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Asimov’s New Guide to Science
Asimov’s New Guide to Science
Isaac Asimov
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Asimov tells the stories behind the science: the men and women who made the important discoveries and how they did it. Ranging from Galilei, Achimedes, Newton and Einstein, he takes the most complex concepts and explains it in such a way that a firsttime reader on the subject feels confident on his/her understanding.
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Year:
1993
Publisher:
Penguin Books
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english
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896
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0140172130
ISBN 13:
9780140172133
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Penguin Press Science
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EPUB, 9.76 MB
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Appendix Mathematics in Science Gravitation As I explained in chapter 1, Galileo initiated science in its modern sense by introducing the concept of reasoning back from observation and experiment to basic principles. In doing so, he also introduced the essential technique of measuring natural phenomena accurately and abandoned the practice of merely describing them in general terms. In short, he turned from the qualitative description of the universe by the Greek thinkers to a quantitative description. Although science depends so much on mathematical relationships and manipulations, and could not exist in the Galilean sense without it, I have nevertheless written this book non mathematically, and have done so deliberately. Mathematics, after all, is a highly specialized tool. To have discussed the developments in science in mathematical terms would have required a prohibitive amount of space, as well as a sophisticated knowledge of mathematics on the part of the reader. But in this appendix, I would like to present an example or two of the way in which simple mathematics has been fruitfully applied to science. How better to begin than with Galileo himself? THE FIRST LAW OF MOTION Galileo (like Leonardo da Vinci nearly a century earlier) suspected that falling objects steadily increase their velocity as they fall. He set out to measure exactly by how much and in what manner the velocity increases. The measurement was anything but easy for Galileo, with the tools he had at his disposal in 1600. To measure a velocity requires the measurement of time. We speak of velocities of 60 miles an hour, of 13 feet a second. But there were no clocks in Galileo’s time that could do more than strike the hour at approximately equal intervals. Galileo resorted to a crude water clock. He let water trickle slowly from a small spout, assuming, hopefully, that it dripped at a constant rate. This water he caught in a cup; and, by the weight of water caught during the interval in which an event took place, Galileo measure; d the elapsed time. (He also used his pulse beat for the purpose on occasion.) One difficulty was, however, that a falling object dropped so rapidly that Galileo could not collect enough water, in the interval of falling, to weigh accurately. What he did, then, was to dilute the pull of gravity by having a brass ball roll down a groove in an inclined plane. The more nearly horizontal the plane, the more slowly the ball moved. Thus Galileo was able to study falling bodies in whatever degree of slow motion he pleased. Galileo found that a ball rolling on a perfectly horizontal plane moves at constant speed. (This supposes a lack of friction, a condition that could be assumed within the limits of Galileo’s crude measurements.) Now a body moving on a horizontal track is moving at right angles to the force of gravity. Under such conditions, the body’s velocity is not affected by gravity either way. A ball resting on a horizontal plane remains at rest, as anyone can observe. A ball set to moving on a horizontal plane moves at a constant velocity, as Galileo observed. Mathematically, then, it can be stated that the velocity v of a body, in the absence of any external force, is constant k, or: v = k. If k is equal to any number other than zero, the ball is moving at constant velocity. If k is equal to zero, the ball is at rest; thus, rest is a “special case” of constant velocity. Nearly a century later, when Newton systemized the discoveries of Galileo in connection with falling bodies, this finding became the First Law of Motion (also called the principle of inertia). This law can be stated: Every body persists in a state of rest or of uniform motion in a straight line unless compelled by external force to change that state. When a ball rolls down an inclined plane, however, it is under the continuous pull of gravity. Its velocity then, Galileo found, is not constant but increases with time. Galileo’s measurements showed that the velocity increases in proportion to the lapse of time t. In other words, when a body is under the action of constant external force, its velocity, starting at rest, can be expressed as: v = kt. What is the value of k? That, it was easy to find by experiment, depends on the slope of the inclined plane. The more nearly vertical the plane, the more quickly the rolling ball gains velocity and the higher the value of k. The maximum gain in speed comes when the plane is vertical—in other words, when the ball drops freely under the undiluted pull of gravity. The symbol g (for “gravity”) is used where the undiluted force of gravity is acting, so that the velocity of a ball in free fall, starting from rest, was: v = gt. Let us consider the inclined plane in more detail. In the diagram: the length of the inclined plane is AB, while its height at the upper end is AC. The ratio of AC to AB is the sine of the angle x, usually abbreviated as “sin x.” The value of this ratio—that is, of sin x—can be obtained approximately by constructing triangles with particular angles and actually measuring the height and length involved. Or it can be calculated by mathematical techniques to any degree of precision, and the results can be embodied in a table. By using such a table, we can find, for instance, that sin 10° is approximately equal to 0.17365, that sin 45° is approximately equal to 0.70711, and so on. There are two important special cases. Suppose that the “inclined” plane is precisely horizontal. Angle x is then zero, and as the height of the inclined plane is zero, the ratio of its height to its length is also zero. In other words, sin 0° = 0. When the “inclined” plane is precisely vertical, the angle it forms with the ground is a right angle, or 90°. Its height is then exactly equal to its length, so that the ratio of one to the other is just 1. Consequently, sin 90° = 1. Now let us return to the equation showing that the velocity of a ball rolling down an inclined plane is proportional to time: v = kt. It can be shown by experiment that the value of k changes with the sine of the angle so that: k = k' sin x (where k' is used to indicate a constant that is different from k). (As a matter of fact, the role of the sine in connection with the inclined plane was worked out somewhat before Galileo’s time by Simon Stevinus, who also performed the famous experiment of dropping different masses from a height—an experiment traditionally, but wrongly, ascribed to Galileo. Still, if Galileo was not the very first to experiment and measure, he was the first to impress the scientific world, indelibly, with the necessity to experiment and measure, and that is glory enough.) In the case of a completely vertical inclined plane, sin x becomes sin 90°, which is 1, so that in free fall k = k'. It follows that k' is the value of k in free fall under the undiluted pull of gravity, which we have already agreed to symbolize as g. We can substitute g for k' and, for any inclined plane: k = g sin x. The equation for the velocity of a body rolling down an inclined plane is, therefore: v = (g sin x) t. On a horizontal plane with sin x =0°, the equation for velocity becomes: v = 0. This is another way of saying that a ball on a horizontal plane, starting from rest, will remain motionless regardless of the passage of time. An object at rest tends to remain at rest, and so on. That is part of the First Law of Motion, and it follows from the inclined plane equation of velocity. Suppose that a ball does not start from rest but has an initial motion before it begins to fall. Suppose, in other words, you have a ball moving along a horizontal plane at 5 feet per second, and it suddenly finds itself at the upper end of an inclined plane and starts rolling downward. Experiment shows that its velocity thereafter is 5 feet per second greater, at every moment, than it would have been if it had started rolling down the plane from rest. In other words, the equation for the motion of a ball down an inclined plane can be expressed more completely as follows: v = (g sin x) t + V where V is the original starting velocity. If an object starts at rest, then V is equal to 0 and the equation becomes as we had it before: v = (g sin x) t. If we next consider an object with some initial velocity on a horizontal plane, so that angle x is 0°, the equation becomes: v = (g sin 0°) + V or, since sin 0° is 0: v = V. Thus the velocity of such an object remains its initial velocity, regardless of the lapse of time. That is the rest of the First Law of Motion, again derived from observed motion on an inclined plane. The rate at which velocity changes is called acceleration. If, for instance, the velocity (in feet per second) of a ball rolling down an inclined plane is, at the end of successive seconds, 4, 8, 12, 16… then the acceleration is 4 feet per second per second. In a free fall, if we use the equation: v = gt, each second of fall brings an increase in velocity of g feet per second. Therefore, g represents the acceleration due to gravity. The value of g can be determined from inclinedplane experiments. By transposing the inclinedplane equation, we get: g = v / (t sin x). Since v, t, and x can all be measured, g can be calculated, and it turns out to be equal to 32 feet per second per second at the earth’s surface. In free fall under normal gravity at earth’s surface, then, the velocity of fall is related to time thus: v = 32t. This is the solution to Galileo’s original problem—namely, determining the rate of fall of a falling body and the manner in which that rate changes. The next question is: How far does a body fall in a given time? From the equation relating the velocity to time, it is possible to relate distance to time by the process in calculus called integration. It is not necessary to go into that, however, because the equation can be worked out by experiment; and, in essence, Galileo did this. He found that a ball rolling down an inclined plane covers a distance proportional to the square of the time. In other words, doubling the time increases the distance fourfold; tripling it increases the distance ninefold; and so on. For a freely falling body, the equation relating distance d and time is: d = ½gt2 or, since g is equal to 32: d = 16t2. Next, suppose that instead of dropping from rest, an object is thrown horizontally from a position high in the air. Its motion would then be a compound of two motions—a horizontal one and a vertical one. The horizontal motion, involving no force other than the single original impulse (if we disregard wind, air resistance, and so on), is one of constant velocity, in accordance with the First Law of Motion, and the distance the object covers horizontally is proportional to the time elapsed. The vertical motion, however, covers a distance, as I have just explained, that is proportional to the square of the time elapsed. Prior to Galileo, it had been vaguely believed that a projectile such as a cannon ball travels in a straight line until the impulse that drives it is somehow exhausted, after which it falls straight down. Galileo, however, made the great advance of combining the two motions. The combination of these two motions (proportional to time horizontally, and proportional to the square of the time vertically) produces a curve called a parabola. If a body is thrown, not horizontally, but upward or downward, the curve of motion is still a parabola. Such curves of motion, or trajectories, apply, of course, to a projectile such as a cannon ball. The mathematical analysis of trajectories, stemming from Galileo’s work, made it possible to calculate where a cannon ball would fall when fired with a given propulsive force and a given angle of elevation of the cannon. Although people had been throwing objects for fun, to