The dependence of the angle of refraction on color has the unfortunate consequence that the glass lenses in telescopes like those of Galileo, Kepler, and Huygens focus the different colors in white light differently, blurring the images of distant objects. To avoid this chromatic aberration Newton in 1669 invented a telescope in which light is initially focused by a curved mirror rather than by a glass lens. (The light rays are then deflected by a plane mirror out of the telescope to a glass eyepiece, so not all chromatic aberration was eliminated.) With a reflecting telescope only six inches long, he was able to achieve a magnification by 40 times. All major astronomical light-gathering telescopes are now reflecting telescopes, descendants of Newton’s invention. On my first visit to the present quarters of the Royal Society in Carlton House Terrace, as a treat I was taken down to the basement to look at Newton’s little telescope, the second one he made.
In 1671 Henry Oldenburg, the secretary and guiding spirit of the Royal Society, invited Newton to publish a description of his telescope. Newton submitted a letter describing it and his work on color to Philosophical Transactions of the Royal Society early in 1672. This began a controversy over the originality and significance of Newton’s work, especially with Hooke, who had been curator of experiments at the Royal Society since 1662, and holder of a lectureship endowed by Sir John Cutler since 1664. No feeble opponent, Hooke had made significant contributions to astronomy, microscopy, watchmaking, mechanics, and city planning. He claimed that he had performed the same experiments on light as Newton, and that they proved nothing—colors were simply added to white light by the prism.
Newton lectured on his theory of light in London in 1675. He conjectured that light, like matter, is composed of many small particles—contrary to the view, proposed at about the same time by Hooke and Huygens, that light is a wave. This was one place where Newton’s scientific judgment failed him. There are many observations, some even in Newton’s time, that show the wave nature of light. It is true that in modern quantum mechanics light is described as an ensemble of massless particles, called photons, but in the light encountered in ordinary experience the number of photons is enormous, and in consequence light does behave as a wave.
In his 1678 Treatise on Light, Huygens described light as a wave of disturbance in a medium, the ether, which consists of a vast number of tiny material particles in close proximity. Just as in an ocean wave in deep water it is not the water that moves along the surface of the ocean but the disturbance of the water, so likewise in Huygens’ theory it is the wave of disturbance in the particles of the ether that moves in a ray of light, not the particles themselves. Each disturbed particle acts as a new source of disturbance, which contributes to the total amplitude of the wave. Of course, since the work of James Clerk Maxwell in the nineteenth century we have known that (even apart from quantum effects) Huygens was only half right—light is a wave, but a wave of disturbances in electric and magnetic fields, not a wave of disturbance of material particles.
Using this wave theory of light, Huygens was able to derive the result that light in a homogeneous medium (or empty space) behaves as if it travels in straight lines, as it is only along these lines that the waves produced by all the disturbed particles add up constructively. He gave a new derivation of the equal-angles rule for reflection, and of Snell’s law for refraction, without Fermat’s a priori assumption that light rays take the path of least time. (See Technical Note 30.) In Huygens’ theory of refraction, a ray of light is bent in passing at an oblique angle through the boundary between two media with different light speeds in much the way the direction of march of a line of soldiers will change when the leading edge of the line enters a swampy terrain, in which their marching speed is reduced.
To digress a bit, it was essential to Huygens’ wave theory that light travels at a finite speed, contrary to what had been thought by Descartes. Huygens argued that effects of this finite speed are hard to observe simply because light travels so fast. If for instance it took light an hour to travel the distance of the Moon from the Earth, then at the time of an eclipse of the Moon the Moon would be seen not directly opposite the Sun, but lagging behind by about 33°. From the fact that no lag is seen, Huygens concluded that the speed of light must be at least 100,000 times as fast as the speed of sound. This is correct; the actual ratio is about 1 million.
Huygens went on to describe recent observations of the moons of Jupiter by the Danish astronomer Ole Rømer. These observations showed that the period of Io’s revolution appears shorter when Earth and Jupiter are approaching each other and longer when they are moving apart. (Attention focused on Io, because it has the shortest orbital period of any of Jupiter’s Galilean moons—only 1.77 days.) Huygens interpreted this as what later became known as a “Doppler effect”: when Jupiter and the Earth are moving closer together or farther apart, their separation at each successive completion of a whole period of revolution of Io is respectively decreasing or increasing, and so if light travels at a finite speed, the time interval between observations of complete periods of Io should be respectively less or greater than if Jupiter and the Earth were at rest. Specifically, the fractional shift in the apparent period of Io should be the ratio of the relative speed of Jupiter and the Earth along the direction separating them to the speed of light, with the relative speed taken as positive or negative if Jupiter and the Earth are moving farther apart or closer together, respectively. (See Technical Note 31.) Measuring the apparent changes in the period of Io and knowing the relative speed of Earth and Jupiter, one could calculate the speed of light. Because the Earth moves much faster than Jupiter, it is chiefly the Earth’s velocity that dominates the relative speed. The scale of the solar system was then not well known, so neither was the numerical value of the relative speed of separation of the Earth and Jupiter, but using Rømer’s data Huygens was able to calculate that it takes 11 minutes for light to travel a distance equal to the radius of the Earth’s orbit, a result that did not depend on knowing the size of the orbit. To put it another way, since the astronomical unit (AU) of distance is defined as the mean radius of the Earth’s orbit, the speed of light was found by Huygens to be 1 AU per 11 minutes. The modern value is 1 AU per 8.32 minutes.
