Archimedes, the most famous mathematician of classical antiquity, was killed in 212 BC, as a small piece of collateral damage in the Roman sack of the Greek city of Syracuse. Syracuse itself was a rather larger piece of collateral damage, having picked the wrong side in Rome’s second war with Carthage. It was not a good year for the ancient Greek cities of the western Mediterranean. Hannibal, still prowling around southern Italy picking off Roman allies, attacked the city of Tarentum. The operation was botched. The Roman garrison held onto the citadel and then itself sacked the lower city. As the Second Punic War drew to a close, Rome was poised to leap across the Adriatic. By the middle of the second century Carthage and Corinth would both be smouldering ruins. Archimedes’ contemporaries must have felt the net already closing around them.
Archimedes’ death is the one fixed point in his biography: the 12th-century Byzantine scholar John Tzetzes gives a birth date of 287 BC. That may be correct, and it may also be correct that his father was an astronomer. But everything else is uncertain. We can, however, say something about the world he lived in. Syracuse and Tarentum were among the greatest of the dozens of Greek cities founded in the eighth century bc and after on the southern Italian coast, in Sicily and as far afield as southern France. By the fifth century, these cities were famous for their wealth, their wars against various barbarians – Etruscans and Lucanians, Gauls and Bruttians as well as Carthaginians and Romans – and each other. Treasuries were built at Delphi to house offerings from western cities. Pindar’s victory odes celebrate many western victors at the Olympics and other sacred games where the Greeks gathered. Thucydides’ account of the fall of Athens culminates in the folly of the democracy in insisting on war with Syracuse while Sparta remained undefeated.
The Greek cities of the west were also renowned as the home of tyrants, rich kings whose cruelty often took bizarre forms. Phalaris had a bronze bull in which his enemies were roasted alive, and it was a Sicilian tyrant who allegedly hung a sword on a slender thread above the head of the courtier Damocles, to show what it was like to be a ruler living in constant peril. And from the same cities came a series of astonishing intellectuals. Pythagoras was born in Samos, but moved to the Italian city of Croton in the late sixth century, where his followers formed some kind of religious sect. Parmenides and his student Zeno, of the paradoxes, came from Elea. One of the greatest early mathematicians was the Pythagoraean Archytas of Tarentum, who served his city as a general and whose work was an important influence on Plato. And then there was Archimedes.
Archimedes was not among the first generation of Greek mathematicians, but became the most celebrated in antiquity. Tradition had him as a relative of the royal house of Syracuse, but all the stories about his life cast him in the more conventional role of learned courtier. The Greek world he was born into was now full of kings, most of them descended from Alexander the Great’s generals, who had divided up his empire. One of the ways these semi-barbarous Macedonian warlords made themselves seem more regal was to patronise scholars, just as Alexander had patronised his teacher, Aristotle. Alexandria, the new capital of Egypt, had the greatest concentration of scholars, including the mathematicians Euclid, Eratosthenes, Conon, Apollonius and Diocles, with several of whom Archimedes corresponded. It’s quite possible that he himself studied in Alexandria – ancient intellectuals moved around a lot – but there is no proof either way. Instead, what we have is a series of stories all set in Syracuse, many revolving around services he performed for the king. Later traditions represent him in terms of the (already) stock figure of the distracted philosopher. The story that he ran naked down the street shouting ‘Eureka! Eureka!’ is a case in point. Some ancient writers even claim that Archimedes regarded his most famous feats of engineering as mere trifles. If so, he was in a minority of one.
