Archimedes: An Ancient Greek Genius Ahead of His Time

Archimedes: An Ancient Greek Genius Ahead of His Time

Archimedes was a Greek mathematician, scientist, mechanical engineer, and inventor who is considered one of the greatest mathematicians of the ancient world. The father of simple machines, he introduced the concept of the lever and the compound pulley, as well as inventions ranging from water clocks to the famous Archimedes screw. He also designed devices to be used in warfare such as the catapult, the iron hand, and the death ray.

The Life of Archimedes: Syracuse and Alexandria

Born in Syracuse on the island of Sicily in 287 BC, Archimedes was the son of an astronomer and mathematician named Phidias. Very little is known about his family, early life, and schooling other than that he was educated in Alexandria, Egypt - the chief center of Greek learning at that time. Alexandria is where Archimedes studied with disciples of Euclid, a famous Greek mathematician, before he returned to Syracuse for the remainder of his life.

In the third century BC, Syracuse was a hub of commerce, art, and science. The ancient Greek biographer, Plutarch, mentions that while in Syracuse, Archimedes offered his services to King Hiero II. It was due to his relationship with the king, and his son Gelon, that Archimedes achieved fame.

Engraving of Archimedes (1584). ( )

The Archimedes Screw

Archimedes is best known for his inventions created during the reign of King Hiero II, such as the Archimedes screw . Originally developed by the ancient Egyptians, it was a device used to raise water from a lower to higher level. Archimedes improved upon that creation.

The machine consists of a hollow tube with a spiral that can be turned by a handle at one end. When the lower end of the tube is placed in the hull and the handle turned, water is carried up the tube. Today, the Archimedes screw is still in use as a method of irrigation in developing countries. It is also used to lift loose materials, such as grains.

The Archimedes Screw.

War Comes to Syracuse and Archimedes' Inventions Help Protect the City

Situated between Rome and Carthage during the Punic wars (264 BC to 146 BC), Syracuse proved to be in the way of Roman expansion. In 214 BC, pro-Carthaginian factions within the city sided with Carthage against Rome. Not long after this, the Roman army sailed to Syracuse with the intention of destroying the city.

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Archimedes helped repel the Romans with his brilliant inventions. He fortified the city walls with military contraptions such as catapults and ballistas, which could fire projectiles long distances and attack enemy ships. These weapons were used in battle and enabled Syracuse to hold out against Rome for about three years.

One of the most famous machines invented by Archimedes and used against Roman ships during the siege of the city was the stone throwing crane. Consisting of a rotating beam that sat on a platform, it had a counterweight at one end (i.e. a large stone) and it was suspended by a rope at the other end. As an enemy ship approached the wall, operators of the device released the winch, enabling the load to pass over the wall by rotating the balance beam. When the load hovered over the ship, the rope was cut so that it would fall and cause substantial damage.

The Claw of Archimedes

A similar invention was the Claw of Archimedes , also known as the Iron Hand. A kind of ancient crane, it had a metal grappling hook at the end of it, could reach over city walls, grab enemy Roman ships, and destroy them on the rocks. The Claw of Archimedes was reportedly used in defense of Syracuse - although no one knows exactly how it looked. Later Greek and Roman historians, such as Plutarch, Polybius, and Livy spoke of the device in their writings. Here is a description of the Claw, from Plutarch’s Lives:

At the same time huge beams were run out from the walls so as to project over the Roman ships: some of them were then sunk by great weights dropped from above, while others were seized at the bows by iron claws or by beaks like those of cranes, hauled into the air by means of counterweights until they stood upright upon their sterns, and then allowed to plunge to the bottom, or else they were spun round by means of windlasses situated inside the city and dashed against the steep cliffs and rocks which jutted out under the walls, with great loss of life to the crews. Often there would be seen the terrifying spectacle of a ship being lifted clean out of the water into the air and whirled about as it hung there, until every man had been shaken out of the hull and thrown in different directions, after which it would be dashed down empty upon the walls.

The Archimedes Claw lifting a ship (1599), Giulio Parigi. ( )

Archimedes' Most Controversial and Terrifying Invention: The Death Ray

The most controversial and arguably the most terrifying of Archimedes' inventions was his famous heat or death ray. Sometimes referred to as “the burning mirror”, it was supposedly a device which used mirrors on the bluffs of Syracuse to focus sunlight onto wooden ships, causing them to burst into flames.

The device consisted of a large array of bronze or copper shields arranged in a parabola. When the Roman fleet approached, legend says that Archimedes set fire to the enemy galleys using this weapon. However, historical accounts of this death ray did not appear in texts until much later, and it is not mentioned by the ancient historians of the era.

Contemporary writers such as like Plutarch, Polybius, and Livy did not mention the use of mirrors setting ships on fire , though they did discuss some of the defensive devices created by Archimedes.

Illustration of the Burning Mirror setting a ship on fire

The earliest source which mentions Archimedes employing burning mirrors was written by Anthemius of Tralles in 500 AD, some 700 years after the fact. In his treatise entitled, On Burning-Glasses , he mentioned how Archimedes may have used a parabolic mirror to focus the sun’s rays on invading Roman ships.

Lucian (120-180 AD) and Galen (130-200 AD) reported that Archimedes set fire to Roman ships through artificial means, but they don’t exactly state how. In 1100 AD, the writers Zonares and Tzetzes quoted use of the death ray from an earlier work (now lost) called the Siege of Syracuse , which stated:

When Marcellus [The Roman General] had placed the ships a bow shot off, the old man [Archimedes] constructed a sort of hexagonal mirror. He placed at proper distances from the mirror other smaller mirrors of the same kind, which were moved by means of their hinges and certain plates of metal. He placed it amid the rays of the sun at noon, both in summer and winter. The rays being reflected by this, a frightful fiery kindling was excited on the ships, and it reduced them to ashes, from the distance of a bow shot. Thus the old man baffled Marcellus, by means of his inventions.

What is known is that the principles of Archimedes' heat ray are understood today and it is possible to replicate the burning mirror using modern technology. Possibly, Archimedes would have known about these same principles when he was alive, but whether or not he could have actually built such a weapon is a different story. Interestingly, the television show MythBusters devoted three episodes to testing the myth of the Death Ray by using 500 large, flat, modern mirrors. In all three episodes it was seen as implausible.

