A more convincing derivation of the law of refraction was given by the mathematician Pierre de Fermat (1601–1665), along the lines of the derivation by Hero of Alexandria of the equal-angles rule governing reflection, but now making the assumption that light rays take the path of least time, rather than of least distance. This assumption (as shown in Technical Note 28) leads to the correct formula, that n is the ratio of the speed of light in medium A to its speed in medium B, and is therefore greater than 1 when A is air and B is glass or water. Descartes could never have derived this formula for n, because for him light traveled instantaneously. (As we will see in Chapter 14, yet another derivation of the correct result was given by Christiaan Huygens, a derivation based on Huygens’ theory of light as a traveling disturbance, which did not rely on Fermat’s a priori assumption that the light ray travels the path of least time.)
Descartes made a brilliant application of the law of refraction: in his Meteorology he used his relation between angles of incidence and refraction to explain the rainbow. This was Descartes at his best as a scientist. Aristotle had argued that the colors of the rainbow are produced when light is reflected by small particles of water suspended in the air.6 Also, as we have seen in Chapters 9 and 10, in the Middle Ages both al-Farisi and Dietrich of Freiburg had recognized that rainbows are due to the refraction of rays of light when they enter and leave drops of rain suspended in the air. But no one before Descartes had presented a detailed quantitative description of how this works.
Descartes first performed an experiment, using a thin-walled spherical glass globe filled with water as a model of a raindrop. He observed that when rays of sunlight were allowed to enter the globe along various directions, the light that emerged at an angle of about 42° to the incident direction was “completely red, and incomparably more brilliant than the rest.” He concluded that a rainbow (or at least its red edge) traces the arc in the sky for which the angle between the line of sight to the rainbow and the direction from the rainbow to the sun is about 42°. Descartes assumed that the light rays are bent by refraction when entering a drop, are reflected from the back surface of the drop, and then are bent again by refraction when emerging from the drop back into the air. But what explains this property of raindrops, of preferentially sending light back at an angle of about 42° to the incident direction?
To answer this, Descartes considered rays of light that enter a spherical drop of water along 10 different parallel lines. He labeled these rays by what is today called their impact parameter b, the closest distance to the center of the drop that the ray would reach if it went straight through the drop without being refracted. The first ray was chosen so that if not refracted it would pass the center of the drop at a distance equal to 10 percent of the drop’s radius R (that is, with b = 0.1 R), while the tenth ray was chosen to graze the drop’s surface (so that b = R), and the intermediate rays were taken to be equally spaced between these two. Descartes worked out the path of each ray as it was refracted entering the drop, reflected by the back surface of the drop, and then refracted again as it left the drop, using the equal-angles law of reflection of Euclid and Hero, and his own law of refraction, and taking the index of refraction n of water to be 4/3. The following table gives values found by Descartes for the angle φ (phi) between the emerging ray and its incident direction for each ray, along with the results of my own calculation using the same index of refraction:
b/R
φ (Descartes)
φ (recalculated)
0.1
5° 40'
5° 44'
0.2
11° 19'
11° 20'
0.3
17° 56'
17° 6'
0.4
22° 30'
22° 41'
0.5
27° 52'
28° 6'
0.6
32° 56'
33° 14'
0.7
37° 26'
37° 49'
0.8
40° 44'
41° 13'
0.9
40° 57'
41° 30'
1.0
13° 40'
14° 22'
The inaccuracy of some of Descartes’ results can be set down to the limited mathematical aids available in his time. I don’t know if he had access to a table of sines, and he certainly had nothing like a modern pocket calculator. Still, Descartes would have shown better judgment if he had quoted results only to the nearest 10 minutes of arc, rather than to the nearest minute.
As Descartes noticed, there is a relatively wide range of values of the impact parameter b for which the angle φ is close to 40°. He then repeated the calculation for 18 more closely spaced rays with values of b between 80 percent and 100 percent of the drop’s radius, where φ is around 40°. He found that the angle φ for 14 of these 18 rays was between 40° and a maximum of 41° 30'. So these theoretical calculations explained his experimental observation mentioned earlier, of a preferred angle of roughly 42°.
Technical Note 29 gives a modern version of Descartes’ calculation. Instead of working out the numerical value of the angle φ between the incoming and outgoing rays for each ray in an ensemble of rays, as Descartes did, we derive a simple formula that gives φ for any ray, with any impact parameter b, and for any value of the ratio n of the speed of light in air to the speed of light in water. This formula is then used to find the value of φ where the emerging rays are concentrated.* For n equal to 4/3 the favored value of φ, where the emerging light is somewhat concentrated, turns out to be 42.0°, just as found by Descartes. Descartes even calculated the corresponding angle for the secondary rainbow, produced by light that is reflected twice within a raindrop before it emerges.