It was his war machines above all that gripped the imagination both of his fellow citizens and his city’s enemies. The Roman siege of Syracuse, like the Athenian siege two centuries earlier, was a protracted affair. From the fifth century, major cities had begun to develop extensive fortifications. Siege technology was developed to counter this, and an arms race began. When the Athenians besieged Syracuse, the main action revolved around the building of walls and counter walls. During the fourth century, however, more and more complex means were developed to undermine or breach walls, to keep defenders away from the ramparts, to protect sappers and soldiers and to bombard or batter fortifications. Sicilian tyrants played a part in this: Dionysius I used siege towers and artillery against the Carthaginian town of Motya. Accounts of sieges became a staple of historiography. As well as Thucydides’ dramatic account of the siege of Plataea, or Caesar’s of his campaigns at Alesia, there are surviving manuals of siegecraft, some of which collect siege accounts from earlier histories.
The war machines Archimedes built to see off the Romans seem to have been unlike anything seen before: they were credited with delaying the fall of Syracuse for nearly two years. Naturally, it is difficult to be sure of their exact impact. Was the enemy’s possession of a military genius a convenient alibi for the Roman general Marcellus as the siege dragged on? Some of the stories – for example, that giant reflective mirrors were used to burn up enemy vessels – are implausible. Perfectly plausible, however, are the stories of a device known as the Claw. Accounts vary, but in essence this seems to have consisted of grappling hooks mounted on a heavy beam which could snatch hold of entire ships, and by the use of counterweights lift them into the air and smash them against the cliffs or drop them in the sea. The same (or similar) devices were used to snatch up clusters of men and swing them over the city walls, where the defenders would finish them off. Counterweights were well known in other spheres, and siege warfare already employed swinging beams. Several versions survive of Archimedes’ use of levers and pulleys to launch an enormous vessel. While military innovation seems for the most part to have progressed by imitating and adapting existing technology – quite often the enemy’s – Archimedes is presented as working from first principles. The ancient world was not short of terrifying siege machines, and it was not at all unusual for them to have nicknames, but the speed and power of Archimedes’ ‘toys’ was exceptional. Roman writers clearly admired him, and Marcellus is reputed to have claimed he tried to save Archimedes when Syracuse finally fell.
Archimedes has had a double afterlife. First, there are his works. On Spirals, On Spheres, On Conoids and Spheroids and On the Quadrature of the Parabola are among those that survived. On Plane Equilibriums applies geometry to mechanical problems; On Floating Objects deals with hydrostatics. By constructing polygons within a circle he calculated that the value of pi lay between 3 1/7 and 3 10/71. He is also credited with inventing what became known as the ‘Archimedean screw’, in which a corkscrew is turned within a tube to bring water up from a lower to a higher level.
Archimedes excelled within a tradition of Greek science that wasn’t opposed to experiment. Alexandria, where Euclid had created the foundations of classical geometry and where Hero built his steam engine, was one of the few places in antiquity where medical, specifically anatomical, knowledge advanced by experiment rather than by observation alone, and Archimedes’ surviving texts show his wide awareness of many fields of scientific investigation. He knew that Aristarchus had argued for a heliocentric universe, in which the Earth was just one planet circling the Sun. His Alexandrian contemporary Eratosthenes measured the curvature of the Earth with some precision by measuring the shadow cast by a post at the same moment the Sun was visible in the bottom of a distant well located on the same line of longitude.
The more abstruse problems that Archimedes set out to solve had been set by earlier geometricians. No doubt he expected his own work to be read and responded to by his contemporaries and successors. Ancient mathematics had, from the very first, a shared technical language; there were conventions about how to lay out proofs. The preface to On the Quadrature of the Parabola is addressed to a fellow mathematician, Dositheus, and begins by consoling him on the loss of a mutual friend, Conon. Archimedes had intended to send Conon the proof of a theorem that he had first devised using a mechanical method Conon had advocated. That he now sent it to Dositheus suggests a community of scholars maintaining contact by the exchange of ideas, drafts and personal reminiscences.
