In the 1640s Torricelli set out on a series of experiments to prove this idea. He reasoned that since the weight of a volume of mercury is 13.6 times the weight of the same volume of water, the maximum height of a column of mercury in a vertical glass tube closed on top that can be supported by the air—whether by the air pressing down on the surface of a pool of mercury in which the tube is standing, or on the open bottom of the tube when exposed to the air—should be 18 braccia divided by 13.6, or using more accurate modern values, 33.5 feet/13.6 = 30 inches = 760 millimeters. In 1643 he observed that if a vertical glass tube longer than this and closed at the top end is filled with mercury, then some mercury will flow out until the height of the mercury in the tube is about 30 inches. This leaves empty space on top, now known as a “Torricellian vacuum.” Such a tube can then serve as a barometer, to measure changes in ambient air pressure; the higher the air pressure, the higher the column of mercury that it can support.
The French polymath Blaise Pascal is best known for his work of Christian theology, the Pensées, and for his defense of the Jansenist sect against the Jesuit order, but he also contributed to geometry and to the theory of probability, and explored the pneumatic phenomena studied by Torricelli. Pascal reasoned that if the column of mercury in a glass tube open at the bottom is held up by the pressure of the air, then the height of the column should decrease when the tube is carried to high altitude on a mountain, where there is less air overhead and hence lower air pressure. After this prediction was verified in a series of expeditions from 1648 to 1651, Pascal concluded, “All the effects ascribed to [the abhorrence of a vacuum] are due to the weight and pressure of the air, which is the only real cause.”12
Pascal and Torricelli have been honored by having modern units of pressure named after them. One pascal is the pressure that produces a force of 1 newton (the force that gives a mass of 1 kilogram an acceleration of 1 meter per second in a second) when exerted on an area of 1 square meter. One torr is the pressure that will support a column of 1 millimeter of mercury. Standard atmospheric pressure is 760 torr, which equals a little more than 100,000 pascals.
The work of Torricelli and Pascal was carried further in England by Robert Boyle. Boyle was a son of the earl of Cork, and hence an absentee member of the “ascendancy,” the Protestant upper class that dominated Ireland in his time. He was educated at Eton College, took a grand tour of the Continent, and fought on the side of Parliament in the civil wars that raged in England in the 1640s. Unusually for a member of his class, he became fascinated by science. He was introduced to the new ideas revolutionizing astronomy in 1642, when he read Galileo’s Dialogue Concerning the Two Chief World Systems. Boyle insisted on naturalistic explanations of natural phenomena, declaring, “None is more willing [than myself] to acknowledge and venerate Divine Omnipotence, [but] our controversy is not about what God can do, but about what can be done by natural agents, not elevated above the sphere of nature.”13 But, like many before Darwin and some even after, he argued that the wonderful capabilities of animals and men showed that they must have been designed by a benevolent creator.
Boyle’s work on air pressure was described in 1660 in New Experiments Physico-Mechanical Touching the Spring of the Air. In his experiments, he used an improved air pump, invented by his assistant Robert Hooke, about whom more in Chapter 14. By pumping air out of vessels, Boyle was able to establish that air is needed for the propagation of sound, for fire, and for life. He found that the level of mercury in a barometer drops when air is pumped out of its surroundings, adding a powerful argument in favor of Torricelli’s conclusion that air pressure is responsible for phenomena previously attributed to nature’s abhorrence of a vacuum. By using a column of mercury to vary both the pressure and the volume of air in a glass tube, not letting air in or out and keeping the temperature constant, Boyle was able to study the relation between pressure and volume. In 1662, in a second edition of New Experiments, he reported that the pressure varies with the volume in such a way as to keep the pressure times the volume fixed, a rule now known as Boyle’s law.
Not even Galileo’s experiments with inclined planes illustrate so well the new aggressive style of experimental physics as these experiments on air pressure. No longer were natural philosophers relying on nature to reveal its principles to casual observers. Instead Mother Nature was being treated as a devious adversary, whose secrets had to be wrested from her by the ingenious construction of artificial circumstances.
