## Achievements

Archimedes is considered the greatest mathematician of all the ancient age, because he opened new paths in solid geometry, laid the foundations for integral calculus and created a new system to represent large numbers. For some historians, he is the true father of mathematical physics.

In On the balance of planes, he described the method for determining the centre of gravity of geometric bodies and outlined the principles of the lever, about which the famous phrase said: “Give me a foothold and I will move the world”.

Archimedes is considered the founder of hydrostatics, a branch of physics that studies liquid bodies at rest, whose principles he enunciated in the book Treatise on floating bodies.

Archimedes' principle can be stated in two parts.

1st - All bodies submerged in a liquid, displaces a determined quantity of that liquid, whose volume is exactly equal to the volume of the submerged body.

2nd - The body submerged in the liquid "loses" its weight an amount equal to the weight of the volume of liquid equal to the submerged volume of the body.

Other contributions

Even though Archimedes is better known for the principle that bears his name, his investigations into the square of the circle are more notable, leading to the discovery of the relationship between the circumference and its diameter.

Archimedes' mathematical works were the ones he considered most important.

He stated the relationship between area and volume of geometric solids; established the relationship between the length of the circumference and its diameter (number pi); demonstrated that the area of a parabola segment is equal to 4/3 of the area of a triangle with base and height equal to that of the segment; and determined the area of the ellipse and the volumes of the ellipsoids and paraboloids of revolution.

In a book, Archimedes used a numerical notation system based on exponents, which avoided the disadvantages of the Greek numbering system and allowed to operate with large quantities.

In response to his wishes, the figure of a sphere inscribed on a cylinder served as an epitaph, in honour of what he considered his most important discovery: the relationship between the volumes of the two solids.

From the current that combined research with practice, he was essentially a mathematician, but he was distracted by resolving mechanical emergencies, from which he always came out boasting of the solution.

He became known for his studies of hydrostatics and for his inventions, such as the pointless screw for raising water. He also gained fame during the second Punic War, where he devoted himself to invent ingenious warlike devices based on pulleys, ropes and hooks, to defend his city against the siege imposed by the Romans, due to her support for the Carthaginians.

He built a giant mirror that reflected the sun's rays and burned enemy ships from a distance. Many of the discoveries were fundamental to mechanics, such as the lever principle. Based on this principle, catapults were built that also helped to resist the Romans.

Archimedes' conclusions were basic to the evolution of Greek mechanics and to the development of studies of the behaviour of fluids in general.

Mathematical works were the ones he considered most important. Mathematically, his studies were based on Euclidean geometry and in these, the study of the spiral in the work about the spiral and of the parable in the Quadrature of the Parabola stands out, where he recorded the famous axiom about Archimedes' areas.

In About the Sphere and the Cylinder, it seems that this was his favourite work, he proved that the area of a sphere is four times the area of its maximum circle, among other important deductions such as the calculation of its volume.

He stated the relationship between area and volume of geometric solids, established the relationship between the length of the circumference and its diameter, where he defined the value of “pi” as 22/7.

He demonstrated that the area of a segment of a parabola is equal to 4/3 of the area of a triangle with base and height equal to that of the segment and determined the area of the ellipse and the volumes of the ellipsoids and paraboloids of revolution.

In a book, he used a numerical notation system based on exponents, which avoided the disadvantages of the Greek numbering system and allowed to operate with large quantities.

In mechanics, his most notable publication was On the Balance of Planes, where Archimedes' axiom of symmetry (law of the lever by static principles) stands out. In the field of fluids, the publication in two volumes, About Floating Bodies, stands out, where it deals with principles of buoyancy and fluctuation of solid and paraboloid bodies, such as, for example, which states that a body submerged in a fluid at rest, suffers a bottom-up thrust equal to the weight of an equal volume of the same fluid (the famous Eureka story!).

It was probably still in Alexandria, interested in the technical problem that was raising water from the Nile River to irrigate the valleys, that this sage developed a device made of tubes in propellers attached to an inclined axis, today called the Archimedes screw (260 BC ). ), this equipment, the origin of what are now called screw pumps.

At least two sciences owe their foundations to this scientist: static (the study of the balance of bodies) and hydrostatics (the study of the balance of liquids).

While, for example, many of the Elements of Euclid were compilations, extensions and improvements of works done by others, while each work by the engineer from Syracuse was an original contribution to physical or mathematical knowledge and some very important ones have arrived almost intact until today and several others were lost.

One of his most important inventions was the so-called Archimedes screw. Designed as a rotary pump to send water from streams to irrigation ditches, it was immersed in a water source with a slight slope, so that the bottom of any thread was lower than the top of the subsequent thread. When the screw rod was turned on its axis, so that the threads rotated in the water, it was raised in the spiral and discharged from the top of the thread.

When, in 214 BC, the Romans attacked Syracuse, Archimedes designed a series of weapons for the defence of the city, from long-range catapults to mirrors that used the Sun to set Roman ships on fire, although this seems very unlikely. Two years later, when the Romans finally managed to invade Syracuse, there were orders that Archimedes be spared. But that, unfortunately, did not happen. When being interrupted by a Roman soldier while doing some calculations with a stick on the floor, the mathematician shouted at the invader. The soldier had no doubt and killed Archimedes right there. Upon learning of what happened, the Roman general Marcelo decided to erect a tomb in his honour.