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The Great Physicists from Galileo to Einstein
By George Gamow
Dover Publications, Inc.Copyright © 1961 George Gamow
All rights reserved.
The Dawn of Physics
IT IS VERY difficult to trace the origin of the science of physics, just as it is difficult to trace the origin of many great rivers. A few tiny springs bubbling from under the green foliage of tropical vegetation, or trickling out from beneath the moss-covered rocks in the barren northern country; a few small brooks and creeks running gaily down the mountain slopes and uniting to form rivulets which in turn unite to form streams big enough to deserve the name "river." The rivers grow broader and broader, being strengthened by numerous tributaries, and finally develop into mighty flows—be it the Mississippi or the Volga, the Nile or the Amazon—carrying their waters into the ocean.
The springs which gave birth to the great river of physical science were scattered all over the surface of the earth inhabited by Homo sapiens, i.e., thinking man. It seems, however, that most of them were concentrated on the southern tip of the Balkan peninsula inhabited by the people now known as "ancient Greeks"; or, at least, it seems so to us who inherited the culture of these early "intellectuals." It is interesting to notice that whereas other ancient nations, such as Babylonia and Egypt, contributed a great deal to the early development of mathematics and astronomy, they were completely sterile in the development of physics. The possible explanation of that deficiency as compared with Greek science is that Babylonian and Egyptian gods lived high up among the stars, while the gods of the ancient Greeks lived at the elevation of only about 10,000 feet, at the top of Mount Olympus, and so were much closer to down-to-earth problems. According to a legend the term "magnetism" originated from the name of a Greek shepherd, [TEXT NOT REPRODUCIBLE IN ASCII], who was surprised to notice that the tip of his iron-shod staff was attracted by a stone (magnetic iron ore) lying along the roadside. Similarly the term "electricity" comes from the Greek word [TEXT NOT REPRODUCIBLE IN ASCII], for amber, maybe because some other Hellenic shepherd, trying to polish a piece of amber by rubbing it against the coat of one of his sheep, noticed that it acquired a mysterious property to attract loose pieces of wood.
PYTHAGOREAN LAW OF STRINGS
Whereas these legendary discoveries would hardly stand their ground in any legal priority suit, the discovery of the Greek philosopher Pythagoras, who lived in the middle of the 6th century B.C., is well documented. Persuaded that the world is governed by number, he was investigating the relation between the lengths of the strings in the musical instruments which produce harmonic combinations of sound. For this purpose he used the so-called monochord, i.e., a single string which can be varied in length and subjected to different tensions caused by a suspended weight. Using the same weight, and varying the length of the string, he found that the pairs of harmonic tones were obtained when the string's lengths stood in simple numerical relations. The length ratio 2: 1 corresponded to what was known as "octave," the ratio 3: 2 to a "fifth," and the ratio 4: 3 to a "fourth." This discovery was probably the first mathematical formulation of a physical law, and can be well considered as the first step in the development of what is now known as theoretical physics. In modern physical terminology we may reformulate Pythagoras' discovery by saying that the frequency, i.e., the number of vibrations per second, of a given string subjected to a given tension is inversely proportional to its length. Thus, if the second string (Fig. I-1b) is half as long as the first one (Fig. I-1a), its vibration frequency will be twice as high. If the two string lengths stand in the ratio of 3: 2 or 4: 3, their vibration frequencies will stand in the ratios 2: 3 or 3: 4 (Fig. I-1c,d). Since the part of the human brain which receives the nerve signals from the ear is built in such a way that simple frequency ratios, as 3: 4, give "pleasure" whereas the complex ones, as 137: 171, "displeasure" (it is for the brain physiologists of the future to explain that fact!), the length of strings giving perfect accord must stand in simple numerical ratios.
Pythagoras tried to go one step further by suggesting that, since the motion of planets "must be harmonious," their distances from the earth must stand in the same ratios as the lengths of the strings (under equal tension) which produce the seven basic tones produced by the lyre, the national Greek musical instrument. This proposal was probably the first example of what is now often called "pathological physical theory."