Descartes saw a connection between the separation of colors that is characteristic of the rainbow and the colors exhibited by refraction of light in a prism, but he was unable to deal with either quantitatively, because he did not know that the white light of the sun is composed of light of all colors, or that the index of refraction of light depends slightly on its color. In fact, while Descartes had taken the index for water to be 4/3 = 1.333 . . . , it is actually closer to 1.330 for typical wavelengths of red light and closer to 1.343 for blue light. One finds (using the general formula derived in Technical Note 29) that the maximum value for the angle φ between the incident and emerging rays is 42.8° for red light and 40.7° for blue light. This is why Descartes saw bright red light when he looked at his globe of water at an angle of 42° to the direction of the Sun’s rays. That value of the angle φ is above the maximum value 40.7° of the angle that can emerge from the globe of water for blue light, so no light from the blue end of the spectrum could reach Descartes; but it is just below the maximum value 42.8° of φ for red light, so (as explained in the previous footnote) this would make the red light particularly bright.
The work of Descartes on optics was very much in the mode of modern physics. Descartes made a wild guess that light crossing the boundary between two media behaves like a tennis ball penetrating a thin screen, and used it to derive a relation between the angles of incidence and refraction that (with a suitable choice of the index of refraction n) agreed with observation. Next, using a globe filled with water as a model of a raindrop, he made observations that suggested a possible origin of rainbows, and he then showed mathematically that these observations followed from his theory of refraction. He didn’t understand the colors of the rainbow, so he sidestepped the issue, and published what he did understand. This is just about what a physicist would do today, but aside from its application of mathematics to physics, what does it have to do with Descartes’ Discourse on Method? I can’t see any sign that he was following his own prescriptions for “Rightly Conducting One’s Reason and of Seeking Truth in the Sciences.”
I should add that in his Principles of Philosophy Descartes offered a significant qualitative improvement to Buridan’s notion of impetus.7 He argued that “al
l movement is, of itself, along straight lines,” so that (contrary to both Aristotle and Galileo) a force is required to keep planetary bodies in their curved orbits. But Descartes made no attempt at a calculation of this force. As we will see in Chapter 14, it remained for Huygens to calculate the force required to keep a body moving at a given speed on a circle of given radius, and for Newton to explain this force, as the force of gravitation.
In 1649 Descartes traveled to Stockholm to serve as a teacher of the reigning Queen Christina. Perhaps as a result of the cold Swedish weather, and having to get up to meet Christina at an unwontedly early hour, Descartes in the next year, like Bacon, died of pneumonia. Fourteen years later his works joined those of Copernicus and Galileo on the Index of books forbidden to Roman Catholics.
The writings of Descartes on scientific method have attracted much attention among philosophers, but I don’t think they have had much positive influence on the practice of scientific research (or even, as argued above, on Descartes’ own most successful scientific work). His writings did have one negative effect: they delayed the reception of Newtonian physics in France. The program set out in the Discourse on Method, of deriving scientific principles by pure reason, never worked, and never could have worked. Huygens when young considered himself a follower of Descartes, but he came to understand that scientific principles were only hypotheses, to be tested by comparing their consequences with observation.8
On the other hand, Descartes’ work on optics shows that he too understood that this sort of scientific hypothesis is sometimes necessary. Laurens Laudan has found evidence for the same understanding in Descartes’ discussion of chemistry in the Principles of Philosophy.9 This raises the question whether any scientists actually learned from Descartes the practice of making hypotheses to be tested experimentally, as Laudan thought was true of Boyle. My own view is that this hypothetical practice was widely understood before Descartes. How else would one describe what Galileo did, in using the hypothesis that falling bodies are uniformly accelerated to derive the consequence that projectiles follow parabolic paths, and then testing it experimentally?
According to the biography of Descartes by Richard Watson,10 “Without the Cartesian method of analyzing material things into their primary elements, we would never have developed the atom bomb. The seventeenth-century rise of Modern Science, the eighteenth-century Enlightenment, the nineteenth-century Industrial Revolution, your twentieth-century personal computer, and the twentieth-century deciphering of the brain—all Cartesian.” Descartes did make a great contribution to mathematics, but it is absurd to suppose that it is Descartes’ writing on scientific method that has brought about any of these happy advances.
Descartes and Bacon are only two of the philosophers who over the centuries have tried to prescribe rules for scientific research. It never works. We learn how to do science, not by making rules about how to do science, but from the experience of doing science, driven by desire for the pleasure we get when our methods succeed in explaining something.
14
The Newtonian Synthesis
With Newton we come to the climax of the scientific revolution. But what an odd bird to be cast in such a historic role! Newton never traveled outside a narrow strip of England, linking London, Cambridge, and his birthplace in Lincolnshire, not even to see the sea, whose tides so much interested him. Until middle age he was never close to any woman, not even to his mother.* He was deeply concerned with matters having little to do with science, such as the chronology of the Book of Daniel. A catalog of Newton manuscripts put on sale at Sotheby’s in 1936 shows 650,000 words on alchemy, and 1.3 million words on religion. With those who might be competitors Newton could be devious and nasty. Yet he tied up strands of physics, astronomy, and mathematics whose relations had perplexed philosophers since Plato.