13
Method Reconsidered
By the end of the sixteenth century the Aristotelian model for scientific investigation had been severely challenged. It was natural then to seek a new approach to the method for gathering reliable knowledge about nature. The two figures who became best known for attempts to formulate a new method for science are Francis Bacon and René Descartes. They are, in my opinion, the two individuals whose importance in the scientific revolution is most overrated.
Francis Bacon was born in 1561, the son of Nicholas Bacon, Lord Keeper of the Privy Seal of England. After an education at Trinity College, Cambridge, he was called to the bar, and followed a career in law, diplomacy, and politics. He rose to become Baron Verulam and lord chancellor of England in 1618, and later Viscount St. Albans, but in 1621 he was found guilty of corruption and declared by Parliament to be unfit for public office.
Bacon’s reputation in the history of science is largely based on his book Novum Organum (New Instrument, or True Directions Concerning the Interpretation of Nature), published in 1620. In this book Bacon, neither a scientist nor a mathematician, expressed an extreme empiricist view of science, rejecting not only Aristotle but also Ptolemy and Copernicus. Discoveries were to emerge directly from careful, unprejudiced observation of nature, not by deduction from first principles. He also disparaged any research that did not serve an immediate practical purpose. In The New Atlantis, he imagined a cooperative research institute, “Solomon’s House,” whose members would devote themselves to collecting useful facts about nature. In this way, man would supposedly regain the dominance over nature that was lost after the expulsion from Eden. Bacon died in 1626. There is a story that, true to his empirical principles, he succumbed to pneumonia after an experimental study of the freezing of meat.
Bacon and Plato stand at opposite extremes. Of course, both extremes were wrong. Progress depends on a blend of observation or experiment, which may suggest general principles, and of deductions from these principles that can be tested against new observations or experiments. The search for knowledge of practical value can serve as a corrective to uncontrolled speculation, but explaining the world has value in itself, whether or not it leads directly to anything useful. Scientists in the seventeenth and eighteenth centuries would invoke Bacon as a counterweight to Plato and Aristotle, somewhat as an American politician might invoke Jefferson without ever having been influenced by anything Jefferson said or did. It is not clear to me that anyone’s scientific work was actually changed for the better by Bacon’s writing. Galileo did not need Bacon to tell him to do experiments, and neither I think did Boyle or Newton. A century before Galileo, another Florentine, Leonardo da Vinci, was doing experiments on falling bodies, flowing liquids, and much else.1 We know about this work only from a pair of treatises on painting and on fluid motion that were compiled after his death, and from notebooks that have been discovered from time to time since then, but if Leonardo’s experiments had no influence on the advance of science, at least they show that experiment was in the air long before Bacon.
René Descartes was an altogether more noteworthy figure than Bacon. Born in 1596 into the juridical nobility of France, the noblesse de robe, he was educated at the Jesuit college of La Flèche, studied law at the University of Poitiers, and served in the army of Maurice of Nassau in the Dutch war of independence. In 1619 Descartes decided to devote himself to philosophy and mathematics, work that began in earnest after 1628, when he settled permanently in Holland.
Descartes put his views about mechanics into Le Monde, written in the early 1630s but not published until 1664, after his death. In 1637 he published a philosophical work, Discours de la méthode pour bien conduire sa raison, et chercher la vérité dans les sciences (Discourse on the Method of Rightly Conducting One’s Reason and of Seeking Truth in the Sciences). His ideas were further developed in his longest work, the Principles of Philosophy, published in Latin in 1644 and then in a French translation in 1647. In these works Descartes expresses skepticism about knowledge derived from authority or the senses. For Descartes the only certain fact is that he exists, deduced from the observation that he is thinking about it. He goes on to conclude that the world exists, because he perceives it without exerting an effort of will. He rejects Aristotelian teleology—things are as they are, not because of any purpose they might serve. He gives several arguments (all unconvincing) for the existence of God, but rejects the authority of organized religion. He also rejects occult forces at a distance—things interact with each other through direct pulling and pushing.
