When a surveyor cannot measure a certain distance directly, say the width of a river or the height of a tree, either by counting paces or by laying out lengths of a standard measure, he can use the properties of triangles to determine the distance. This idea, which goes back to Thales of Miletus (624–565 BCE), is one of the first in the history of physics and mathematics.
Miletus was, in the sixth century BCE, a Greek port on an island off the west coast of Asia Minor, now modern Turkey, and Thales was an early philosopher or “lover of wisdom.” Thales traveled far from Miletus in his search for wisdom—to Babylon and across the eastern Mediterranean to Egypt. Egypt, even in the sixth century BCE, was known for its ancient civilization. After all, the great pyramids were built in approximately 2500 BCE. What Thales found in Egypt, if not wisdom, was the practical knowledge of local Egyptian land measurers or geometers who were skilled at measuring the position, size, and shape of agricultural plots, presumably so they would not be lost or confused with neighboring plots after an episode of Nile flooding.
How did Thales convert the practical knowledge of the Egyptian land measurers to the universally applicable principles of geometrical surveying we now call triangulation? He may have been helped by a diagram such as that of the tree and the rod (figure 1). When an upright rod casts a shadow equal in length to its height, one can expect that every other upright object will also cast a shadow equal in length to its height. Thus, when the height of the rod is equal to the length of its shadow, the unknown height of the tree is equal to the length of its easily measured shadow.
Such an inference requires that different rays of sunlight are all straight and parallel to one another. This supposition also allows us to use a rod and its shadow to determine the height of the tree at any time of day since the sides of all similarly shaped triangles stand in the same relation to one another. For whenever an upright object casts a shadow, a right triangle is formed out of the object, its shadow, and a line connecting the top of the object to the top of its shadow. Thus, the ratio of the height of the object to its shadow is the same for all upright objects at any one time and location.
Figure 2 shows two such triangles: one formed by a taller object and its shadow and the other formed by a shorter one and its shadow. Both shadows are shorter than the objects are tall. Since the two triangles have the same shape, the ratios of their heights to the length of their shadows, and , must be the same, that is, . Therefore, the presumably unknown height of the taller object H can be expressed in terms of the directly measurable quantities, , , and , in the formula . Note that one need not measure an angle in order to use this method.
Figure 2
Although no record exists in Thales’s own words, secondary sources credit him with measuring the height of the great pyramid at Giza and with determining the distance from the shore to ships at sea—possibly with methods similar to those described here. Today our smart phones and global positioning satellites also exploit the properties of similar triangles.
Thales is also said to have discovered how to inscribe a right triangle within a circle—a discovery in gratitude for which he sacrificed an ox. He speculated that the principle, or source, of all things is water; he shifted the course of a river; and he correctly predicted, within a year, the occurrence of a relatively rare event: the moon completely obscuring the sun in a solar eclipse. For these feats of skill and wisdom, Thales was honored as one of the seven wise men of antiquity.
Thales, unlike those who seek only practical mastery, sought universal truths among the diversity of particular facts. He was a philosopher. And for his application of mathematical truths to the natural world, he could also be called the first physicist.
2. Pythagorean Monochord (500 BCE)
Figure 3
One of the simplest musical instruments imaginable, the Pythagorean monochord, is a single stretched string fixed at each end. When plucked, the string vibrates and produces a tone of a particular pitch. Longer and heavier strings produce lower tones just as longer and larger wind and percussion instruments do. These facts must have been known before the time Pythagoras flourished around 525 BCE. After all, musical instruments with several strings of different lengths, such as the oud and the lyre, are depicted on Greek vases that date from the seventh century BCE. But Pythagoras, or one of his followers, may have been the first to quantify the relationship between the length of a string and the tone it produces.
Around 525 BCE Pythagoras emigrated from his native island of Samos, near the west coast of Asia Minor in the Aegean Sea, to the Dorian Greek colony of Croton (modern-day Crotone) near the ball of the boot that outlines the coast of southern Italy. There he founded a brotherhood of scholars who practiced a discipline whose object was to care for and purify the soul. The brotherhood also aspired to be the beneficent if austere political leaders of Croton. Around 450 BCE the original brotherhood was overthrown and broken up, but mystics and scholars called Pythagoreans were prominent for at least another hundred years. Some of them were mathematically talented investigators who attributed their own discoveries to their leader Pythagoras.
