ARCHIMIDES

By: Kallimachos Savvas Grigoropoulos

Archimedes, born in Syracuse around 287 BC, emerged as a leading mathematician and inventor during the Hellenistic period. Renowned for inventions like the Archimedes screw, he left an indelible mark on both engineering and theoretical sciences.

His mathematical breakthroughs included hydrostatic principles, allowing the measurement of an object’s volume when submerged in water, and advancements in calculating surface areas and volumes. The method of exhaustion, a precursor to integral calculus, showcased his analytical brilliance.

Archimedes’ genius extended to geometry, introducing the fundamental Archimedes’ principle in buoyancy. Surviving works such as “On the Sphere and Cylinder” reflect his profound exploration of mathematical concepts.

Beyond his intellectual pursuits, Archimedes played a crucial role in practical engineering, designing war machines instrumental during the Roman siege of Syracuse. Tragically, his life ended heroically during the sack of Syracuse in 212/211 BC, defending the city with inventive war machines. His legacy endures as a symbol of brilliance and innovation, influencing fields from mathematics to engineering.

Why Archimedes was the greatest scientist of the Classical age

A mathematician, physicist, astronomer, inventor and engineer, the ancient Greek thinker Archimedes left a legacy much greater than a story about bathtubs, yelling “Eureka” and streaking nude down the street – even if, as Jonny Wilkes explores, it took a while for the rest of the world to catch up to him

Study: Archimedes Set Roman Ships Afire with Cannons The steam cannons could have fired hollow balls made of clay and filled with something similar to an incendiary chemical mixture known as Greek fire in order to set Roman ships ablaze. A heated cannon barrel would have converted barely more than a tenth of a cup of water (30 grams) into enough steam to hurl the projectiles.

A legend begun in the Medieval Ages tells of how Archimedes used mirrors to concentrate sunlight as a defensive weapon during the siege of Syracuse, then a Greek colony on the island of Sicily, from 214 to 212 B.C.

Channeling steam power

Italian inventor Leonardo da Vinci sketched a steam cannon in the late 15th century, which he credited to Archimedes, and several other historical accounts mention the device in connection with Archimedes.

Indirect evidence for the steam cannon also comes from the Greek-Roman historian Plutarch, who tells of a pole-shaped device that forced besieging Roman soldiers to flee at one point from the walls of Syracuse.

The Greek-Roman physician and philosopher Galen similarly mentioned a burning device used against the Roman ships, but used words that Rossi said cannot translate into “burning mirror.”

Rossi calculated that such cannons could have fired a cannonball weighing roughly 13 pounds (6 kilograms) at speeds of roughly 134 miles per hour (60 meters per second). That allowed the cannons to possibly target troops or ships at distances of approximately 492 feet (150 m) while firing at a fairly flat trajectory to make aiming easier.

“As far as I know, it is the first paper about that use of a steam cannon by Archimedes,” Rossi told LiveScience.

Past investigations by Greek engineer Joannis Stakas and Evanghelos Stamatis, a historian, showed that a parabolic mirror can set small, stationary wooden ships on fire. MIT researchers carried out a similar demonstration more than three decades later in 2005.

But whether mirrors could have maintained a constantly changing curvature to keep the right burning focus on moving ships seems doubtful, Rossi noted. He added that ancient sailors could have easily put out any fires that started from a slow-burning mirror.

By contrast, Greek fire emerged in many historical accounts as a deadly threat for ancient warships. The unknown chemical mixture reportedly burned underwater, and saw most use by the Byzantine Empire that dominated the Eastern Mediterranean starting in A.D. 330. Other records mention earlier versions of the burning mixture.

Recreating the past

The steam cannons only represent the latest historical investigation by Rossi. He previously coauthored the book “Ancient Engineers’ Inventions: Precursors of the Present” (Springer, 2009), along with military historians Flavio Russo and Ferruccio Russo.

The trio plan to meet up with other historians in the future and possibly reconstruct versions of the ancient weapons. Flavio previously built several working reconstructions of ancient Roman artillery weapons, and Ferruccio specializes in 3-D virtual reconstructions of mechanical devices.