get food, to attack, and to defend, for uncounted thousands of years, it was only due to Galileo that for the first time, thanks to experiment and measurement, there was a science of ballistics. As it happened, then, the very first achievement of modern experimental science proved to have a direct and immediate military application. It also had an important application in theory. The mathematical analysis of combinations of more than one motion answered several objections to the Copernican theory. It showed that an object thrown upward will not be left behind by the moving earth, since the object will have two motions: one imparted to it by the impulse of throwing, and one that it shares along with the moving earth. This analysis also made it reasonable to expect the earth to have two motions at once: rotation about its axis and revolution about the sun—a situation that some of the nonCopernicans insisted was unthinkable. THE SECOND AND THIRD LAWS Isaac Newton extended the Galilean concepts of motion to the heavens and showed that the same set of laws of motion apply to the heavens and the earth alike. He began by considering that the moon might be falling toward the earth in response to the earth’s gravity but never struck the earth’s surface because of the horizontal component of its motion. A projectile fired horizontally, as I said, follows a parabolically curved path downward to intersection with the earth’s surface. But the earth’s surface curves downward, too, since the earth is a sphere. A projectile given a sufficiently rapid horizontal motion might curve downward no faster than the earth’s surface and would therefore eternally circle the earth. Now the moon’s elliptical motion around the earth can be split into horizontal and vertical components. The vertical component is such that, in the space of a second, the moon falls a trifle more than 1/20 inch toward the earth. In that time, it also moves about 3,300 feet in the horizontal direction, just far enough to compensate for the fall and carry it around the earth’s curvature. The question was whether this 1/20inch fall of the moon is caused by the same gravitational attraction that causes an apple, falling from a tree, to drop 16 feet in the first second of its fall. Newton visualized the earth’s gravitational force as spreading out in all directions like a vast, expanding sphere. The surface area A of a sphere is proportional to the square of its radius r: A = 4πr2. He therefore reasoned that the gravitational force, spreading out over the spherical area, must weaken as the square of the radius. The intensity of light and of sound weakens as the square of the distance from the source. Why not the force of gravity as well? The distance from the earth’s center to an apple on its surface is roughly 4,000 miles. The distance from the earth’s center to the moon is roughly 240,000 miles. Since the distance to the moon was 60 times greater than to the apple, the force of the earth’s gravity at the moon must be 602, or 3,600, times weaker than at the apple. Divide 16 feet by 3,600, and you come out with roughly 1/20 of an inch. It seemed clear to Newton that the moon does indeed move in the grip of the earth’s gravity. Newton was persuaded further to consider mass in relation to gravity. Ordinarily, we measure mass as weight. But weight is only the result of the attraction of the earth’s gravitational force. If there were no gravity, an object would be weightless; nevertheless, it would still contain the same amount of matter. Mass, therefore, is independent of weight and should be capable of measurement by a means not involving weight. Suppose you tried to pull an object on a perfectly frictionless surface in a direction horizontal to the earth’s surface, so that there was no resistance from gravity. It would take effort to set the body in motion and to accelerate its motion, because of the body’s inertia. If you measured the applied force accurately—say, by pulling on a spring balance attached to the object—you would see that the force f required to bring about a given acceleration a would be directly proportional to the mass m. If you doubled the mass, it would take double the force. For a given mass, the force required would be directly proportional to the acceleration desired. Mathematically, this is expressed in the equation: f = ma. The equation is known as Newton’s Second Law of Motion. Now, as Galileo had found, the pull of the earth’s gravity accelerates all bodies, heavy or light, at precisely the same rate. (Air resistance may slow the fall of very light bodies; but in a vacuum, a feather will fall as rapidly as a lump of lead, as can easily be demonstrated.) If the Second Law of Motion is to hold, one must conclude that the earth’s gravitational pull on a heavy body must be greater than on a light body, in order to produce the same acceleration. To accelerate a mass that is eight times as great as another, for instance, takes eight times as much force. It follows that the earth’s gravitational pull on any body must be exactly proportional to the mass of that body. (That, in fact, is why mass on the earth’s surface can be measured quite accurately as weight.) Newton evolved a Third Law of Motion, too: “For every action there is an equal and opposite reaction.” This law applies to force. In other words, if the earth pulls at the moon with a certain force, then the moon pulls on the earth with an equal force. If the moon were suddenly doubled in mass, the earth’s gravitational force upon it would also be doubled, in accordance with the Second Law; of course, the moon’s gravitational force on the earth would then have to be doubled in accordance with the Third Law. Similarly, if it were the earth rather than the moon that doubled in mass, it would be the moon’s gravitational force on the earth that would double, according to the Second Law, and the earth’s gravitational force on the moon that would double, in accordance with the Third. If both the earth and the moon were to double in mass, there would be a doubled doubling, each body doubling its gravitational force twice, for a fourfold increase all told. Newton could only conclude, by this sort of reasoning, that the gravitational force between any two bodies in the universe was directly proportional to the product of the masses of the bodies. And, of course, as he had decided earlier, it is inversely proportional to the square of the distance (center to center) between the bodies. This is Newton’s Law of Universal Gravitation. If we let f represent the gravitational force, m1 and m2 the masses of the two bodies concerned, and d the distance between them, then the law can be stated: f = Gm1m2 d2 G is the gravitational constant; the determination of which made it possible to “weigh the earth” (see chapter 4). It was Newton’s surmise that G has a fixed value throughout the universe. As time went on, it was found that new planets, undiscovered in Newton’s time, temper their motions to the requirements of Newton’s law; even double stars incredibly far away dance in time to Newton’s analysis of the universe. All this came from the new quantitative view of the universe pioneered by Galileo. As you see, much of the mathematics involved was really very simple. Those parts of it I have quoted here are highschool algebra. In fact, all that was needed to introduce one of the greatest intellectual revolutions of all time was: 1. A simple set of observations any highschool student of physics might make with a little guidance. 2. A simple set of mathematical generalizations at high school level. 3. The transcendent genius of Galileo and Newton, who had the insight and originality to make these observations and generalizations for the first time. Relativity The laws of motion as worked out by Galileo and Newton depended on the assumption that such a thing as absolute motion exists—that is, motion with reference to something at rest. But everything that we know of in the universe is in motion: the earth, the sun, our galaxy, the systems of galaxies. Where in the universe, then, can we find absolute rest against which to measure absolute motion? THE MICHELSONMORLEY EXPERIMENT It was this line of thought that led to the MichelsonMorley experiment, which in turn led to a scientific revolution as great, in some respects, as that initiated by Galileo (see chapter 8). Here, too, the basic mathematics is rather simple. The experiment was an attempt to detect the absolute motion of the earth against an ether that was supposed to fill all space and to be at rest. The reasoning behind the experiment was as follows. Suppose that a beam of light is sent out in the direction in which the earth is traveling through the ether; and that at a certain distance in that direction, there is a fixed mirror which reflects the light back to the source. Let us symbolize the velocity of light as c, the velocity of the earth through the ether as v, and the distance of the mirror as d. The light starts with the velocity c + v: its own velocity plus the earth’s velocity. (It is traveling with a tail wind, so to speak.) The time it takes to reach the mirror is d divided by (c + v). On the return trip, however, the situation is reversed. The reflected light now is bucking the head wind of the earth’s velocity, and its net velocity is c − v. The time it takes to return to the source is d divided by (c − v). The total time for the round trip is: d + d c + v c − v Combining the terms algebraically, we get: d(c − v) + d(c + v) (c + v) (c − v) = dc − dv + dc + dv = 2dc c2 − v2 c2 − v2 Now suppose that the lightbeam is sent out to a mirror at the same distance in a direction at right angles to the earth’s motion through the ether. The beam of light is aimed from S (the source) to M (the mirror) over the distance d. However, during the time it takes the light to reach the mirror, the earth’s motion has carried the mirror from M to M', so that the actual path traveled by the light beam is from S to M'. This distance we call x, and the distance from M to M' we call y (see diagram above). While the light is moving the distance x at its velocity c, the mirror is moving the distance y at the velocity of the earth’s motion Y. Since both the light and the mirror arrive at M' simultaneously, the distances traveled must be exactly proportional to the respective velocities. Therefore: y = v x c or: y = vx c Now we can solve for the value of x by use of the Pythagorean theorem, which states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. In the right triangle SMM', then, substituting vx / c for y: x2 = d2 + ( vx ) 2 c x2 − ( vx ) 2 = d2 c x2 − v2x2 = d2 c2 c2x2 − v2x2 = d2 c2 (c2x2 − v2)x2 = d2c2 x2 = d2c2 c2 − v2 x = d2c2 √c2 − v2 The light is reflected from the mirror at M to the source, which meanwhile has traveled on to S'. Since the distance S'S" is equal to SS', the distance MS" is equal to x. The total path traveled by the light beam is therefore 2x, or 2dc/√c2 − v2. The time taken by the light beam to cover this distance at its velocity c is: 2dc ÷ c = 2d √c2 − v2 √c2 − v2 How does this compare with the time that light takes for the round trip in the direction of the earth's motion? Let us divide the time in the parallel case (2dc/(c2 − v2) ) by the time in the perpendicular case (2d/√c2 − v2): 2dc ÷ 2d c2 − v2 √c2 − v2 = 2dc × √c2 − v2 = c√c2 − v2 c2 − v2 2d c2 − v2 Now any number divided by its square root gives the same square root as a quotient, that is, x / √x = √x. Conversely, √x / x = 1 / √x. So the last equation simplifies to: c √c2 − v2 This expression can be further simplified if we multiply both the numerator and the denominator by √1 /c2 (which is equal to 1/c): c√1 /c2 √c2 − v2 √1 /c2 = c/c = 1 √c2/c2 − v2/c2 √1 − v2/c2 And there you are. That is the ratio of the time that light should take to travel in the direction of the earth’s motion as compared with the time it should take in the direction perpendicular to the earth’s motion. For any value of y greater than zero, the expression 1/√1 − v2/c2 is greater than 1. Therefore, if the earth is moving through a motionless ether, it should take longer for light to travel in the direction of the earth’s motion than in the perpendicular direction. (In fact, the parallel motion should take the maximum time and the perpendicular motion the minimum time.) Michelson and Morley set up their experiment to try to detect the directional difference in the travel time of light. By trying their beam of light in all directions, and measuring the time of return by their incredibly delicate interferometer, they felt they ought to get differences in apparent velocity. The direction in which they found the velocity of light to be at a minimum should be parallel to