Figure 3 depicts a Pythagorean monochord. We now know that the tone produced by a monochord is determined by the dominant frequency of its vibration and that this frequency is in turn determined by the length of the string (the longer the string, the lower the pitch) and by the speed of a disturbance on the string (the faster the disturbance, the higher the pitch). We also know that strings of lesser density and strings under greater tension produce more quickly traveling disturbances and, therefore, produce higher frequencies and higher pitches. Yet in no way does this knowledge reduce the mystery of the relation, discovered by the Pythagoreans, between whole numbers and pleasing sounds.
Imagine two such monochords with strings of identical composition and tension but unequal in length. When simultaneously plucked or struck, the two strings produce two different tones. This humble arrangement allowed the Pythagoreans to make a discovery that filled them, and today fills us, with wonder. When one monochord is twice as long as a similarly constructed monochord or, more generally, when the lengths of the monochord strings are to one another as two small whole numbers are, such as 2 to 1, 3 to 2, or 4 to 3, then when plucked at the same time they produce, respectively, the pleasing sound of an octave, a perfect fifth, or a perfect fourth. Otherwise the tones are not so pleasing, but rather inharmonious or discordant.
That the small whole numbers, 1, 2, 3, and 4, should correspond to pleasing sounds became for the Pythagoreans emblematic of the numeric nature of our world. According to the Pythagoreans, both the form and the substance of the world are composed of whole numbers. Thus, for instance, the soul is a numerical harmony of the parts of the body. Even particular qualities such as “maleness” and “femaleness” are associated with numbers—in this case, odd and even numbers respectively. Today we find such ideas both vague and arbitrary. But the idea of finding common ratios and numerical forms in various phenomena is consistent with modern physics.
The Pythagorean monochord is the most basic of stringed instruments—hardly a musical instrument at all. But it demonstrates the principle behind the way violins, harps, and other stringed instruments make pleasing sounds. Flutes and other wind instruments also produce musical sounds—in their case by causing a column of air to vibrate. Drums produce sound when the membrane of the drumhead vibrates. Supposedly Pythagoras’s last words to his disciples were “Work the monochord.” Was this his way of saying “Become a musician” or of saying “Investigate the nature of the universe”? We understand Pythagoras better if we realize that for him these are the same vocation.
3. Phases of the Moon (448 BCE)
Figure 4
The various appearances of the moon—new moon (or no moon), tiny crescent moon, quarter moon, gibbous moon (partway between quarter and full), and full moon—are so familiar we may wonder why they need to be explained at all. Yet a certain kind of mind strives to explain complex phenomena, whether familiar or unfamiliar, in terms of simple concepts. These simple concepts should themselves be plausible and explain other phenomena. If successful, the explanation becomes part of a coherent outlook or theory.
All we need in order to explain the progress of the moon through an ordered series of phases is to assume that (1) the moon gives off no light of its own but reflects the light of the sun, (2) the moon travels around the earth in an approximately circular orbit, and (3) the rays of sunlight reaching the moon and earth travel along parallel lines. These ideas are illustrated in figure 4.
But beware. Figure 4 necessarily distorts other aspects of reality. The moon is neither so large compared to the earth nor so close to it. Nor, as is implied, does the moon pass into the earth’s shadow every month and cause a lunar eclipse or pass between the sun and earth and cause a solar eclipse, for the plane of the moon’s orbit around the earth is slightly tilted with respect to the plane of the earth’s orbit around the sun.
The sun’s rays illumine, at any one time, only half the moon’s surface and half the earth’s surface. The rest is in shadow—a shadow that on earth we call night. An observer, located in the diagram at the point of contact between the (larger) circle of the earth and the dotted line, has just been carried into this shadow by the earth’s daily counterclockwise rotation. This observer can see only that region of the sky above his local horizon, here indicated with the dotted line. The bulk of the earth blocks the rest of his view. What this observer sees of the moon is a relatively thin sliver of reflected light. This sliver appears as a crescent with its horns pointing away from the sun. As the earth’s rotation continues to carry the observer in a counterclockwise direction, the moon drops below the observer’s horizon.
Figure 5 shows several positions of the moon, each about seven days apart, as it moves around the earth. This cyclic motion takes about 29.5 days—a moneth in Middle English or, as we now say, a month. In each position a different portion of the moon’s illuminated surface is visible to observers on the night side of the earth. These different appearances are the moon’s phases: waxing half moon, full moon, waning half moon, new moon, and all those in between. Since the monthly motion of the moon around the earth is slow compared to the daily rotation of the earth on its axis, an observer on Earth sees very much the same phase of the moon throughout any one night.