Some of Rossi’s other work looked at ancient motors that may have moved siege towers used by the Greeks and Romans. The likeliest motors may have relied on counterweights, and emerged in records as the invention of Heron of Alexandria in the first century.

Such devices could have been placed inside the protection of the towers themselves, Rossi noted. He pointed to an account by the Roman general Julius Caesar, who told of using such towers against a town defended by Gallic tribes in modern-day France. The sight of towers appearing to move by themselves frightened the defenders into negotiating for surrender.

A research paper on the siege towers was presented alongside Rossi’s recent work entitled “Archimedes’ Cannons against the Roman Fleet?” at the International World Conference held in Syracuse, Italy from June 8-10. The conference proceedings appear in a book titled “The Genius of Archimedes — 23 Centuries of Influence on Mathematics, Science and Engineering” (Springer, 2010).

In the end, the engineering talents of Archimedes did not save him from death when the Romans finally stormed Syracuse. But at least a love of history among Rossi and his colleagues may lead to the resurrection of some of his ancient devices.

It was the time of the Second Punic War and the city of Syracuse in Sicily had made the fatal mistake of siding with the Carthaginians against Rome. So, in 216 BC, a large Roman fleet under the command of Consul Claudius Marcellus set out to besiege Syracuse and bring it to its knees.

It ought to have gone smoothly and swiftly enough for the well-drilled Romans. But they had not reckoned on the intervention of a remarkable, 70-year-old man. Archimedes, the greatest (and virtually the only) Greek technologist, was a Syracusian, born and bred. And though he despised violence and had no interest at all in military campaigns, in his home town’s hour of need he turned his ingenuity upon the invaders and helped to hold them at bay for almost four years.

The Romans had never seen the like of some of the war-machines he unleashed upon them. A huge pair of pincers appeared from the city walls, grasped the hull of a ship between its jaws, shook it, and sent it crashing in ruins into the sea. Giant catapults hurled showers of heavy spears and stones onto vessels at anchor some distance from the shore. But there is one weapon that Archimedes is said to have used which is so fantastic that it defies credibility. In fairness, one of the few ancient writers to mention it is Plutarch and he had a vested interest in making Greek brains seem superior to Roman brawn. According to Plutarch, Archimedes reflected the Sun’s rays onto the Roman galleys and set them alight. Such a story, true or not, was too good for the medieval chroniclers over a thousand years later to resist. Like the tabloid journalists of their day, they embellished the tale with details of their own until history and legend became hopelessly intertwined. Some of these later writers say that Archimedes used the polished round shields of the Greek troops to concentrate the sunlight, while others insist that he focused the rays with a giant single mirror. Joannes Zonaras, a Byzantine historian of the 12th century, wrote:”At last, in an incredible manner, he burned up the whole Roman fleet. For by tilting a kind of mirror he ignited the air from the beam and kindled a great flame, the whole of which he directed at the ships at anchor in the path of the fire, until he consumed them all.

“Evidently, Zonaras had only a tenuous grasp on the principles of optics. Archimedes, on the other hand, understood reflection as well as anyone up until the time of Newton. Yet even he would have struggled to exploit it as an incendiary weapon.

One recent piece of research looked at the effect of 440 men each wielding a polished metal mirror almost three feet across. The calculations showed that even if the men worked in concert, all pointing their mirrors precisely the same way at the same time – a difficult enough feat – the best they could hope for would be to ignite a small patch of wood 50 yards away. Such a capability would hardly have struck terror into the Romans who, in any case, were not short of a drop or two of water with which to douse any smoldering timber.

Conceivably, mirrors could have been used as an anti-personnel weapon. Fifty mirrors, say, accurately trained on the steersman of a Roman galley might have given him a nasty burn, though the prospect of the entire fleet being disabled in this way seems a bit far-fetched.

