The greek **Eudox** (408 BC - 355 BC) of Cynide was the inventor of the celestial spheres and one of the first to describe the motion of the planets. There is little information available about it. He is known to have been in the city of Tarento, Italy, to study with a disciple of Pythagoras named Arquitas. He also studied medicine in Sicily before traveling to Athens, where he spent two months attending philosophy seminars with Plato and other academics.

Son of a family of great doctors, he graduated in medicine and practiced for a few years until discovering astronomy, which he learned from the Egyptians in the city of Heliopolis. He then did his first historical work, recording for the first time that the length of the year is not only 365 days but 365 days and six hours. Eudoxo was also the father of the idea of explaining the motion of the planets and stars, imagining that the stars were attached to transparent celestial spheres, all revolving around the earth. This kind of cosmic structure would peak almost half a millennium later with the studies of another famous Greek, Ptolemy of Alexandria.

Although the book Elements (written by Euclid of Alexandria around the 3rd century BC) was for a long time the most important text for the development of science, many of the statements contained in it had already been presented by older masters, especially Eudoxo.

Around 350 BC Eudoxo moves to the city of Cinido, where he finds the democratic regime, which replaces the former oligarchy. With this, it receives the task of writing the new constitution, which should govern the new political system. A contemporary of the philosopher Plato, Eudoxo became one of the best-known mathematicians of his day, for mastering the techniques of prevailing geometry. Your work deserves our attention when you study a mathematical procedure for calculating surface area. Thus, through his technique, which he called the Exhaustion Method, he articulates the concepts of the infinitesimal, the concept of Higher Sum (Sup) and Lower Sum (Inf), which would greatly influence the creators of integral calculus.

We can illustrate the Exhaustion Method by calculating the circle area. For this we have to inscribe and circumscribe regular polygons in the geometric figure under study. As the sides of the polygons increase, we converge to the actual area of the circle. Eudoxo would draw a map of the sky. He studied the calendars and carefully recorded the times when the stars rise and set. In addition, it would mark the days of the Nile ebb and seek to gather indications of weather variations, with which to predict the changing seasons of the year. These data were released to the Greek people and passed on from generation to generation. From the observations of this great mathematician we can read:

*"March 12, the Pleiades goes down. The star of Hera turns red, we will have signs of change in temperature. The south wind is blowing, and if it blows stronger it will burn the fruits of the soil."*

He violently fought the horoscopes, always saying to everyone: "When the Chaldean people want to make predictions and predictions about a citizen's life with their horoscopes based on their birthday, we should not give any credit, because the influences of the stars are so complicated. to calculate, that there is no man on earth who can do it yet. " It is interesting to note the power of an idea, for Eudoxo would not write his conclusions about geometry. He would transmit his results orally. However, these conclusions went from word of mouth, from generation to generation, reaching us, men of the twentieth century. Thus Eudoxo, through his genius, his intuition in having created primarily the method of exhaustion, contributed in a definitive way to the advent of the ideas of Newton, Leibniz and Riemann, in the conception of the most important work of the last centuries: the development of integrals.

In mathematics, Eudoxo also created formulas that are still used today to calculate the volume of cones and pyramids. But most of his talent was devoted to making comparisons between the numbers. He then elaborated a theory of proportions in which he included for the first time the so-called irrational numbers that so much headache had given to mathematicians of the past. Since irrationals often appear in terms of areas and volumes - that is, in accounts that are currently made through integral calculus, Eudoxo is considered one of the creators of this discipline. Note that integral calculus was not definitively established until the late 19th century, 2200 years after its time.

Regarding the theory of proportions, the definition created by Eudoxo allowed the comparison of irrational lengths analogously to the current multiplication in cross. One of the great difficulties of mathematics at the time was that certain lengths were not comparable. The method of comparing two lengths x and y, looking for a length t such that x = m.t and y = n.t (with m and n integers), did not work for segments of lengths 1 and 2, as shown by the Pythagorean Theorem. With the theory created by Eudoxo, lengths of any kind could be compared.

As quoted at the beginning of this text, one of Eudoxo's most important works was that of his planetary theory, along with the publication of a currently lost book on velocities. Eudoxus was greatly influenced by the philosophy learned from his master Arquitas, creating a totally sphere-based planetary system. The system consists of a number of spheres of equal radius in rotation, with axes passing through the center of the earth. Each axis of rotation, in turn, also rotates through fixed points in another rotating sphere, thus generating a composition of motions.

* Summary created by * Just Math*, based on sources:

- Galileo Special Magazine no. 1, page 6, April / 2003

- MacTutor History of Mathematics archive