Figure 5
We cannot be sure who first explained the existence and succession of the moon’s phases in this way. However, we do know that the Greek philosopher Anaxagoras (~500–428 BCE) was the first to leave a written record suggesting important aspects of this explanation. Anaxagoras was a native of Clazomenae, a city of Greek speakers in the middle of the west coast of Asia Minor, now Turkey. Anaxagoras spent twenty to thirty years of his mature life in Athens and so witnessed firsthand the beginning of the Peloponnesian War and other stirring events and impressive achievements of fifth-century BCE Athens. His time in Athens must have overlapped the lives of near-contemporary Athenian tragic playwrights Sophocles and Euripides and the life of the younger Socrates (469–399 BCE).
We do know that Anaxagoras wrote books because Socrates claimed to have read one of them, though he did not like it very much. Today we have only a few fragments of Anaxagoras’s writing preserved as quotations in other ancient texts. “The Sun puts the shine in the Moon” is one of these fragments. Later commentators, Aetius and Plutarch, writing four and five centuries after Anaxagoras claimed that Anaxagoras was the first to clearly explain the cause of the moon’s phases. Diogenes Laertius, who flourished in the third century CE, also claimed that Anaxagoras was the first to put a diagram in a book. If Anaxagoras did, indeed, explain the phases of the moon with a diagram, I suspect his diagram looked much like figure 5.
Anaxagoras is best known for his creative cosmology, that is, for his way of explaining everything. Anaxagoras’s first cosmological principle was that the mind directs and orders all things. Anaxagoras spoke of mind so often that his contemporaries gave him the nickname Nous, the Greek word for mind, just as today we might, with a little sarcasm, call someone a “brain.” It was Anaxagoras’s failure to follow through with the idea of Mind that so disappointed Socrates. Instead of explaining phenomena in terms of creation for some purpose as a mindful person might create, that is, for the sake of being beautiful or useful, Anaxagoras, in fact, often resorted exclusively to material and mechanical causes. One such materialistic idea, that the sun and all the stars are simply fiery pieces of metal, led to his conviction for impiety and banishment from Athens.
4. Empedocles Discovers Air (450 BCE)
Figure 6
The air that surrounds us is invisible, odorless, and tasteless. Nor does it usually produce sound or resist our movement through it. Of course, sometimes we feel a breeze at our back or a wind in our face. Less frequently tornadoes obliterate solid buildings and gale force winds raise seas that put neighborhoods under water. No doubt our ancestors had been aware of these phenomena for millennia when Empedocles (490–430 BCE), a native of Acragas in Sicily, sought to explain them. He was one of several physically minded, Greek-speaking philosophers or cosmologists who sought the principles or, in Empedocles’s usage, the roots of all phenomena.
For Thales the single principle was water, presumably, because of its ubiquity and because under common conditions water exists in three different phases: solid, liquid, and gas. For the Pythagoreans the single principle was number. For Anaximenes (ca. 500 BCE) it was air. For Heraclitus (ca. 495 BCE) it was fire. In a surviving fragment, Heraclitus suggests that while the logos (or the account) abides, there is no permanent material thing, since, accordingly to Heraclitus, “You cannot step twice into the same river, for other waters and yet others go ever flowing on.”
Empedocles sought to explain both the variableness and the stability of our experience by postulating that everything is composed of just four elements: earth, air, fire, and water. According to Empedocles, these four elements are neither created nor destroyed. The mixture and separation of different quantities of earth, air, fire, and water, brought about by the agencies of love and strife, account for the world of change we experience. Much later Empedocles’s four elements became the basis of Aristotelian and medieval cosmology.
What prompted Empedocles to consider air as one of the four fundamental elements? Air is a peculiar choice for, unlike earth, fire, and water, air seems bereft of qualities. For instance, we can see the water vapor above a boiling pot, but not the air that supports that vapor. Likewise, we can see the fire under the pot, but not the air the fire consumes. One fragment of Empedocles’s poem On Nature, out of the several hundred of its lines that survive, suggests an answer to this question. In describing human respiration, Empedocles compares the pores in our lungs and in our skin with a clepsydra—the main part of an ancient water clock that, in turn, is an open-mouthed jar that can be drained through a small spout at the center of its bottom. He writes: “As when a young girl, playing with a clepsydra of shining bronze, puts the passage of the pipe against her pretty hand and dunks it into the delicate body of silvery water, no liquid enters the vessel, but the bulk of air, pressing from inside on the close-set holes, keeps it out until she uncovers the compressed stream. But then when the air is leaving the water duly enters.”