There already was experimental evidence of the wave nature of light that would have been available to Newton and Huygens: the discovery of diffraction by the Bolognese Jesuit Francesco Maria Grimaldi (a student of Riccioli), published posthumously in 1665. Grimaldi had found that the shadow of a narrow opaque rod in sunlight is not perfectly sharp, but is bordered by fringes. The fringes are due to the fact that the wavelength of light is not negligible compared with the thickness of the rod, but Newton argued that they were instead the result of some sort of refraction at the surface of the rod. The issue of light as corpuscle or wave was settled for most physicists when, in the early nineteenth century, Thomas Young discovered interference, the pattern of reinforcement or cancellation of light waves that arrive at given spots along different paths. As already mentioned, in the twentieth century it was discovered that the two views are not incompatible. Einstein in 1905 realized that although light for most purposes behaves as a wave, the energy in light comes in small packets, later called photons, each with a tiny energy and momentum proportional to the frequency of the light.
Newton finally presented his work on light in his book Opticks, written (in English) in the early 1690s. It was published in 1704, after he had already become famous.
Newton was not only a great physicist but also a creative mathematician. He began in 1664 to read works on mathematics, including Euclid’s Elements and Descartes’ Geometrie. He soon started to work out the solutions to a variety of problems, many involving infinities. For instance, he considered infinite series, such as x − x2/2 + x3/3 − x4/4 + . . . , and showed that this adds up to the logarithm* of 1 + x.
In 1665 Newton began to think about infinitesimals. He took up a problem: suppose we know the distance D(t) traveled in any time t; how
do we find the velocity at any time? He reasoned that in nonuniform motion, the velocity at any instant is the ratio of the distance traveled to the time elapsed in an infinitesimal interval of time at that instant. Introducing the symbol o for an infinitesimal interval of time, he defined the velocity at time t as the ratio to o of the distance traveled between time t and time t + o, that is, the velocity is [D(t + o) − D(t)]/o. For instance, if D(t) = t3, then D(t + o) = t3 + 3t2o + 3to2 + o3. For o infinitesimal, we can neglect the terms proportional to o2 and o3, and take D(t + o) = t3 + 3t2 o, so that D(t + o) − D(t) = 3t2 o and the velocity is just 3t2. Newton called this the “fluxion” of D(t), but it became known as the “derivative,” the fundamental tool of modern differential calculus.*
Newton then took up the problem of finding the areas bounded by curves. His answer was the fundamental theorem of calculus; one must find the quantity whose fluxion is the function described by the curve. For instance, as we have seen, 3x2 is the fluxion of x3, so the area under the parabola y = 3x2 between x = 0 and any other x is x3. Newton called this the “inverse method of fluxions,” but it became known as the process of “integration.”
Newton had invented the differential and integral calculus, but for a long while this work did not become widely known. Late in 1671 he decided to publish it along with an account of his work on optics, but apparently no London bookseller was willing to undertake the publication without a heavy subsidy.2
In 1669 Barrow gave a manuscript of Newton’s De analysi per aequationes numero terminorum infinitas to the mathematician John Collins. A copy made by Collins was seen on a visit to London in 1676 by the philosopher and mathematician Gottfried Wilhelm Leibniz, a former student of Huygens and a few years younger than Newton, who had independently discovered the essentials of the calculus in the previous year. In 1676 Newton revealed some of his own results in letters that were meant to be seen by Leibniz. Leibniz published his work on calculus in articles in 1684 and 1685, without acknowledging Newton’s work. In these publications Leibniz introduced the word “calculus,” and presented its modern notation, including the integration sign ∫.
To establish his claim to calculus, Newton described his own methods in two papers included in the 1704 edition of Opticks. In January 1705 an anonymous review of Opticks hinted that these methods were taken from Leibniz. As Newton guessed, this review had been written by Leibniz. Then in 1709 Philosophical Transactions of the Royal Society published an article by John Keill defending Newton’s priority of discovery, and Leibniz replied in 1711 with an angry complaint to the Royal Society. In 1712 the Royal Society convened an anonymous committee to look into the controversy. Two centuries later the membership of this committee was made public, and it turned out to have consisted almost entirely of Newton’s supporters. In 1715 the committee reported that Newton deserved credit for the calculus. This report had been drafted for the committee by Newton. Its conclusions were supported by an anonymous review of the report, also written by Newton.
The judgment of contemporary scholars3 is that Leibniz and Newton had discovered the calculus independently. Newton accomplished this a decade earlier than Leibniz, but Leibniz deserves great credit for publishing his work. In contrast, after his original effort in 1671 to find a publisher for his treatise on calculus, Newton allowed this work to remain hidden until he was forced into the open by the controversy with Leibniz. The decision to go public is generally a critical element in the process of scientific discovery.4 It represents a judgment by the author that the work is correct and ready to be used by other scientists. For this reason, the credit for a scientific discovery today usually goes to the first to publish. But though Leibniz was the first to publish on calculus, as we shall see it was Newton rather than Leibniz who applied calculus to problems in science. Though, like Descartes, Leibniz was a great mathematician whose philosophical work is much admired, he made no important contributions to natural science.