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"Do Not Disturb My Circles"

Archimedes died in 212 BC at the age of 75, when Syracuse was captured by Roman forces. Legend says that he was working on a math problem when a Roman soldier ordered him to meet with his commander. Archimedes reportedly refused to do so - which angered the soldier who killed Archimedes on the spot.

His last words were reportedly “do not disturb my circles.” Cicero described visiting the tomb of Archimedes , which he said was surmounted by a sphere and a cylinder, representing Archimedes' mathematical discoveries.

Death of Archimedes (1815) by Thomas Degeorge. ( )

The Archimedes Palimpsest

Many say that Archimedes' death brought an end to a golden age of mathematics. His writings were seen as the definitive texts on geometry at the time and held in an almost religious light. Greek mathematics gradually declined with the Dark Ages and an interest in mathematics was lost until the Renaissance.

While the originals have long been lost, many of Archimedes’ writings survived and were copied by scribes who passed his work on down through the generations. In the 10th century, one copy of his most important work, called the Method of Mechanical Theorems was made.

However, in the 1200s, a medieval scribe ran out of parchment and recycled the 300-year-old pages into a book of prayers. He trimmed the parchment, erased the text, and turned the sheets at a right angle before inking on the prayers.

Called a “ palimpsest”, the manuscript began its new life at the Mar Saba monastery in the Judean desert in the Middle East, where Archimedes' work was unread and unknown for centuries. The palimpsest would turn up again rather mysteriously in a library in Constantinople in 1906, before it went missing again - until it was auctioned off in New York in 1998. Today, this book is the sole surviving source for two works by Archimedes, which are now fully legible through imaging technology.

Archimedes' Palimpsest. ( )

Modern Controversy

In a rather unexpected turn of events, in 2017 a statue of Archimedes was deemed by some as inappropriate and of ‘bad taste’ . As you’ve just read, Archimedes was a talented mathematician and inventor, so what could be the issue?

One word: Nudity.

Concerns have been raised against that the statue residing in a village in Hampshire, England. Specifically, it is argued that the Archimedes statue, “[…] is illuminated at night and as such is both a potential distraction for drivers of vehicles driving down College Lane and again completely inappropriate in a rural area in my opinion. The very nature of the statue (a naked man) may seem to represent art to some but could also be seen as offensive to others.”

I bet Archimedes would have never predicted his appearance could be so unsettling.


Archimedes (287-212 BCE) was a Greek mathematician and mechanical engineer, a pioneer in both fields, many centuries ahead of his contemporaries. Today he is best known for formulating Archimedes' Principle, also known as the law of buoyancy, but he observed many other laws of physics and recorded his observations as mathematical theorems.

His works can be categorized into three groups:

  1. Works that prove theorems related to solids and areas bounded by curves and surfaces.
  2. Works that analyse problems in statics and hydrostatics from a geometrical viewpoint.
  3. Miscellaneous works, including some that emphasize counting, such as The Sand Reckoner.

Historical Context

Archimedes' success in applying his mathematical knowledge to weapons of war played a major role during the war between Rome and Syracuse during the Second Punic War. The development of this conflict can be traced back to around 290 BCE when the Romans became the new rulers of central Italy and began to conquer the Greek cities on the Italian coast. In 270 BCE Hiero II (308-215 BCE) became king of Syracuse, located on the island of Sicily, and the city enjoyed a last period of prosperity. In Sicily, Romans and Carthaginians were brought face to face and in 264 BCE, the First Punic War started. The Carthaginians were the masters of the sea, so the Romans relied on help from the Greek cities in the south in order to build their own ships and so were able to fight the Carthaginians at sea. In 241 BCE Rome defeated Carthage and took over Sicily. During his reign, Hieron II remained on peaceful terms with the Romans and when Rome took over Sicily after the First Punic War, Syracuse remained independent.


In 218 BCE the Second Punic War started this was the second major war between Carthage and Rome. In 215 BCE, Hiero II died and his successor Hieronymus made a very poor decision by switching sides and supporting Carthage: He felt the Romans would lose the war. The Romans were not happy about this decision, and they made it clear by besieging the city of Syracuse from 214 to 212 BCE. In the end, the Romans entered the city, slaughtered and enslaved its citizens, and sacked it.

During the time of Archimedes, the centre of Greek culture was Alexandria, the greatest centre of scholarship at this time. Here Archimedes, son of an astronomer named Phidias, received the finest training available in several disciplines, including mathematics under the successors of Euclid. Archimedes' devotion to mathematics has been compared with that of Newton's both often neglected food, drink, and even the basic care of their bodies in order to continue studying mathematics. Plutarch wrote on Archimedes some three centuries later:


It is not possible to find in all geometry more difficult and intricate questions, or more simple and lucid explanations. Some ascribe this to his natural genius while others think that incredible effort and toil produced these, to all appearances, easy and unlabored results.

(Durant, 629)

Archimedes' Principle

Like all important figures in antiquity who were supremely talented, his story became filled throughout the centuries with many myths and other non-historical accounts to sustain his specialness. One of the first details we read about Archimedes in almost every account of his life is the famous scene where he runs wet and naked through the streets of Syracuse shouting “Eureka!, Eureka!” (“I have found it!”). This famous incident started with a gold crown made for Hiero II. The king suspected that the artisan might have kept for himself some of the gold provided for the task and replaced it with a mixture of gold and materials of lower quality. The king wanted to know whether the artisan replaced the gold, but he wanted to find out without damaging the crown, so he requested that many experts test the crown without damaging it.

We are told that Archimedes was among those experts and after several weeks thinking about the matter, he found the answer while stepping into a tub at the public baths. He noticed two things first, that the water overflowed in accordance with the depth of his immersion, and second, that his body appeared to weigh less the deeper it was submerged. Upon this revelation, if we are to believe the legend, Archimedes rushed off down the streets of Syracuse, presumably naked and wet, shouting in excitement that he had found the answer to the king's question.