Figure 6 illustrates the phenomenon Empedocles observed. According to the poem, the young girl inverts the clepsydra, closes its spout or pipe with her finger (here replaced with a plug), and submerges its open mouth into “the delicate body of silvery water.” Interestingly, something prevents the water from rising and assuming the same level inside the clepsydra as outside. That something is air. For when the child lifts her finger, its seal with the pipe is broken, air rushes through the pipe, and the water below the clypsedra pushes upward. If we have ever doubted the substantiality of air, this little demonstration, which can be reproduced with a kitchen funnel and a sink full of water, should dispel our doubt.
Thales, Anaxagoras, Empedocles, and the other Greek philosophers predating Socrates were imaginative and, for the most part, materialistically minded thinkers who sought to explain phenomena with the fewest, self-consistent, and most plausible principles. While these cosmologists appealed to common observations (for instance, that water exists in three phases), their tools of discovery and verification were invariably speculation and argument. They did not perform experiments. But one wonders: Could Empedocles have resisted imitating the child he observed? Did he not also play with the clepsydra? If so, Empedocles did something unusual for a Greek of his time. He not only thought about nature, but manipulated a natural phenomenon with the intention of learning something new. He performed an experiment.
5. Aristotle’s Universe (350 BCE)
Figure 7
Have you ever heard someone say “Eventually scientists will figure out how to do it?” You fill in the reference for “it.” Travel faster than the speed of light? Build a heat engine with 100 percent efficiency? Extract energy from the cosmic microwave background? Indeed, it may be that some things once thought impossible will turn out to be quite possible. But it is not true that everything of which we might dream is possible. After all, we live in a world that has a nature: a characteristic way of being and of becoming, of remaining the same and of changing.
We may learn about that nature and discover ways to employ it, but we have no power to change the nature of things. According to Francis Bacon (1561–1626), “Nature, to be commanded, must be obeyed.” Aristotle (384–321 BCE) transmitted to us this indispensable concept of nature—a concept rejected, perhaps unintentionally, by those who think scientists and engineers can do anything and everything.
The word nature comes to us from a Latin root, the Greek equivalent of which is φύσις or phusis, from which also derives physics. Of course, modern physics was born struggling against certain Aristotelian ideas. Nevertheless, Aristotle’s concept of nature is the bedrock upon which the practice of modern physics stands.
Figure 7 illustrates Aristotle’s universe—not as one would observe it—but rather in the state of perfection toward which Aristotle’s universe tends by virtue of its nature. Earth and water move down toward the center—earth more persistently than water. Air and fire move up away from the center—fire more readily than air. Thus upward and downward motions characterize the region below the sphere of the moon. Objects above the lunar sphere are composed of a fifth substance, the quintessence or ether. The sun and wandering stars or planets (not shown in the diagram) and the fixed stars reside on transparent spheres that carry them around the earth in concentric circles. Circular motion characterizes the region above the sphere of the moon.
Aristotle borrowed many of the features of his universe from his pre-Socratic predecessors, for instance, the four elements (earth, air, fire, and water) and the celestial spheres. Furthermore, the pre-Socratics were the first to formulate the concept of nature. But Aristotle composed these ideas into an ordered whole, a cosmos, that answered the questions of his day and, at the same time, remained consistent with commonplace observations.
That last statement needs to be qualified for Aristotle must have observed sublunary objects that do not always travel up or down. Toss a clod of earth, and it travels along an approximately parabolic arc, at first up, then down, and always in the direction thrown. According to Aristotle, motion requires a mover and, if that mover is not the nature of the moving object, motion must be imparted and maintained externally, that is, unnaturally or by “violence.” Thus, it is the hand that throws the clod and the air through which the clod moves that cause its unnatural horizontal movement.
According to this view, to manipulate objects and study their behavior, that is, to perform experiments, is not a reliable way to study nature. For in doing so, one fruitlessly studies that which has no nature: the whimsy of the human boy, for instance, that tossed the clod in a particular way. To manipulate a natural phenomenon, is to spoil its naturalness—at least according to Aristotle.
Nevertheless, Aristotle was a great observer of nature and, according to the eminent historian of science George Sarton, “one of the greatest philosophers and scientists of all times.” He discovered the law of the lever and was the first to systematically study meteorology. He “carried on immense botanical, zoological, and anatomical investigations [and] clearly recognized the fundamental problems of biology: sex, heredity, nutrition, growth, and adaptation.” He structured the elements of logic and originated the inductive method. Aristotle also wrote ageless treatises on literary criticism, ethics, and metaphysics. Indeed, there is hardly a branch of human knowledge to which Aristotle did not contribute.