Descartes was a leader in bringing mathematics into physics, but like Plato he was too much impressed by the certainty of mathematical reasoning. In Part I of the Principles of Philosophy, titled “On the Principles of Human Knowledge,” Descartes described how fundamental scientific principles could be deduced with certainty by pure thought. We can trust in the “natural enlightenment or the faculty of knowledge given to us by God” because “it would be completely contradictory for Him to deceive us.”2 It is odd that Descartes thought that a God who allowed earthquakes and plagues would not allow a philosopher to be deceived.
Descartes did accept that the application of fundamental physical principles to specific systems might involve uncertainty and call for experimentation, if one did not know all the details of what the system contains. In his discussion of astronomy in Part III of Principles of Philosophy, he considers various hypotheses about the nature of the planetary system, and cites Galileo’s observations of the phases of Venus as reason for preferring the hypotheses of Copernicus and Tycho to that of Ptolemy.
This brief summary barely touches on Descartes’ views. His philosophy was and is much admired, especially in France and among specialists in philosophy. I find this puzzling. For someone who claimed to have found the true method for seeking reliable knowledge, it is remarkable how wrong Descartes was about so many aspects of nature. He was wrong in saying that the Earth is prolate (that is, that the distance through the Earth is greater from pole to pole than through the equatorial plane). He, like Aristotle, was wrong in saying that a vacuum is impossible. He was wrong in saying that light is transmitted instantaneously.* He was wrong in saying that space is filled with material vortices that carry the planets around in their paths. He was wrong in saying that the pineal gland is the seat of a soul responsible for human consciousness. He was wrong about what quantity is conserved in collisions. He was wrong in saying that the speed of a freely falling body is proportional to the distance fallen. Finally, on the basis of observation of several lovable pet cats, I am convinced that Descartes was also wrong in saying that animals are machines without true consciousness. Voltaire had similar reservations about Descartes:3
He erred on the nature of the soul, on the proofs of the existence of God, on the subject of matter, on the laws of motion, on the nature of light. He admitted innate ideas, he invented new elements, he created a world, he made man according to his own fashion—in fact, it is rightly said that man according to Descartes is Descartes’ man, far removed from man as he actually is.
Descartes’ scientific misjudgments would not matter in assessing the work of someone who wrote about ethical or political philosophy, or even metaphysics; but because Descartes wrote about “the method of rightly conducting one’s reason and of seeking truth in the sciences,” his repeated failure to get things right must cast a shadow on his philosophical judgment. Deduction simply cannot carry the weight that Descartes placed on it.
Even the greatest scientists make mistakes. We have seen how Galileo was wrong about the tides and comets, and we will see how Newton was wrong about diffraction. For all his mistakes, Descartes, unlike Bacon, did make significant contributions to science. These were published as a supplement to the Discourse on Method, under three headings: geometry, optics, and meteorology.4 These, rather than his philosophical writings, in my view represent his positive contributions to science.
Descartes’ greatest contribution was the invention of a new mathematical method, now known as analytic geometry, in which curves or surfaces are represented by equations that are satisfied by the coordinates of points on the curve or surface. “Coordinates” in general can be any numbers that give the location of a point, such as longitude, latitude, and altitude, but the particular kind known as “Cartesian coordinates” are the distances of the point from a center along a set of fixed perpendicular directions. For instance, in analytic geometry a circle of radius R is a curve on which the coordinates x and y are distances measured from the center of the circle along any two perpendicular directions, and satisfy the equation x2 + y2 = R2. (Technical Note 18 gives a similar description of an ellipse.) This very important use of letters of the alphabet to represent unknown distances or other numbers originated in the sixteenth century with the French mathematician, courtier, and cryptanalyst François Viète, but Viète still wrote out equations in words. The modern formalism of algebra and its application to analytic geometry are due to Descartes.