the earth’s absolute motion, and the direction in which the velocity would be at a maximum should be perpendicular to the earth’s motion. From the difference in velocity, the amount (as well as the direction) of the earth’s absolute motion could be calculated. They found no differences at all in the velocity of light with changing direction! To put it another way, the velocity of light was always equal to c, regardless of the motion of the source—a clear contradiction of the Newtonian laws of motion. In attempting to measure the absolute motion of the earth, Michelson and Morley had thus managed to cast doubt not only on the existence of the ether, but on the whole concept of absolute rest and absolute motion, and upon the very basis of the Newtonian system of the universe. THE FITZGERALD EQUATION The Irish physicist G. F. FitzGerald conceived a way to save the situation. He suggested that all objects decrease in length in the direction in which they are moving by an amount equal to √1 − v2/c2. Thus: L' = L √1 − v2/c2 where L' is the length of a moving body in the direction of its motion and L is what the length would be if it were at rest. The foreshortening fraction √1 − v2/c2, FitzGerald showed, would just cancel the ratio 1/√1 − v2/c2, which related the maximum and minimum velocities of light in the MichelsonMorley experiment. The ratio would become unity, and the velocity of light would seem to our foreshortened instruments and sense organs to be equal in all directions, regardless of the movement of the source of light through the ether. Under ordinary conditions, the amount of foreshortening is very small. Even if a body were moving at onetenth the velocity of light, or 18,628 miles per second, its length would be foreshortened only slightly, according to the FitzGerald equation. Taking the velocity of light as 1, the equation says: L' = L √(1 − 0.1 ) 2 1 L' = L √1 − 0.01 L' = L √0.99 Thus L' turns out to be approximately equal to 0.995L, a foreshortening of about half of 1 percent. For moving bodies, velocities such as this occur only in the realm of the subatomic particles. The foreshortening of an airplane traveling at 2,000 miles per hour is infinitesimal, as you can calculate for yourself. At what velocity will an object be foreshortened to half its restlength? With L' equal to onehalf L, the FitzGerald equation is: L/2 = L √1 − v2/c2 or, dividing by L: ½ = √1 − v2/c2 Squaring both sides of the equation: ¼ = 1 − v2/c2 v2/c2 = ¾ v = √3c / 4 = 0.866c Since the velocity of light in a vacuum is 186,282 miles per second, the velocity at which an object is foreshortened to half its length is 0.866 times 186,282, or roughly 161,300 miles per second. If a body moves at the speed of light, so that v equals c, the FitzGerald equation becomes: L' = L √1 − c2/c2 = L √0 = 0 At the speed of light, then, length in the direction of motion becomes zero. It would seem, therefore, that no velocity faster than that of light is possible. THE LORENTZ EQUATION In the decade after FitzGerald had advanced his equation, the electron was discovered, and scientists began to examine the properties of tiny charged particles. Lorentz worked out a theory that the mass of a particle with a given charge is inversely proportional to its radius. In other words, the smaller the volume into which a particle crowds its charge, the greater its mass. Now if a particle is foreshortened because of its motion, its radius in the direction of motion is reduced in accordance with the FitzGerald equation. Substituting the symbols R and R' for L and L', we write the equation: R' = R √1 − v2/c2 R'/R = √1 − v2/c2 The mass of a particle is inversely proportional to its radius. Therefore: R' = M R M' where M is the mass of the particle at rest and M' is its mass when in motion. Substituting M/M' for R'/R in the preceding equation, we have: M/M' = √1 − v2/c2 M' = M √1 − v2/c2 The Lorentz equation can be handled just as the FitzGerald equation was. It shows, for instance, that for a particle moving at a velocity of 18,628 miles per second (onetenth the speed of light), the mass M would appear to be 0.5 percent higher than the restmass M. At a velocity of 161,300 miles per second, the apparent mass of the particle would be twice the restmass. Finally, for a particle moving at a velocity equal to that of light, so that v is equal to c, the Lorentz equation becomes: M' = M = M √1 − c2/c2 0 Now as the denominator of any fraction with a fixed numerator becomes smaller and smaller (approaches zero), the value of the fraction itself becomes larger and larger without limit. In other words, from the equation preceding, it would seem that the mass of any object traveling at a velocity approaching that of light becomes infinitely large. Again, the velocity of light would seem to be the maximum possible. All this led Einstein to recast the laws of motion and of gravitation. He considered a universe, in other words, in which the results of the MichelsonMorley experiments were to be expected. Yet even so we are not quite through. Please note that the Lorentz equation assumes some value for M that is greater than zero. This is true for most of the particles with which we are familiar and for all bodies, from atoms to stars, that are made up of such particles. There are, however, neutrinos and antineutrinos for which M, the mass at rest, or restmass, is equal to zero. This is also true of photons. Such particles travel at the speed of light in a vacuum, provided they are indeed in a vacuum. The moment they are formed they begin to move at such a velocity without any measurable period of acceleration. We might wonder how it is possible to speak of the restmass of a photon or a neutrino, if they are never at rest but can only exist while traveling (in the absence of interfering matter) at a constant speed of 186,280 miles per second. The physicists OlexaMyron Bilaniuk and Ennackal Chandy George Sudarshan have therefore suggested that M be spoken of as proper mass. For a particle with mass greater than zero, the proper mass is equal to the mass measured when the particle is at rest relative to the instruments and observer making the measurement. For a particle with mass equal to zero, the proper mass is obtained by indirect reasoning. Bilaniuk and Sudarshan also suggest that all particles with a proper mass of zero be called luxons (from the Latin word for “light”) because they travel at lightspeed, while particles with a proper mass greater than zero be called tardyons because they travel at less than lightspeed, or at subluminal velocities. In 1962, Bilaniuk and Sudarshan began to speculate on the consequences of fasterthanlight velocities (superluminal velocities). Any particle traveling with fasterthanlight velocities would have an imaginary mass. That is, the mass would be some ordinary value multiplied by the square root of −1. Suppose, for instance, a particle were going at twice the speed of light, so that in the Lorentz equation v = 2c. In that case: M' = M √1 − (2c)2/c2 = M = M √1 − 4c2/c2 √–3 This works out to the fact that its mass while in motion would be some proper mass (M) divided by √–3. But √–3 is equal to √3 × √–1 and therefore to 1.74 √−1. The proper mass M is therefore equal to M' × 1.74 × √−1. Since any quantity that includes √−1 is called imaginary, we conclude that particles at superluminal velocities must have imaginary proper masses. Ordinary particles in our ordinary universe always have masses that are zero or positive. An imaginary mass can have no imaginable significance in our universe. Does this mean that fasterthanlight particles cannot exist? Not necessarily. Allowing the existence of imaginary proper masses, we can make such fasterthanlight particles fit all the equations of Einstein’s Special Theory of Relativity. Such particles, however, display an apparently paradoxical property: the more slowly they go, the more energy they contain. This is the precise reverse of the situation in our universe and is perhaps the significance of the imaginary mass. A particle with an imaginary mass speeds up when it meets resistance and slows down when it is pushed ahead by a force. As its energy declines, it moves faster and faster, until when it has zero energy it is moving at infinite speed. As its energy increases, it moves slower and slower until, as its energy approaches the infinite, it slows down to approach the speed of light. Such fasterthanlight particles have been given the name of tachyons from the Greek word for “speed,” by the American physicist Gerald Feinberg. We may imagine, then, the existence of two kinds of universes. One, our own, is the tardyonuniverse, in which all particles go at subluminal velocities and may accelerate to nearly the speed of light as their energy increases. The other is the tachyonuniverse, in which all particles go at superluminal velocities and may decelerate to nearly the speed of light as their energy increases. Between is the infinitely narrow luxon wall in which there are particles that go at exactly luminal velocities. The luxon wall can be considered as being held by both universes in common. If a tachyon is energetic enough and therefore moving slowly enough, it might have sufficient energy and remain in one spot for a long enough period of time to give off a detectable burst of photons. (Tachyons would leave a wake of photons even in a vacuum as a kind of Cerenkov radiation.) Scientists are watching for those bursts, but the chance of happening to have an instrument in just the precise place where one of those (possibly very infrequent) bursts appears for a trillionth of a second or less, is not very great. There are those physicists who maintain that “anything that is not forbidden is compulsory.” In other words, any phenomenon that does not actually break a conservation law must at some time or another take place; or, if tachyons do not actually violate special relativity, they must exist. Nevertheless, even physicists most convinced of this as a kind of necessary “neatness” about the universe, would be rather pleased (and perhaps relieved) to obtain some evidence for the nonforbidden tachyons. So far, they have not been able to. EINSTEIN’S EQUATION One consequence of the Lorentz equation was worked out by Einstein to produce what has become perhaps the most famous scientific equation of all time. The Lorentz equation can be written in the form: M' = M (1 − v2/c2)−½ since in algebraic notation l/√x can be written x−½. This puts the equation into a form that can be expanded (that is, converted into a series of terms) by a formula discovered by, of all people, Newton. The formula is the binomial theorem. The number of terms into which the Lorentz equation can be expanded is infinite, but since each term is smaller than the one before, if you take only the first two terms you are approximately correct, the sum of all the remaining terms being small enough to be neglected. The expansion becomes: > (1 − v2/c2)−½ = 1 + ½v2 … c2 Substituting that in the Lorentz equation, we get: M' = M (1 + ½v2 ) c2 = M + ½Mv2 c2 Now, in classical physics, the expression Y2My2 represents the energy of a moving body. If we let the symbol e stand for energy, the equation above becomes: M' = M + e/c2 or: M' − M = e/c2 The increase in mass due to motion (M' − M) can be represented as m, so: m = e/c2 or: e = mc2 It was this equation that for the first time indicated mass to be a form of energy. Einstein went on to show that the equation applies to all mass, not merely to the increase in mass due to motion. Here again, most of the mathematics involved is only at the highschool level. Yet it presented the world with the beginnings of a view of the universe greater and broader even than that of Newton, and also pointed the way to concrete consequences. It pointed the way, for instance, to the nuclear reactor and the atom bomb. Bibliography A guide to science would be incomplete without a guide to more reading. I am setting down here a brief selection of books. The list is miscellaneous and does not pretend to be a comprehensive collection of the best modern books about science, but I have read most or all of each of them myself and can highly recommend all of them, even my own. General ASIMOV, ISAAC. A Choice of Catastrophes. New York: Simon & Schuster, 1979. ASIMOV, ISAAC. Asimov’s Biographical Encyclopedia of Science and Technology (and rev. ed.). New York: Doubleday, 1982. ASIMOV, ISAAC. Exploring the Earth and the Cosmos. New York: Crown Publishers, 1982. ASIMOV, ISAAC. Measure of the Universe. New York: Harper & Row, 1983. ASIMOV, ISAAC. Understanding Physics (1vol. ed.). New York: Walker, 1984. CABLE, E. J., et al. The Physical Sciences. New York: PrenticeHall, 1959. GAMOW, GEORGE. Matter, Earth, and Sky. New York: PrenticeHall, 1958. HUTCHINGS, EDWARD, JR., ed. Frontiers in Science. New York: Basic Books, 1958. SAGAN, CARL. Cosmos. New York: Random House, 1980. SHAPLEY, HARLOW; RAPPORT, SAMUEL; and WRIGHT, HELEN, eds. A Treasury of Science (4th ed.). New York: Harper, 1958. SLABAUGH, W. H.; and BUTLER, A. B. College Physical Science. New York: PrenticeHall, 1958. WATSON, JANE WERNER. The World of Science. New York: Simon & Schuster, 1958. Chapter 1: What Is Science? BERNAL, J. D. Science in History. New York: Hawthorn Books, 1965. CLAGETT, MARSHALL. Greek Science in Antiquity. New York: AbelardSchuman, 1955. CROMBIE, A. C. Medieval and Early Modem Science (2 vols.). New York: Doubleday, 1959. DAMPIER, SIR WILLIAM CECIL. A History of Science. New York: Cambridge University Press, 1958. DREYER, 1. L. E. A History of Astronomy from Thales to Kepler. New York: Dover Publications, 1953. FORBES, R. J.; and DIJKSTERHUIS, E. J. A History of Science and Technology (2 vols.). Baltimore: Penguin Books, 1963. RONIN, COLIN A. Science: Its History and Development among the World’s Cultures. New York: Facts on File Publications, 1982. TATON, R., ed. History of Science (4 vols.). New York: Basic Books, 196366. Chapter 2: The Universe ABELL, GEORGE O. Exploration of the Universe (4th ed.). Philadelphia: Saunders College Publishing, 1982. ASIMOV, ISAAC. The Collapsing Universe. New York: Walker, 1977. ASIMOV, ISAAC. The Universe (new rev. ed.). New York: Walker, 1980. BURBIDGE, G.; and BURBIDGE, M. QuasiStellar Objects. San Francisco: W. H. Freeman, 1967. FUMMARION, G. G, et al. The Flammarion Book of Astronomy. New York: Simon & Schuster, 1964. GOLDSMITH, DONALD. The Universe. Menlo Park, Calif.: W. A. Benjamin, 1976. HOYLE, FRED. Astronomy. New York: Doubleday, 1962. KIPPENHAHN, RUDOLF. 