In 335 BCE Aristotle established a school of philosophy and science in Athens called the Lyceum. Those who studied with and followed Aristotle became known as peripatetics, that is, those who study while walking from place to place. Aristotle’s most famous pupil, Alexander the Great, the son of Phillip II of Macedon, conquered the known world.
According to Aristotle, the realm of the celestial spheres is perfect. Its motions, unlike those of the sublunary realm, are completely natural, manifestly beautiful, and ultimately caused only by the desire for the good. It is not hard to see why Aristotle’s view of the universe has influenced thought and literature for over two thousand years. It is, after all, a privilege and delight to look at the heavens each night and be inspired by their perfection.
6. Relative Distance of the Sun and the Moon (280 BCE)
Figure 8
Aristarchus of Samos (310–230 BCE) was the first to determine the relative distance of the sun and the moon from the earth. His method, like that of Thales, depends on the properties of similar or same-shaped triangles. But his application—to the relative distances of heavenly bodies—is much bolder. Aristarchus assumed only that the moon receives its light from the sun.
Figure 8 suggests Aristarchus’s argument at the cost of making the sun too close relative to the earth-moon separation. When the moon appears, to an observer on the earth, exactly half-dark and half-light, that is, appears as a half moon, Aristarchus knew that a line from the sun to the moon must meet a line from the earth to the moon at a right angle. Thus, if at half moon we are able to measure the size of the slightly-less-than-right angle between the earth-sun and earth-moon lines (admittedly a difficult measurement), we know all there is to know about the shape of the right triangle formed by Earth, Moon, and Sun. All there is to know about the shape of the right triangle would allow us, for instance, to reproduce a similarly shaped triangle on papyrus and in this way determine the ratio of the two sides representing the sun-earth and moon-earth distances, that is, the ratio .
Aristarchus’s procedure led to a ratio according to which the sun is about twenty times more distant from the earth than the moon is from the earth. The ratio is, in fact, closer to 400. However, Aristarchus’s method is sound. At least one scholar has claimed that the techniques available to Aristarchus should have allowed him to be more accurate. If so, perhaps his interest was more with pioneering a new method than with its careful application.
Aristarchus is better known for his teaching that the earth rotates daily on its axis and revolves yearly around the sun. But he convinced few of his contemporaries. The problem is that a yearly motion of the earth around a presumably stationary sun implies that the relative positions of the stars as observed from the earth should change during the year. We now know that the closest stars are too distant for this effect of observer motion, the stellar parallax effect, to be seen with the naked eye.
Note that figure 8 makes no distinction between Earth- and Sun-centered planetary systems. Both are consistent with Aristarchus’s method of determining relative distances since in each case the moon revolves around the earth. Thus Aristarchus’s contemporaries could, with perfect consistency, accept his determination of relative distances and reject his Sun-centered planetary system.
The manuscript in which Aristarchus made his argument, On the Sizes and Distances of the Sun and Moon, refers not only to the distances of the sun and moon but also to their sizes. His determination of the relative size of the sun and moon is also insightful. Aristarchus observed that during a total eclipse of the sun the disc of the moon completely obscures the disc of the sun but just so—as shown in figure 9. Because the larger right triangle appearing in this figure is similar to the smaller one, the ratio of the distances of Sun and Moon from Earth must equal the ratio of their radii . Therefore, if the sun is, as Aristarchus believed, twenty times more distant than the moon, the sun must be twenty times larger than the moon. Since the sun is, in fact, about four hundred times more distant than the moon, the sun is actually about four hundred times larger than the moon.
Figure 9
Aristarchus forged yet one more link in his chain of arguments. He noticed that the time required for the moon to pass into the earth’s shadow during a lunar eclipse is close to the time the moon stays completely obscured by the earth’s shadow. If so, the earth’s radius must be twice that of the moon—assuming the earth’s shadow from Earth to Moon is approximately cylindrical. And if the earth is two times larger than the moon and the sun is twenty times larger than the moon, then the sun must be must be ten times larger than the earth. Again, Aristarchus’s argument is valid even if his data are not accurate. According to modern measurements the earth is about four times larger than the moon and the sun is some one hundred times larger than the earth.
Aristarchus’s methods illustrate the intellectual trends of his time. Since he was a younger contemporary of Euclid (the latter flourished around 300 BCE), Aristarchus lived during a period in which the propositions of geometry had become widely known. Subsequently, astronomical knowledge was more frequently framed in geometrical language than before and, in this way, physical science began to distinguish itself from philosophical wisdom. At the same time, the center of Greek learning was migrating from Athens to the newly founded city of Alexandria near the mouth of the Nile. While Aristarchus may or may not have traveled from his native Samos to Alexandria, his life falls within the initial stages of a period of cultural ferment set in motion by the conquests and foundations of Alexander the Great.