Perhaps, after all, the tale of Archimedes and his military mirrors is a product of overwrought imagination. The real damage to the Roman ships, as both Thucydides and Aeneas Tacitus claimed, may have been caused by burning gobbets of sulfurpitch, and charcoal, lobbed from shore-based catapults. This could well have inspired the legend. As O. N. Stavroudis, of the University of Arizona, pointed out: “To the terrified Roman seamen such an attack would have seemed like great bolts of fire descending from the Sun itself. And the few surviving Greeks would perhaps allow, with half a grin, that indeed it

A wall painting from the Uffizi Gallery, Stanzino delle Matematiche, in Florence, Italy, shows the Greek mathematician Archimedes’ mirror burning Roman military ships. Painted in 1600 by Gieulio Parigi. (Image credit: Giulio Parigi)

Greek inventor Archimedes is said to have used mirrors to burn ships of an attacking Roman fleet. But new research suggests he may have used steam cannons and fiery cannonballs instead.

A legend begun in the Medieval Ages tells of how Archimedes used mirrors to concentrate sunlight as a defensive weapon during the siege of Syracuse, then a Greek colony on the island of Sicily, from 214 to 212 B.C. No contemporary Roman or Greek accounts tell of such a mirror device, however.

Both engineering calculations and historical evidence support use of steam cannons as “much more reasonable than the use of burning mirrors,” said Cesare Rossi, a mechanical engineer at the University of Naples “Federico II,” in Naples, Italy, who along with colleagues analyzed evidence of both potential weapons.

Channeling steam power

Italian inventor Leonardo da Vinci sketched a steam cannon in the late 15th century, which he credited to Archimedes, and several other historical accounts mention the device in connection with Archimedes.

Indirect evidence for the steam cannon also comes from the Greek-Roman historian Plutarch, who tells of a pole-shaped device that forced besieging Roman soldiers to flee at one point from the walls of Syracuse.

The Greek-Roman physician and philosopher Galen similarly mentioned a burning device used against the Roman ships, but used words that Rossi said cannot translate into “burning mirror.”

Rossi calculated that such cannons could have fired a cannonball weighing roughly 13 pounds (6 kilograms) at speeds of roughly 134 miles per hour (60 meters per second). That allowed the cannons to possibly target troops or ships at distances of approximately 492 feet (150 m) while firing at a fairly flat trajectory to make aiming easier.

“As far as I know, it is the first paper about that use of a steam cannon by Archimedes,” Rossi told LiveScience.

Past investigations by Greek engineer Joannis Stakas and Evanghelos Stamatis, a historian, showed that a parabolic mirror can set small, stationary wooden ships on fire. MIT researchers carried out a similar demonstration more than three decades later in 2005.

But whether mirrors could have maintained a constantly changing curvature to keep the right burning focus on moving ships seems doubtful, Rossi noted. He added that ancient sailors could have easily put out any fires that started from a slow-burning mirror.

By contrast, Greek fire emerged in many historical accounts as a deadly threat for ancient warships. The unknown chemical mixture reportedly burned underwater, and saw most use by the Byzantine Empire that dominated the Eastern Mediterranean starting in A.D. 330. Other records mention earlier versions of the burning mixture.

Recreating the past

The steam cannons only represent the latest historical investigation by Rossi. He previously coauthored the book “Ancient Engineers’ Inventions: Precursors of the Present” (Springer, 2009), along with military historians Flavio Russo and Ferruccio Russo.

The trio plan to meet up with other historians in the future and possibly reconstruct versions of the ancient weapons. Flavio previously built several working reconstructions of ancient Roman artillery weapons, and Ferruccio specializes in 3-D virtual reconstructions of mechanical devices.

Some of Rossi’s other work looked at ancient motors that may have moved siege towers used by the Greeks and Romans. The likeliest motors may have relied on counterweights, and emerged in records as the invention of Heron of Alexandria in the first century.

Such devices could have been placed inside the protection of the towers themselves, Rossi noted. He pointed to an account by the Roman general Julius Caesar, who told of using such towers against a town defended by Gallic tribes in modern-day France. The sight of towers appearing to move by themselves frightened the defenders into negotiating for surrender.

A research paper on the siege towers was presented alongside Rossi’s recent work entitled “Archimedes’ Cannons against the Roman Fleet?” at the International World Conference held in Syracuse, Italy from June 8-10. The conference proceedings appear in a book titled “The Genius of Archimedes — 23 Centuries of Influence on Mathematics, Science and Engineering” (Springer, 2010).