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Archimedes' Principle, also known as the law of buoyancy, states that any object fully or partially immersed in a fluid will experience an upward force equal to the weight of the displaced fluid. This principle offered Archimedes a test for the material make-up of the crown. Back home he discovered that a given weight of silver, when immersed, displaced water which was more than an equal weight of gold. The reason for this is that silver has more volume per weight in comparison to gold. He then proceeded to submerge the crown and compared the water displaced by it with a quantity of gold equal to the crown in weight. Archimedes concluded that the crown was not made entirely of gold, confirming the king's suspicions, and so he was able to tell exactly how much gold was missing.

Other Discoveries

In his work On the Measurement of the Circle, Archimedes arrives at the logical conclusion that the ratio of a circle's circumference to its diameter, the mathematical constant we today call “pi” (π), is greater than 3 1/7 but less than 3 10/71 a very good approximation.


In a lost treatise which we know only through summaries, Archimedes formulated the Law of the Lever and Balance. He did it so accurately that no advancement was made until the 16th century CE. He also discovered the benefits of the pulley for lifting large weights. He was so amazed by the mechanical advantages provided by both the lever and the pulley that he famously stated, “give me a place to stand, and I will move the Earth”. King Hiero challenged Archimedes to put his claim to the test, so Archimedes arranged a cleverly designed series of cogs and pulleys in such a manner that he alone, sitting on one end of the mechanism, managed to draw a fully loaded vessel out of the water and place it onto the land, a task that a hundred men could barely accomplish.

Despite all of the physical laws he discovered, Archimedes never actually referred to them as laws, nor did he describe them in reference to observation and measurement he instead treated them as pure mathematical theorems, within the logic of a system similar to the one Euclid developed for geometry. Greek science during Archimedes' day had a tendency to undervalue observations and favour logical arguments: Greeks believed that the highest knowledge was based on deductive reasoning. This, however, did not prevent Archimedes from experimenting in fact, he stands out from his contemporaries because he successfully applied his theoretical knowledge into practice. But the way he presents his discoveries is always from a mathematical perspective, and he never attempted to offer a systematic description from an engineering viewpoint. Moreover, when he refers to mechanical experiments he is actually using them to help the understanding of mathematics. This shows a key difference in approach between ancient science, where experimentation was used to help theoretical understanding, and modern science, where theory is used to pursue practical results.

Death & Legacy

After the death of Hiero II, war began between Syracuse and the Romans. The city was attacked by both land and sea. 75 years of age were no obstacle for Archimedes in playing a central role defending the city. Applying his skills as an engineer, he developed and arranged catapults that hurled heavy stones to a great distance, pierced holes in the city walls for bowmen to shoot their arrows, and set up cranes that were able to release a large weight of stones on the Roman ships when they came within reach. These inventions were so effective that Marcus Claudius Marcellus, the Roman commander, abandoned the idea of attacking Syracuse and decided that a siege was the only way of breaking the city. In 212 BCE, the starving city surrendered and the Romans captured Syracuse.


Marcellus was so impressed by the genius of Archimedes that he ordered that the talented Greek should be captured alive. Nonetheless, when the Roman soldiers located Archimedes, he was on the beach drawing geometrical figures in the sand and working on one of his many theorems. He ignored the soldiers' orders and requested some extra time to finish his work. The furious soldiers, probably feeling a little insulted, immediately killed one of the greatest minds of all history.

Archimedes died, but his ideas could not be killed, and Archimedes' works, after many adventures and translations during the Middle Ages, have survived in an accessible form. During the Renaissance, the work of Archimedes gained a wide interest in the developing scientific movement. Galileo was very interested in Archimedes due to the application of mathematics to physics. observation of the Heavenly bodies, and many of his clever experiments. The western world would have to wait until Leonardo Da Vinci to see a greater mechanical genius.

Quick Guide – Archimedes’ Greatest Achievements

In the 3rd Century BC, Archimedes:

• invented the sciences of mechanics and hydrostatics.

• discovered the laws of levers and pulleys, which allow us to move heavy objects using small forces.

• invented one of the most fundamental concepts of physics – the center of gravity.

• calculated pi to the most precise value known. His upper limit for pi was the fraction 22 &frasl7. This value was still in use in the late 20th century, until electronic calculators finally laid it to rest.

• discovered and mathematically proved the formulas for the volume and surface area of a sphere.

• showed how exponents could be used to write bigger numbers than had ever been thought of before.

• proved that to multiply numbers written as exponents, the exponents should be added together.

• infuriated mathematicians who tried to replicate his discoveries 18 centuries later – they could not understand how Archimedes had achieved his results.

• directly inspired Galileo Galilei and Isaac Newton to investigate the mathematics of motion. Archimedes’ surviving works (tragically, many have been lost) finally made it into print in 1544. Leonardo da Vinci was lucky enough to see some of the hand-copied works of Archimedes before they were eventually printed.

• was one of the world’s first mathematical physicists, applying his advanced mathematics to the physical world.

• was the first person to apply lessons from physics – such as the law of the lever – to solve problems in pure mathematics.

• invented war machines such as a highly accurate catapult that stopped the Romans conquering Syracuse for years. He may have done this by understanding the mathematics of projectile trajectory.

• became famous throughout the ancient world for his brilliant mind – so famous that we cannot be sure that everything he is said to have done is true. One example of this, the Archimedean screw or cochlias is discussed below.

• inspired what we now believe are myths including a mirror system to burn attacking ships using the sun’s rays, and jumping from his bath, then running naked through the streets of Syracuse shouting ‘Eureka’ meaning ‘I’ve found it’ after realizing how he could prove whether the king’s gold crown had silver in it.

Lifetimes of Selected Ancient Greek Scientists and Philosophers

Early Days and Greek Culture

The ancient Greeks were the first people to do real science and recognize science as a discipline to pursue for its own sake.

Although other cultures had made scientific discoveries, these were made for thoroughly practical reasons, such as how to build stronger temples or predict when the heavens would be right for planting crops or getting married.

Today, we would describe the Ancient Greeks’ work as blue skies scientific research.

They investigated the world for the sheer pleasure of adding to their knowledge. They studied geometry for its logic and its beauty. With no practical purpose in mind, Democritus proposed that all matter was made of tiny particles called atoms and that these atoms could not be split into smaller particles and were in constant motion and colliding with one another. He produced logical arguments for his idea.

Archimedes was born into this Greek scientific culture. In his work The Sand Reckoner he tells us that his father was an astronomer.