7. Archimedes’s Balance (250 BCE)
Figure 10
Around 300 BCE, Euclid organized the mathematical knowledge of his time into definitions, common notions, postulates, and the demonstrations of propositions. Some of the definitions are familiar, for instance, “A line is breadthless length,” and some seem a little mysterious, “A straight line is a line that lies evenly with the points on itself.” The common notions are self-evident statements common to all kinds of reasoning such as “Things which are equal to the same thing are also equal to one other.” The postulates are a small group of unproven statements, assumed to be true, such as “All right angles are equal to one other,” and the propositions are statements whose truth Euclid demonstrates by valid argument from the postulates, the common notions, and previously demonstrated propositions. The result, contained within the thirteen books of Euclid’s Elements, is an extended deductive system that has, for 2,300 years, been a model of rigorous thinking. The outstanding lesson of Euclid’s system is that many truths can be demonstrated and not merely asserted.
Euclid’s Elements astonishes and charms its readers. It is said that Sir Thomas Hobbes, on first picking up Book I of the Elements and reading Proposition 47, the Pythagorean theorem, exclaimed, “By God, this is impossible.” Hobbes then read its demonstration and then the demonstrations of the propositions used to demonstrate Proposition 47 and so on until he had read a good part of Book I in reverse order—a method of reading Euclid I do not recommend. On the other hand, I do commend Edna St. Vincent Millay’s response to Euclid, a fourteen-line Shakespearean sonnet, “Euclid alone has looked on Beauty bare,” whose middle verses are as follows:
... let geese
Gabble and hiss, but heroes seek release
From dusty bondage into luminous air.
O blinding hour, O holy, terrible day,
When first the shaft into his vision shone
Of light anatomized! Euclid alone
Has looked on Beauty bare. ...
Archimedes (287–212 BCE), certainly the most original mathematician and physicist of antiquity, also fell under Euclid’s spell. We know this because he followed Euclid in organizing what he had discovered about the equilibrium of heavy bodies into a system of postulates, propositions, and demonstrations.
Figure 10 illustrates Propositions 6 and 7 of Archimedes’s On the Equilibrium of Planes that together compose his law of the balance: two objects balance at distances inversely proportional to their weights—a law illustrated each time two children of unequal weight balance themselves on a teeter-totter. The dark line in the diagram is the balance beam, the triangle is the beam’s support or pivot, short, light lines mark the beam at equally spaced intervals on either side of the pivot, and the blocks stand for units of weight. In the left panel a weight of two units is one unit to the right of the pivot and a weight of one unit is two units to the left of the pivot, so that each weight is at a distance from the pivot inversely proportional to its magnitude.
Figure 10 illustrates a demonstration of a particular case of the law of the balance. The demonstration requires only two premises, both quite reasonable. One of these is Archimedes’s Postulate 1, equal weights at equal distances (from the pivot) balance, and the other is a previously demonstrated proposition, Proposition 4, the center of gravity of two equal weights taken together is in the middle of a line joining their centers. The phrase center of gravity refers to the location at which a pair of identical weights can be replaced by a single weight equal in magnitude to the total weight of the pair. Thus, Proposition 4 justifies the transition, shown in the center panel that makes the stability of the left and right panels equivalent. Of course, according to Postulate 1, the weights in the right panel balance. Therefore, the sequence of panels from left to right (or from right to left) demonstrates that a weight of two units located one unit to the right of the pivot balances a weight of one unit located two units to the left of the pivot.
Archimedes’s demonstration is more general than ours. Nevertheless, our demonstration exploits the rule justified by his Proposition 4: a weight at a particular location can be replaced by two weights, each equal to half the original weight, placed equal distances on either side of their original location. By applying this rule several times one can show that the two balance beams illustrated in figure 11 are physically equivalent. Give it a try. And remember: You are allowed to place blocks on top of the pivot.
Figure 11
Archimedes may have sojourned for a while in Alexandria, and if so, he may have known the somewhat younger Eratosthenes (276–194 BCE) who plays a role in a later essay. Even so Archimedes lived the greater part of his life in his native Syracuse, a Greek city on the island of Sicily. Greek colonists had inhabited Sicily and the southeastern coast of the Italian mainland since the eighth century BCE. During Archimedes’s lifetime, the Romans extended their dominion over the Italian peninsula and Sicily and engaged the North African city of Carthage in a life-and-death struggle. Syracuse, and therefore Archimedes, stood directly in the path of this Roman expansion.