In the end, the engineering talents of Archimedes did not save him from death when the Romans finally stormed Syracuse. But at least a love of history among Rossi and his colleagues may lead to the resurrection.

Published: December 7, 2023 at 11:43 AM

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Everywhere you look in ancient Greece, you are sure to find not only pioneers, but forefathers of their fields; famous Greeks who advanced their discipline leaps and bounds, and whose influence can be felt to this day. Medicine had Hippocrates, politics had Solon, history had Herodotus, and philosophy had the likes of Socrates, Plato and Aristotle. When it comes to mathematics, one name stands above all others: Archimedes.

His discoveries and writings shaped mathematical thought for millennia, from his plethora of geometrical findings to his accurate approximation of pi. He experimented with calculus before it even existed and laid out a law of the lever that, he purportedly declared, would allow him to “move the Earth”.

Archimedes’ genius stretched far beyond theory, though. He was an inventor and engineer, who conceived of machines still in use and weapons powerful enough to give the Roman military cause to worry. That is, of course, if all the stories are to be believed.

Who was Archimedes?

As next to nothing is known about the life of Archimedes, stories written by historians long after his death comprise the only biographical information we have – other than his written works, many of which survive only in fragments.

From what can be surmised, Archimedes was born c287 BC in Syracuse, a Greek-speaking city on the south-east coast of Sicily. In one of his works, he named his father as Phidias, an astronomer, but nothing else is known about his family or upbringing.

It is possible that he travelled to Alexandria, Egypt, and studied there for a spell since he seemed to form relationships with the polymath Eratosthenes of Cyrene (chief of the Library of Alexandria, one of the Seven Wonders of the Ancient World) and the astronomer Conon of Samos, a great intellect of the city.

For nearly all of his life, Archimedes lived and worked in Syracuse, often in the employ of the king, which led to perhaps his most famous story.

What was Archimedes’ ‘Eureka moment’ and did it really happen?

According to the first-century BC Roman architect Vitruvius, Archimedes was approached by Hieron (or Hiero) II of Syracuse to determine whether the golden crown that had just been made for the king was, indeed, solid gold. Hieron suspected the goldsmith of substituting silver.

This posed a tricky problem for Archimedes, as he had to reveal the truth without causing any damage to the crown.

Initially stumped for a solution, inspiration finally came to Archimedes when he settled into a relaxing bath: observing how the water rose as he lowered himself into the tub, he realised the answer might lie in water displacement.

The amount displaced was equal to the amount of his body in the water. So astounded was he that he leaped out of the bath and ran down the street, still naked, yelling “Eureka!”, or “I have found it!”

By placing the crown into water and measuring the displacement, Archimedes could work out the volume and, therefore, the density. Gold is denser than silver, so he could compare the results to a solid lump of gold the same weight as the crown.

Eureka, it turned out that the king had been cheated.

While the story is certainly apocryphal, the great mathematician is credited with a significant discovery concerning buoyancy. He just did not necessarily make it while naked in the bath.

Archimedes’ Principle – which states that an object immersed in fluid will be buoyed upwards by a force equal to the weight of the fluid being displaced – secured his reputation as a founding father of hydrostatics, the study of fluids at rest.

What were some of Archimedes’ greatest discoveries and inventions?

Archimedes laid out his Principle on hydrostatics in his work On Floating Bodies, which survives partly in Greek and in a medieval Latin translation.

This was just one of the many solutions to mathematical questions that he got through in his illustrious career. He proved the law of the lever in On the Equilibrium of Planes, and, the work of which he was most proud – On the Sphere and Cylinder – unpicked the relationship between a sphere and a cylinder.

In geometry, Archimedes proved a number of theorems by finding the areas and volumes of an array of shapes, including cones, parabolas and spheres, using methods that would later develop into what became calculus.

The same techniques helped Archimedes reach an approximation for the value of pi between 3.1408 and 3.1429. The true value is 3.14159 (and so on).

As well as mathematics and physics, Archimedes wrote on astronomy too. In The Sand Reckoner, he attempted to calculate the number of grains of sand needed to fill the universe.

