# Archimedes

Created | Updated Apr 22, 2012

Archimedes was an ancient Greek mathematician and scientist. He was without a doubt the greatest mathematician of the ancient Greek world, and probably one of the greatest of all time. Working entirely within the confines of Greek mathematics, he discovered many new and interesting results, including the volume of a sphere and the area under a parabola^{1}. Within physics, he is credited with the principles of buoyancy and flotation and with the law of the lever. Indeed, he is supposed to have said 'Give me a place to stand on, and I will move the Earth'. In the field of practical engineering, he invented the Archimedes' Screw, a device for pumping water uphill which is still used in Egypt to this day.

### The Life of Archimedes

We know very few facts about Archimedes, but there are many stories. We know that he lived in the Greek city-state of Syracuse (modern Siracusa) on the island of Sicily, now a part of Italy. The Romans were at war against Carthage at the time (the Second Punic War), and Syracuse joined forces with Carthage, unfortunately picking the losing side. Archimedes is credited with leading the defence of Syracuse against the Romans in 212 BC, and died in the attack. He was an old man at the time. That's basically all we know about him other than his work.

Archimedes uses the phrase 'tou akoupater' in one of his works, referring to an astronomer by the name of Pheidias. The phrase is meaningless as written, but the hand-written Greek letter k is very similar to the letter m. If it is a misprint for 'tou amou pater', then it means 'my father', implying that Archimedes was the son of an astronomer called Pheidias. This would perhaps explain his fascination with the stars and with the science of measurement.

Archimedes wrote many works on mathematics and physics and a lot of them have surived to this day by being copied again and again. The works were in the form of letters to eminent people explaining his mathematics. There was no system for publishing mathematical works at the time. The letters tell very little about the man himself - each starts with an introduction in which he addresses the recipient of the letter, then he launches straight into the mathematics. We can glean a few facts from these introductions, such as that he had a friend called Conon of Samos who was a keen mathematician but died suddenly. We can also see that Archimedes was very proud of his own work. He regularly mentions that the things he has proved have never been proved before by anyone else. In one work he says that he has thrown in a couple of obvious mistakes to prevent people from claiming his work as their own without understanding it.

Lacking any other solid facts, we must rely on the stories told by his biographers. These were written down after his death, so they should be taken with a grain of salt.

#### Archimedes and the Bath

The most famous story is that King Hieron of Syracuse had a golden crown which had been made specially for him. He suspected that the crown was not pure gold and that the craftsman had mixed silver in with the gold. He gave the problem to Archimedes to think about. The scientist realised that silver and gold had different densities, so he needed to measure the density of the crown. This is calculated from the weight and the volume, but Archimedes couldn't think of any way of measuring the volume of the crown, as it was not a simple geometrical shape. Later he was taking a bath and noticed that as he got in, the level of the water in the bath rose. He suddenly realised in a flash of insight that the amount by which it rose could be used to calculate the displaced volume. He jumped out of the bath and ran down the street shouting 'Eureka, eureka' which means 'I have found it!', oblivious to the fact that he was stark naked.

#### The Siege of Syracuse

When the Romans were besieging Archimedes's city, he is said to have produced many inventions for the defence of the city. Some of these may have existed while some are definitely only in the imagination of the chroniclers:

A device which could grab the fronts of ships in the harbour and lift them up, dropping them onto the rocks.

A type of catapult which could adjust its range to strike the enemy whatever their distance from the city walls; normal catapults have a fixed range and are useless when the enemy gets too close.

A set of mirrors which could focus the sun's rays onto the ships and set fire to them - this story was first reported by the Byzantine historian Tzetzes about 1,300 years after the event. It has been shown to be possible but extremely unlikely by the MythBusters team. The Byzantines themselves were experts at burning ships but there's no evidence that the ancient Greeks did it.

Eventually the Roman soldiers became reluctant to go near the city, not knowing what the scientist would come up with next:

The Romans had become so scared that if only a rope or small beam were seen over the wall they would turn and fly, crying out that Archimedes was bringing some engine to bear upon them- Plutarch,The Life of Marcellus

#### The Death of Archimedes

Archimedes died in 212 BC when the Romans finally invaded Syracuse after the long siege. The Roman historian Plutarch, writing in about 100 AD, recorded three different versions of his death.