100 Billion Suns. New York: Basic Books, 1983. LEY, WILLY. Watchers of the Skies. New York: Viking Press, 1966. MCLAUGHLIN, DEAN B. Introduction to Astronomy. Boston: Houghton MifHin, 1961. MITTON, SIMON, edinchief, The Cambridge Encyclopaedia of Astronomy. New York: Crown, 1977. SHKLOVSKII, l. S.; and SAGAN, CARL. Intelligent Life in the Universe. San Francisco: HoldenDay, 1966. SMITH, F. GRAHAM. Radio Astronomy. Baltimore: Penguin Books, 1960. STRUVE, OTTO; and ZEBERGS, VELTA. Astronomy of the 20th Century. New York: Macmillan, 1962. Chapter 3: The Solar System BEATTY, J. KELLY; O’LEARY, BRIAN; and CHAIKIN, ANDREW, eds. The New Solar System. Cambridge, Mass.: Sky Publishing, and Cambridge, England: Cambridge University Press, 1981. RYAN, PETER; and PESEK, LUDEK. Solar System. New York: Viking Press, 1978. Chapter 4: The Earth ADAMS, FRANKDAWSON. The Birth and Development of the Geological Sciences. New York: Dover Publications, 1938. ASIMOV, ISAAC. The Ends of the Earth. New York: Weybright & Talley, 1975. ASIMOV, ISAAC. Exploring the Earth and the Cosmos. New York: Crown Publishers, 1982. BURTON, MAURICE. Life in the Deep. New York: Roy Publishers, 1958. GAMOW, GEORGE. A Planet Called Earth. New York: Viking Press, 1963. GILLULY, J.; WATERS, A. G.; and WOODFORD, A. O. Principles of Geology. San Francisco: W. H. Freeman, 1958. JACKSON, DONALD DALE. Underground Worlds. Alexandria, Va.: TimeLife Books, 1982. KUENEN, P. H. Realms of Water. New York: John Wiley, 1963. MASON, BRIAN. Principles of Geochemistry. New York: John Wiley, 1958. MOORE, RUTH. The Earth We Live On. New York: Alfred A. Knopf, 1956. SCIENTIFIC AMERICAN, eds. The Planet Earth. New York: Simon & Schuster, 1957. SMITH, DAVID G., edinchief. The Cambridge Encyclopaedia of Earth Sciences. New York: Crown, 1981. SULLIVAN, WALTER. Continents in Motion. New York: McGrawHill, 1974. TIMELIFE BOOKS, eds. Volcano. Alexandria, Va.: TimeLife Books, 1982. Chapter 5: The Atmosphere BATES, D. R., ed. The Earth and Its Atmosphere. New York: Basic Books, 1957. GLASSTONE, SAMUEL. Sourcebook on the Space Sciences. New York: Van Nostrand, 1965. LEY, WILLY. Rockets, Missiles, and Space Travel. New York: Viking Press, 1957. LOEBSACK, THEO. Our Atmosphere. New York: New American Library, 1961. NEWELL, HOMER E., JR. Window in the Sky. New York: McGrawHill, 1959. NININGER, H. H. Out of the Sky. New York: Dover Publications, 1952. ORR, CLYDE, JR. Between Earth and Space. New York: Collier Books, 1961. YOUNG, LOUISE B. Earth’s Aura. New York: Alfred A. Knopf, 1977. Chapter 6: The Elements ALEXANDER, W.; and STREET, A. Metals in the Service of Man. New York: Penguin Books, 1954. ASIMOV, ISAAC. A Short History of Chemistry. New York: Doubleday, 1965. ASIMOV, ISAAC. The Noble Gases. New York: Basic Books, 1966. DAVIS, HELEN MILES. The Chemical Elements. Boston: Ballantine Books, 1959. HOLDEN, ALAN; and SINGER, PHYLIS. Crystals and Crystal Growing. New York: Doubleday, 1960. IHDE, AARON J. The Development of Modern Chemistry. New York: Harper & Row, 1964. LEICESTER, HENRY M. The Historical Background of Chemistry. New York: John Wiley, 1956. PAULING, LINUS. College Chemistry (3rd ed.), San Francisco: W. H. Freeman, 1964. PRYDE, Lucy T. Environmental Chemistry: An Introduction. Menlo Park, Calif.: Cummings Publishing, 1973. WEEKS, MARYE.; and LEICESTER, H. M. Discovery of the Elements (7th ed.). Easton, Pa.: Journal of Chemical Education, 1968. Chapter 7: The Particles ALFREN, HANNES. Worlds Antiworlds. San Francisco: W. H. Freeman, 1966. ASIMOV, ISAAC. The Neutrino. New York: Doubleday, 1966. FEINBERG, GERALD. What Is the World Made Of? Garden City, N.Y.: Anchor Press/Doubleday, 1977. FORD, KENNETH W. The World of Elementary Particles. New York: Blaisdell Publishing, 1963. FRIEDLANDER, G.; KENNEDY, J. W.; and MILLER, J. M. Nuclear and Radiochemistry (2nd ed.), New York: John Wiley, 1964. GARDNER, MARTIN. The Ambidextrous Universe (2nd rev. ed.), New York: Charles Scribner’s, 1979. GLASSTONE, SAMUEL. Sourcebook on Atomic Energy (3rd ed.). Princeton: Van Nostrand, 1967. HUGHES, DONALD J. The Neutron Story. New York: Doubleday, 1959. MASSEY, SIR HARRIE. The New Age in Physics. New York: Harper, 1960. PARK, DAVID. Contemporary Physics. New York: Harcourt, Brace & World, 1964. WEINBERG, STEVEN. The Discovery of Subatomic Particles. New York: Scientific library, 1983. Chapter 8: The Waves BENT, H. A. The Second Law. New York: Oxford University Press, 1965. BERGMANN, P. G. The Riddle of Gravitation. New York: Charles Scribner’s, 1968. BLACK, N. H.; and LITTLE, E. P. An Introductory Course in College Physics. New York: Macmillan, 1957. FREEMAN, IRA M. Physics Made Simple. New York: Made Simple Books, 1954. GARDNER, MARTIN. Relativity for the Million. New York: Macmillan, 1962. HOFFMAN, BANESH. The Strange Story of the Quantum. New York: Dover Publications, 1959. ROUSE, ROBERT S.; and SMITH, ROBERT O. Energy: Resource, Slave Pollutant. New York: Macmillan, 1975. SCHWARTZ, JACOB T. Relativity in Illustrations. New York: New York University Press, 1962. SHAMOS, MORRIS H. Great Experiments in Physics. New York: Henry Holt, 1959. Chapter 9: The Machine BITTER, FRANCIS. Magnets. New York: Doubleday, 1959. CLARKE, DONALD, ed. The Encyclopedia of How It Works. New York: A & W Publishers, 1977. DE CAMP, L. SPRAGUE. The Ancient Engineers. New York: Doubleday, 1963. KOCK, W. E. Lasers and Holography. New York: Doubleday, 1969. LARSEN, EGON. Transport. New York: Roy Publishers, 1959. LEE, E. W. Magnetism. Baltimore: Penguin Books, 1963. LENGYEL, BELA A. Lasers. New York: John Wiley, 1962. NEAL, HARRYEDWARD. Communication. New York: Julius Messner, 1960. PIERCE, JOHN R. Electrons, Waves and Messages. New York: Doubleday, 1956. PIERCE, JOHN R. Symbols, Signals and Noise. New York: Harper, 1961. SINGER, CHARLES; HOLMYARD, E. J.; and HALL. A. R., eds. A History of Technology (5 vols.). New York: Oxford University Press. 1954— TAYLOR, F. SHERWOOD. A History of Industrial Chemistry. New York: AbelardSchuman, 1957. UPTON, MONROE. Electronics for Everyone (2nd rev. ed.). New York: New American Library, 1959. USHER, ABBOTT PAYSON. A History of Mechanical Inventions. Boston: Beacon Press, 1959. WARSCHAUER, DOUGLAS M. Semiconductors and Transistors. New York: McGrawHill, 1959. Chapter 10: The Reactor ALEXANDER, PETER. Atomic Radiation and Life. New York: Penguin Books. 1957. BISHOP, AMASA S. Project Sherwood. Reading. Mass: AddisonWesley. 1958. FOWLER. JOHN M. Fallout: A Study of Superbombs, Strontium 90, and Survival. New York: Basic Books. 1960. JUKES, JOHN. ManMade Sun. New York: AbelardSchuman, 1959. JUNGK, ROBERT. Brighter Than a Thousand Suns. New York: Harcourt. Brace. 1958. PURCELL. JOHN. The BestKept Secret. New York: Vanguard Press. 1963. RIEDMAN, SARAH R. Men and Women behind the Atom. New York: AbelardSchuman, 1958. SCIENTIFIC AMERICAN, eds. Atomic Power. New York: Simon & Schuster. 1955. WILSON. ROBERT R.; and LITTAUER, R. Accelerators. New York: Doubleday. 1960. Chapter 11: The Molecule FIESER, L. F.; and FIESER, M. Organic Chemistry. Boston: D. C. Heath. 1956. GIBBS, F. W. Organic Chemistry Today. Baltimore: Penguin Books, 1961. HUTTON. KENNETH. Chemistry. New York: Penguin Books. 1957. PAULING, LINUS. The Nature of the Chemical Bond (3rd ed.). Ithaca. N.Y.: Cornell University Press, 1960. PAULING, LINUS; and HAYWARD, R. The Architecture of Molecules. San Francisco: W. H. Freeman, 1964. Chapter 12: The Proteins ASIMOV, ISAAC. Photosynthesis. New York: Basic Books, 1969. BALJ)WIN, ERNEST. Dynamic Aspects of Biochemistry (5th ed.). New York: Cambridge University Press, 1967. BALDWIN, ERNEST. The Nature of Biochemistry. New York: Cambridge University Press,) 962. HARPER, HAROLD A. Review of Physiological Chemistry (8th ed.). Los Altos, Calif.: Lange Medical Publications, 1961. KAMEN, MARTIN D. Isotopic Tracers in Biology. New York: Academic Press, 1957. KARLSON, P. Introduction to Modern Biochemistry. New York: Academic Press, 1963. LEHNINGER, ALBERT L. Biochemistry (2nd ed.), New York: Worth Publishers, 1975. LEHNINGER, ALBERT L. Bioenergetics. New York: Benjamin Company, 1965. Chapter 13: The Cell ANFINSEN, CHRISTIAN B. The Molecular Basis of Evolution. New York: John Wiley, 1959. ASIMOV, ISAAC. Extraterrestrial Civilizations. New York: Crown Publishers, 1979. ASIMOV, ISAAC. The Genetic Code. New York: Orion Press, 1962. ASIMOV, ISAAC. A Short History of Biology. New York: Doubleday, 1964. BUTLER, J. A. V. Inside the Living Cell. New York: Basic Books, 1959. CARR, DONALD E. The Deadly Feast of Life. Garden City, N.Y.: Doubleday, 1971. FEINBERG, GERALD; and SHAPIRO, ROBERT. Life Beyond Earth. New York: William Morrow, 1980. HARTMAN, P. E.; and SUSKIND, S. R. Gene Action. Englewood Cliffs, N.J.: PrenticeHall, 1965. HUGHES, ARTHUR. A History of Cytology. New York: AbelardSchuman, 1959. NEEL, J. V.; and SCHULL, W. J. Human Heredity. Chicago: University of Chicago Press, 1954. OPARIN, A. I. The Origin of Life on the Earth. New York: Academic Press, 1957. SINGER, CHARLES. A Short History of Anatomy and Physiology from the Greeks to Harvey. New York: Dover Publications, 1957. SULLIVAN, WALTER. We Are Not Alone. New York: McGrawHill, 1964. TAYLOR, GORDON R. The Science of Life. New York: McGrawHill, 1963. WALKER, KENNETH. Human Physiology. New York: Penguin Books, 1956. Chapter 14: The Microorganism BURNET, F. M. Viruses and Man (2nd ed.), Baltimore: Penguin Books, 1955. CURTIS, HELENA. The Viruses. Garden City, N.Y.: Natural History Press, 1965. DE KRUIF, PAUL. Microbe Hunters. New York: Harcourt, Brace, 1932. DUBOS, RENE. Louis Pasteur. Boston: Little, Brown, 1950. LUDOVICI, L. J. The World of the Microscope. New York: G. P. Putnam, 1959. McGRADY, PAT. The Savage Cell. New York: Basic Books, 1964. RIEDMAN, SARAH R. Shots without Guns. Chicago: Rand, McNally, 1960. SINGER, CHARLES; and UNDERWOOD, E. ASHWORTH. A Short History of Medicine (2nd ed.). New York: Oxford University Press, 1962. SMITH, KENNETH M. Beyond the Microscope. Baltimore: Penguin Books, 1957. WILLIAMS, GREER. Virus Hunters. New York: Alfred A. Knopf, 1959. ZINSSER, HANS. Rats, Lice and History. Boston: Little, Brown, 1935. Chapter 15: The Body ASIMOV, ISAAC. The Human Body. Boston: Houghton Mifflin, 1963. CARLSON, ANTON J.; and JOHNSON, VICTOR. The Machinery of the Body. Chicago: University of Chicago Press, 1953. CHANEY, MARGARETS. Nutrition. Boston: Houghton Mifflin, 1954. MCCOLLUM, ELMER VERNER. A History of Nutrition. Boston: Houghton Mifflin, 1957. SMITH, ANTHONY. The Body. London: George Allen & Unwin, 1968. TANNAHILL, REAY. Food in History. New York: Stein & Day, 1973. WILLIAMS, ROGER J. Nutrition in a Nutshell. New York: Doubleday, 1962. WILLIAMS, SUE RODWELL. Essentials of Nutrition and Diet Therapy. St. Louis: C. V. Mosby, 1974. Chapter 16: The Species ASIMOV, ISAAC. Wellsprings of Life. New York: AbelardSchuman, 1960. BOULE, M.; and VALLOIS, H. V. Fossil Men. New York: Dryden Press, 