8. Archimedes’s Principle (250 BCE)
Figure 12
The story goes that when the solution to a particularly challenging problem came to Archimedes in his bath, he leapt from the tub shouting “Eureka! Eureka!” (I have found it! I have found it!) But what had Archimedes found? According to Vitruvius (ca. 75–15 BCE), the Roman military engineer who told the story almost two centuries after the event, Archimedes had discovered a method for determining whether a crown that had been made for King Hieron of Syracuse was of pure gold, as per instructions, or mixed with silver. The story is a good one—almost too good to be true—for the method Archimedes discovered concerned bodies submerged in a fluid just as Archimedes’s body was submerged in his bathwater. However, Vitruvius does not tell us the details of Archimedes’s method.
Figure 12 illustrates the physics behind what physics teachers call Archimedes’s principle—the content of which Archimedes outlined in Propositions 3–7 of Book I of his text On the Equilibrium of Floating Bodies. Archimedes’s principle is simply stated and elegantly proved and could have been used to determine the composition of King Hieron’s crown.
The left panel shows a container filled with water, or any other fluid, at rest. The dashes outline a region of the fluid while the arrows represent the direction and magnitude of the pressure exerted by the fluid outside the outlined region on the fluid inside the outlined region. (The longer the arrows, the larger the pressure.) Note that, as one might expect, the magnitude of this pressure increases with depth. Therefore, the upward push, on the fluid in the outlined region, is larger than the downward push. In fact, the net upward push must be just enough to support the weight of the fluid in the outlined region in order to keep the fluid at rest.
In the right panel of figure 12 the fluid that was in the outlined region has been replaced with an identically shaped object. A string attached to the object keeps it from sinking. Since the fluid outside the object in the right panel is identical to the fluid outside the region outlined in the left panel, the net force the outside fluid exerts is also identical. Thus, the fluid outside an object exerts a net upward force on the object equal in magnitude to the weight of the fluid the object displaces. This is Archimedes’s principle. The argument with which we have reached Archimedes’s principle applies to floating as well as to fully submerged objects.
Once we understand Archimedes’s principle we can use it to solve problems. Here is one problem physics teachers sometimes assign their students. A cargo ship containing iron ore is in a watertight lock as illustrated in figure 13. The captain orders his crew to dump the iron ore into the bottom of the lock. Does the water level in the lock rise, fall, or stay the same when the ore is dumped? The answer (the water level falls) requires the creative use of Archimedes’s principle.
Figure 13
Archimedes also proved that the surface of a fluid at rest is part of the surface of a sphere whose center is at the center of the earth. Stop a moment and consider this claim. The surface of every glass of water, every cup of coffee, and every farmer’s pond is curved, concave downward, with its center of curvature at the center of the earth! Of course, Archimedes’s claim ignores the distorting effect of the water’s surface tension. But, still, the claim is amazing. The bulk of the earth pulls on a fluid in such a way as to shape it into a section of a sphere. Figure 14 illustrates this claim and provides the seed of Archimedes’s proof, which, however, I do not spell out.
Figure 14
Although Archimedes was primarily a mathematician and physicist, he also invented devices that exploit physical principles: the so-called Archimedean screw that could pump water from a lower to a higher level and the compound pulley that could, in principle, allow one person to, very slowly, lift a massive ship. Some of Archimedes’s inventions were weapons of war, for instance, burning mirrors and catapults, which he devised in order to defend his native Syracuse from the Roman army that besieged it in 212 BCE.
But the Romans prevailed and Archimedes died as he had lived—absorbed in a problem of mathematical science. The Roman commander of the besieging army, Marcellus, had given orders that the famous Archimedes, then seventy-five years old, be spared. But when confronted by an armed Roman soldier Archimedes, who had been studying some figures drawn in the sand, brusquely demanded, “Stand back from my diagrams!” Those were his last words. Evidently, this was no way to address an armed Roman soldier.
9. The Size of the Earth (225 BCE)
Figure 15
Eratosthenes (276–194 BCE) was born in the North African town of Cyrene (in modern Libya) and educated in Athens, but lived the greater part of his life in Alexandria where around 244 BCE he became the head of its great library. The wealth of this library’s holdings can be inferred from the task given Callimachus, a contemporary of Eratosthenes—to catalog the library’s books—and the result of his effort: 120 volumes of bibliography. Thus, we can understand Callimachus’s famous complaint, “A great [or large] book is a great evil.” Nevertheless, the library and its great books made Eratosthenes’s career possible.