Archimedes spent most of his life in Syracuse. As a young man he spent time in the Egyptian city of Alexandria, where Alexander the Great’s successor, Ptolemy Lagides, had built the world’s greatest library.

The Library of Alexandria, with its meeting rooms and lecture halls, had become the focal point for scholars in the ancient world.

Some of Archimedes’ work is preserved in copies of the letters he sent from Syracuse to his friend Eratosthenes. Eratosthenes was in charge of the Library of Alexandria, and was no mean scientist himself. He was the first person to calculate the size of our planet accurately.

An artist’s view of Archimedes’ friend Eratosthenes teaching in the Library of Alexandria. Of course, the books in the library would have been scrolls, rather than the codex style shown here.

Immersed in the scientific culture of Ancient Greece, Archimedes blossomed into one of the finest minds our world has known. He was the Einstein of his time, or perhaps we should say that Einstein was the Archimedes of his time.

An Annoying Mathematician Ignites Curiosity Far into the Future

Two thousand years after Archimedes’ time, during the Renaissance and 1600s, mathematicians looked again at his work.

They knew Archimedes’ results were correct, but they couldn’t figure out how the great man had found them.

Archimedes was very frustrating, because he gave clues, but did not reveal his full methods. In truth, Archimedes enjoyed teasing other mathematicians. He would tell them the correct answer to problems, then see if they could solve the problems for themselves.

A Real Life Indiana Jones Style Discovery

The mystery of Archimedes’ mathematics wasn’t solved until 1906, when Professor Johan Heiberg discovered a book in the city of Constantinople, Turkey. (The city is now, of course, called Istanbul.)

The book was a Christian prayer book written in the thirteenth century, when Constantinople was the last outpost of the Roman Empire. Within Constantinople’s walls were stored many of the great works of Ancient Greece. The book Heiberg found is now called the Archimedes Palimpsest.

Heiberg discovered that the book’s prayers had been written on top of mathematics. The monk who wrote the prayers had tried to remove the original mathematical work only faint traces of it remained.

It turned out that the traces of mathematics were actually copies of Archimedes’ work – a momentous discovery. The Archimedes text had been copied in the 10th century.

A false color view of a page from the Archimedes Palimpsest, showing some of the recovered mathematics. Courtesy of The Walters Museum.

Archimedes Revealed

The book contained seven treatises from Archimedes including The Method, which had been lost for many centuries.

Archimedes had written The Method to reveal how he did mathematics. He sent it to Eratosthenes to be lodged in the Library of Alexandria. Archimedes wrote:

“I presume there will be some current as well as future generations who can use The Method to find theorems which we have not discovered.”

And so by reading The Method, twentieth century mathematicians learned just how far ahead of his time Archimedes was and the techniques he used to solve problems. He summed series he used his discoveries in physics – the law of the lever, and how to find centers of gravity – to discover new theorems in pure mathematics and he used infinitesimals to do work as close to integral calculus as anyone would get for 1,800 years.

Archimedes: An Ancient Greek Genius Ahead of His Time - History

An ancient mathematician from the coastal city of Syracuse, Archimedes is largely considered one of the most prolific and most brilliant scientific minds of antiquity.
His work focused on, but was not limited to, applying the concept of infinitesimals and the method of exhaustion to prove a number of geometrical theorems.

He might have also been a super villain. I don’t know. Maybe. I’m just saying.

All of the weapons that are mentioned in this article were said to have found use during the siege of Syracuse in 214 BCE.
It was the height of the Second Punic War and it was feared by the Roman Republic that the Kingdom of Syracuse might ally with their enemy, the Carthaginian Empire.

“But Archimedes had constructed artillery which could cover a whole variety of ranges, so that while the attacking ships were still at a distance he scored so many hits with his catapults and stone-throwers that he was able to cause them severe damage and harass their approach.” -Polybius (Universal Histories)

It was said that across the city wall, there were a series of holes that had been drilled through. These loopholes within the walls were said to have the breadth of a palms width. Behind these peepholes and within the city walls were stationed a number of archers with rows of the so-called “scorpions”.

A smaller catapult, or possibly a very large crossbow, the weapon discharged iron darts at the invading mariners. Deadly and impossible to counterattack, the projectile weapon would be the woe of General Marcellus. In the words of Polybius, the scorpion “put many mariners out of action”.
And if the ships still managed to out maneuver the long range artillery and the deadly “scorpions”, they still had to contend with…

Sometimes referred to as “The Iron Hand” or “The Sinker of Ships”, this weapon was said to have been a massive grappling hook. It was said to have been dropped from the top of the city fortifications onto an enemy ship. From there, the claw would be lifted back up, bringing the vessel and the entire crew with it. The ship would be dashed against the rocks or simply capsized. The crew, weighted down by heavy armor, might have sunk beneath the waves and drowned.

“A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.” -Plutarch (Parallel Lives:Marcellus)

“At last in an incredible manner he burned up the whole Roman fleet. For by tilting a kind of mirror toward the sun he concentrated the sun’s beam upon it and owing to the thickness and smoothness of the mirror he ignited the air from this beam and kindled a great flame, the whole of which he directed upon the ships that lay at anchor in the path of the fire, until he consumed them all.” -Dio Cassius (Roman History)

Okay, you may have put your doubts aside to accept the claw of Archimedes, but surely the creation of a deadly heat ray in 200 BCE is pure myth, right?
The existence of such a weapon has been a matter of some debate over the centuries. Several scientists have attempted to recreate the machine with varying success.

The story goes that a Roman soldier came upon Archimedes in his home where the scientist was busying himself with his work. Flustered that somebody had interrupted him, Archimedes ordered the soldier to leave. The Roman either did not recognize Archimedes or realized that he was the man responsible for hundreds of Roman deaths. Either way, the story ends the same. Archimedes, now in his late eighties, was slain in his workshop by the invaders.

8. Euclid

One of the earliest mathematicians to have ever lived, Euclid of Alexandria, is often regarded as the father of geometry. Due to the lack of early records, and the fact that most of the documents on the life of Euclid have perished with time, very little is known about his life. However, he was mentioned by the ancient Greek philosopher Proclus in a report aptly named the Summary of Greek Mathematicians. According to this, Euclid was an influential and active mathematician involved in the library of Alexandria around the time of Ptolemy I. This puts him at a much earlier time than another famous Greek – Archimedes.