1957. CALVIN, MELVIN. Chemical Evolution. New York: Oxford University Press, 1969. CAMPBELL, BERNARD. Human Evolution (2nd ed.). Chicago: Aldine Publishing, 1974. CARRINGTON, RICHARD. A Biography of the Sea. New York: Basic Books, 1960. DARWIN, FRANCIS, ed. The Life and Letters of Charles Darwin (2 vols.). New York: Basic Books, 1959. DE BELL, G. The Environmental Handbook. New York: Ballantine Books, 1970. EHRLICH, PAUL; and EHRLICH, ANN. Extinction. New York: Random House, 1981. HANRAHAN, JAMES S.; and BUSHNELL, DAVID. Space Biology. New York: Basic Books, 1960. HARRISON, R. J. Man, the Peculiar Animal. New York: Penguin Books, 1958. HEPPENHEIMER, T. A. Colonies in Space. Harrisburg, Pa.: Stackpole Books, 1977. HOWELLS, WILLIAM. Mankind in the Making. New York: Doubleday, 1959. HUXLEY, T. H. Man’s Place in Nature. Ann Arbor: University of Michigan Press, 1959. LEWONTIN, RICHARD. Human Diversity. New York: Scientific American Library, 1982. MEDAWAR, P. B. The Future of Man. New York: Basic Books, 1960. MILNE, L. J.; and MILNE, M. J. The Biotic World and Man. New York: PrenticeHall, 1958. MONTAGU, ASHLEY. The Science of Man. New York: Odyssey Press, 1964. MOORE, RUTH. Man, Time, and Fossils (2nd ed.). New York: Alfred A. Knopf, 1963. O’NEILL, GERARD K. 2081. New York: Simon & Schuster, 1981. ROMER, A. S. Man and the Vertebrates (2 vols.). New York: Penguin Books, 1954. ROSTAND, JEAN. Can Man Be Modified? New York: Basic Books, 1959. SAX, KARL. Standing Room Only. Boston: Beacon Press, 1955. SIMPSON, GEORGE G.; PRITENDRIGH, C. S.; and TIFFANY, L. H. Life: An Introduction to College Biology (2nd ed.). New York: Harcourt, Brace, 1965. TAYLOR, GORDON RATTRAY. The Doomsday Book. New York: World Publishing, 1970. TINBERGEN, NIKO. Curious Naturalists. New York: Basic Books, 1960. UBBELOHDE, A. R. Man and Energy. Baltimore: Penguin Books, 1963. Chapter 17: The Mind ASIMOV, ISAAC. The Human Brain. Boston: Houghton Mifflin, 1964. BERKELEY, EDMUND C. Symbolic Logic and Intelligent Machines. New York: Reinhold Publishing, 1959. EVANS, CHRISTOPHER. The Making of the Micro, A History of the Computer. New York: Van Nostrand Reinhold, 1981. FACKLAM, MARGERY; and FACKLAM, HOWARD. The Brain. New York: Harcourt, Brace, Jovanovich, 1982. GARDNER, HOWARD. Frames of Mind. New York: Basic Books, 1983. GOULD, STEPHEN JAY. The Mismeasure of Man. New York: W. W. Norton, 1981. JONES, ERNEST. The Life and Work of Sigmund Freud (3 vols.). New York: Basic Books, 1957. LASSEK, A. M. The Human Brain. Springfield, Ill.: Charles C Thomas, 1957. MENNINGER, KARL. Theory of Psychoanalytic Technique. New York: Basic Books, 1958. MURPHY, GARDNER. Human Potentialities. New York: Basic Books, 1958. RAWCLIFFE, D. H. Illusions and Delusions of the Supernatural and Occult. New York: Dover Publications, 1959. SANDERS, DONALD H. Computers Today. New York: McGrawHill, 1983. SCIENTIFIC AMERICAN, eds. Automatic Control. New York: Simon & Schuster, 1955. SCOTT, JOHN PAUL. Animal Behavior. Chicago: University of Chicago Press, 1957. THOMPSON, CLARA; and MULLAHY, PATRICK. Psychoanalysis: Evolution and Development. New York: Grove Press, 1950. Appendix: Mathematics in Science COURANT, RICHARD; and ROBBINS, HERBERT. What Is Mathematics? New York: Oxford University Press, 1941. DANTZIG, TOBIAS. Number, the Language of Science. New York: Macmillan, 1954. FELIX, LUCIENNE. The Modern Aspect of Mathematics. New York: Basic Books, 1960. FREUND, JOHN E. A Modern Introduction to Mathematics. New York: PrenticeHall, 1956. KLINE, MORRIS. Mathematics and the Physical World. New York: Thomas Y. Crowell, 1959. KLINE, MORRIS. Mathematics in Western Culture. New York: Oxford University Press, 1953. NEWMAN, JAMES R. The World of Mathematics (4 vols.), New York: Simon & Schuster, 1956. STEIN, SHERMAN K. Mathematics, the ManMade Universe. San Francisco: W. H. Freeman, 1963. VALENS, EVANS G. The Number of Things. New York: Dutton, 1964. Chapter 1 What is Science? Almost in the beginning was curiosity. Curiosity, the overwhelming desire to know, is not characteristic of dead matter. Nor does it seem to be characteristic of some forms of living organism, which, for that very reason, we can scarcely bring ourselves to consider alive. A tree does not display curiosity about its environment in any way we can recognize; nor does a sponge or an oyster. The wind, the rain, the ocean currents bring them what is needful, and from it they take what they can. If the chance of events is such as to bring them fire, poison, predators, or parasites, they die as stoically and as undemonstratively as they lived . Early in the scheme of life, however, independent motion was developed by some organisms. It meant a tremendous advance in their control of the environment. A moving organism no longer had to wait in stolid rigidity for food to come its way, but went out after it. Thus, adventure entered the world—and curiosity. The individual that hesitated in the competitive hunt for food, that was overly conservative in its investigation, starved. Early on, curiosity concerning the environment was enforced as the price of survival. The onecelled paramecium, moving about in a searching way, cannot have conscious volitions and desires in the sense that we do, but it has a drive, even if only a “simple” physicalchemical one, which causes it to behave as if it were investigating its surroundings for food or safety, or both. And this “act of curiosity” is what we most easily recognize as being inseparable from the kind of life that is most akin to ours. As organisms grew more intricate, their sense organs multiplied and became both more complex and more delicate. More messages of greater variety were received from and about the external environment. At the same time, there developed (whether as cause or effect we cannot tell), an increasing complexity of the nervous system, the living instrument that interprets and stores the data collected by the sense organs. THE DESIRE TO KNOW There comes a point where the capacity to receive, store, and interpret messages from the outside world may outrun sheer necessity. An organism may be sated with food, and there may, at the moment, be no danger in sight. What does it do then? It might lapse into an oysterlike stupor. But the higher organisms at least still show a strong instinct to explore the environment. Idle curiosity, we may call it. Yet, though we may sneer at it, we judge intelligence by it. The dog, in moments of leisure, will sniff idly here and there, pricking up its ears at sounds we cannot hear; and so we judge it to be more intelligent than the cat, which in its moments of leisure grooms itself or quietly and luxuriously stretches out and falls asleep. The more advanced the brain, the greater the drive to explore, the greater the “curiosity surplus.” The monkey is a byword for curiosity. Its busy little brain must and will be kept going on whatever is handy. And in this respect, as in many others, man is a supermonkey. The human brain is the most magnificently organized lump of matter in the known universe, and its capacity to receive, organize, and store data is far in excess of the ordinary requirements of life. It has been estimated that, in a lifetime, a human being can learn up to 15 trillion items of information. It is to this excess that we owe our ability to be afflicted by that supremely painful disease, boredom. A human being, forced into a situation where one has no opportunity to utilize one’s brain except for minimal survival, will gradually experience a variety of unpleasant symptoms, up to and including serious mental disorganization. The fact is that the normal human being has an intense and overwhelming curiosity. If one lacks the opportunity to satisfy it in immediately useful ways, one will satisfy it in other ways—even regrettable ways to which we have attached admonitions such as “Curiosity killed the cat,” and “Mind your own business.” The overriding power of curiosity, even with harm as the penalty, is reflected in the myths and legends of the human race. The Greeks had the tale of Pandora and her box. Pandora, the first woman, was given a box that she was forbidden to open. Quickly and naturally enough she opened it and found it full of the spirits of disease, famine, hate, and all kinds of evil—which escaped and have plagued the world ever since. In the Biblical story of the temptation of Eve, it seems fairly certain (to me, at any rate) that the serpent had the world’s easiest job and might have saved his words: Eve’s curiosity would have driven her to taste the forbidden fruit even without external temptation. If you are of a mind to interpret the Bible allegorically, you may think of the serpent as simply the representation of this inner compulsion. In the conventional cartoon picturing Eve standing under the tree with the forbidden fruit in her hand, the serpent coiled around the branch might be labeled “Curiosity.” If curiosity can, like any other human drive, be put to ignoble use—the prying invasion of privacy that has given the word its cheap and unpleasant connotation—it nevertheless remains one of the noblest properties of the human mind. For its simplest definition is “the desire to know.” This desire finds its first expression in answers to the practical needs of human life: how best to plant and cultivate crops, how best to fashion bows and arrows, how best to weave clothing—in short, the “applied arts.” But after these comparatively limited skills have been mastered, or the practical needs fulfilled, what then? Inevitably the desire to know leads on to less limited and more complex activities. It seems clear that the “fine arts” (designed to satisfy inchoate and boundless and spiritual needs) were born in the agony of boredom. To be sure, one can easily find more mundane uses and excuses for the fine arts. Paintings and statuettes were used as fertility charms and as religious symbols, for instance. But one cannot help suspecting that the objects existed first and the use second. To say that the fine arts arose out of a sense of the beautiful may also be putting the cart before the horse. Once the fine arts were developed, their extension and refinement in the direction of beauty would have followed inevitably, but even if this had not happened, the fine arts would have developed nevertheless. Surely the fine arts antedate any possible need or use for them, other than the elementary need to occupy the mind as fully as possible. Not only does the production of a work of fine art occupy the mind satisfactorily; the contemplation or appreciation of the work supplies a similar service to the audience. A great work of art is great precisely because it offers a stimulation that cannot readily be found elsewhere. It contains enough data of sufficient complexity to cajole the brain into exerting itself past the usual needs; and, unless a person is hopelessly ruined by routine or stultification, that exertion is pleasant. But if the practice of the fine arts is a satisfactory solution to the problem of leisure, it has this disadvantage: it requires, in addition to an active and creative mind, physical dexterity. It is just as interesting to pursue mental activities that involve only the mind, without the supplement of manual skill. And, of course, such activity is available. It is the pursuit of knowledge itself, not in order to do something with it but for its own sake. Thus, the desire to know seems to lead into successive realms of greater etherealization and more efficient occupation of the mind—from knowledge of accomplishing the useful, to knowledge of accomplishing the esthetic, to “pure” knowledge. Knowledge for itself alone seeks answers to such questions as How high is the sky? or, Why does a stone fall? This is sheer curiosity—curiosity at its idlest and therefore perhaps at its most peremptory. After all, it serves no apparent purpose to know how high the sky is or why the stone falls. The lofty sky does not interfere with the ordinary business of life; and, as for the stone, knowing why it falls does not help us to dodge it more skillfully or soften the blow if it happens to hit us. Yet there have always been people who ask such apparently useless questions and try to answer them out of the sheer desire to know—out of the absolute necessity of keeping the brain working. The obvious method of dealing with such questions is to make up an esthetically satisfying answer: one that has sufficient analogies to what is already known to be comprehensible and plausible. The expression “to make up” is rather bald and unromantic. The ancients liked to think of the process of discovery as the inspiration of the muses or as a revelation from heaven. In any case, whether it was inspiration, revelation, or