He is also credited – in the writings of the great Roman orator Marcus Tullius Cicero and mathematician Pappus of Alexandria – with a device to show the positions of the Sun, Moon and planets. This may be the legendary Antikythera Mechanism, the world’s first analogue computer (and the MacGuffin, dubbed the Archimedes Dial, in the fifth Indiana Jones movie, Indiana Jones and the Dial of Destiny).

Among Archimedes’ inventions are war machines used during the siege of Syracuse, an early odometer (a small cart where a series of gears dropped a pebble in a box every mile) and the so-called Archimedes Screw, a device to draw water upwards.

According to an anecdotal story, it had been invented in conjunction with another of his achievements: the Syracusia, one of the largest ships in antiquity.

Built for the king of Syracuse, it was large enough for more than 1,900 people and featured among its hundreds of rooms a temple, library and gymnasium. But its size increased the risk of leaks in the hull, leading to Archimedes developing the screw to remove water from the lower decks.

In truth, something similar may have existed earlier in Egypt, but it still bears Archimedes’ name and remains in widespread use around the world today.

What did Archimedes do during the Siege of Syracuse?

During the Second Punic War (218-201 BC), Syracuse incurred the wrath of Rome by switching allegiance to Carthage, resulting in what was expected to be a short and straightforward military campaign under the command of general Marcus Claudius Marcellus and Appius Claudius Pulcher.

Yet the siege of Syracuse lasted for two years, from 214-212 BC, thanks in no small part to Archimedes.

As related in The Histories by second-century BC historian Polybius, the mathematician designed a number of weapons and machines to defend the walls of Syracuse from sea attack, including improved forms of the catapult.

Chief among them was the Archimedes Claw, a crane-like contraption with a hook at the end that could be dropped onto Roman ships and lift them out of the water, and his alleged heat ray (also known as the Archimedes Death Ray).

While debate rages over whether this existed at all, the idea was to angle a series of mirrors to focus the Sun’s rays onto the wood of the enemy’s hulls, which would then catch on fire.

How did Archimedes die?

Syracuse did eventually fall to the Romans in 212 BC, while the defenders were distracted with the celebrations of a religious festival.

Marcellus had demanded that Archimedes be taken alive, knowing what his genius could be worth to Rome, but, as the story goes, a soldier approached the mathematician while absorbed in some calculations.

With his mind elsewhere, he failed to heed what the soldier was saying, and the angered Roman stabbed and killed him.

Where is Archimedes’ tomb?

Today, visitors to the Parco Archeologico della Neapolis in Syracuse will be able to visit what’s called the ‘Tomb of Archimedes’.

However, this is not where the great man was buried, but rather a Roman construction from a couple of centuries later. Archimedes’ true resting place was lost to time, neglected and left to decay.

That said, the Roman statesman and orator Cicero claimed he discovered its whereabouts while serving as an official in Sicily.

He knew it to be the right place from the stone adornments of a sphere and cylinder, placed there in honour of Archimede’s mathematical achievements. Cicero had the tomb restored, but again it would be lost over time.

How has Archimedes’ legacy changed over history?

Despite his contributions both in the realms of mathematical theory and in defensive weaponry, Syracuse quickly forgot Archimedes after his death (which goes to explain how his tomb became lost).

It is not known if he had children, or followers, but there is little evidence to suggest that anyone took up the mantle of carrying on his work.

Only in the sixth century AD did Archimedes’ legacy start to be formed when Isidore of Miletus, an architect on the Hagia Sophia, compiled his works for the first time.

Then Eutocius of Ascalon wrote commentaries on Archimedes, which were translated into Arabic in the eighth and ninth centuries, bringing the great mathematician of Syracuse to the attention of the great mathematicians of medieval Islam.

Archimedes’ reputation would flourish, making him a constant source of inspiration during the Middle Ages, the Renaissance and the Scientific Revolution.

The real Archimedes Dial? The Antikythera mechanism explained

The archaeological macguffin at the heart of Indiana Jones and Dial of Destiny is based on a very real treasure: the Antikythera mechan

A floating object’s weight Fp and its buoyancy Fa (Fb in the text) must be equal in size.

