The Story of Western Science

by Susan Wise Bauer

  So the unphilosophical man must be brought slowly out of the “cave” into which he was born, into the full sunlight of philosophical knowledge. He must be carefully taught to understand the Ideal, so that he will no longer be satisfied with the Copy. Arithmetic and geometry are tools of the transformation. Arithmetic, because it begins by distinguishing between the concepts of singular and many, unity and plurality, the one and the infinite, has “a power of drawing and converting the mind to the contemplation of true being.” Geometry, likewise, leads the soul toward truth, and creates the “spirit of philosophy” where none had before existed.6

  But both arithmetic and geometry have this power only when they deal with abstractions. The study of arithmetic, Plato cautions, must be “pursued in the spirit of a philosopher, and not of a shopkeeper . . . arithmetic has a very great and elevating effect, compelling the soul to reason about abstract numbers, and rebelling against the introduction of visible or tangible objects into the argument.” Geometry must be practiced, not as most geometricians do (“They have in view practice only, and are always speaking, in a narrow and ridiculous manner, of squaring and extending and applying and the like—they confuse the necessities of geometry with those of daily life”), but as a method of understanding Ideal forms and shapes. “The knowledge at which geometry aims,” he explains, “is knowledge of the eternal, and not of aught perishing and transient.”

  In fact, because Plato’s philosophy dictated that reason, untainted by the input of the senses, can yield truth, Plato was suspicious of any arithmetical conclusions that were related to observation. This made him wary, for example, of astronomy. Astronomers observed past movements of heavenly bodies, analyzed them, and used mathematics to calculate their future positions. But such calculation incorporated observation—the senses—into the practice of mathematics, thus bringing it from the realm of the ideal into the realm of the shadow. So, while Plato acknowledged the value of astronomical calculations, he also warned astronomers not to assume that their theories actually described the universe. The astronomer’s conclusions might be likely, but in no way could they be characterized as true.7

  This dismissal of the real world still echoes down to us in the language used by math departments; applied mathematics is no longer scorned as “shopkeeping,” but theoreticians still claim the snobby title of “pure mathematics” for their own discipline.8

  •

  Despite Plato’s scorn, mathematicians were, increasingly, trying to find the intersection between maths and natural science.

  Indeed, it seems likely that Plato’s snarky remark about “shopkeeper” mathematics was directed at one of his contemporaries, a Pythagorean mathematician named Archytas who dared to depart from the mysticism of his colleagues in order to apply mathematics to real problems in the real world. The third-century biographer Diogenes Laertius calls Archytas “the first who methodically applied the principles of mathematics to mechanics.” Tradition held that Archytas had invented a wood dove that actually flew. And in his Politics, Aristotle remarks offhand that since young children are unable to be quiet, they should be given a toy rattle designed by Archytas, to occupy them and “stop them from breaking things in the house.” But only fragments of Archytas’s own writings remain, and the scraps shed no light on his scientific pursuits.9

  Not until the middle of the third century did a piece of scientific writing make use of mathematics for its investigation—and survive. The text is called “The Sand-Reckoner,” and it was written by Archimedes, a native of the Sicilian city of Syracuse.10

  Archimedes had an advantage that Archytas lacked: a handbook, written by Euclid (ca. 325–265 BC), assembling the geometric knowledge that had been floating around Pythagorean circles for centuries into thirteen nonmystical, entirely unreligious books. The Elements begins with a list of definitions (“A point is that which has no part. . . . An obtuse angle is an angle greater than a right angle”) and continues with what Euclid calls “postulates” and “common notions”—both being statements so self-evident that they do not need to be proven. Postulates have to do with geometry in particular (“All right angles are equal to one another”), and common notions apply to both geometry and other fields of study (“The whole is greater than the part”).

  But the meat of Euclid’s book is in his geometric proofs—problems and solutions showing that the rules of geometry work in all places, for all times, for everyone. There is nothing less mystical and obscure than Euclid’s proofs. Given his starting assumptions, and perhaps a ruler and compass, everyone (not just the initiates) could comprehend his system.