the kind of creative thinking that goes into storytelling, the explanations depended heavily on analogy. The lightning bolt is destructive and terrifying but appears, after all, to be hurled like a weapon and does the damage of a hurled weapon—a fantastically violent one. Such a weapon must have a wielder similarly enlarged in scale, and so the thunderbolt becomes the hammer of Thor or the flashing spear of Zeus. The morethannormal weapon is wielded by a morethannormal man. Thus a myth is born. The forces of nature are personified and become gods. The myths react on one another, are built up and improved by generations of myth tellers until the original point may be obscured. Some myths may degenerate into pretty stories (or ribald ones), whereas others may gain an ethical content important enough to make them meaningful within the framework of a major religion. Just as art may be fine or applied, so may mythology. Myths may be maintained for their esthetic charm or bent to the physical uses of human beings. For instance, the earliest farmers were intensely concerned with the phenomenon of rain and why it fell capriciously. The fertilizing rain falling from the heavens on the earth presented an obvious analogy to the sex act; and, by personifying both heaven and earth, human beings found an easy explanation of the release or the withholding of the rains. The earth goddess, or the sky god, was either pleased or offended, as the case might be. Once this myth was accepted, farmers had a plausible basis for the art of bringing rain—namely, appeasing the god by appropriate rites. These rites might well be orgiastic in nature—an attempt to influence heaven and earth by example. THE GREEKS The Greek myths are among the prettiest and most sophisticated in our Western literary and cultural heritage. But it was the Greeks also who, in due course, introduced the opposite way of looking at the universe—that is, as something impersonal and inanimate. To the myth makers, every aspect of nature was essentially human in its unpredictability. However mighty and majestic the personification, however superhuman the powers of Zeus, or Ishtar or Isis or Marduk or Odin, they were also—like mere humans—frivolous, whimsical, emotional, capable of outrageous behavior for petty reasons, susceptible to childish bribes. As long as the universe was in the control of such arbitrary and unpredictable deities, there was no hope of understanding it, only the shallow hope of appeasing it. But in the new view of the later Greek thinkers, the universe was a machine governed by inflexible laws. The Greek philosophers now devoted themselves to the exciting intellectual exercise of trying to discover just what the laws of nature might be. The first to do so, according to Greek tradition, was Thales of Miletus, about 600 B.C. He was saddled with an almost impossible number of discoveries by later Greek writers, and it may be that he first brought the gathered Babylonian knowledge to the Greek world. His most spectacular achievement is supposed to have been predicting an eclipse for 585 B.C.—which actually occurred. In engaging in this intellectual exercise, the Greeks assumed, of course, that nature would play fair; that, if attacked in the proper manner, it would yield its secrets and would not change position or attitude in midplay. (Over two thousand years later, Albert Einstein expressed this feeling when he said, “God may be subtle, but He is not malicious.”) There was also the feeling that the natural laws, when found, would be comprehensible. This Greek optimism has never entirely left the human race. With confidence in the fair play of nature, human beings needed to work out an orderly system for learning how to determine the underlying laws from the observed data. To progress from one point to another by established rules of argument is to use “reason.” A reasoner may use “intuition” to guide the search for answers, but must rely on sound logic to test particular theories. To take a simple example: if brandy and water, whiskey and water, vodka and water, and rum and water are all intoxicating beverages, one may jump to the conclusion that the intoxicating factor must be the ingredient these drinks hold in common—namely, water. There is something wrong with this reasoning, but the fault in the logic is not immediately obvious; and in more subtle cases, the error may be hard indeed to discover. The tracking down of errors or fallacies in reasoning has amused thinkers from Greek times to the present. And we owe the earliest foundations of systematic logic to Aristotle of Stagira who in the fourth century B.C. first summarized the rules of rigorous reasoning. The essentials of the intellectual game of managainstnature are three. First, you must collect observations about some facet of nature. Second, you must organize these observations into an orderly array. (The organization does not alter them but merely makes them easier to handle. This is plain in the game of bridge, for instance, where arranging the hand in suits and order of value does not change the cards or show the best course of play, but makes it easier to arrive at the logical plays.) Third, you must derive from your orderly array of observations some principle that summarizes the observations. For instance, we may observe that marble sinks in water, wood floats, iron sinks, a feather floats, mercury sinks, olive oil floats, and so on. If we put all the sinkable objects in one list and all the floatable ones in another and look for a characteristic that differentiates all the objects in one group from all in the other, we will conclude: Objects denser than water sink in water, and objects less dense than water, float. The Greeks named their new manner of studying the universe philosophia (“philosophy”), meaning “love of knowledge” or, in free translation, “the desire to know.” GEOMETRY AND MATHEMATICS The Greeks achieved their most brilliant successes in geometry. These successes can be attributed mainly to the development of two techniques: abstraction and generalization. Here is an example. Egyptian land surveyors had found a practical way to form a right angle: they divided a rope into twelve equal parts and made a triangle in which three parts formed one side, four parts another, and five parts the third side—the right angle lay where the threeunit side joined the four— unit side. There is no record of how the Egyptians discovered this method, and apparently their interest went no further than to make use of it. But the curious Greeks went on to investigate why such a triangle should contain a right angle. In the course of their analysis, they grasped the point that the physical construction itself was only incidental; it did not matter whether the triangle was made of rope or linen or wooden slats. It was simply a property of “straight lines” meeting at angles. In conceiving of ideal straight lines, which are independent of any physical visualization and can exist only in imagination, the Greeks originated the method called abstraction—stripping away nonessentials and considering only those properties necessary to the solution of the problem. The Greek geometers made another advance by seeking general solutions for classes of problems, instead of treating individual problems separately. For instance, one might have discovered by trial that a right angle appeared in triangles, not only with sides 3, 4, and 5 feet long, but also in triangles of 5, 12, and 13 feet and of 7, 24, and 25 feet. But these were merely numbers without meaning. Could some common property be found that would describe all right triangles? By careful reasoning, the Greeks showed that a triangle is a right triangle if, and only if, the lengths of the sides have the relation x² + y² = z², z being the length of the longest side. The right angle lies where the sides of length x and y meet. Thus for the triangle with sides of 3,4, and 5 feet, squaring the sides gives 9 + 16 = 25; similarly, squaring the sides of 5, 12, and 13 gives 25 + 144 = 169; and squaring 7, 24, and 25 gives 49 + 576 = 625. These are only three cases out of an infinity of possible ones and, as such, trivial. What intrigued the Greeks was the discovery of a proof that the relation must hold in all cases. And they pursued geometry as an elegant means of discovering and formulating such generalizations. Various Greek mathematicians contributed proofs of relationships existing among the lines and points of geometric figures. The one involving the right triangle was reputedly worked out by Pythagoras of Samos about 525 B.C. and is still called the Pythagorean theorem in his honor. About 300 B.C., Euclid gathered the mathematical theorems known in his time and arranged them in a reasonable order, such that each theorem could be proved through the use of theorems proved previously. Naturally, this system eventually worked back to something unprovable: if each theorem had to be proved with the help of one already proved, how could one prove theorem no. 1? The solution was to begin with a statement of truths so obvious and acceptable to all as to need no proof. Such a statement is called an “axiom.” Euclid managed to reduce the accepted axioms of the day to a few simple statements. From these axioms alone, he built an intricate and majestic system of “Euclidean geometry.” Never before was so much constructed so well from so little, and Euclid’s reward is that his textbook has remained in use, with but minor modification, for more than 2,000 years. THE DEDUCTIVE PROCESS Working out a body of knowledge as the inevitable consequence of a set of axioms (“deduction”) is an attractive game. The Greeks fell in love with it, thanks to the success of their geometry—sufficiently in love with it to commit two serious errors. First, they came to consider deduction as the only respectable means of attaining knowledge. They were well aware that, for some kinds of knowledge, deduction was inadequate; for instance, the distance from Corinth to Athens could not be deduced from abstract principles but had to be measured. The Greeks were willing to look at nature when necessary; however, they were always ashamed of the necessity and considered that the highest type of knowledge was that arrived at by cerebration. They tended to undervalue knowledge directly involved with everyday life. There is a story that a student of Plato, receiving mathematical instruction from the master, finally asked impatiently, “But what is the use of all this?” Plato, deeply offended, called a slave and, ordering him to give the student a coin, said, “Now you need not feel your instruction has been entirely to no purpose.” With that, the student was expelled. There is a wellworn belief that this lofty view arose from the Greek’s slavebased culture, in which all practical matters were relegated to the slaves. Perhaps so, but I incline to the view that the Greeks felt that philosophy was II sport, an intellectual game. Many people regard the amateur in sports as a gentleman socially superior to the professional who makes his living at it. In line with this concept of purity, we take almost ridiculous precautions to make sure that the contestants in the Olympic games are free of any taint of professionalism. The Greek rationalization for the “cult of uselessness” may similarly have been based on a feeling that to allow mundane knowledge (such liS the distance from Athens to Corinth) to intrude on abstract thought was 10 allow imperfection to enter the Eden of true philosophy. Whatever the rationalization, the Greek thinkers were severely limited by their attitude. Greece was not barren of practical contributions to civilization, but even its great engineer, Archimedes of Syracuse, refused to write about his practical inventions and discoveries; to maintain his amateur status, he broadcast only his achievements in pure mathematics. And lack of interest in earthly things—in invention, in experiment, in the study of nature—was but one of the factors that put bounds on Greek thought. The Greeks’ emphasis on purely abstract and formal study—indeed, their very success in geometry—led them into a second great error and, eventually, to a dead end. Seduced by the success of the axioms in developing a system of geometry, the Greeks came to think of the axioms as “absolute truths” and to suppose that other branches of knowledge could be developed from similar “absolute truths.” Thus in astronomy they eventually took as selfevident axioms the notions that (l) the earth was motionless and the center of the universe, and (2) whereas the earth was corrupt and imperfect, the heavens were eternal, changeless, and perfect. Since the Greeks considered the circle the