Archimedes’ works were read and commentated on by successive generations of Alexandrine mathematicians. They were as difficult for most ancients to follow as they are for most of us, and were probably read by very few until the early Byzantine period. (Euclid’s Elements were much more widely known as they had a place in the educational canon.) Yet, thanks to Byzantine editions and Arabic translations, a proportion of Archimedes’ oeuvre has survived. Research continues into his writings, notably in Baltimore, where a palimpsest containing two of his works overwritten by a medieval prayer book has been intensively studied for a decade now by Reviel Netz and his colleagues. The next decade is likely to see a major surge in Archimedean scholarship, with their completion of an authoritative translation and edition which will include the new material.
Yet alongside Archimedes’ conventional scientific legacy, there is a second afterlife composed primarily of anecdotes. It’s almost as if one were to reconstruct Einstein’s life and personality from a collection of posters, T-shirts, pop songs and quotations. Einstein in the modern imagination would be wise, irreverent, Jewish, a genius (of course), with an idiosyncratic appearance (of course), linked for ever with E = mc2 and (perhaps paradoxically) the most famous quantum-theory sceptic. None of this is false, but it hardly amounts to a coherent life story; it would be as difficult to read against the General and Special Theories of Relativity as Eureka-man is to read against On Conoids and Spheroids. It is on this anecdotal Archimedes that Mary Jaeger fastens her attention in Archimedes and the Roman Imagination, painstakingly reinserting each story about him into the text from which it is usually extracted and asking how it fitted the aims of the work, the interests of the author and the wider Roman (rather than modern) imagination. For Jaeger’s analysis it is the use Cicero makes of Archimedes that is crucial in shaping his later reception and the subsequent retellings of his fragmented story all the way to Petrarch. En route we encounter the Augustan architect Vitruvius, the Greek philosopher-biographer Plutarch, the fourth-century ad astrologer Firmicus Maternus and various other witnesses. For almost none of these was the actual mathematical work of Archimedes of much interest.
Archimedes has walk-on parts in various narratives of Rome’s second great war against Carthage. Polybius, Livy, Silius Italicus (who made an epic poem out of the war) and Plutarch (who wrote a biography of Marcellus) all dealt with Archimedes as cunning artificer and accidental victim of the Roman troops. So, in passing, did Cicero. The facts vary little. Archimedes builds his engines, the Romans are terrified and hold off, they prevail nonetheless, and in the chaos that follows Archimedes is killed, while engaged on his geometry. Sometimes the distracted scientist, sometimes the nutty professor, in each version he dies a mathematician rather than an engineer.
Jaeger explores the many differences in nuance. Plutarch, being both Greek and a Roman citizen, must reconcile his admiration of Marcellus the philhellenic Roman with his admiration of Archimedes the Greek intellectual. The death is a tragic accident that Marcellus strove to avoid. As a philosopher and a Platonist, Plutarch also emphasises Archimedes the mathematician, hence the apparently dismissive way he has Archimedes approach his siege engines. Silius the poet will use Archimedes’ death to tragic effect, Valerius the moralist strongly hints that a little less academic monomania might have saved him, and so on. Only very occasionally can these anecdotes be linked to Archimedes’ work as it is known from his texts. What Archimedes ‘found’ in the eureka story was the utility of specific gravity to solve the king’s problem – how to establish whether the gold in his crown had been adulterated without destroying it. Realising that the volume of the crown could be estimated from the amount of water it displaced was useful only when combined with the realisation that the same volume of different metals weighs different amounts. Yet it is the naked mathematician we remember.
One striking feature of many of the shorter pieces of testimony is that Archimedes needs no introduction. Silius does not even name him, but refers obliquely to his inventions and his drawing of geometrical figures in the dust. Classicists sometimes write as if a relay race played by literary texts was the primary means through which the past was remembered. In some cases this was probably true, but the past was constituted by much more than a chain of intertexts: there was also a common stock of knowledge, common at least to the educated, which was passed on orally or rehearsed endlessly as stock examples in speeches practised and performed at school as well as in public. Archimedes, like Einstein or Darwin today, had entered this common stock. Like them, he was known through anecdotes by far more people than had ever read his forbiddingly titled works. Indeed many of the stories present his mathematical work as so remote from everyday experience that it was virtually unknowable.