Despite the fact that little is known about his life, his contributions have had a great impact on the history of geometry and mathematics as a whole. His main work is the Elements, which gave birth to basic geometry in concept and essence. Originally written as a set of 13 books, his famous work is used even today as a textbook in mathematics and is second only to the Bible in terms of the number of reprints sold. His collection of definitions, postulations, propositions, and proofs created the basis of today’s modern mathematics.

Archimedes’ legacy: inventions and discoveries

Archimedes is the perfect embodiment of a man ahead of his time. Even amon gst p eers that practice d p hilosophy and the arts as well as established democrac y, Archimedes of Syracuse outshined them all. A true polymath, Archimedes was active in the fields of astronomy, geometry, logic, physics, and mathematics , and was recognized as the best engineer and inventor of his time. As a part of his grand legacy, many of his inventions and discoveries from over 2,000 years ago are still in use toda y.

Archimedes’ screw

This ingeniously contrived device was invented by Archimedes to help poor farmers irrigate their crops. The device consists of a screw mechanism inside a hollow casing. When the screw is rotated, either by windmill or manual labour, the bottom end of the screw scoop s water, then move s it through the casing against gravity until it escape s through the last thread to reach irrigation canals.

A model of Archimedes’ screw, probably of the late Ptolemaic period, has been found in Lower Egypt.Credit: The New York Times, June 18, 1898

To day, the same principle is used in modern machinery for drainage and irrigation, and also in some types of high-speed tools. It can also be applied for handling light, loose materials such as grain, sand, and ashes. Of course, these look more impressive. Since 1980, Texas City, TX, USA uses eight 12-ft.-diameter Archimedes screws to manage rainstorm runoff. Each screw is powered by a 750-hp diesel engine and can pump up to 125,000 gallons per minute. The SS Archimedes was a ship named after the great inventor, which was the first steamship to come with a screw propelle r.

One of eight 12-ft.-diameter Archimedes screws in Texas CIty, Texas, USA. Credit: Popular Mechanics (April 1980, page 62).

Burning mirrors

Wall painting from the Stanzino delle Matematiche in the Galleria degli Uffizi (Florence, Italy). Painted by Giulio Parigi (1571-1635) in the years 1599-1600.

Th roughout his career as an inventor, Archimedes would frequently be commissioned by the rulers of Syracuse to invent war machines to protect their fair city. Such is the case with his “burning mirrors” – a system of large mirrors placed on the walls of the city that concentrate d s olar power in order to burn any ships foolish enough to sail against Syracuse. The story is extremely controversial, and even to this day historians and engineers alike debate whether this is a fact or myth.

The earliest account of Archimedes’ ancient death ray was written in the 12th century by Zonares and Tzetzes who were quoting an earlier, but now lost work called The Siege of Syracuse.

When Marcellus [The Roman General] had placed the ships a bow shot off, the old man [Archimedes] constructed a sort of hexagonal mirror. He placed at proper distances from the mirror other smaller mirrors of the same kind, which were moved by means of their hinges and certain plates of metal. He placed it amid the rays of the sun at noon, both in summer and winter. The rays being reflected by this, a frightful fiery kindling was excited on the ships, and it reduced them to ashes, from the distance of a bow shot. Thus the old man baffled Marcellus, by means of his inventions.

Crafty old man, indeed, but did it really happen? The ability of mirrors to concentrate the sun and obtain high temperatures is no myth, as any kid who used a magnifying glass to burn scraps can attest. This year, Morocco opened the largest concentrated solar power (CSP) plant in the world which will generate enough electricity to power the homes of one million people. CSP plants typically use 12m high parabolic mirrors that reflect sunlight onto pipework that contains a heat transfer fluid (HTF), typically thermal oil. This increases the temperature of the fluid to almost 400°C. The HTF is then used to heat steam in a standard turbine generator. Some CSPs heat the target tower to temperatures in excess of 1,000 degrees Fahrenheit (537 degrees Celsius), so it’s easy to imagine how Archimedes might have pulled something similar to burn enemy ships.

The real question isn’t whether it’s possible per se, but whether Archimedes actually made a burning mirror system using the tools and resources at his disposal two thousand years ago.

Apparently , in 1973 a Greek scientist, Dr. Ioannis Sakkas, became curious about whether Archimedes could really have used a “burning glass” to destroy the Roman fleet , so he set up an experiment involving 60 Greek sailors each using an oblong 3′ by 5′ flat mirror to focus light on a wooden rowboat 160 feet awa y. Th e boat was set on fire fairly quickly, though it’s worth mentioning the boat was coated in tar paint , which is highly flammable. Tar paint was used frequently to coat ships back in Archimedes’ time . However, more recently, when the Mythbusters made their own reenactment, things didn’t go quite as smoothly. In 2010, 500 flat mirrors controlled by 500 volunteer middle and high school students were focused on the sail of a ship, which should have combusted at 500 °F . After an hour, no more than 230 °F could be reached, so the team classified this as ‘inconclusive’. Jamie Hyneman, who was stationed on the moc k b oat for the duration of the experiment, did say that he could barely see, however . He suggests that Archimedes’ burning mirrors might have been real, but perhaps was used more for dazzling enemies than burning boats.

The gold crown and “Eureka!”

According to the Roman architect Vitruvius, the Syracusan king Hiero II commissioned a gold crown shaped like a laurel wreath to be placed in a temple. The king himself weighed the gold and gave the goldsmith the material to turn it into a piece of art. At the appointed day, the goldsmith presented his masterpiece — a gold crown shaped like a laurel wreath, exactly as the king ordered. When it was weighed, it had exactly the same mass as measured earlier. The king was pleased, but only days before the temple ceremony, he heard rumors that the goldsmith had cheated him and given him a crown not of pure gold, but of gold that had silver mixed with it.

Hiero believed there was only one man in Syracuse capable of discovering the truth and solving his problem — his cousin, Archimedes, a young man of 22 who already distinguished himself in the fair city for his work in mathematics, physics and engineering.

