On Archimedes and His Legacy

Archimedes. Domenico Fetti, Public domain, via Wikimedia Commons

We have all come across the name Archimedes in our schooling years. I believe (and I may be wrong) that in all parts of the world, wherever science and mathematics are taught, the education system makes it a point to dedicate some space in their textbooks to Archimedes and his work, either in the field of mathematics, or physics in particular, or science in general.

Where I grew up we certainly learned about Archimedes and his famous Eureka! Eureka! moment, and some of his other discoveries as well, almost all of which I do not remember today.

However, for the sake of this essay, I will try to look into those other discoveries once again, just so that I can lay them down here for your benefit. I will also try to give a brief account of his life, something that our textbooks never did.

Needless to say, being a man from ancient history, very little is actually known about Archimedes’ life, and out of what is known, very little can be considered wholly accurate. A lot of the details about his life have been arrived at through speculation and guesswork done by very reliable historians and scholars over the course of several centuries.

But obviously, this does not come as a surprise to you, so let us begin with who was Archimedes.

Archimedes was a Greek mathematician, physicist, astronomer, inventor, and engineer, born sometime around 287 BC in the ancient and historic city of Syracuse on the Italian island of Sicily. The city of Syracuse is known for its rich Greek and Roman history, culture, and architecture.

However, the most well-known element of Syracuse is Archimedes himself, so much so that Archimedes is often referred to as Archimedes of Syracuse, and Syracuse has come to be known as the birthplace of Archimedes.

Archimedes went on to accomplish several great things in his various fields of interest, leading him to be regarded as one of the greatest and leading scientists of the classical antiquity period.

Although the exact date of birth of Archimedes is not known, the year of his birth is calculated based on a statement by the Byzantine-Greek historian and poet John Tzetzes that Archimedes lived for 75 years before his death in 212 BC. Therefore, unfortunately, all we have is an approximate idea of his birth year on the basis of an account we cannot confirm or prove.

Archimedes’ father, Phidias, was also an astronomer according to Archimedes’ work Sand-Reckoner. However, nothing else is known of him.

Unfortunately, we do not have much information about Archimedes’ childhood and youth, and very little is known to us about his adult life.

The earliest known reference made to Archimedes is found in The Histories, written by the Greek historian Polybius about 70 years after the death of Archimedes. Polybius’ work does not say much about Archimedes as a person but instead focuses on the war machines that Archimedes was said to have built to defend the city from the Romans during the siege of Syracuse.

Polybius mentions many machines designed by Archimedes such as improved versions of stone-throwers, improved catapults, and multiple crane-like machines that could be swung around in an arc. Although the Romans succeeded in capturing Syracuse, they suffered more losses than they expected mainly due to the inventions of Archimedes.

Archimedes is also mentioned in some of the works of Roman statesman Cicero, who claimed to have visited Archimedes’ tomb in Syracuse while serving as a public official in Sicily more than a hundred years after Archimedes’ death. Cicero even claimed to have gotten his unkempt tomb cleaned, enabling him to see some carvings and verses inscribed on it, including a sculpture illustrating Archimedes’ favorite mathematical proof, that the volume and surface area of the sphere is two-thirds that of the cylinder including its bases.

In one of his works Cicero also mentions that Roman general Marcus Claudius Marcellus brought back to Rome two planetariums built by Archimedes.

The next significant mention of Archimedes comes from the Greek historian Plutarch in his famous work Parallel Lives. In the work, Plutarch claims that Archimedes was related to the King of Syracuse, Hiero II. Hiero II features in the famous story that ends with Archimedes running naked through Syracuse shouting Eureka! Eureka!

But more on that later.

Plutarch also mentions two accounts of how Archimedes dies as Syracuse is captured by the Romans. In one account, as the city was being captured, Archimedes was contemplating a mathematical diagram when a Roman soldier confronted and ordered him to come and meet General Marcellus. Archimedes refused to go on the pretext that he had to finish working on the problem, thereby angering the soldier who then proceeded to kill Archimedes with his sword.