DEMOCRITUS, THE ATOMIST
Another important physical theory, which in modern terminology could be called "a theory without any experimental foundation" but which turned out to be a "dream that came true," was proposed by another ancient Greek philosopher, Democritus, who lived, thought, and taught around the year 400 B.C. Democritus conceived the idea that all material bodies are aggregates of innumerable particles so small as to be unnoticeable by the human eye. He called these particles atoms, or indivisibles ([TEXT NOT REPRODUCIBLE IN ASCII]) in Greek, because he believed that they represented the ultimate stage of the division of material bodies into smaller and smaller parts. He believed that there are four different kinds of atoms: the atoms of stone, dry and heavy; the atoms of water, heavy and wet; the atoms of air, cold and light; and the atoms of fire, slippery and hot. By a combination of these four different kinds of atoms all known materials were supposed to be made. The soil was a combination of stone and water atoms. A plant growing from the soil under the influence of sun rays consisted of the stone and water atoms from the soil and the fire atoms from the sun. That is why dry wood logs which have lost all their water atoms would burn, liberating fire atoms (flame) and leaving behind stone atoms (ashes). When certain kinds of stones (metallic ores) were put in the flame, stone atoms united with the fire atoms producing the substances known as metals. Cheap metals like iron contained very small amounts of fire atoms and therefore looked rather dull. Gold had the maximum amount of fire atoms and thus was brilliant and valuable. Consequently, if one could add more fire atoms to the plain iron, one should be able to make precious gold!
A student who would tell all that in his introductory chemistry examination would certainly get an F grade. But, although these particular examples of the nature of chemical transformation were certainly wrong, the fundamental idea of obtaining almost unlimited numbers of different substances by a combination of only a few basic chemical elements was undoubtedly correct and now represents the foundation of chemistry today. It took, however, twenty-two centuries from the time of Democritus to the time of Dalton to make things right.
One of the giants of the ancient Greek world was a man named Aristotle, who became famous on two counts: first, because he was a real genius; second, because he was a tutor and later a protégé of Alexander the Great of Macedonia. He was born in 384 B.C. in the Greek colonial town Stagira on the Aegean Sea to a former court physician of the Macedonian royal family. At the age of 17 he came to Athens and joined the philosophical school of Plato, remaining an ardent student of Plato until his (Plato's) death in 347 B.C. After this, there followed a period of extensive travel until he finally returned to Athens and founded a philosophical school known as Peripatetic which was held at the Lyceum. Most of the works of Aristotle preserved until our time are the "treatises" which probably represent the texts of lectures which he delivered in the Lyceum on various branches of science. There are the treatises on logic and psychology of which he was the inventor, the treatises on political science, and on various biological problems, especially on classification of plants and animals. But, whereas in all these fields Aristotle made tremendous contributions which influenced human thought for two millennia after his death, probably his most important contribution in the field of physics was the invention of the name of that science which was derived by him from the Greek word [TEXT NOT REPRODUCIBLE IN ASCII] which means nature. The shortcomings of Aristotelian philosophy in the study of physical phenomena should be ascribed to the fact that Aristotle's great mind was not mathematically inclined as were the minds of many other ancient Greek philosophers. His ideas concerning the motion of terrestrial objects and celestial bodies did probably more harm than service to the progress of science. At the rebirth of scientific thinking during the Renaissance, people like Galileo had to struggle hard to throw off the yoke of Aristotelian philosophy, which was generally considered at that time as "the last word in knowledge," making further inquiries into the nature of things quite unnecessary.
ARCHIMEDES' LAW OF LEVER
Another great Greek of the ancient period who lived about a century after the time of Aristotle was Archimedes (Fig. 1-2), father of the science of mechanics, who lived in Syracuse, the capital of a Greek colony in Sicily. Being the son of an astronomer, he early acquired mathematical interests and skill, and in the course of his life made a number of very important contributions to different branches of mathematics. His most important work in pure mathematics was the discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder; in fact, in accordance with his desire, his tomb was marked with a sphere inscribed into a cylinder. In his book entitled [TEXT NOT REPRODUCIBLE IN ASCII] (Psammites or Sandreckoner), he develops the method of writing very large numbers by ascribing to each figure in the row different "order" according to its position, and applying it to the problem of writing down the number of grains of sand contained within a sphere of the size of the earth.
In his famous book On the Equilibrium of Planes (in two volumes) Archimedes develops the laws of lever and discusses the problem of finding the center of gravity of any given body. For a modern reader, the style of Archimedes' writings will sound rather heavy and long-winded, resembling in many respects the style of Euclid's books on geometry. In fact, at the time of Archimedes, Greek mathematics was almost entirely limited to geometry, algebra being invented much later by the Arabs. Thus, various proofs in the field of mechanics and other branches of physics were carried out by considering geometrical figures rather than, as we do now, writing algebraic equations. As in Euclid's Geometry, over which many a reader had sweated in his or her school days, Archimedes formulates the basic laws of "statics" (i.e., the study of equilibrium) by formulating the "postulates" and deriving from them a number of "propositions." We reproduce here the beginning of the first volume.
