ravity pulls the Earth away from the water. This explained why in some locations high tides are separated by roughly 12 rather than 24 hours. But the effect is too complicated for this theory of tides to have been verified in Newton’s time. Newton knew that the Sun as well as the Moon plays a role in raising the tides. The highest and lowest tides, known as spring tides, occur when the Moon is new or full, so that the Sun, Moon, and Earth are on the same line, intensifying the effects of gravitation. But the worst complications come from the fact that any gravitational effects on the oceans are greatly influenced by the shape of the continents and the topography of the ocean bottom, which Newton could not possibly take into account.
This is a common theme in the history of physics. Newton’s theory of gravitation made successful predictions for simple phenomena like planetary motion, but it could not give a quantitative account of more complicated phenomena, like the tides. We are in a similar position today with regard to the theory of the strong forces that hold quarks together inside the protons and neutrons inside the atomic nucleus, a theory known as quantum chromodynamics. This theory has been successful in accounting for certain processes at high energy, such as the production of various strongly interacting particles in the annihilation of energetic electrons and their antiparticles, and its successes convince us that the theory is correct. We cannot use the theory to calculate precise values for other things that we would like to explain, like the masses of the proton and neutron, because the calculation is too complicated. Here, as for Newton’s theory of the tides, the proper attitude is patience. Physical theories are validated when they give us the ability to calculate enough things that are sufficiently simple to allow reliable calculations, even if we can’t calculate everything that we might want to calculate.
Book III of Principia presents calculations of things already measured, and new predictions of things not yet measured, but even in the final third edition of Principia Newton could point to no predictions that had been verified in the 40 years since the first edition. Still, taken all together, the evidence for Newton’s theories of motion and gravitation was overwhelming. Newton did not need to follow Aristotle and explain why gravity exists, and he did not try. In his “General Scholium” Newton concluded:
Thus far I have explained the phenomena of the heavens and of our sea by the force of gravity, but I have not yet assigned a cause to gravity. Indeed, this force arises from some cause that penetrates as far as the centers of the Sun and planets without any diminution of its power to act, and that acts not in proportion to the quantity of the surfaces of the particles on which it acts (as mechanical causes are wont to do), but in proportion to the quantity of solid matter, and whose action is extended everywhere to immense distances, always decreasing as the inverse squares of the distances. . . . I have not as yet been able to deduce from phenomena the reasons for these properties of gravity, and I do not “feign” hypotheses.
Newton’s book appeared with an appropriate ode by Halley. Here is its final stanza:
Then ye who now on heavenly nectar fare,
Come celebrate with me in song the name
Of Newton, to the Muses dear; for he
Unlocked the hidden treasuries of Truth:
So richly through his mind had Phoebus cast
The radius of his own divinity,
Nearer the gods no mortal may approach.
The Principia established the laws of motion and the principle of universal gravitation, but that understates its importance. Newton had given to the future a model of what a physical theory can be: a set of simple mathematical principles that precisely govern a vast range of different phenomena. Though Newton knew very well that gravitation was not the only physical force, as far as it went his theory was universal—every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to the square of their separation. The Principia not only deduced Kepler’s rules of planetary motion as an exact solution of a simplified problem, the motion of point masses in response to the gravitation of a single massive sphere; it went on to explain (even if only qualitatively in some cases) a wide variety of other phenomena: the precession of equinoxes, the precession of perihelia, the paths of comets, the motions of moons, the rise and fall of the tides, and the fall of apples.16 By comparison, all past successes of physical theory were parochial.
After the publication of the Principia in 1686–1687, Newton became famous. He was elected a member of parliament for the University of Cambridge in 1689 and again in 1701. In 1694 he became warden of the Mint, where he presided over a reform of the English coinage while still retaining his Lucasian professorship. When Czar Peter the Great came to England in 1698, he made a point of visiting the Mint, and hoped to talk with Newton, but I can’t find any account of their actually meeting. In 1699 Newton was appointed master of the Mint, a much better-paid position. He gave up his professorship, and became rich. In 1703, after the death of his old enemy Hooke, Newton became president of the Royal Society. He was knighted in 1705. When in 1727 Newton died of a kidney stone, he was given a state funeral in Westminster Abbey, even though he had refused to take the sacraments of the Church of England. Voltaire reported that Newton was “buried like a king who had benefited his subjects.”17
Newton’s theory did not meet universal acceptance.18 Despite Newton’s own commitment to Unitarian Christianity, some in England, like the theologian John Hutchinson and Bishop Berkeley, were appalled by the impersonal naturalism of Newton’s theory. This was unfair to the devout Newton. He even argued that only divine intervention could explain why the mutual gravitational attraction of the planets does not destabilize the solar system,* and why some bodies like the Sun and stars shine by their own light, while others like the planets and their satellites are themselves dark. Today of course we understand the light of the Sun and stars in a naturalistic way—they shine because they are heated by nuclear reactions in their cores.