Cicero is an exception. No mathematician himself – very few Romans were – he nevertheless knew a certain amount about Archimedes’ real interests. There were personal connections too. Cicero’s first overseas command had been in Sicily, and he afterwards made his name as an orator prosecuting a rapacious Roman governor of the island. Already fascinated by Greek culture (he had studied in Athens and on the island of Samos), he had taken advantage of his posting to visit Syracuse, where (according to his own account) he had rediscovered Archimedes’ tomb, recognising it by the sphere and cylinder set up as a memorial. How much can this memoir be trusted? Cicero wrote down the anecdote late in life, when returning to intellectual pursuits after an at first successful and then disastrous political career that had left him in effect an exile in his suburban villa. Engaged in creating a native Roman philosophy in Latin, Cicero had other reasons to accuse the Greeks of neglecting their native talent. His salad days in Sicily were long past. Besides, the story of dumb and feckless natives not knowing the meaning of the monuments among which they live is all too familiar. Romans in Greek lands wondered at the past and disparaged the present. Another Roman general, Sulla, sacking another Greek city, Athens, called off the assault in the end, saying: ‘I spare the few for the sake of the many, the living for the dead.’
All the same, Cicero’s recognition of the sphere and cylinder shows some deeper knowledge of a less stereotypical Archimedes. A passage in the Republic goes even further, recalling that among the spoils of Syracuse, Marcellus brought back two spheres, one a solid celestial globe based on a prototype invented hundreds of years before, the other a sort of orrery, devised by Archimedes himself, a mechanical model of the heavens in which the planets could be moved against the fixed stars. The uses Cicero makes of the story of the two spheres might seem rather banal and introverted, a set of metaphors for new v. old knowledge, or everyday knowledge v. elite understanding. Although he understood Archimedes’ achievement this failed to excite him enough to make him engage with the science involved.
Yet there is something fascinating and not at all implausible about the idea that Archimedes made a mechanical model of the heavens. Babylonian science had long before identified the seven planets, literally the wanderers, comprising the Sun and Moon, Mercury, Venus, Mars, Jupiter and Saturn, the only bodies that – before Galileo turned his telescope on the heavens – could be observed to move back and forth against the fixed stars. The seven days of the Hebrew creation reflect this knowledge, and astrology made use of it across the Mediterranean world in Archimedes’ day, and even more in Cicero’s. Could a physical model really have been made? No spherical orrery has ever been recovered from antiquity. But more than a hundred years ago, sponge-divers discovered the wreck of a ship dating to the early first century bc – to Cicero’s lifetime, that is – lying off Antikythera at the southern tip of Greece. The ship was laden with Greek bronzes, presumably either plunder or artworks purchased by Roman connoisseurs in Italy. Among the statues was an enigmatic metal object made up of more than 30 fused gear-wheels. The Antikythera mechanism has been restored, scanned and X-rayed – the latest paper on it was published in Nature in July 2008. It is clear that it was, at least in part, a device that could model the movements of the heavens, and relate them to precise historical chronologies. No text describes it, and we don’t even know where it was made – Rhodes and Alexandria are the current best guesses – or who designed it. But it gives a tantalising glimpse of that rich scientific tradition to which the historical Archimedes belonged.
Compared to this, the imaginary Archimedes seems all the more impoverished. Stripped of his interests in mechanics and hydraulics because Plutarch felt that philosophers should not be interested in mere technical matters; rendered eccentric because great minds are like that; attributed with eureka moments of brilliant inspiration because solid mathematics is an unglamorous activity. One last story makes this point neatly. Euclid was once asked by Ptolemy, king of Alexandria, for an easier way to learn mathematics. ‘There is no royal road to geometry,’ he replied. At least in this anecdote the scientist has the last word.