Eratosthenes’s writings include the text Geographica, now lost but often cited in antiquity. This book gathered together what was then known about geography—a word he was the first to use in its modern sense. Eratosthenes’s compilation of the geographical wisdom of the past into a single, expansive treatise had many imitators during the centuries of Roman domination that followed his death. Pliny’s Natural History, for instance, included all that was known of the natural world. However, even the best Roman scholars were more concerned with the utility and entertainment value of the learning inherited from the Greeks than with creatively understanding or extending that learning.
But Eratosthenes was an Alexandrian Greek who not only preserved but also built upon the wisdom of the past. For instance, he was the first to add lines of longitude to a map of the known world. On a globe these lines are great circles that pass through both poles. The particular line of longitude or meridian that connects Alexandria and Syene (modern Aswan) plays a role in Eratosthenes’s determination of the circumference of the earth.
The diagram illustrates Eratosthenes’s method. Eratosthenes noticed that when rays of sunlight reach the bottom of a deep well at Syene, rays of sunlight at Alexandria make an angle equal to 1/50 of a circle with a straight vertical pole or gnomon. Eratosthenes also knew that, so distant was the sun from the earth, the different rays of sunlight striking the earth are essentially parallel, and that, according to Euclid, a line falling upon two parallel lines makes the alternate interior angles equal—as illustrated in figure 15. Therefore, according to the geometry of the diagram and Eratosthenes’s measurement, an angle equal to 1/50 of a circle (about 7 degrees) with vertex at the center of the earth subtends or encompasses the meridian connecting Syene and Alexandria along the surface of the earth. Consequently, the distance between Syene and Alexandria is 1/50 the circumference of the earth. It only remained for Eratosthenes to determine this distance and multiply by 50.
As it happens, Syene is located near the first waterfall upstream from the Nile’s mouth at Alexandria. Between Syene and Alexandria, the Nile flooded often, and, consequently, was measured frequently by the cadre of Egyptian geometers, literally “land measurers,” whose job was to preserve the identity of property along the Nile. Eratosthenes had access to the land measurers’ records and from them inferred that the distance from Syene to Alexandria was about 5,000 stadia. Therefore, according to Eratosthenes’s method, the circumference of the earth is about 250,000 stadia.
But how long is a single stade? Ancient documents provide at least two different answers to this question. The Egyptian stade is 158 meters and the more commonly used Greek stade is 185 meters. The first produces a circumference within one percent of the modern value, 40,000 kilometers, while the second produces one 17 percent too large. But comparing Eratosthenes’s value to that determined by modern methods teaches us little. It is more important to understand that Eratosthenes’s method is sound and based on measurements rather than on speculations.
Eratosthenes also understood that his measurements were uncertain. We know this because Eratosthenes attempted to quantify their uncertainty. For instance, he determined that on the longest day of the year, sunlight reaches the bottom of any well at Syene within a circle with radius of about 300 stadia. This effect alone limits the accuracy of Eratosthenes’s determination of the circumference of the earth to plus or minus 6 percent.
That Eratosthenes assumed the earth is spherical is unremarkable. For by his time it had long been known that (1) as we travel north the southern constellations sink toward the horizon and the Pole Star rises higher in the nighttime sky, and (2) during a lunar eclipse the shadow of the earth on the moon is a section of a circle. No observant person could argue with these facts. Only a desire to make sense of them was needed. We know from his determination of the size of the earth that Eratosthenes had that desire.
and 
, must be the same, that is, 
. Therefore, the presumably unknown height of the taller object H can be expressed in terms of the directly measurable quantities, 
, 
, and 
, in the formula 
. Note that one need not measure an angle in order to use this method.
between the earth-sun and earth-moon lines (admittedly a difficult measurement), we know all there is to know about the shape of the right triangle formed by Earth, Moon, and Sun. All there is to know about the shape of the right triangle would allow us, for instance, to reproduce a similarly shaped triangle on papyrus and in this way determine the ratio of the two sides representing the sun-earth 
and moon-earth 
distances, that is, the ratio 
.
according to which the sun is about twenty times more distant from the earth than the moon is from the earth. The ratio 
is, in fact, closer to 400. However, Aristarchus’s method is sound. At least one scholar has claimed that the techniques available to Aristarchus should have allowed him to be more accurate. If so, perhaps his interest was more with pioneering a new method than with its careful application.
from Earth must equal the ratio of their radii 
. Therefore, if the sun is, as Aristarchus believed, twenty times more distant than the moon, the sun must be twenty times larger than the moon. Since the sun is, in fact, about four hundred times more distant than the moon, the sun is actually about four hundred times larger than the moon.
