Archimedes was what we today call a physicist. He was deeply interested in the way simple machines work: levers, balances, inclined planes. He saw the same or similar principles in operation in many different situations. Today we would say he was a master experimenter with gravity.
Perhaps his most famous discovery had to do with the specific gravity of various substances, which he rightly understood differs from substance to substance. His friend, King Hieron of Syracuse, had received as a gift a certain amount of gold. Hieron has his goldsmiths fashion it into a crown. When he received the crown, he began to suspect the goldsmiths had alloyed the original gold with silver, which of course had less value. But how was he to prove this? He asked Archimedes, who puzzled over the problem for a long time. One day, as he was stepping into the public baths of Syracuse, Archimedes slopped the water in the bath over the edge—and immediately was struck by an idea. His body displaced water from the bath; gold and silver would also displace water if placed in a full basin; and since given amounts of gold and silver were different in weight, they should therefore displace different amounts of water. Archimedes was so excited by this idea that he jumped out of the bath, quite naked (he had forgotten to put his clothes back on), and ran home to experiment, shouting “Eureka!,” which in Greek means “I have found it!”
Archimedes’ hunch turned out to be correct, and so did the king’s. When Archimedes tested the metals in a water bath he found that a certain weight of silver displaces more water from the bath than the same weight of gold, because the gold is heavier, for the same amount of volume, than the silver. Archimedes therefore carefully weighed the crown and measured out an equal weight of pure gold. The gold displaced less water than the crown; thus, the crown contained silver (or some other metal, not gold). What happened to the goldsmiths is not, as far as I know, a matter of history. They certainly came to no good.
Archimedes was intrigued by this problem, but not because of the king’s concern about his crown. Rather, Archimedes was interested in the mathematical and physical principles involved, and after further study he discovered the important principle that is named after him: that a body immersed in a fluid loses as much in weight as the weight of the fluid it displaces. He wrote a treatise on this, On Floating Bodies, and I suggest that you read Book I of this work.
Archimedes’ most important discoveries, in his own view, involved the so-called method of exhaustion, whereby a sought-after goal is achieved by a succession of approximations, each closer than the last. This method, which he carried further than anyone before him, is closely akin to integral calculus. He wrote two treatises on this subject, The Method and Measurement of a Circle. Read the latter and the first two propositions of the former.
I recognize that you may not find it easy to read these and other short works of Archimedes, either in the famous edition of Sir Thomas Heath, in the slender paperback from St. John’s College Press (Annapolis, Maryland), or in the volume of Greek mathematicians and scientists in Great Books of the Western World. But I recommend that you try hard to read one short treatise by Archimedes that is highly suggestive in its implications. Called The Sand-Reckoner, it survives in the form of a letter to one King Gelon, who has asked Archimedes how many grains of sand would be required to fill the entire universe. Archimedes takes the question seriously but immediately recognizes the fundamental problem, which is not to estimate the number of grains of sand but instead to create a number system capable of expressing such large numbers.
The Greeks, with all their wit, were crippled in their scientific work by the lack of a convenient number notation, which made it not only impossible to express large numbers but also very hard to conceive of them. The Romans were similarly hampered. The creation of a workable number system is an achievement of the past few centuries to which we owe much of the scientific progress that is our civilization’s proudest boast. But Archimedes was on the track of a workable number system nearly two thousand years before the modern scientific period.
He had made significant progress toward it, as shown in The SandReckoner, when he died. His death was a ridiculous accident. Syracuse fell to the Romans in 212 after a protracted siege. A general massacre followed, but the Roman general Marcellus gave specific orders that the great sage Archimedes should not be harmed. However, during the rout, a Roman soldier came upon an old man drawing a mathematical figure in the sand. Exactly what happened between them we shall never know, but according to a two-thousand-year-old story, the soldier asked Archimedes what he was doing. The old man didn’t look up at him, saying only, “You are standing in my light,” whereupon the soldier stabbed him. That was the demise of Archimedes.
chapter three
The Silver Age of Tyranny
Virgil tells us in the Aeneid that, after the victory of the Achaeans and the destruction of Troy, Aeneas, fruit of the illicit coupling of the divine Aphrodite and the human Anchises, escaped from the burning city. After many adventures he reached Italy, where he had been fated to found a city and eventually a new Troy. This happened at the time of the Trojan War—and many interesting events followed in the next ten centuries. Things became even more interesting during the last century before our era, and during the first after it began; and this period we know more about.
After the defeat of Carthage and the final defeat of what remained of Greece (although Greeks continued to be the teachers of the Romans), civil war broke out over the question of which should rule Rome, a Senatus of wealthy and powerful men or a Popolus of lesser but more numerous persons. (The famous motto of the Republic, SPQR, stood for “Senatus Popolusque Romanus,” which, translated, means the Roman Senate and People.) There were several factions, one led by Julius Caesar and Mark Anthony, another by Pompey, a third by Brutus and Cassius and other senators. Caesar and Pompey were both generals, but Caesar captured Pompey and killed him. Brutus and Cassius then assassinated Caesar, whereupon Antony and Octavian, the young adopted son of Caesar, and a third man, Lepidus, who was not very powerful but was very rich, “elected” themselves a Triumvirate ruler of what was by now the largest, richest empire in the world. But Anthony and Octavian could not rule together, especially after Anthony went to Egypt, fell in love with its queen, divorced his wife (who was Octavian’s sister), and married Cleopatra—incidentally obtaining control of her enormous treasure. This turned out to be just what Octavian needed, and with a ruse he trapped Anthony’s and his wife’s separate navies at Actium, a harbor on the western coast of Greece. Cleopatra abandoned her lover suddenly, for a reason that has never been clear. Anthony, defeated, committed suicide; Cleopatra did the same.