In one, the scientist is working on some thorny problem by sketching diagrams in the dust on the ground. A soldier commands Archimedes to follow him to Marcellus, the leader of the Romans. The scientist insists that he should finish the problem before he goes, and the soldier loses his temper and kills him on the spot.

The second account is identical except that the soldier is not taking Archimedes to his leader, he just runs up shouting that he is going to kill the Greek, who responds that he has to finish the problem, and is slaughtered for his trouble.

In the third version, Archimedes is walking through the city carrying the tools of his trade, drawing equipment and measuring devices. The soldier thinks that he is holding something worth money, and kills him for it.

In the most famous version, Archimedes said to the soldier, 'Do not touch my diagrams', causing the soldier to lose his temper and kill him. This version is recorded by the Byzantine Tzetzes in about 1150 AD, so it is potentially suspect.

### The Archimedes Palimpsest

The works of Archimedes were copied and kept in the library in Alexandria. The library did not survive the general downfall of civilisation that occurred in the first few centuries AD, but there are tantalising hints that much of the contents were taken to the capital of the Roman empire, Constantinople (now Istanbul). Many of the works of Archimedes survived, eventually being copied into books hand-written on parchment, but some were believed to be lost. Among these was a work called 'The Method', known from references by other writers, but thought no longer to exist in the original or in any copy.

Then in 1906, a Danish scholar by the name of Heiber discovered a prayerbook in a monastery in Constantinople. It was a palimpsest - this means that the book was made by recycling parchment from an earlier volume by scratching off the original text. Parchment, made from cured animal hide, is an expensive commodity, so books that were not considered important were often erased and the parchment re-used. In this particular case, the erasure had been done rather crudely and Heiber was able to read the original text. He found it was a collection of the works of Archimedes! Many of the works were ones which were already known from other sources, but in there among them was a copy of 'The Method'.

Heiber copied as much of the Archimedes text as he was able to read and added one more work to the collection of the works of Archimedes. But he was not allowed to interfere with the prayerbook in any way, and some parts of the partially erased book were hidden in the binding of the prayerbook, because the book had been rebound at the time it was erased.

The 20th Century was a time of war. Not only were there two world wars, but Constantinople was involved in a civil war and a war against Greece. Somewhere along the way, the prayerbook was lost and only turned up in the late 1990s, when it was bought by a rich philanthropist (for a sum of about $2,000,000). He chose to hand it over to scientists for analysis. The book was dismantled and every page treated with the best possible restoration work to counteract the effects of time. Experts in ancient Greek manuscripts and mathematics have now reconstructed portions of 'The Method' which were unavailable to Heiber as well as correcting some mistakes he made. This work was still in progress in 2009.

### The Works of Archimedes

Archimedes wrote a large number of mathematical works and the following have survived:

**On the Equilibrium of Planes, Vols 1 and 2**- this work explores the concept of 'centre of gravity'.**Quadrature of the Parabola**- Archimedes works out the area enclosed by a parabola and a straight line, predating integral calculus by about 1,800 years.**On the Sphere and Cylinder, Vols 1 and 2**- these calculate such values as the volume of a sphere, the surface area of a cone and the volume of a cylinder.**On Spirals**- Archimedes investigates a type of spiral now known as an Archimedean or arithmetic spiral, which grows by a constant amount each turn around the centre.**On Conoids and Spheroids**- a treatment of ellipses, ellipsoids, paraboloids^{2}and hyperboloids.**On Floating Bodies, Vols 1 and 2**- Archimedes is famous for his discovery of the principle of flotation, and it is presented in this work. It also investigates more elaborate topics such as the stability of floating bodies with various cross-sections.**Measure of a Circle**- by calculating the perimeter of a 96-sided polygon, Archimedes proves that π lies between 22/7 and 223/71. These limits are within 0.04% of the true value.**The Sand-Reckoner**- Archimedes devises a method of writing down extremely large numbers, and as a demonstration of their use, calculates the number of grains of sand needed to fill the known universe. In passing, he mentions another Greek philosopher, Aristarchus, who believed that the universe was much bigger than commonly believed, that the Sun was much bigger than the Earth, and that the Earth revolves around the Sun rather than the other way around.**Book of Lemmas**- although normally included within the works of Archimedes, this seems to have been mainly written by someone else, as it quotes Archimedes in certain parts of the work.**The Problem of the Cattle of the Sun**(attributed to Archimedes) - a fiendishly difficult problem stated in the form of a 44-line poem. It involves the number of cattle owned by one of the gods. The original poem does not reveal the solution. Modern analysis has shown that there is a solution but it seems unlikely that Archimedes could have worked it out, as the problem is an extremely tough one.**The Method**- this work, unlike most of his mathematical treatises, demonstrates a method for tackling unsolved problems, rather than just presenting a solution to an existing problem. It includes something remarkably like integral calculus, predating Newton and Leibniz by more than 1,800 years.**The Stomachion**(mostly lost) - a discussion of a puzzle in which a square is dissected into a number of shapes which are then re-arranged to make other patterns, something similar to the Tangram.