According to the second account, Archimedes was carrying some mathematical instruments when he was killed by a soldier who thought the instruments were valuable items.

Archimedes’ last words were said to be Do not disturb my circles! in reference to the mathematical diagram he was working on when confronted by the Roman soldier.

It is said that General Marcellus had given instructions not to harm Archimedes as he considered Archimedes a valuable scientific asset, referring to him as a geometrical Briareus. Hence, when Marcellus discovered that Archimedes had been killed, he became very angry.

No doubt, there is probably some myth and fiction involved in both these accounts of his death. The truth is that we will never know how he died or what his last words were. Therefore, not wasting more time speculating on this matter, let us now move on to his scientific and mathematical achievements.

Obviously, we must begin with the most famous one that has come to be known as Archimedes’ Principle. According to an account by the Roman architect and engineer Vitruvius, King Hiero suspected a goldsmith (to whom he had supplied gold to make a votive crown for a temple) of cheating him. To confirm his suspicions, he asked Archimedes to find out, without damaging the crown, if the goldsmith had used all the gold he had provided or not.

As Archimedes had to solve the problem without damaging the crown, he could not melt it down into a regularly-shaped body to calculate its density. And that was when, so the legend goes, while bathing Archimedes noticed that the level of the water in the tub rose when he got in. It struck him that he could use this effect to determine the crown’s volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be calculated, and this density would be lower than that of gold if cheaper and less dense metals had been added while making the crown.

As per Vitruvius’ account, this was how Archimedes invented a method for determining the volume of irregularly-shaped objects. And after conducting this experiment with the crown, he discovered that silver had indeed been added while making the crown. He was so ecstatic with his discovery that he rose from the tub and, forgetting to dress up, ran through the streets of Syracuse naked shouting Eureka! Eureka!

Now although this whole account sounds very fascinating and even amusing, the practicality of the method used by Archimedes in this story has been questioned due to the extreme accuracy that would be required while measuring the displacement of water. Also, this account and the method used in it are not found in any known works of Archimedes.

A more plausible method used by him would be the method he describes in his treatise On Floating Bodies. As per this principle, a body immersed in fluid experiences a buoyant force equal to the weight of the fluid it displaces.

By using this principle, Archimedes would have been able to compare the density of the crown to that of pure gold by balancing the crown on a scale with a pure gold reference sample of the same weight, and then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.

It is this principle that would come to be known as Archimedes’ Principle.

Galileo himself would go on to invent a hydrostatic balance in 1586 for weighing metals in air and water after being inspired by Archimedes’ work. Galileo thought it more probable that Archimedes used this particular method rather than the one described in Vitruvius’ story, as this method was not only accurate but was also based on demonstrations found by Archimedes himself.

It is said that a lot of inventions of Archimedes were for the sake of Syracuse, either to enhance its protection or improve it in some way or the other.

For instance, it is said that Archimedes may have used mirrors collectively as a parabolic reflector in order to burn ships attacking Syracuse. This method would come to be known as the Archimedes heat ray, wherein the device was used to focus sunlight onto approaching ships, resulting in them catching fire. This device was said to be used during the siege of Syracuse.

The basic principle of this device has been used in the modern era to design similar devices such as solar furnaces and heliostats.

However, modern researchers, who have attempted to recreate the effect using just the means that would have been available to Archimedes, have only met with unsatisfactory and negative results, thereby casting some doubts on the credibility of Archimedes’ invention. They found that at most this device would serve to blind or distract the crew of the enemy ships, but not burn it.

Another device he was said to have designed to defend Syracuse was Archimedes’ Claw, also known as the ship shaker, which was used to defend the sea-facing part of the city’s wall from any attack from enemy ships. This device consisted of a crane-like arm from which a large metal grappling hook was suspended. And when the claw was dropped onto an attacking ship, the arm would swing upwards and lift the ship out of the water and possibly sink it.

Interestingly, this device has been proven by modern researchers to work.

