ARCHIMEDES
(c. 287 BCE–c. 212 BCE)
Archimedes has been called the greatest scientist of antiquity and, perhaps, the greatest scientist who ever lived. He lived, and he died, in Syracuse, Sicily, where it is said (said, because no one has ever found it) that his tomb was decorated with a sculpture of a sphere and a cylinder.
Perhaps this seems a ludicrously modest monument to the father of physics. We do know that Archimedes’s own favorite among his many discoveries and inventions was his elegant mathematical proof that the volume and surface area of a sphere is two-thirds that of a cylinder of the same height and diameter. But, favorite or not, does such a mundane mathematical proof support the lofty appraisal of history—that Archimedes was the progenitor of modern science? Yes.
Consider the most celebrated story about Archimedes, which comes down to us from the great Roman architect Vitruvius (c. 80–70 BCE–after 15 BCE). It seems that King Hiero II of Syracuse had paid a certain goldsmith to fashion a votive crown for a temple. The king, who had personally supplied the required gold, was concerned that the goldsmith might have purloined some of the precious metal and melted in silver to make up the volume. He asked Archimedes to determine if the golden crown had been adulterated with silver.
In the ancient world, merchants and bankers knew the weight per volume of pure gold. The standard method of solving the problem, therefore, would have been to melt down the crown and pour the metal into a mold, so that the volume could be measured precisely, and the density of metal could then be calculated. But King Hiero II insisted that Archimedes solve the problem without destroying the crown. At the time, there was simply no known way to do this. No one could imagine how to calculate the volume of an irregular object. In fact, it seemed impossible because, as everyone knew, numbers were about regular shapes and could not be applied to irregular ones.
Apparently, Archimedes had something in common with at least two other famous men, men from the future. The late nineteenth- and early twentieth-century composer Gustav Mahler and the World War II prime minister of Great Britain Winston Churchill habitually settled into a nice hot bath when they needed to think through an intractable problem. Archimedes did the same thing. This time, while taking his bath, he noticed that the water level in the tub rose as he got in. Archimedes looked at the water rise, and he suddenly realized that this effect—the volume of water displaced by the volume of his body—could be used to calculate the volume and therefore the density of the crown. He understood that water, unlike air, cannot be compressed. Because of its incompressibility, a volume of water equal to the mass of his body was displaced by his presence in that tub. Put the crown in a known volume of water, measure the volume of water displaced by the crown, and you could determine the mass of the crown. Divide this figure by the volume of water displaced, and the density of the crown would be obtained. Since pure gold is of greater density than gold alloyed with cheaper metals, the displacement would solve the problem without harming the crown.
So thrilled was Archimedes by this scientific epiphany that he leaped out of the tub, neglected to dress, and ran out into the streets shouting “Eureka!” (“I’ve found it!”). As Vitruvius relates the conclusion of the anecdote, the measurement of displaced water proved that the gold used in the crown had been, in fact, adulterated. (What consequences the larcenous goldsmith suffered as a result of this scientific discovery, we do not know.)
The Eureka! story is a good one, though we cannot know whether it is true or not. We do have a treatise by Archimedes titled “On Floating Bodies,” which describes the principle that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces, and no less a scientist—and disruptor—than Galileo believed it probable that Archimedes was able to compare the density of the crown to the density of a known sample of pure gold of equal weight. Archimedes would have balanced the two on a scale and then lowered the scale, with the two samples, into water. Any difference in density would have caused the balanced scale to tip.
Fact or fiction, the Eureka! anecdote is one of only two fragments of anecdotal biography that have come down to us concerning history’s greatest scientist. The other concerns the manner of his death. About 212 BCE, during the Second Punic War, the city of Syracuse finally fell to Roman legions after a two-year siege. A Roman legionnaire approached Archimedes, who (according to the historian Plutarch) was “working out some problem with the aid of a diagram, and having fixed his thoughts and his eyes as well upon the matter of his study, he was not aware of the incursion of the Romans or of the capture of the city.” The legionnaire demanded that he go with him to his commander, Marcellus. “This Archimedes refused to do until he had worked out his problem and established his demonstration, whereupon the soldier flew into a passion, drew his sword, and dispatched him.”
Plutarch noted that there was an alternative version of this account, which said “that the Roman came upon him with drawn sword threatening to kill him at once, and that Archimedes, when he saw him, earnestly besought him to wait a little while, that he might not leave the result that he was seeking incomplete . . . but the soldier paid no heed to him and made an end of him.”
The generations that followed Plutarch eagerly adopted both of these possible accounts as stories worthy of Archimedes’s end, for both depict him as having, in effect, died nobly for science. Sometime after Plutarch, last words were even ascribed to Archimedes. They were in Latin, because Roman authors put them into the scientist’s mouth: “Noli turbare circulos meos”—“Do not disturb my circles!”
Those “last words” make two good stories even better, but Plutarch mentions a third story, which is that Archimedes was simply robbed and killed. It is the most plausible of the tales, but the least popular, because it fails to support the same conclusion as the first two versions, that Archimedes died for science. Archimedes would have vehemently disapproved of the bias in favor of the first two versions. For the objective of science is to find truth, regardless of sentiment, rhetoric, or even the satisfaction of hearing a good story. Human beings believe or perceive many things to be true. Science and mathematics, Archimedes showed, can achieve a truth beyond human belief and perception.