1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium, but incline towards the weight which is at the greater distance.
2. If, when weights at certain distances are in equilibrium, something is added to one of the weights, they are not in equilibrium, but incline towards the weight to which the addition is made.
3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken.
4. If equal and similar plane figures coincide if applied to one another, their centers of gravity similarly coincide.
5. If figures are unequal but similar, the centers of gravity will be similarly situated. By points similarly situated in relation to similar figures I mean points such that, if straight lines be drawn through them to the equal angles, they make equal angles with the corresponding sides.
6. If two weights at certain distances be in equilibrium, other two weights equal to them will be also in equilibrium at the same distances. [Ain't it clear?]
7. In any figure whose perimeter is concave in the same direction the center of gravity must be within the figure.
These postulates are followed by fifteen propositions derived from them by straightforward logical arguments. We give here the first five propositions, omitting their proof, and quote the exact proofs of the sixth proposition since it involves the fundamental law of lever.
1. Weights that balance at equal distances are equal....
2. Unequal weights at equal distances will not balance but will incline towards greater weight....
3. Unequal weights will (or rather may) balance at unequal distances, the greater weight being at the lesser distance....
4. If two equal weights have not the same center of gravity, the center of gravity of both taken together is the middle point of the line joining their centers of gravity....
5. If three equal weights have their centers of gravity on the straight line at equal distances, the center of gravity of the system will coincide with that of the middle weight....
We now turn to the proof of the sixth proposition, modernizing it slightly for the sake of the reader:
6. Two weights balance at distances reciprocally proportional to the weights.
Suppose the weights A, B are commeasurable, and the points represent their centers of gravity (Fig. 1-3a):
Draw through αβ a straight line so divided at γ that
A : B = [bar.βγ] : [bar.γα]
We have to prove that γ is the center of gravity of the two taken together. Since A and B are commeasurable, so are [bar.βγ] and [bar.γα]. Let [bar.μν] be a common measure of [bar.βγ] and [bar.γα]. Make [bar.βδ] and [bar.βε] each equal to [bar.αγ], and [bar.αζ] equal to [bar.γβ]. Then [bar.αδ] = [bar.γβ] since [bar.βδ] = [bar.γα]. Therefore, [bar.ζδ] is bisected at α as [bar.δε] is bisected at β. Thus ζδ and [bar.δε] must each contain [bar.μν] even number of times.
Take a weight Ω such that Ω is contained as many times in A as [bar.μν] is contained in [bar.ζδ], whence:
A : Ω = [bar.ζδ] : [bar.μν]
But, B : A = [bar.γα] : [bar.βγ] = [bar.δε] : [bar.ζδ]
Hence, ex aequalis B: Ω = δε: [bar.μν] or Ω is contained in B as many times as [bar.μν] is contained in [bar.δε]. Thus Ω is a common measure of A and B.
Divide [bar.ζδ] and [bar.δε] into parts each equal to μν and A and B into parts each equal to Ω. The parts of A will therefore be equal in number to those of [bar.ζδ] and parts of B equal in number to those of [bar.δε]. Place one of the parts of A in the middle point of each of the parts [bar.μν] of [bar.ζδ], and one of the parts of B in the middle point of each of the parts [bar.μν] of [bar.δε] (Fig. 1-4b).
Then the center of gravity of the parts of A placed on equal distances of [bar.ζδ] will be at α the middle point of [bar.ζγ], and the center of gravity of the parts of B placed at equal distances along [bar.δε] will be at B the middle point of [bar.δε]. But the system formed by the parts Ω of A and B together is a system of equal weights even in number at places at equal distances along [bar.ζε]. And, since [bar.ζα] = [bar.γβ] and [bar.αγ] = [bar.βε], [bar.ζγ] = [bar.γε], so that γ is the middle point of [bar.ζε]. Therefore γ is the center of gravity of the system ranged along [bar.ζε]. Therefore, A acting in a and B acting in β balance about the point γ.
This proposition is followed by proposition seven in which the same statement is proved when the weights A and B are incommeasurable.
Excerpted from The Great Physicists from Galileo to Einstein by George Gamow. Copyright © 1961 George Gamow. Excerpted by permission of Dover Publications, Inc..
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