At the beginning of the 3rd century BC. C. Greek science was almost exclusively concerned with the study of geometry. At that time, this discipline accumulated nearly three hundred years of joint evolution of mathematics and philosophy. An interval of time in which the problems raised and resolved had shown an increasingly greater degree of complexity.
That wealth of knowledge had just been collected by the Greek mathematician Euclid in Elements. But there was still much to do. The greatest achievements would be achieved by someone who is considered one of the most important scientists of all time: Archimedes.
Despite his popularity, there is little information we know about his life. It is known that he was born in Syracuse in 287 BC. C. and that his father was an astronomer. Also that he received part of his training in Alexandria, then the capital of the Hellenistic world and the center of an intense cultural life. And when he returned to his hometown he dedicated himself to the study of mathematics and its applications, a task that he was able to carry out thanks to his privileged position, close to the circle of power.
Archimedes produced a large collection of scientific treatises, which he would make known by sending them to the mathematicians with whom he had shared years of learning in the Egyptian city. Precisely, this dissemination has allowed a good number of his works to reach us through copies or Arabic and Latin translations. The wise man explained his work in such a perfect way that it continues to amaze even today. In them he covered issues of mathematics, astronomy and physics, and included extraordinary contributions to the science of the time.
Archimedes deduced some of the notions of geometry that are studied today in the first mathematics courses. In On the Sphere and the Cylinder he explained how to calculate areas and volumes of round bodies (including the sphere, the cone or the cylinder) and introduced the concept of concavity, which Euclid never mentioned.
The first postulate of this book says that the line is the shortest line between two points. Today this definition seems most intuitive to us, but the fact that it appeared to Archimedes for the first time after three centuries of study by Greek mathematicians shows that it may not be so obvious.
In another treatise he presented a very precise approximation of the number π. He gave it a value ranging between 3.1408 and 3.1424 (today we know that its real value is 3.14159265...). He also devised a numbering system that allowed him to handle numbers with many digits (today expressed in powers).
The wise man from Syracuse was interested in more complex geometric figures. He set out (and, of course, succeeded) in calculating the volumes of objects generated by rotating ellipses, parabolas and hyperbolas around their axes. Likewise, he dedicated a treatise, On Spirals, to the study of a curve never before approached from a mathematical point of view: the spiral.
The particular type he considered is today called the Archimedean spiral, and is characterized by uniform width between its turns. The complexity of On Spirals is such that mathematicians of the 16th and 17th centuries, the first to begin studying his work, were forced to accept that they were not capable of fully understanding this book.
In fact, the works of Archimedes are difficult to encompass, despite the fact that the exposition of ideas always presents the same structure. It corresponds to what Euclid had imposed: hypotheses, theorems and demonstrations. The difficulty of understanding lies in the fact that the mathematician does not show the way by which he obtains the theorems.
Did Archimedes hide a method that allowed him to add so many results, all of them brilliant? Modern scientists who rescued his works from oblivion suggested so. In fact, the wise man had a method, but it was not secret, but unknown.
In 1906, a Danish Hellenist discovered an unpublished work hidden in a palimpsest (reused manuscript) from the 10th century. It was titled, precisely, Method on Mechanical Theorems dedicated to Eratosthenes. It is the only book by Archimedes, and practically all of Greek science, that shows the path of research to discover some theorems.
In it, Archimedes explains how, starting from material reality, or mechanical considerations, mathematical knowledge can be acquired. It is a procedure that breaks with the scientific tradition of the moment, highly influenced by Platonic philosophy, opposed to any practical application that strayed from the abstract and ideal reasoning of pure mathematics.
Sometimes Archimedes follows the opposite path: he applies mathematical properties to areas related to material reality. Among them, physics. In this sense, he is considered one of the creators of statics. In On the balance of planes he calculates the centers of gravity of figures such as the triangle or the parallelogram and exposes what is known as the law of the lever, which states that a small force can be in balance with a large one if the proportion between the arms lever of both forces is adequate.
Today this postulate maintains as much validity as the so-called Archimedes principle, one of the fundamental laws of hydrostatics, with which the mathematician inaugurated, precisely, this branch of physics.
Like his father, Archimedes dealt with questions of astronomy, and left a record of this in The Sand Teller. In this work he describes a way to calculate the angle that the Sun presents at a given moment from the position of the observer, the so-called apparent diameter.
Fifteen centuries after Archimedes exploited his talent, his works began to be translated into Latin. And the scientists who consulted them from the 16th and 17th centuries onwards – Galileo, Kepler, Pascal, Fermat... – unanimously recognized their value. But long before that, the genius's contemporaries already appreciated his talent when it came to creating all kinds of practical inventions.
Of those attributed to him, the best known is what is now called Archimedes' screw, or worm screw, a device capable of raising water simply by turning a crank. Other devices have been lost, and we know of their existence thanks to third-party descriptions. This is the case of a planetarium, a mobile sphere built using complicated gears in which Archimedes represented the known universe – the Sun, the Moon and the planets – and reproduced the formation of atmospheric phenomena, such as thunder and lightning.
The wise man is also said to have created optical, surgical and musical instruments. And if we pay attention to Leonardo da Vinci, Renaissance artist and author of a large number of inventions, a kind of cannon that worked... by steam!
Astonished by his fertile imagination, Hiero II, king of Syracuse, commissioned him to build all types of war devices, which would serve to defend the city in the event of a siege by Roman troops. At that time Rome was in full expansion and disputed with Carthage for control of the Mediterranean. Syracuse, in Sicily, was a city coveted for its strategic location.
It was not easy for the Roman troops to subdue it. The city had not only a fortification on top of a promontory facing the sea, but also the fearsome inventions of Archimedes. His catapult allowed heavy stones to be thrown over great distances; a mechanism based on a large lever, pulleys and hooks could have lifted enemy ships and then sunk them; and it is possible that enormous parabolic mirrors had concentrated solar rays, projecting them towards the ships until they caught fire.
After three years of fighting, in 212 BC. C., while the Syracusans celebrated the festivals of Artemis, the troops of the Roman general Marcellus managed to conquer Syracuse and sack it. The old Archimedes was not spared from the fury. According to the Greek historian Plutarch, Marcellus asked his soldiers to bring him alive the genius who had caused him so much misfortune. However, his order was not carried out.
A soldier entered Archimedes' home and ordered him to follow him. But he ignored it because he was focused on solving some problems. It is said that before the soldier pierced him with his sword, Archimedes told him: "Do not disturb my circles." There are several versions of his death, but they all coincide in this violent episode.
The wise man was buried in Syracuse and, fulfilling his wishes, an image was engraved on his tombstone that alluded to the discovery that had most captivated him: the sphere, both in volume and surface, is one and a half times smaller than the cylinder that contained it. circumscribe.
This text is part of an article published in number 460 of the magazine Historia y Vida. Do you have something to contribute? Write to us at redaccionhyv@historiayvida.com.