  The fifth-century Greek philosopher Proclus, who wrote an extensive commentary on Euclid’s Elements some seven hundred years later, relays a well-known story about Euclid and Ptolemy I, the king of Egypt. Ptolemy, an ex-general of Alexander the Great, was no scholar. But he knew the value of learning and tackled the Elements, only to find it a little too meaty for his taste. So he asked Euclid for a simpler way to understand its principles. “Sir,” Euclid responded, “there is no royal road to geometry.”

  The story may be apocryphal, but it reveals the new truth of geometry. There were no shortcuts, no divine revelations or ritual sacrifices required, and also no privilege. The Elements rescued geometry from the Pythagoreans and gave it to the world.11

  Almost immediately afterward, Archimedes connected this new tool to the contemplation of the universe.

  Archimedes is best remembered for a discovery that might not have happened. According to the Roman biographer Vitruvius, two hundred years later, Archimedes had been tasked by his king with the job of figuring out whether a dishonest goldsmith had stolen some of the gold intended to go into the royal crown and replaced it with cheaper silver. The crown looked gold, and it was the correct weight. But was it pure?

  “While the case was still on his mind, he happened to go to the bath,” writes Vitruvius, “and on getting into a tub observed that the more his body sank into it, the more water ran out over the tub. As this pointed out the way to explain the case in question, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, ‘Eureka, eureka’ [I have found (it)].” He had just realized that, since silver is lighter than gold, a crown of pure gold would have to be slightly smaller than a crown of equal weight made up of a gold and silver combination. So, if the adulterated (and thus larger) crown were submerged into a jar of liquid, it would displace more water than the pure, microscopically smaller crown. Calculating how much water the unadulterated crown should have displaced, and comparing it with the actual displacement, Archimedes was able to measure the extent of the silver substitution, and manifestum furtum redemptoris: the stolen property was revealed.12

  Vitruvius is notoriously unreliable, and more than one experimenter has pointed out that it would be almost impossible to measure the tiny difference in water displacement with enough accuracy to predict the exact makeup of the crown. But there is no question that Archimedes understood the science behind the story. In his own On Floating Bodies, he describes the Principle of Buoyancy (“Archimedes’ Principle”), which is simply this: “A body partly or completely immersed in a fluid is lifted up by a force equal to the weight of the displaced fluid.”13

  Archimedes wrote several essays expanding on Euclidean geometry, and he is generally credited with a whole series of inventions, including Archimedes’ screw (a pump used to raise water from a lower to a higher level), the ship shaker (a mechanical claw that could lift an attacking ship out of the water), a planetarium, and various sorts of levers. The proof for most of these is weak or nonexistent; Archimedes may well have improved them, but his own writings don’t show that he invented them.

  What they do show is a mathematician who knew how to apply his knowledge to scientific questions. In his essay “The Sand-Reckoner,” Archimedes finally brought geometry to bear on the study of the natural world.

&nb
sp; The premise of “The Sand-Reckoner” is fairly straightforward: How many grains of sand would it take to fill the universe? Although this may seem like a mere thought experiment, remember that the Greeks were accustomed to measuring in terms of ratios. Archimedes’s question was not merely “How big is the universe?” It was, instead, “Is it possible to measure the universe using the mathematical tools that we possess?” Ratio was the tool he had in mind; and his quest was to discover whether a meaningful relationship existed between two natural objects of vastly different sizes: a speck of sand and the whole of physical reality.

  For the purposes of his essay, Archimedes decided to adopt a less-than-widely-accepted model of the universe—one with the sun at its center. It was much more popular, in ancient times, to see the universe as a relatively compact set of interlocking spheres centered around the earth. But Archimedes thought that this smallish universe wouldn’t set him much of a challenge.