### An Example of Archimedes's Mathematical Work

As a demonstration of the methods of Archimedes, we'll present his proof that the surface area of a cone is given by the formula π.r.s where r is the radius of the circular base and s is the length of the slope from the apex to the circle around the base.

Archimedes was not the first to prove this, but his proof is probably the most elegant one ever devised, and is a good example of the 'method of exhaustion', which is closely related to the modern concept of a limit.

First, Archimedes does something which appears to be unrelated. He draws a regular polygon outside a circle, so that the circle touches the middle of each side, and a similar polygon inside the circle so that the vertices of the polygon lie on the circle. He has sandwiched the circle between two polygons. Archimedes shows a method for increasing the number of sides of the inner and outer polygons, and shows that the ratio of the perimeter of the outer to that of the inner polygon can be made as close to 1 as you like by increasing the number of sides of the polygons.

Now Archimedes builds two such polygons around the circular base of the cone. He joins the corners of the polygons to the apex of the cone to make two many-sided pyramids. One of these pyramids is inside the cone and the other is outside the cone. He shows that:

- the surface area of the outer pyramid is greater than that of the cone
- the surface area of the inner pyramid is less than that of the cone
- the surface areas of the outer and inner pyramids can each be made as close as you like to the value of π.r.s by increasing the number of sides of the polygons at the base

Archimedes now states that the surface area of the cone is also equal to π.r.s (let's call this value A) and proves it by showing that if it was more than this by any finite amount a, then by increasing the number of sides of the outer pyramid, we could make its surface area between A and A + a, which would contradict 1 above. And if the area of the cone was less than A, for example being equal to A - a for some small a, then we could make the inner pyramid have a surface area between A - a and A, which would contradict 2 above. Therefore the surface area of the cone must equal A = π.r.s.

Archimedes has used a concept very close to the modern concept of a limit, without ever mentioning infinity. His proof is completely watertight, and doesn't involve anything as bizarre as Newton's infinitesimals: infinitely small quantities which are bigger than zero. In fact it wasn't until the 19th Century that the calculus of Newton and Leibniz was put on a completely sound theoretical basis by adopting a definition of a limit similar to the one used by Archimedes.

### Remembering Archimedes

Archimedes's tomb was inscribed with a picture of a sphere inside a cylinder. This was done at his request because he was very proud of his discovery and proof that such a sphere and cylinder will have volumes in the ratio of 2:3. The Roman orator Cicero found the grave many years later and cleaned it up, but it did not survive into modern times.

Archimedes is generally remembered in popular culture today as the scatter-brained Greek scientist in the bath. But the scientific and mathematical communities hold him in more reverence, as one of the first true scientists and one of the greatest mathematicians. A crater and a line of mountains on the Moon are both named after him, as well as one of the asteroids. The Archimedes Principle is still the name by which the principle of flotation is best known. And the Fields Medal, one of the greatest honours which can be awarded to a mathematician, bears a picture of Archimedes and a quote attributed to him:

Rise above oneself and grasp the world.

^{1}A parabola is the curve made when you cut a slice off a cone parallel to the axis. It's also almost exactly the path that an object takes when thrown.

^{2}This is the shape of the reflector in a car headlamp or a torch. It projects the light from the bulb in a parallel beam.