It was Newton’s theories of motion and gravitation that had the greatest historical impact. It was an old idea that the force of gravity that causes objects to fall to the Earth decreases with distance from the Earth’s surface. This much was suggested in the ninth century by a well-traveled Irish monk, Duns Scotus (Johannes Scotus Erigena, or John the Scot), but with no suggestion of any connection of this force with the motion of the planets. The suggestion that the force that holds the planets in their orbits decreases with the inverse square of the distance from the Sun may have been first made in 1645 by a French priest, Ismael Bullialdus, who was later quoted by Newton and elected to the Royal Society. But it was Newton who made this convincing, and related the force to gravity.
Writing about 50 years later, Newton described how he began to study gravitation. Even though his statement needs a good deal of explanation, I feel I have to quote it here, because it describes in Newton’s own words what seems to have been a turning point in the history of civilization. According to Newton, it was in 1666 that:
I began to think of gravity extending to the orb of the Moon & (having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere) from Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs, I deduced that the forces which keep the Planets in their Orbs must [be] reciprocally as the squares of their distances from the centers about which they revolve & thereby compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly. All this [including his work on infinite series and calculus] was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention and minded Mathematicks and Philosophy more than at any time since.5
As I said, this takes some explaining.
First, Newton’s parenthesis “having found out how to estimate the force with which [a] globe revolving within a sphere presses the surface of the sphere” refers to the calculation of centrifugal force, a calculation that had already been done (probably unknown to Newton) around 1659 by Huygens. For Huygens and Newton (as for us), acceleration had a broader definition than just a number giving the change of speed per time elapsed; it is a directed quantity, giving the change per time elapsed in the direction as well as in the magnitude of the velocity. There is an acceleration in circular motion even at constant speed—it is the “centripetal acceleration,” consisting of a continual turning toward the center of the circle. Huygens and Newton concluded that a body moving at a constant speed v around a circle of radius r is accelerating toward the center of the circle, with acceleration v2/r, so the force needed to keep it moving on the circle rather than flying off in a straight line into space is proportional to v2/r. (See Technical Note 32.) The resistance to this centripetal acceleration is experienced as what Huygens called centrifugal force, as when a weight at the end of a cord is swung around in a circle. For the weight, the centrifugal force is resisted by tension in the cord. But planets are not attached by cords to the Sun. What is it that resists the centrifugal force produced by a planet’s nearly circular motion around the Sun? As we will see, the answer to this question led to Newton’s discovery of the inverse square law of gravitation.
Next, by “Kepler’s rule of the periodical times of the Planets being in sesquialterate proportion of their distances from the center of their Orbs” Newton meant what we now call Kepler’s third law: that the square of the periods of the planets in their orbits is proportional to the cubes of the mean radii of their orbits, or in other words, the periods are proportional to the 3/2 power (the “sesquialterate proportion”) of the mean radii.* The period of a body moving with speed v around a circle of radius r is the circumference 2πr divided by the speed v, so for circular orbits Kepler’s third law tells us that r2/v2 is proportional to r3, and therefore their inverses are proportional: v2/r2 is proportional to 1/r3. It follows that the force keeping the planets in orbit, which is proportional to v2/r, must be prop
ortional to 1/r2. This is the inverse square law of gravity.
This in itself might be regarded as just a way of restating Kepler’s third law. Nothing in Newton’s consideration of the planets makes any connection between the force holding the planets in their orbits and the commonly experienced phenomena associated with gravity on the Earth’s surface. This connection was provided by Newton’s consideration of the Moon. Newton’s statement that he “compared the Moon in her Orb with the force of gravity at the surface of the Earth & found them answer pretty nearly” indicates that he had calculated the centripetal acceleration of the Moon, and found that it was less than the acceleration of falling bodies on the surface of the Earth by just the ratio one would expect if these accelerations were inversely proportional to the square of the distance from the center of the Earth.
To be specific, Newton took the radius of the Moon’s orbit (well known from observations of the Moon’s diurnal parallax) to be 60 Earth radii; it is actually about 60.2 Earth radii. He used a crude estimate of the Earth’s radius,* which gave a crude value for the radius of the Moon’s orbit, and knowing that the sidereal period of the Moon’s revolution around the Earth is 27.3 days, he could estimate the Moon’s velocity and from that its centripetal acceleration. This acceleration turned out to be less than the acceleration of falling bodies on the surface of the Earth by a factor roughly (very roughly) equal to 1/(60)2, as expected if the force holding the Moon in its orbit is the same that attracts bodies on the Earth’s surface, but reduced in accordance with the inverse square law. (See Technical Note 33.) This is what Newton meant when he said that he found that the forces “answer pretty nearly.”
To Explain the World: The Discovery of Modern Science Page 23