When faced with the challenge, Archimedes devised a clever science experiment to get to the bottom of things, but not until after thoroughly pondering the situation.
Legend has it that Archimedes was thinking about the golden crown while bathing in the public baths one day. As he began to enter a cold bathtub for his final dip, he noticed water started dripping on the sides. As he continued to lower his body into the bath, even more water ran out over the sides of the tub. In this instant, he recognized the solution to Hiero’s problem, jumped out of the tub at once, and ran all the way home without remembering to put his clothes on, all the while shouting, ‘Eureka, Eureka!’ – which in Greek means, ‘I have found it! I have found it!’

Alas, the “Eureka!” story itself is likely a fabrication, but Archimedes is genuinely credited as the first to state the laws of buoyancy.

Archimedes' Principle

He knew that if the crown was pure gold, its volume would be the same as that of the lump of gold (which he had made sure weighed the same as the crown), regardless of shape , an d i t would displace the same amount of water as the gold. If the goldsmith had indeed cheated and replaced some of the gold with silver, then the volume of gold and silver would be greater, and thus the crown would displace more water. According to Vitruvius, Archimedes used this method and found the goldsmith had indeed cheated.

Skeptics weren’t convinced, however . As far back as 1586, Galileo wrote a short treatise called La Bilancetta, or The Little Balance, in which he argued this method could not be work because the differences in gold and silver volumes are too small. Instead, he suggest ed Archimedes used a similar, but more crafty technique. In short, Archimedes probably suspended the gold crown on one end of a scale, and a lump of gold of equal mass on the other end.

The scale would have been then submerged in water, with both contents still on the ends of the scale. Since a body immersed in water is buoyed up by a force equal to the weight of the water displaced by the body, the denser body, which has a smaller volume for the same weight, would sink lower in the water than the less dense one. If the crown was pure gold, the scales would continue to balance even under water.

The Iron Claw

We continue with yet another war machine designed by Archimedes: the so-called Iron Claw. True to its name, this mechanical device was installed on the walls of the old city of Syracuse. The exact design has been lost in time, but we know its purpose was to topple eager Roman ships. Once the claw fastened itself to a ship’s underbelly, it would be tugged in an upward fashion and then released from a distance. In 2005, the producers of Discovery Channel’s Superweapons of the Ancient World challenged engineers to replicate this arcane device on the condition they’d use only techniques and materials known to be available in the 3rd century BC. Within seven days , they were able to test their creation, and they did succeed in tipping over a model of a Roman ship to make it sink.

The Odometer

The same Vitruvius who accounted Archimedes’ “Eureka!” moment also reported Archimedes to have “mounted a large wheel of known circumference in a small frame, in much the same fashion as the wheel is mounted on a wheelbarrow when it was pushed along the ground by hand it automatically dropped a pebble into a container at each revolution, giving a measure of the distance traveled. It was, in effect, the first odometer,” according to Encyclopedia Britannia. This mechanism is said to have been invented by Archimedes during the First Punic War. It seems to have been used until the time of Emperor Commodus (192A.D.) and then was lost in Europe until the middle of the fifteenth century.

The block and tackle pulley system

“Give me a place to stand on, and I can move the earth,” Archimedes once said speaking of the power of the lever. While he did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes.

Archimedes' law of the lever

Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance.

If, when weights at certain distances are in equilibrium, something is added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition was made.

Similarly, if anything is taken away from one of the weights, they are not in equilibrium but incline towards the weight from which nothing was taken.
When equal and similar plane figures coincide if applied to one another, their centers of gravity similarly coincide.

The familiar king Hieron was very impressed by this statement and asked Archimedes to prove it. The occasion seemed very fitting because Syracuse at the time was biting off more than it could chew. The city built a magnificent 55-meter-long ship called the Syracusia packed with a sumptuous decor of exotic woods and marble along with towers, statues, a gymnasium, a library, and even a temple. Oh, and the ship was designed by Archimedes. According to Plutarch, Archimedes managed to set the Syracuse out of harbor using an intricate system of pulleys, although his account seems a bit too poetic.

“[Archimedes] had stated [in a letter to King Hieron] that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king’s arsenal, which could not be drawn out of the dock without great labour and many men and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavour, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.”

Artist impression of the Syracusia.

“Archimedes chose for his demonstration a three-masted merchantman of the royal fleet, which had been hauledashore with immense labour by a large gang of men, and he proceeded to have the ship loaded with her usual freight and embarked a large number of passengers. He then seated himself at some distance away and without using any noticeable force, but merely exerting traction with his hand through a complex system of pulleys, he drew the vessel towards him with as smooth and even a motion as if she were gliding through the water.,” Plutarch.

Geometry of spheres and cylinders

According to Plutarch, the famous Greek biographer, Archimedes had a low opinion of the mechanical contraptions he invented and for which he was recognized in the entire ancient world. Instead, he relished in his theoretical explorations of mathematics and physics. Archimedes is credited for nine extant treatises, among which is the two-volume On the Sphere and Cylinder. In this fantastic work, Archimedes determined the surface area of any sphere of radius r is four times that of its greatest circle (in modern notation, S = 4πr 2 ) and that the volume of a sphere is two-thirds that of the cylinder in which it is inscribed ( V = 4 /3 πr 3 ). Archimedes was so proud of this achievement that he left instructions for his tomb to be inscribed with “a sphere inscribed in a cylinder.” Marcus Tullius Cicero (106–43 bce) found the tomb, overgrown with vegetation, a century and a half after Archimedes’ death.

The measurement of the circle

D etermining the area of a circle was once considered a great mathematical challenge. Archimedes found a way to approximate it with a method called “squaring the circle”. He first created a square inscribed inside of the circle (inscribed means that it exactly fits inside, with its vertices just touching the edge of the circle). Since he kn ew t he area of the square is (the product of two sides), it was clear that the area of the circle is bigger than the area of that inscribed square. He then fitted a polygon with six sides instead of four within the circle and computed its area he gradually worked his way up with more complex polygons to get even closer to the circle’s true area .

Eventually, Archimedes got really good at this and discovered π (pi) — the ratio of the circumference to the diameter of a circle. His calculations using an astonishing 96 – sided polyg on to suggest that pi lies “between the limits of 3 and 10/71 and 3 and 1/7”. In other words, he calculated an estimate that was equal to pi to two digits (3.14). Until the advent of calculus and computing infinite series 1,500 years later , no t m any digits were added to the ones found by Archimede s. A major breakthrough was made in 1655 when the English mathematician derived a formula for pi as the product of an infinite series of ratios.