  Instead, he chose to work out his calculations as they would apply to another model of the universe, one proposed by his contemporary Aristarchus. “You are aware,” “The Sand-Reckoner” begins, “that ‘universe’ is the name given by most astronomers to the sphere whose centre is the centre of the earth. . . . But Aristarchus of Samos brought out a book consisting of [the] hypotheses . . . that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit.”14

  According to Aristarchus, the stars were the equivalent of 100 million earth-diameters away from the center of the universe—much, much farther away than the stars of the geocentric model. And to make the universe even larger, Archimedes decreed that “earth-diameter” (an unknown quantity) should be understood as 1 million stadia across.

  One million stadia is almost 100,000 miles, which is way too huge; the earth is really only about 7,900 miles across, depending on which bulges you measure. Even so, Archimedes’s calculations of the size of the universe yielded a ridiculously small number. His universe turned out to be about 10 trillion miles across, which we now know to be less than 2 light-years; a single light-year is about 6 trillion miles, and the Milky Way galaxy alone is perhaps 120,000 light-years in diameter.15

  But the actual dimensions were beside the point. What mattered was the larger question: “Is it possible to use mathematical language to describe a reality that is bigger than anything we have been able to measure in the past?”

  And to this, Archimedes was able to answer a resounding yes.

  Expressing the answer was slightly complicated, though. Greek numbers couldn’t count that many grains of sand. The largest number in the Greek system was the myriad, or 10,000, written as

  (capital mu). The myriad could be combined with other numbers: so, for example, since ε (epsilon) represented 5, the compound symbol Με stood for 5 × 10,000, or 50,000.

  The greatest quantity that could be expressed using this system was the myriad myriad, or the myriad times itself: 100 million (100,000,000). Archimedes needed more numbers. So when he reached 100,000,000, he designated the myriad myriad as a single number, written βΜ (β, beta, represented 2). This meant he could now write multiples of 100,000,000; 500,000,000, for example, was βΜε.16

  This was not the most elegant number system in the world (just by way of illustration, 785,609,574,104 had to be written as βΜ,ζωνς, αΜνξ, δρδ),† but it allowed Archimedes to figure that it would take 1051 grains of sand to fill up the heliocentric universe of Aristarchus. For the first time, a scientist had forced mathematics to serve the purposes of science, rather than the other way around. Instead of shaping the universe to the perfect, abstract, Ideal of mathematical knowledge, Archimedes had molded the language of mathematics so that it fit the reality of the universe.

  And he had conveyed another very clear message as well. For centuries, sand had represented the uncountable; like sand and like the stars meant, very simply, “That which is beyond our numbering.” Choosing sand as his measure of the star-filled sky, Archimedes had made a new assertion: There is nothing in the universe that cannot be counted, and understood, by man.

  To read relevant excerpts from “The Sand-Reckoner,” visit http://susanwisebauer.com/story-of-science.

  ARCHIMEDES

  (287–212 BC)

  “The Sand-Reckoner”

  The classic nineteenth-century translation by Thomas Heath is available as a free e-book.

  Archimedes, The Works of Archimedes, trans. T. L. Heath, Cambridge University Press (e-book, 1897).

  A print version published by Dover includes not only “The Sand-Reckoner” but also Heath’s introductory essay and eight other short works of Archimedes, including “On the Sphere and Cylinder” and “Measurement of a Circle.”

  Archimedes, The Works of Archimedes on Mathematics, trans. Thomas L. Heath, Dover Publications (paperback, 2013, ISBN 978-0486420844).

  Heath’s translations transform Archimedes’s Greek numeral system into exponential numbers readable by English speakers. In places, he retains the Greek in brackets.

  * * *

  * For example, the fraction , because it is the ratio of 4 to 9 (or, 4 divided by 9); or 71, because it is the ratio of 71 to 1 (or, 71 divided by 1); or –11, because it is the ratio of –11 to 1 (–11 divided by 1). Another way to look at it: a rational number can always be put into the form a/b, as long as a and b are both integers and b does not equal zero.

  † My thanks to Russell Cottrell’s online script “The Greek Number Converter” for converting Arabic to Greek numbers.