• • •
What, then, do we know for certain? Archimedes was born in Syracuse in or about 287 BCE, the son of Phidias, an astronomer and mathematician. He may have been distantly related to the Syracusan king Hiero II. We also know that Syracuse at that time was a major trading town, as well as an intersection of art and science. It was a lively city, both commercially and intellectually, and therefore a wonderful place for the inquisitive son of an astronomer and mathematician to feed his hunger for problem-solving and insight. Nevertheless, it is reported that he very quickly exhausted the knowledge of the city’s local teachers. He set off for Egypt to study in Alexandria, at that time the very zenith of learning in the Hellenistic world and home of the renowned library built by Alexander the Great himself. Euclid (born 300 BCE) had died years before Archimedes arrived in Alexandria, but his Elements, the great collation of the entire sum of Greek geometry, was available in the city, and Archimedes doubtless absorbed it all before he returned to Syracuse.
From his return to his native city until his death, all that we know of Archimedes was what he left the world in the form of eleven treatises, the most important of which are preserved in The Archimedes Palimpsest, a document discovered in Constantinople in 1906 by a Danish professor of philology, Johan Ludvig Heiberg (1854–1928). More precisely, it was not the document that he discovered, but an aspect of that document. The document was a 174-page goatskin parchment of prayers, which had been written in the 13th century CE. Over some seven centuries, many had read it before Heiberg got hold of it. But no one before him had realized that it was a palimpsest—a document on which the later text (those prayers) had been written over earlier writing, which had been erased. In an era when parchment, or vellum, was scarce and very valuable, it was common practice to scrape the ink away from existing works and write on the material again.
Heiberg could barely make out what lay beneath the prayers, but he recognized just enough to conclude that it contained mathematical works by Archimedes. A few years after Heiberg’s discovery, in the 1920s, a private collector purchased the palimpsest, and on October 29, 1998, it was sold at a Christie’s auction to an anonymous bidder for $2 million. Modern technologies, including the use of ultraviolet and other light sources as well as X-rays, have made the original overwritten text as clear as it must have been in the tenth century when a Byzantine Greek scribe, working from much earlier texts, copied it down. The seven foundational treatises are
* On the Equilibrium of Planes
* On Spirals
* Measurement of a Circle
* On the Sphere and Cylinder
* On Floating Bodies
* The Method of Mechanical Theorems
* The Ostomachion
Although most of what was discovered in the palimpsest had been known in less complete versions derived from Archimedes during Roman times and in the Renaissance, the original document contains the sum and substance of all that he introduced into the world:
• The laws of levers and pulleys, which proved fundamental to civilization as a means of moving heavy objects using small forces. In these laws, Archimedes provided the key to multiplying human strength exponentially for the purposes of building and—in a military context—destroying. Archimedes’s genius was in discovering and formulating the principles of physics and taking them from the realm of theory to application in a variety of machines. Although Plutarch and others make mention of numerous “engines” of war (catapults and the like), the most celebrated of Archimedes’s machines is the Archimedean Screw, which enables water to be extracted from the ground. The principle and the device itself are still in use today.
• The concept of the center of gravity. Among the foundational concepts of physics, calculations employing the center of gravity are essential to engineering; the design of all kinds of vehicles; aeronautics and astronautics; and the study of astronomy and the solution to all problems relating to bodies in motion.
• The most precise calculation of the value of pi prior to the invention of electronic calculators and digital computers. Through the millennia, applications of pi—the ratio of a circle’s circumference to its diameter—have emerged in every field of endeavor involving mathematics. It is among the numbers basic to civilization itself.
• Mathematical proofs for formulas used to determine the volume and surface area of a sphere. Archimedes made the connection between mathematics and the physical world seamless, thereby establishing that the universe is capable of being described mathematically. This is the foundation for classical physics and also, even more, for modern physics—the physics of Einstein, Planck, and Heisenberg, among numerous others.
• The use of exponents to express numbers far greater than had ever been imagined before. Archimedes gave untold generations of mathematicians, scientists, statisticians, engineers, and social and political leaders the tools to imagine, contemplate, and manipulate vast numerical values. He also provided the mathematical proof that to multiply numbers written as exponents, the exponents had to be added together, thereby greatly facilitating calculations using very large numbers. This is analogous in the realm of mathematics to his work with levers and pulleys in the physical world. As pulleys exponentially multiply the physical power of human beings, so exponents exponentially multiply their intellectual power.
Archimedes was the first physicist, in that he applied an advanced mathematics (largely of his own creation) to describe and manipulate the physical world. At the same time, he also reversed this process, becoming the first physicist to import lessons from physics—the law of leverage, for example—back into pure mathematics in order to solve problems in that intellectual realm. Thus, Archimedes effectively wedded the universe of mind to the physical universe of matter and energy. Without this union, our modern world and everything in it—virtually every invention, every item of technology—would be utterly unimaginable and therefore simply impossible.
This can be seen most explicitly in the achievements of Leonardo da Vinci, Galileo Galilei, and Isaac Newton, all of whom built key insights directly on the work of Archimedes. As Marshall Clagett (1916–2005), the foremost American historian of medieval science, wrote: “The importance of the role played by Archimedes in the history of science can scarcely be exaggerated. . . . His name appears on the pages of the works of the great figures that fashioned the beginnings of modern mechanics. . . . Galileo mentions Archimedes by actual count over one hundred times and in almost Homeric hyperbole. . . .”
Clagett credits Archimedes with having invented “a variety of machines and fields of science like statics, hydrostatics, combinatorics, and mathematical physics.” In the “rebirth” of ancient learning that gave the Renaissance its name, Archimedes was not merely a source of knowledge, he was the model of a modern scientist—a human being who dares to challenge mere faith with mathematics, a symbolic logic that moves human expression beyond the limitations of language that relies on the distortions wrought by assertion, intuition, belief, faith, hope, desire, fear, religion, and political coercion.