perfect curve, and since the heavens were perfect, it followed that all the heavenly bodies must move in circles around the earth. In time, their observations (arising from navigation and calendar making) showed that the planets do not move in perfectly simple circles, and so the Greeks were forced to allow planets to move in ever more complicated combinations of circles, which, about 150 A.D., were formulated as an uncomfortably complex system by Claudius Ptolemaeus (Ptolemy) at Alexandria. Similarly, Aristotle worked up fanciful theories of motion from “selfevident” axioms, such as the proposition that the speed of an object’s fall was proportional to its weight. (Anyone could see that a stone fell faster than a feather.) Now this worship of deduction from selfevident axioms was bound to wind up at the edge of a precipice, with no place to go. After the Greeks had worked out all the implications of the axioms, further important discoveries in mathematics or astronomy seemed out of the question. Philosophic knowledge appeared complete and perfect; and for nearly 2,000 years after the Golden Age of Greece, when questions involving the material universe arose, there was a tendency to settle matters to the satisfaction of all by saying, “Aristotle says…” or, “Euclid says…” THE RENAISSANCE AND COPERNICUS Having solved the problems of mathematics and astronomy, the Greeks turned to more subtle and challenging fields of knowledge, One was the human soul. Plato was far more interested in such questions as What is justice? or, What is virtue? than in why rain falls or how the planets move, As the supreme moral philosopher of Greece, he superseded Aristotle, the supreme natural philosopher. The Greek thinkers of the Roman period found themselves drawn more and more to the subtle delights of moral philosophy and away from the apparent sterility of natural philosophy, The last development in ancient philosophy was an exceedingly mystical “neoPlatonism” formulated by Plotinus about 250 A.D. Christianity, with its emphasis on the nature of God and His relation to man, introduced an entirely new dimension into the subject matter of moral philosophy that increased its apparent superiority as an intellectual pursuit over natural philosophy. From 200 A.D, to 1200 A.D., Europeans concerned themselves almost exclusively with moral philosophy, in particular with theology. Natural philosophy was nearly forgotten. The Arabs, however, managed to preserve Aristotle and Ptolemy through the Middle Ages; and, from them, Greek natural philosophy eventually filtered hack to Western Europe. By 1200, Aristotle had been rediscovered. Further infusions came from the dying Byzantine empire, which was the last area in Europe to maintain a continuous cultural tradition from the great days of Greece. The first and most natural consequence of the rediscovery of Aristotle was the application of his system of logic and reason to theology, About 1250, the Italian theologian Thomas Aquinas established the system called “Thomism,” based on Aristotelian principles, which still represents the basic theology of the Roman Catholic Church. But Europeans soon began to apply the revival uf Greek thought to secular fields as well, Because the leaders of the Renaissance shifted emphasis from matters concerning God to the works of humanity, they were called “humanists,” and the study of literature, art, and history is still referred to as the “humanities.” To the Greek natural philosophy, the Renaissance thinkers brought a fresh outlook, for the old views no longer entirely satisfied. In 1543, the Polish astronomer Nicolaus Copernicus published a book that went so far as to reject a basic axiom of astronomy: he proposed that the sun, not the earth, be considered the center of the universe, (He retained the notion of circular orbits for the earth and other planets, however.) This new axiom allowed a much simpler explanation of the observed motions of heavenly bodies, Yet the Copernican axiom of a moving earth was far less “selfevident” than the Greek axiom of a motionless earth, and so it is not surprising that it took more than half a century for the Copernican theory to be accepted. In a sense, the Copernican system itself was not a crucial change, Copernicus had merely switched axioms; and Aristarchus of Samos had already anticipated this switch to the sun as the center 2,000 years earlier. I do not mean to say that the changing of an axiom is a minor matter. When mathematicians of the nineteenth century challenged Euclid’s axioms and developed “nonEuclidean geometries” based on other assumptions, they influenced thought on many matters in a most profound way: today the very history and form of the universe are thought to conform to a nonEuclidean geometry rather than the “commonsense” geometry of Euclid. But the revolution initiated by Copernicus entailed not just a shift in axioms but eventually involved a whole new approach to nature, This revolution was carried through in the person of the Italian Galileo Galilei toward the end of the sixteenth century. EXPERIMENTATION AND INDUCTION The Greeks, by and large, had been satisfied to accept the “obvious” facts of nature as starting points for their reasoning. It is not on record that Aristotle ever dropped two stones of different weight to test his assumption that the speed of fall is proportional to an object’s weight. To the Greeks, experimentation seemed irrelevant. It interfered with and detracted from the beauty of pure deduction. Besides, if an experiment disagreed with a deduction, could one be certain that the experiment was correct? Was it likely that the imperfect world of reality would agree completely with the perfect world of abstract ideas; and if it did not, ought one to adjust the perfect to the demands of the imperfect? To test a perfect theory with imperfect instruments did not impress the Greek philosophers as a valid way to gain knowledge. Experimentation began to become philosophically respectable in Europe with the support of such philosophers as Roger Bacon (a contemporary of Thomas Aquinas) and his later namesake Francis Bacon. But it was Galileo who overthrew the Greek view and effected the revolution. He was a convincing logician and a genius as a publicist. He described his experiments and his point of view so clearly and so dramatically that he won over the European learned community. And they accepted his methods along with his results. According to the bestknown story about him, Galileo tested Aristotle’s theories of falling bodies by asking the question of nature in such a way that all Europe could hear the answer. He is supposed to have climbed to the top of the Leaning Tower of Pisa and dropped a 10pound sphere and a lpound sphere simultaneously; the thump of the two balls hitting the ground in the same split second killed Aristotelian physics. Actually Galileo probably did not perform this particular experiment, but the story is so typical of his dramatic methods that it is no wonder it has been widely believed through the centuries. Galileo undeniably did roll balls down inclined planes and measured the distance that they traveled in given times. He was the first to conduct time experiments and to use measurement in a systematic way. His revolution consisted in elevating “induction” above deduction as the logical method of science. Instead of building conclusions on an assumed set of generalizations, the inductive method starts with observations and derives generalizations (axioms, if you will) from them. Of course, even the Greeks obtained their axioms from observation; Euclid’s axiom that a straight line is the shortest distance between two points was an intuitive judgment based on experience. But whereas the Greek philosopher minimized the role played by induction, the modern scientist looks on induction as the essential process of gaining knowledge, the only way of justifying generalizations. Moreover, the scientist realizes that no generalization can be allowed to stand unless it is repeatedly tested by newer and still newer experiments—the continuing test of further induction. The present general viewpoint is just the reverse of the Greeks. Far from considering the real world an imperfect representation of ideal truth, we consider generalizations to be only imperfect representatives of the real world. No amount of inductive testing can render a generalization completely and absolutely valid. Even though billions of observations tend to bear out a generalization, a single observation that contradicts or is inconsistent with it must force its modification. And no matter how many times a theory meets its tests successfully, there can be no certainty that it will not be overthrown by the next observation. This, then, is a cornerstone of modern natural philosophy. It makes no claim of attaining ultimate truth. In fact, the phrase “ultimate truth” becomes meaningless, because there is no way in which enough observations can be made to make truth certain and, therefore, “ultimate.” The Greek philosophers recognized no such limitation. Moreover, they saw no difficulty in applying exactly the same method of reasoning to the question What is justice? as to the question What is matter? Modern science, on the other hand, makes a sharp distinction between the two types of question. The inductive method cannot make generalizations about what it cannot observe; and, since the nature of the human soul, for example, is not observable by any direct means yet known, this subject lies outside the realm of the inductive method. The victory of modern science did not become complete until it established one more essential principle—namely, free and cooperative communication among all scientists. Although this necessity seems obvious now, it was not obvious to the philosophers of ancient and medieval times. The Pythagoreans of ancient Greece were a secret society who kept their mathematical discover ies to themselves. The alchemists of the Middle Ages deliberately obscured their writings to keep their socalled findings within as small an inner circle as possible. In the sixteenth century, the Italian mathematician Niccolo Tartaglia, who discovered a method of solving cubic equations, saw nothing wrong in attempting to keep it a secret. When Geronimo Cardano, a fellow mathematician, wormed the secret out of Tartaglia on the promise of confidentiality and published it, Tartaglia naturally was outraged; but aside from Cardano’s trickery in breaking his promise, he was certainly correct in his reply that such a discovery had to be published. Nowadays no scientific discovery is reckoned a discovery if it is kept secret. The English chemist Robert Boyle, a century after Tartaglia and Cardano, stressed the importance of publishing all scientific observations in full detail. A new observation or discovery, moreover, is no longer considered valid, even after publication, until at least one other investigator has repeated the observation and “confirmed” it. Science is the product not of individuals but of a “scientific community.” One of the first groups (and certainly the most famous) to represent such a scientific community was the Royal Society of London for Improving Natural Knowledge, usually called simply the “Royal Society.” It grew out of Informal meetings, beginning about 1645, of a group of gentlemen interested in the new scientific methods originated by Galileo. In 1660, the society was formally chartered by King Charles II. The members of the Royal Society met and discussed their findings openly, wrote letters describing them in English rather than Latin, and pursued their experiments with vigor and vivacity. Nevertheless, through most of the seventeenth century, they remained in a defensive position. The attitude of many of their learned contemporaries might be expressed by a cartoon, after the modern fashion, showing the lofty shades of Pythagoras, Euclid, and Aristotle staring down haughtily at children playing with marbles and labeled “Royal Society.” All this was changed by the work of Isaac Newton, who became a member of the society. From the observations and conclusions of Galileo, of the Danish astronomer Tycho Brahe, and of the German astronomer Johannes Kepler, who figured out the elliptical nature of the orbits of the planets, Newton arrived by induction at his three simple laws of motion and his great fundamental generalization—the law of universal gravitation. (Nevertheless, when he published his findings, he used geometry and the Greek method of deductive explanation.) The educated world was so impressed with this discovery that Newton was idolized, almost deified, in his own lifetime. This majestic new universe, built upon a few simple