Archimedes is also credited to have designed block-and-tackle pulley systems that allowed sailors to use the principle of leverage to lift objects that would have been too heavy to lift. This is not to say that he invented the lever, but that he designed an improved version of the lever and gave mathematical proof of the principle in his treatise On the Equilibrium of Planes.

Greek mathematician Pappus of Alexandria claimed that Archimedes’ work and knowledge of levers led him to remark, Give me a place to stand on and I will move the Earth.

Archimedes was also credited to have invented the odometer, also known as the odograph, during the First Punic War fought between Carthage and Rome. The instrument was used to measure distances traveled by a vehicle. The one designed by Archimedes was described as a cart with a gear mechanism that dropped a ball into a container after every mile traveled.

He also designed what is now known as the Archimedes Screw, which is one of the earliest hydraulic machines. After being asked by king Hiero II to design the giant ship, Syracusia, which is said to have been the largest ship built in classical antiquity. Serving as a multipurpose ship, it could be used for carrying supplies, luxury travel, and as a naval warship as well.

And since such a large ship would leak a considerable amount of water through the hull, Archimedes designed the screw to remove the bilge water. It was basically a device with a revolving screw-shaped blade inside a cylinder that was turned by hand. It could also be used for transferring water from a low-lying body of water into irrigation canals. The Archimedes Screw is still used to this very day.

Archimedes has also gained a considerable reputation as an astronomer after having discussed the astronomical measurement of the sun, earth, and moon, as well as ancient Greek astronomer Aristarchus‘ heliocentric model of the universe (which was the first-known model that placed the sun at the center of the known universe with the earth revolving around it once a year and rotating about its axis once a day), in his treatise Sand-Reckoner.

He made these observations without the use of a table of chords or trigonometry, instead using a straight rod with pegs or grooves. Then he applied correction factors to these measurements and arrived at the result in the form of lower and upper bounds to account for observational error.

These observations lead us to believe that Archimedes was the first-known Greek to have recorded multiple solstice dates and times in successive years.

Archimedes also went on to make a great contribution to the field of mathematics, so great that he is often regarded as the greatest mathematician of ancient history and one of the greatest of all times.

He derived an approximation of pi, designed a system using exponentiation for expressing large numbers, and defined and investigated the Archimedean spiral. He was one of the first ones to apply mathematics to physical phenomena, founding statics and hydrostatics. Doing this, he came up with proof of the principle of the lever, enunciated the law of buoyancy, and made widespread use of the concept of gravity.

More importantly, Archimedes is credited to have anticipated modern analysis and calculus by applying the concept of the infinitely small and the method of exhaustion to derive and prove a range of geometrical theorems such as the area of a parabola, the area of an ellipse, the area of a parabola, the area of a spiral, the volume of a segment of a paraboloid of revolution, the surface area and volume of a sphere, and the volume of a segment of a hyperboloid of revolution.

Unfortunately, during his lifetime Archimedes did not receive as much recognition for his mathematical achievements as he did for his inventions for the protection of his city. In fact, his mathematical writings and treatises were barely known in antiquity.

It was only much later after his death that his mathematical writings began to be read and quoted in Alexandria, Egypt. And it was only sometime around 530 AD that the first comprehensive compilation of his mathematical writings was made by Byzatine-Greek architect Isidore of Miletus.

In the 6th century, Palestinian-Greek mathematician Eutocius of Ascalon wrote the first commentaries on many of Archimedes’ treatises, making them available to a wider readership.

Archimedes’ influence on subsequent generations has been massive and unparalleled. Many regard him as the father of mathematics and mathematical physics. Mathematical and scientific scholars almost unanimously consider him the greatest mathematician from antiquity, and one of the greatest and most influential ones of all time.

His works in all the fields he dabbled in went on to inspire and influence great scientists and mathematicians of subsequent generations such as da Vinci, Galileo, Leibniz, Huygens, Newton, Gauss, Tesla, and several others.

His works also greatly influenced scientists during the Renaissance and again in the 17th century. Through his work and influence, his legacy shall continue to live on for centuries to come.