Archimedes’ principle allows the buoyancy of any floating object partially or fully immersed in a fluid to be calculated. The downward force on the object is simply its weight. The upward, or buoyant, force on the object is that stated by Archimedes’ principle above. Thus, the net force on the object is the difference between the magnitudes of the buoyant force and its weight. If this net force is positive, the object rises; if negative, the object sinks; and if zero, the object is neutrally buoyant—that is, it remains in place without either rising or sinking. In simple words, Archimedes’ principle states that, when a body is partially or completely immersed in a fluid, it experiences an apparent loss in weight that is equal to the weight of the fluid displaced by the immersed part of the body(s).

Formula

A floating object’s weight Fp and its buoyancy Fa (Fb in the text) must be equal in size.

Consider a cuboid immersed in a fluid, its top and bottom faces orthogonal to the direction of gravity (assumed constant across the cube’s stretch). The fluid will exert a normal force on each face, but only the normal forces on top and bottom will contribute to buoyancy. The pressure difference between the bottom and the top face is directly proportional to the height (difference in depth of submersion). Multiplying the pressure difference by the area of a face gives a net force on the cuboid ⁠ ⁠—  the buoyancy ⁠ ⁠—  equaling in size the weight of the fluid displaced by the cuboid. By summing up sufficiently many arbitrarily small cuboids this reasoning may be extended to irregular shapes, and so, whatever the shape of the submerged body, the buoyant force is equal to the weight of the displaced fluid. weight of displaced fluid=weight of object in vacuum−weight of object in fluid{\displaystyle {\text{ weight of displaced fluid}}={\text{weight of object in vacuum}}-{\text{weight of object in fluid}}\,}

The weight of the displaced fluid is directly proportional to the volume of the displaced fluid (if the surrounding fluid is of uniform density). The weight of the object in the fluid is reduced, because of the force acting on it, which is called upthrust. In simple terms, the principle states that the buoyant force (Fb) on an object is equal to the weight of the fluid displaced by the object, or the density (ρ) of the fluid multiplied by the submerged volume (V) times the gravity (g)[1][3]

We can express this relation in the equation:��=���{\displaystyle F_{a}=\rho gV}

where ��{\displaystyle F_{a}} denotes the buoyant force applied onto the submerged object, �\rho  denotes the density of the fluid, �V represents the volume of the displaced fluid and �g is the acceleration due to gravity. Thus, among completely submerged objects with equal masses, objects with greater volume have greater buoyancy.

Suppose a rock’s weight is measured as 10 newtons when suspended by a string in a vacuum with gravity acting on it. Suppose that, when the rock is lowered into the water, it displaces water of weight 3 newtons. The force it then exerts on the string from which it hangs would be 10 newtons minus the 3 newtons of buoyant force: 10 − 3 = 7 newtons. Buoyancy reduces the apparent weight of objects that have sunk completely to the sea-floor. It is generally easier to lift an object through the water than it is to pull it out of the water.

For a fully submerged object, Archimedes’ principle can be reformulated as follows:apparent immersed weight=weight of object−weight of displaced fluid{\text{apparent immersed weight}}={\text{weight of object}}-{\text{weight of displaced fluid}}\,

then inserted into the quotient of weights, which has been expanded by the mutual volumedensity of objectdensity of fluid=weightweight of displaced fluid{\frac  {{\text{density of object}}}{{\text{density of fluid}}}}={\frac  {{\text{weight}}}{{\text{weight of displaced fluid}}}}

yields the formula below. The density of the immersed object relative to the density of the fluid can easily be calculated without measuring any volume isdensity of objectdensity of fluid=weightweight−apparent immersed weight.{\frac  {{\text{density of object}}}{{\text{density of fluid}}}}={\frac  {{\text{weight}}}{{\text{weight}}-{\text{apparent immersed weight}}}}.\,

Authors

Jonny Wilkes

Jonny WilkesFreelance writer

Jonny Wilkes is a former staff writer for BBC History Revealed, and he continues to write for both the magazine and HistoryExtra. He has BA in History from the University of York.

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