Using analytic geometry, we can find the coordinates of the point where two curves intersect, or the equation for the curve where two surfaces intersect, by solving the pair of equations that define the curves or the surfaces. Most physicists today solve geometric problems in this way, using analytic geometry, rather than the classic methods of Euclid.
In physics Descartes’ significant contributions were in the study of light. First, in his Optics, Descartes presented the relation between the angles of incidence and refraction when light passes from medium A to medium B (for example, from air to water): if the angle between the incident ray and the perpendicular to the surface separating the media is i, and the angle between the refracted ray and this perpendicular is r, then the sine* of i divided by the sine of r is an angle-independent constant n:
sine of i / sine of r = n
In the common case where medium A is the air (or, strictly speaking, empty space), n is the constant known as the “index of refraction” of medium B. For instance, if A is air and B is water then n is the index of refraction of water, about 1.33. In any case like this, where n is larger than 1, the angle of refraction r is smaller than the angle of incidence i, and the ray of light entering the denser medium is bent toward the direction perpendicular to the surface.
Unknown to Descartes, this relation had already been obtained empirically in 1621 by the Dane Willebrord Snell and even earlier by the Englishman Thomas Harriot; and a figure in a manuscript by the tenth-century Arab physicist Ibn Sahl suggests that it was also known to him. But Descartes was the first to publish it. Today the relation is usually known as Snell’s law, except in France, where it is more commonly attributed to Descartes.
Descartes’ derivation of the law of refraction is difficult to follow, in part because neither in his account of the derivation nor in the statement of the result did Descartes make use of the trigonometric concept of the sine of an angle. Instead, he wrote in purely geometric terms, though as we have seen the sine had been introduced from India almost seven centuries earlier by al-Battani, whose work was well known in medieval Europe. Descartes’ derivation is based on an analogy with what Descartes imagined would happen when a tennis ball is hit through a thin fabric; the ball will lose some speed, but the fabric can have no effect on the component of the ball’s velocity along the fabric. This assumption leads (as shown in Technical Note 27) to the result cited above: the ratio of the sines of the angles that the tennis ball makes with the perpendicular to the screen
before and after it hits the screen is an angle-independent constant n. Though it is hard to see this result in Descartes’ discussion, he must have understood this result, because with a suitable value for n he gets more or less the right numerical answers in his theory of the rainbow, discussed below.
There are two things clearly wrong with Descartes’ derivation. Obviously, light is not a tennis ball, and the surface separating air and water or glass is not a thin fabric, so this is an analogy of dubious relevance, especially for Descartes, who thought that light, unlike tennis balls, always travels at infinite speed.5 In addition, Descartes’ analogy also leads to a wrong value for n. For tennis balls (as shown in Technical Note 27) his assumption implies that n equals the ratio of the speed of the ball vB in medium B after it passes through the screen to its speed vA in medium A before it hits the screen. Of course, the ball would be slowed by passing through the screen, so vB would be less than vA and their ratio n would be less than 1. If this applied to light, it would mean that the angle between the refracted ray and the perpendicular to the surface would be greater than the angle between the incident ray and this perpendicular. Descartes knew this, and even supplied a diagram showing the path of the tennis ball being bent away from the perpendicular. Descartes also knew that this is wrong for light, for as had been observed at least since the time of Ptolemy, a ray of light entering water from the air is bent toward the perpendicular to the water’s surface, so that the sine of i is greater than the sine of r, and hence n is greater than 1. In a thoroughly muddled discussion that I cannot understand, Descartes somehow argues that light travels more easily in water than in air, so that for light n is greater than 1. For Descartes’ purposes his failure to explain the value of n didn’t really matter, because he could and did take the value of n from experiment (perhaps from the data in Ptolemy’s Optics), which of course gives n greater than 1.
To Explain the World: The Discovery of Modern Science Page 21