Writers about Newton sometimes stress that he was not a modern scientist. The best-known statement along these lines is that of John Maynard Keynes (who had bought some of the Newton papers in the 1936 auction at Sotheby’s): “Newton was not the first of the age of reason. He was the last of the magicians, the last of the Babylonians and Sumerians, the last great mind which looked out on the visible and intellectual world with the same eyes as those who began to build our intellectual inheritance rather less than 10,000 years ago.”* But Newton was not a talented holdover from a magical past. Neither a magician nor an entirely modern scientist, he crossed the frontier between the natural philosophy of the past and what became modern science. Newton’s achievements, if not his outlook or personal behavior, provided the paradigm that all subsequent science has followed, as it became modern.
Isaac Newton was born on Christmas Day 1642 at a family farm, Woolsthorpe Manor, in Lincolnshire. His father, an illiterate yeoman farmer, had died shortly before Newton’s birth. His mother was higher in social rank, a member of the gentry, with a brother who had graduated from the University of Cambridge and become a clergyman. When Newton was three his mother remarried and left Woolsthorpe, leaving him behind with his grandmother. When he was 10 years old Newton went to the one-room King’s School at Grantham, eight miles from Woolsthorpe, and lived there in the house of an apothecary. At Grantham he learned Latin and theology, arithmetic and geometry, and a little Greek and Hebrew.
At the age of 17 Newton was called home to take up his duties as a farmer, but for these he was found to be not well suited. Two years later he was sent up to Trinity College, Cambridge, as a sizar, meaning that he would pay for his tuition and room and board by waiting on fellows of the college and on those students who had been able to pay their fees. Like Galileo at Pisa, he began his education with Aristotle, but he soon turned away to his own concerns. In his second year he started a series of notes, Questiones quandam philosophicae, in a notebook that had previously been used for notes on Aristotle, and which fortunately is still extant.
In December 1663 the University of Cambridge received a donation from a member of Parliament, Henry Lucas, establishing a professorship in mathematics, the Lucasian chair, with a stipend of £100 a year. Beginning in 1664 the chair was occupied by Isaac Barrow, the first professor of mathematics at Cambridge, 12 years older than Newton. Around then Newton began his study of mathematics, partly with Barrow and partly alone, and received his bachelor of arts degree. In 1665 the plague struck Cambridge, the university largely shut down, and Newton went home to Woolsthorpe. In those years, from 1664 on, Newton began his scientific research, to be described below.
Back in Cambridge, in 1667 Newton was elected a fellow of Trinity College; the fellowship brought him £2 a year and free access to the college library. He worked closely with Barrow, helping to prepare written versions of Barrow’s lectures. Then in 1669 Barrow resigned the Lucasian chair in order to devote himself entirely to theology. At Barrow’s suggestion, the chair went to Newton. With financial help from his mother, Newton began to spread himself, buying new clothes and furnishings and doing a bit of gambling.1
A little earlier, immediately after the restoration of the Stuart monarchy in 1660, a society had been formed by a few Londoners including Boyle, Hooke, and the astronomer and architect Christopher Wren, who would meet to discuss natural philosophy and observe experiments. At the beginning it had just one foreign member, Christiaan Huygens. The society received a royal charter in 1662 as the Royal Society of London, and has remained Britain’s national academy of science. In 1672 Newton was elected to membership in the Royal Society, which he later served as president.
In 1675 Newton faced a crisis. Eight years after beginning his fellowship, he had reached the point at which fellows of a Cambridge college were supposed to take holy orders in the Church of England. This would require swearing to belief in the doctrine of the Trinity, but that was impossible for Newton, who rejected the decision of the Council of Nicaea that the Father and the Son are of one substance. Fortunately, the deed that had established the Lucasian chair included a stipulation that its holder should not be active in th
e church, and on that basis King Charles II was induced to issue a decree that the holder of the Lucasian chair would thenceforth never be required to take holy orders. So Newton was able to continue at Cambridge.
Let’s now take up the great work that Newton began at Cambridge in 1664. This research centered on optics, mathematics, and what later came to be called dynamics. His work in any one of these three areas would qualify him as one of the great scientists of history.
Newton’s chief experimental achievements were concerned with optics.* His undergraduate notes, the Questiones quandam philosophicae, show him already concerned with the nature of light. Newton concluded, contrary to Descartes, that light is not a pressure on the eyes, for if it were then the sky would seem brighter to us when we are running. At Woolsthorpe in 1665 he developed his greatest contribution to optics, his theory of color. It had been known since antiquity that colors appear when light passes through a curved piece of glass, but it had generally been thought that these colors were somehow produced by the glass. Newton conjectured instead that white light consists of all the colors, and that the angle of refraction in glass or water depends slightly on the color, red light being bent somewhat less than blue light, so that the colors are separated when light passes through a prism or a raindrop.* This would explain what Descartes had not understood, the appearance of colors in the rainbow. To test this idea, Newton carried out two decisive experiments. First, after using a prism to create separate rays of blue and red light, he directed these rays separately into other prisms, and found no further dispersion into different colors. Next, with a clever arrangement of prisms, he managed to recombine all the different colors produced by refraction of white light, and found that when these colors are combined they produce white light.
To Explain the World: The Discovery of Modern Science Page 22