Though unfair to Newton, Hutchinson and Berkeley were not entirely wrong about Newtonianism. Following the example of Newton’s work, if not of his personal opinions, by the late eighteenth century physical science had become thoroughly divorced from religion.
Another obstacle to the acceptance of Newton’s work was the old false opposition between mathematics and physics that we have seen in a comment of Geminus of Rhodes quoted in Chapter 8. Newton did not speak the Aristotelian language of substances and qualities, and he did not try to explain the cause of gravitation. The priest Nicolas de Malebranche (1638–1715) in reviewing the Principia said that it was the work of a geometer, not of a physicist. Malebranche clearly was thinking of physics in the mode of Aristotle. What he did not realize is that Newton’s example had revised the definition of physics.
The most formidable criticism of Newton’s theory of gravitation came from Christiaan Huygens.19 He greatly admired the Principia, and did not doubt that the motion of planets is governed by a force decreasing as the inverse square of the distance, but Huygens had reservations about whether it is true that every particle of matter attracts every other particle with such a force, proportional to the product of their masses. In this, Huygens seems to have been misled by inaccurate measurements of the rates of pendulums at various latitudes, which seemed to show that the slowing of pendulums near the equator could be entirely explained as an effect of the centrifugal force due to the Earth’s rotation. If true, this would imply that the Earth is not oblate, as it would be if the particles of the Earth attract each other in the way prescribed by Newton.
Starting already in Newton’s lifetime, his theory of gravitation was opposed in France and Germany by followers of Descartes and by Newton’s old adversary Leibniz. They argued that an attraction operating over millions of miles of empty space would be an occult element in natural philosophy, and they further insisted that the action of gravity should be given a rational explanation, not merely assumed.
In this, natural philoso
phers on the Continent were hanging on to an old ideal for science, going back to the Hellenic age, that scientific theories should ultimately be founded solely on reason. We have learned to give this up. Even though our very successful theory of electrons and light can be deduced from the modern standard model of elementary particles, which may (we hope) in turn eventually be deduced from a deeper theory, however far we go we will never come to a foundation based on pure reason. Like me, most physicists today are resigned to the fact that we will always have to wonder why our deepest theories are not something different.
The opposition to Newtonianism found expression in a famous exchange of letters during 1715 and 1716 between Leibniz and Newton’s disciple, the Reverend Samuel Clarke, who had translated Newton’s Opticks into Latin. Much of their argument focused on the nature of God: Did He intervene in the running of the world, as Newton thought, or had He set it up to run by itself from the beginning?20 The controversy seems to me to have been supremely futile, for even if its subject were real, it is something about which neither Clarke nor Leibniz could have had any knowledge whatever.
In the end the opposition to Newton’s theories didn’t matter, for Newtonian physics went from success to success. Halley fitted the observations of the comets observed in 1531, 1607, and 1682 to a single nearly parabolic elliptical orbit, showing that these were all recurring appearances of the same comet. Using Newton’s theory to take into account gravitational perturbations due to the masses of Jupiter and Saturn, the French mathematician Alexis-Claude Clairaut and his collaborators predicted in November 1758 that this comet would return to perihelion in mid-April 1759. The comet was observed on Christmas Day 1758, 15 years after Halley’s death, and reached perihelion on March 13, 1759. Newton’s theory was promoted in the mid-eighteenth century by the French translations of the Principia by Clairaut and by Émilie du Châtelet, and through the influence of du Châtelet’s lover Voltaire. It was another Frenchman, Jean d’Alembert, who in 1749 published the first correct and accurate calculation of the precession of the equinoxes, based on Newton’s ideas. Eventually Newtonianism triumphed everywhere.
To Explain the World: The Discovery of Modern Science Page 25