assumptions derived from inductive processes, now made the Greek philosophers look like boys playing with marbles. The revolution that Galileo had initiated at the beginning of the seventeenth century was triumphantly completed by Newton at the century’s end. MODERN SCIENCE It would be pleasant to be able to say that science and human beings have lived happily ever since. But the truth is that the real difficulties of both were only beginning. As long as science remained deductive, natural philosophy could be part of the general culture of all educated men (women, alas, being rarely educated until recent times). But inductive science became an immense labor—of observation, learning, and analysis. It was no longer a game for amateurs. And the complexity of science grew with each decade. During the century after Newton, it was still possible for a man of unusual attainments to master all fields of scientific knowledge. But, by 1800, this had become entirely impracticable. As time went on, it was increasingly necessary for a scientist to limit himself to a portion of the field with which he was intensively concerned. Specialization was forced on science by its own inexorable growth. And with each generation of scientists, specialization has grown more and more intense. The publications of scientists concerning their individual work have never been so copious—and so unreadable for anyone but their fellow specialists. This has been a great handicap to science itself, for basic advances in scientific knowledge often spring from the crossfertilization of knowledge from differ ent specialties. Even more ominous, science has increasingly lost touch with nonscientists. Under such circumstances, scientists come to be regarded al most as magicians—feared rather than admired. And the impression that science is incomprehensible magic, to be understood only by a chosen few who are suspiciously different from ordinary mankind, is bound to turn many youngsters away from science. Since the Second World War, strong feelings of outright hostility toward science were to be found among the young—even among the educated young in the colleges. Our industrialized society is based on the scientific discoveries of the last two centuries, and our society finds it is plagued by undesirable side effects of its very success. Improved medical techniques have brought about a runaway increase in population; chemical industries and the internalcombustion engine arc fouling our water and our air; the demand for materials and for energy is depleting and destroying the earth’s crust. And this is all too easily blamed on “science” and “scientists” by those who do not quite understand that while knowledge can create problems, it is not through ignorance that we can solve them. Yet modern science need not be so complete a mystery to nonscientists. Much could be accomplished toward bridging the gap if scientists accepted the responsibility of communication—explaining their own fields of work as simply and to as many as possible—and if nonscientists, for their part, accepted the responsibility of listening. To gain a satisfactory appreciation of the developments in a field of science, it is not essential to have a total understand ing of the science. After all, no one feels that one must be capable of writing a great work of literature in order to appreciate Shakespeare. To listen to a Beethoven symphony with pleasure does not require the listener to be capable of composing an equivalent symphony. By the same token, one can appreciate and take pleasure in the achievements of science even though one does not oneself have a bent for creative work in science. But what, you may ask, would be accomplished? The first answer is that no one can really feel at home in the modern world and judge the nature of its problems—and the possible solutions to those problems—unless one has some intelligent notion of what science is up to. Furthermore, initiation into the magnificent world of science brings great esthetic satisfaction, inspiration to youth, fulfillment of the desire to know, and a deeper appreciation of the wonderful potentialities and achievements of the human mind. It is to provide such initiation that I have undertaken to write this book. Chapter 2 The Universe The Size of the Universe There is nothing about the sky that makes it look particularly distant to a casual observer. Young children have no great trouble in accepting the fantasy that “the cow jumped over the moon”—or “he jumped so high, he touched the sky.” The ancient Greeks, in their myth telling stage, saw nothing ludicrous in allowing the sky to rest on the shoulders of Atlas, Of course, Atlas might have been astronomically tall, but another myth suggests otherwise, Atlas was enlisted by Hercules to help him with the eleventh of his famous twelve labors—fetching the golden apples (oranges) of the Hesperides (“the far west”—Spain?), While Atlas went off to fetch the apples, Hercules stood on a mountain and held up the sky, Granted that Hercules was a large specimen, he was nevertheless not a giant. It follows then that the early Greeks took quite calmly to the notion that the sky cleared the mountaintops by only a few feet II is natural to suppose, to begin with, that the sky is simply a hard canopy in which the shining heavenly bodies are set like diamonds. (Thus the Bible refers to the sky as the “firmament,” from the same Latin root as the word firm.) As early as the sixth to the fourth centuries B.C., Greek astronomers realized that there must be more than one canopy, For while the “fixed” stars moved around Earth in a body, apparently without changing their relative positions, this was not true of the sun, the moon, and five bright starlike objects (Mercury, Venus, Mars, Jupiter, and Saturn): in fact, each moved in a separate path. These seven bodies were called planets (from a Greek word meaning “wanderer”), and it seemed obvious that they could not be attached to the vault of the stars. The Greeks assumed that each planet was set in an invisible spherical vault of its own, and that the vaults were nested one above the other, the nearest belonging to the planet that moved fastest. The quickest motion belonged to the moon, which circled the sky in about twentyseven and a third days. Beyond it lay in order (so thought the Greeks) Mercury, Venus, our sun, Mars, Jupiter, and Saturn. EARLY MEASUREMENTS The first scientific measurement of any cosmic distance came about 240 B.C. Eratosthenes of Cyrene, the head of the Library at Alexandria, then the most advanced scientific institution in the world, pondered the fact that on 21 June, when the noonday sun was exactly overhead at the city of Syene in Egypt, it was not quite at the zenith at noon in Alexandria, 500 miles north of Syene. Eratosthenes decided that the explanation must be that the surface of the earth curved away from the sun. From the length of the shadow in Alexandria at noon on the solstice, straightforward geometry could yield the amount by which the earth’s surface curved in the 500mile distance from Syene to Alexandria. From that one could calculate the circumference and the diameter of the earth, assuming it to be spherical in shape—a fact Greek astronomers of the day were ready to accept (figure 2.1). Eratosthenes worked out the answer (in Greek units), and, as nearly as we can judge, his figures in our units came out at about 8,000 miles for the diameter and 25,000 miles for the circumference of the earth. These figures, as it happens, are just about right. Unfortunately, this accurate value for the size of the earth did not prevail. About 100 B.C. another Greek astronomer, Posidonius of Aparnea, repeated Eratosthenes’ work but reached the conclusion that the earth was but 18,000 miles in circumference. It was the smaller figure that was accepted throughout ancient and medieval times. Columbus accepted the smaller figure and thought that a 3,000mile westward voyage would take him to Asia. Had he known the earth’s true size, he might not have ventured. It was not until 152123, when Magellan’s fleet (or rather the one remaining ship of the fleet) finally circumnavigated the earth, that Eratosthenes’ correct value was finally established. In terms of the earth’s diameter, Hipparchus of Nicaea, about 150 B.C, worked out the distance to the moon. He used a method that had been suggested a century earlier by Aristarchus of Samos, the most daring of all Greek astronomers. The Greeks had already surmised that eclipses of the moon were caused by the earth coming between the sun and the moon. Aristarchus saw that the curve of the earth’s shadow as it crossed the moon should indicate the relative sizes of the earth and the moon. On this basis, geometric methods offered a way to calculate how far distant the moon was in terms of the diameter of the earth. Hipparchus, repeating this work, calculated that the moon’s distance from the earth was 30 times the earth’s diameter. If Eratosthenes’ figure of 8,000 miles for the earth’s diameter was correct, the moon must be about 240,000 miles from the earth. This figure again happens to be about correct. Figure 2.1. Eratosthenes measured the size of the earth from its curvature. At noon, on 21 June, the sun is directly overhead at Syene, which lies on the Tropic of Cancer. But, at the same time, the sun’s rays, seen from farther north in Alexandria, fall at an angle of 7.S degrees to the vertical and therefore cast a shadow. Knowing the distance between the two cities and the length of the shadow in Alexandria, Eratosthenes made his calculations. But finding the moon’s distance was as far as Greek astronomy managed to carry the problem of the size of the universe—at least correctly. Aristarchus had made a heroic attempt to determine the distance to the sun. The geometric method he used was absolutely correct in theory, but it involved measuring such small differences in angles that, without the use of modern instruments, he was unable to get a good value. He decided that the sun was about 20 times as far as the moon (actually it is about 400 times). Although his figures were wrong, Aristarchus nevertheless did deduce from them that the sun must be at least 7 times larger than the earth. Pointing out the illogic of supposing that the large sun circled the small earth, he decided that the earth must be revolving around the sun. Unfortunately, no one listened to him. Later astronomers, beginning with Hipparchus and ending with Claudius Ptolemy, worked out all the heavenly movements on the basis of a motionless earth at the center of the universe, with the moon 240,000 miles away and other objects an undetermined distance farther. This scheme held sway until 1543, when Nicolaus Copernicus published his book, which returned to the viewpoint of Aristarchus and forever dethroned Earth’s position as the center of the universe. MEASURING THE SOLAR SYSTEM The mere fact that the sun was placed at the center of the solar system did not in itself help determine the distance of the planets. Copernicus adopted the Greek value for the distance of the moon, but he had no notion of the distance of the sun. It was not until 1650 that a Belgian astronomer, Godefroy Wendelin, repeated Aristarchus’ observations with improved instruments and decided that the sun was not 20 times the moon’s distance (5 million miles) but 240 times (60 million miles). The estimate was still too small, but it was much more accurate than before. In 1609, meanwhile, the German astronomer Johannes Kepler had opened the way to accurate distance determinations with his discovery that the orbits of the planets were ellipses, not circles. For the first time, it became possible to calculate planetary orbits accurately and, furthermore, to plot a scale map of the solar system: that is, the relative distances and orbit shapes of all the known planets in the system could be plotted. Thus, if the distance between any two planets in the system could be determined in miles, all the other distances could be calculated at once. The distance to the sun, therefore, need not be calculated directly, as Aristarchus and Wendelin had attempted to do. The determination of the distance of any nearer body, such as Mars or Venus, outside the Earthmoon system would do. One method by which cosmic distances can be calculated involves the use of parallax. It is easy to illustrate what this term means. Hold your finger about 3 inches before your eyes and look at it first with just the left eye and then with just the right. Your finger will shift position against the background, because you have changed your point of view. Now if you repeat this procedure with your finger farther away—say, at arm’s length—the finger again will s