  FIVE

  The Void

  The first treatise on nature to dispense entirely with the divine

  All fails, and in one single moment dies.

  —Lucretius, On the Nature of Things, ca. 60 BC

  While Archimedes calculated, while Aristotle investigated change, while Plato taught of ideal Forms, the atomists continued to insist that physical reality is made up of nothing more than indivisible particles, traveling at random through the infinite void.

  Democritus, who had acknowledged the existence of the gods but taught that they, too, were made of nothing but atoms and “the empty,” had died around 400 BC; a later chronicler records that he was 104 and simply decided to stop eating. His disciples lived on, passing his maxims from one generation to the next. The most successful and notorious of them was Epicurus, founder of a philosophical school that began to meet in the garden of his Athenian home around 307 BC. Epicurus developed atomism into an entire philosophical system—one that would finally lend itself to the cause of science two centuries after his death.1

  Like Thales, Aristarchus of Samos, and the early atomists, Epicurus did not leave us his writings. Only fragments of his works survive, along with a letter to the Greek historian Herodotus that seems to have been written, fortuitously, as a précis of his teachings: a deeply secular philosophy that saw no pattern or design in the world.

  Nothing comes into existence from what does not exist . . . there is no thing outside the whole, there is nothing that could enter the whole and produce a change.

  All things that exist are either bodies or empty space. . . . We call this empty space “void.” . . . Besides these two kinds, bodies and empty space, it is impossible to conceive of anything else.

  The building blocks of bodily natures are the atoms or indivisibles.

  The atoms are perpetually in motion throughout the vast ages.

  There is no absolute beginning of such motions, since both the atoms and the empty space have existed forever.2

  Like Democritus, Epicurus saw around him only an “infinite and mechanical universe of interacting particles.” The physical objects around us, he explained, came into being not by divine intervention, but because atoms—spinning through the void—sometimes give an unpredictable hop, a random jump sideways, slam into each other, and join up to create new objects.3

  Epicurus wasn’t particularly interested in the science of thes
e atoms. He cared nothing for knowledge as knowledge; he had no appreciation for the beauty of a scientific theory, no satisfaction in understanding the workings of the natural world. His preoccupation was ethical: How, living in such a world, should men act? Given that there is no divine plan, no afterlife, no immortal soul, how then should we live? How does man reach ataraxia, peace of mind, when adrift in an uncaring and impersonal universe, with no navigator but chance, with no guarantee of a safe farther shore? “Remember,” he wrote himself to his disciple Pythocles, “that, like everything else, knowledge of celestial phenomena . . . has no other end in view than peace of mind and firm convictions.”4

  This peace of mind was not reached, as Epicurus’s enemies would later insist, through thoughtless indulgence in sensual pleasures. Rather, Epicurus struggled to order his priorities with no reference to the divine. Happiness, he believed, lay in the absence of fear: fear of pain, fear of poverty, fear of death. This fear was overcome through the enjoyment of the senses, yes, but this enjoyment required prudence, moderation, virtue, responsibility.

  Two hundred years after Epicurus’s death, his disciple Lucretius—a Roman educated in Greek philosophy, a writer of astonishing clarity—retold Epicurus’s teachings in the form of a long poem that went beyond the master. De rerum natura (On the Nature of the Universe or, more literally, On the Nature of Things) spells out the implications of Epicurean atomism for the practice of natural science. First and foremost, Lucretius insisted, only the pure materialism of the Epicureans makes rational thought—truly scientific thought—possible. As long as men insist on believing in a supernatural Designer or Mover, even a benevolent one, they will suffer from “terror and darkness of the mind.”

  Clarity of thought, the ability to grasp physical reality as it actually is, and (above all) peace of mind come only when men admit that the universe consists nothing but the material. Nothing else: no hell below us, nothing but sky above. Lucretius was the Richard Dawkins of the ancient world, zealous in his materialism, harsh in his criticisms of those who clung to supernatural explanations. “Nothing is ever created by divine power out of nothing,” he writes in Book I of De rerum natura.