How Archimedes, Thomas Edison, and Elon Musk used First-Principles Thinking to Create World-Changing Technological Breakthroughs

Roshan Thomas was one of the first employees at Tesla, joining the upstart electric car company in 2001. Across from him sat CEO Elon Musk, a tall, energetic engineer who described his dream of replacing the internal combustion engine with a global fleet of electric cars.

He asked Musk whether taking on a problem that automotive giants with their billions of research dollars had failed to solve was too ambitious. The CEO answered that he looked at only two things before embarking on anything. First, can it be done? Are we breaking any laws of physics by doing this? Second, is it important enough for humanity that it would make a major dent? If the answer to 1 and 2 is ‘yes,’ then he would move forward.

Musk describes this approach to problem solving as “First-Principles” Thinking. It is a thought process that allows a designer to innovate in clear leaps instead of incremental gains. With first-principles thinking, an innovator begins at the most fundamental truths and reasons up from there.

Such an approach has allowed Musk to do unprecedented things with the Tesla Model S. Musk is obsessed with each car being perfect. He has told his teams that he wants the cars to be so accurate that they could be used as a calibration device. If he wanted to know how long a meter was, he could measure the car. This approach to design comes from the design book for rocket design, which he uses for his rocket firm SpaceX.

“ This is very extreme for the car business, but for the rocket business it is not, so from my standpoint, when people say you can’t do that, it’s like, ‘I do that every day. What are you talking about? I know it’s possible.’ We’re trying to take the precision of rockets, where fractions of a millimeter can mean the difference between success and failure. We’re applying rocket science to the car business. If you want to make the best car, that’s what you have to do.”

Musk is not the first person to apply first-principles thinking to problem solving. Aristotle said 2,300 years ago that approaching first principles is the key to doing any kind of systematic inquiry. Another ancient Greek inventor also applied this thinking – Archimedes.

Archimedes was the Elon Musk of his day, building technology centuries ahead of his time and discovering scientific proofs that were not rediscovered until the time of Newton. He was such a genius inventor that Roman chroniclers claimed he built a primitive laser out of an array of mirrors. Its bursts of solar energy burned down an entire enemy naval fleet.

Archimedes lived in the city-state of Syracuse, a Mediterranean backwater with little access to technological tools or written works that he could use to carry on his studies. Despite his isolation, Archimedes’s aggressive adherence to logic allowed him to make big discoveries.

Many of Archimedes’s inventions are still in use today: the compound pulley is still the basic mechanical feature of an elevator. He invented a screw that moved water uphill and catapults that defended Syracuse from Roman invaders. He was the first scientist to apply abstract mathematical principles to the world around him.

The third inventor to use first-principles thinking is Thomas Edison. He achieved similar levels of productivity as Musk and Archimedes. He was arguably the most prolific inventor in all of history, with 1,093 patents to his name. His lab in Menlo Park, New Jersey, churned out a minor invention every 10 days and a big thing every six months or so.

While many of Edison’s most famous inventions were more practical versions of things that already existed (such as the light bulb), he applied first-principles thinking to come up with original designs, such as the phonograph. While other inventors had already made devices that recorded sounds, Edison’s invention was the first to reproduce the recorded sound.

The phonograph took nearly a decade to bring to the market. He first conceived of the idea in the 1870s of turning electromagnetic waves into speech. Edison first used grooved paper disks or spools of paper tape. Edison eventually settling on a tinfoil disk. But tinfoil was so delicate it could only be played once or twice before becoming unusable.

Edison spent 10 years testing every substance imaginable until settling on the wax cylinder. His invention spread rapidly and became the dominant audio recording format for most of the 20 th century.

Being a first-order inventor meant having every conceivable material on hand in order to test any theory. Edison’s laboratory materials supply items included over 8,000 kinds of chemicals, every size of needle, every kind of screw made, every kind of cord or wire. It also included hair of humans, horses, hogs, cows, rabbits, goats, minx, camels, silk in every texture, cocoons, ostrich feathers, and even a peacock’s tail.

Whether you are an ancient Greek scientist like Archimedes, an American tinkerer like Thomas Edison, or a CEO of a rocket ship firm like Elon Musk, first-principles thinking can help you overcome problems in a completely different way than those around you.

Eudoxus of Knidos (c. 390–c. 340 BCE)

Thehopads/Wikimedia Commons/CC BY 4.0

Eudoxus improved the sundial (called an Arachne or spider) and made a map of the known stars. He also devised:

  • A theory of proportion, which allowed for irrational numbers
  • A concept of magnitude
  • A method for finding areas and volumes of curvilinear objects

Eudoxus used deductive mathematics to explain astronomical phenomena, turning astronomy into a science. He developed a model in which the earth is a fixed sphere inside a larger sphere of the fixed stars, which rotate around the earth in circular orbits.

“Eureka!” Archimedes’ Moment of Genius

S ome of mankind’s greatest achievements remain shrouded in mystery centuries later. This is the case, for instance, of the Great Pyramids erected by the Egyptians which we barely seem to understand nowadays (and aliens did not take part in building those, but thanks for passing by and saying hi, conspirators).

Surely, science and technology took giant leaps over the Antiquity period. This also happened because some savants shared an enthusiasm for furthering human knowledge and pushed for progress in literally every scientific field. Aristotle, Euclid, Hippocrates, Socrates (among others) have laid the foundation of mathematics, geometry, medicine and philosophy. Without Ancient Greek thinkers, general knowledge games would last about five minutes.

Ever wondered why complex math problems feature Greek letters like alpha ( α ) or omega (ω)? Because Ancient Greeks were at the forefront of mathematical thinking. You’re welcome.

To put that into more tangible perspective, some of their thousand-year-old inventions still form the pattern of your own daily routines (the following illustrations may not apply depending on your country of residence, please cross out irrelevant answers) with things like democracy, the first alarm clock, the art of theater or the Olympic Games… Ancient Greeks also introduced the first historian, Herodotus – hence the curiosity for history you satisfy reading through this post (thanks, by the way) could be another legacy of their inventiveness.

Nevertheless, given the time it took for the Hellenistic civilization to slowly turn into our Western societies, accounts of scientific breakthroughs in Ancient Greece still lie at the boundary between fact and legend. That is perhaps better exemplified with the story of Archimedes, who lived in Syracuse (Sicily) in the 3 rd century B.C.

Before he earned a deserved reputation of brilliant astronomer and mathematician, Archimedes worked at the court of Hiero II, King of Syracuse. Only aged 22, he was personal adviser to the monarch and assisted him in any matter requiring quick-solving skills. This position happened to be a good springboard to his future scientific achievements.

On one occasion, the king ordered a local jeweler to mold a votive crown -a piece of jewelry meant as an offering to the gods- out of pure gold. He then handed the quantity of gold required to do the job to the craftsman, and days later, Hiero received the precious object ceremoniously. (Alike Midas, it seems like Greek monarchs were fascinated by gold-made items.)

But something was not quite right. The king was doubtful about the final result more specifically, he wondered whether the jeweler had followed his instructions to the letter or not. What if the crown had been made out of gold but also less ‘noble’ metals – especially silver – so that the jewelry maker could retain some of the king’s gold for him?

Syracuse, in ruins today. (Photo: Berthold Werner via Wikipedia, CC BY-SA 3.0)

Faced with such insoluble a question, King Hiero looked for advice from his 22-year-old counsellor. He tasked Archimedes with solving the issue and determining whether there had been foul play or not. Most importantly, the young scientist was ordered not to break the crown apart or melt it in an attempt to check its contents – such an offense could cause divine anger.

Despite his fascination for puzzles and riddles, Archimedes stumbled over the problem as he first investigated the issue. But when he went to the public baths days later, he was suddenly struck by the realization that water could be the key to solving the king’s query. Indeed, diving into the steamy waters of the public baths – that was a thing back in the days – he noticed that the water level shot up once he had gotten in. The quantity of water displaced was proportional to the volume of the body placed into it. So he could use a single experiment to figure out whether the crown had been made out of pure gold or some extra, less costly contents had been added.

Statue of Archimedes taking a bath located in Manchester, England. (Photo: Andrew via Flickr)

The scientist knew from experience that silver was less dense than gold. That meant that, for the exact same weight, those two metals did not move the same quantity of water when immersed: silver would sink and raise the water level slightly above gold’s.

Legend has it that a thrilled Archimedes then jumped out of the baths and run naked across the streets of Syracuse, shouting “Eureka!” (“I’ve found it!”). Back home and dressed (much to the relief of the Greek scientific community), he performed the experiment with the dubious votive crown – sinking it into water and measuring the water level – and the amount of gold the monarch had given to mold it. The results were surprising: the crown raised more water in the bath, meaning that it was made using less dense components – some of the king’s gold had been replaced by silver. Archimedes had unmasked the deceptive craftsman.

If you missed the point of the last three paragraphs, here’s a funny comic from Margreet de Heer to get it. On a side note, now you know how to defend yourself when charged with indecent exposure: “Sorry, Your Honour, I was only celebrating a scientific breakthrough following an ancient tradition.”

Upon hearing the news, Hiero’s own level of anger probably rose as well, which one would measure by the fate awaiting the tricky jeweler. Unfortunately, no historical evidence accounts for what happened next. Much alike Newton’s apple, this whole episode still raises historians’ eyebrows to this day. (Though the one about Newton is very likely to have occured.)

Whatever the true story behind Archimedes’ brilliant idea was, the principle he came up with became a cornerstone of hydrostatics, reading (take a deep breath):

“The upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the center of mass of the displaced fluid.”

The young scientist set off for a brilliant career, making scientific discoveries and perfecting his problem-solving skills in the course of the following fifty years. In 214 B.C., the savant had turned into an old man: he was 73 years old. The city of Syracuse, a long-time ally of the Roman Empire, had reshuffled the diplomatic cards and partnered with Hannibal’s troops under the reign of Hieronymus, Hiero II’s grandson.

Thus the outbreak of the Second Punic War, in the course of which both Rome and Carthage fought one another for control in the Mediterranean, posed a direct threat to Syracuse. Roman legions came in great numbers and laid siege to the city under the command of General Marcellus thanks to machines designed by Archimedes himself to protect the city, Roman forces were unable to break Syracuse defenses until, two years later, the city eventually fell into Marcellus’ command.

Thomas Ralph Spence, Archimedes Directing the Defenses of Syracuse, 1895. (Photo: Wikipedia)

The latter expressly ordered his men not to harm Archimedes, whom he considered a previous asset for forthcoming military campaigns or from his sheer scientific genius. Nevertheless, when a Roman legionary came across the 75-year-old scientist, he certainly failed to recognize in him ‘sheer genius’. Indeed, Archimedes was kneeling on the ground, drawing geometric shapes in the sand and probably uttering complex mathematical formulas when the soldier asked him to surrender. Legend has it that the old savant got irritated to be disturbed in the course of an experiment, and replied tit for tat: “Do not disturb my circles.” Furious, the legionary then picked up his sword and killed the old man.

The Greek Archimedes underwent a strange fate. His scientific ‘birth’ involved a fake votive crown, a greedy craftsman and a providential bath, while his deathbed was made up sand covered with geometric shapes. He lived and died in the middle of an experiment.

My Archimedes Report

287-212 B.C.
A famous quote of Archimedes: "Give me a place to stand and a lever long enough and I will move the Earth." This quote may sound crazy but it actually reinforces his brilliance. Read on, and get lost in the great world of the mathematical genius Archimedes.

  1. On plane equilibriums (two books)
  2. Quadrature of the parabola
  3. On the sphere and cylinder (two books)
  4. On spirals
  5. On conoids and spheroids
  6. On floating bodies (two books)
  7. Measurement of a circle
  8. The Sandreckoner

These are only some of the surviving books. Many books were lost through the ages, including some very important ones about Archimedes' life.


1) Keating, Susan and Tartarotti, Stefano. Archimedes: Ancient Greek Mathematician. Pennsylvania: Mason Crest Publishers, 1999.

Watch the video: Αρχιμήδης ο Συρακούσιος: Ο ευφυής εφευρέτης και σπουδαίος μαθηματικός της αρχαιότητας