To sum up, there are mainly the following aspects:

1.

The Ionian school headed by Thales has made great contributions to proposition proof and taken the first step to establish a geometric deduction system. Later, a school headed by Pythagoras, a mysterious political, religious and philosophical group, took "everything counts" as its creed, abstracted mathematical theories from concrete things and gave mathematics a special independent position.

2.

The philosopher Plato founded the famous Plato Academy in Athens, and trained a large number of mathematicians, which became the link between the early Pythagorean school and the long-term active Alexandria school. Eudoxus is one of the most famous figures in this college. He founded the proportional theory which is suitable for incommensurable measure and incommensurable measure. Plato's student Aristotle is the founder of formalism, and his logical thought opens the way for arranging geometry in a strict logical system in the future.

3.

Euclid summed up classical Greek mathematics, arranged geometry with axiomatic method, and wrote 13 volume "The Original". The significance of this epoch-making historical masterpiece lies in that it sets the earliest example of establishing deductive mathematics system by axiomatic method.

4.

Archimedes was the greatest mathematician, mechanic and mechanic in ancient times. He organically combined the empirical research method of experiment with the deductive reasoning method of geometry, making mechanics scientific, with both qualitative analysis and quantitative calculation. Archimedes is also involved in a wide range of pure mathematics. One of his great contributions is to establish accurate quadrature methods for the area of various plane figures and the volume of rotating bodies, which includes the idea of calculus.

Eratosthenes, the librarian of Alexandria Library, was also a famous scholar in this period. "Conic Curve" written by Apolloni Uss systematically sorted out the knowledge of conic curve obtained by predecessors, made new contributions, and had a great influence on the development of mathematics in17th century.

5.

Alexander's later period was under Roman rule. Fortunately, the Greek cultural tradition has not been destroyed, and scholars can continue to study it. However, it has lost the majestic momentum of the previous period. Outstanding mathematicians in this period include Helen, Plume, Diophantine and Pappus. Diophantine algebra is unique in Greek mathematics; Papos's work is a summary and supplement to previous research results. After that, Greek mathematics was at a standstill.

In 4 15 AD, Hipatia, a female mathematician and leader of Neo-Platonism, was brutally killed by Christians. Her death marked the decline of Greek civilization, and the days when Alexandria University was full of creativity are gone forever.

The development history of Greek mathematics can be divided into three periods.

The first period, from Ionian school to Plato school, lasted from the middle of the 7th century BC to the 3rd century BC.

The second period is the pre-Alexandria period, from Euclid to BC 146, Greece was trapped in Rome.

The third period was the later period of Alexandria, which was ruled by Romans and ended when 64 1 Alexandria was occupied by Arabs.

The following is about the mathematical achievements of this period and its influence.

The main achievements of ancient Greece in mathematics and its influence on the development of science The main achievements of ancient Greece in mathematics and its influence on the development of science Although mathematics appeared thousands of years earlier than ancient Greece, it belonged to empirical mathematics, and it was not until ancient Greece that mathematics developed into deductive mathematics. As an independent knowledge system, mathematics originated in ancient Greece. Since its birth, mathematicians have been pursuing truth for more than 2000 years and have achieved brilliant results. Ancient Greece produced many great mathematicians and developed the first golden age of mathematics. Three people contributed a lot. Pythagoras: The founder of the Pythagorean school. It is said that he was the first person to prove Pythagoras theorem, so westerners call him Pythagoras theorem. In 580 BC, Pythagoras was born on Samos Island near Miletus. Pythagoras' father was a rich businessman. At the age of nine, he was sent to Tyre by his father to study with Semite Syrian scholars, where he came into contact with eastern religions and cultures. Later, I went to Asia Minor on business with my father many times. In 55 1 BC, Pythagoras came to Miletus, Delos and other places, visited Thales, anaximander and Philkudes, and became their students. Before that, he studied poetry and music under the Samos poet Clay Filos. In 550 BC, 30-year-old Pythagoras was disliked by the local people for propagating rational theology, wearing oriental clothes and having long hair. Since then, Samos has always been biased against Pythagoras, thinking that he is unconventional and advocates heresy. In 535 BC, Pythagoras was forced to leave his hometown for Egypt. On the way, he stopped in the coastal city of Phoenicia, studied local myths and religions, and then lived in seclusion in a temple in Tyre. After arriving in Egypt, King Amassis recommended him to study in the temple. During the ten years from 535 BC to 525 BC, Pythagoras studied hieroglyphics, Egyptian mythology, history and religion, and publicized Greek philosophy, which was highly respected by many Greeks and many people went to his door to study. Pythagoras returned to his hometown of Samos at the age of 49 and began to give lectures and run schools, but it did not achieve his expected results. Around 520 BC, in order to get rid of the tyranny of the monarch at that time, he left Samos with his mother and only disciple, moved to Sicily, and later settled in Crotone. There, he recruited disciples and established a religious, political and academic group. His speech attracted people from all walks of life, and many upper-class people came to the lecture. According to the custom at that time, women were forbidden to attend public meetings. Pythagoras broke this rule and allowed them to attend lectures. Among the enthusiastic audience was his later wife Siena, who was young and beautiful and wrote a biography for him, but unfortunately it has been lost. Pythagoras established a secret society in Claughton, a Greek territory in southern Italy. There are men and women in this society, they all have equal status, and all property belongs to the public. The organization discipline of the association is very strict, even with a strong religious color. Every student must reach a certain academic level, join an organization and go through a series of mysterious rituals in order to achieve "spiritual purification". They have to accept long-term training and examinations, abide by many norms and commandments, and swear never to reveal the secrets and theories of the school. They believe that relying on mathematics can sublimate the soul and integrate with God. Everything contains numbers, even everything is numbers, and God rules the universe through numbers. This is the main difference between Pythagoras school and other sects. Members of the school share the same philosophical beliefs and political ideals. They eat simple food and receive strict training. School teaching encourages self-control, abstinence, purity and obedience. They began to gain a high reputation and considerable influence in Great Greece (now southern Italy), which also aroused the envy of rival factions. Later, due to the impact of the pro-democracy movement, the venues for community activities in Crotone were severely damaged. Pythagoras was forced to move to Linton (now taranto in southern Italy) and died in 500 BC at the age of 80. Many disciples fled back to Greece and re-established their position in Frios. Others went to taranto to continue their mathematical philosophy research and political activities until the middle of the 4th century BC. The Pythagorean school continued to prosper for two centuries. Euclid: He was admired by later generations for writing The Elements. However, there are only two stories about him: According to Proklos (about 4 10 ~ 485), King Ptolemy once asked Euclid if there were any other shortcuts to learning geometry besides his Elements of Geometry. Euclid replied, "In geometry, there is no paved road for kings." This sentence later became an eternal learning motto. Stobeus (about 500) told another story, saying that a student had just started to learn the first proposition and asked Euclid what he would get after learning geometry. Euclid said: Give him three coins because he wants to get real benefits from his study. "Yuan" is still studied by students from generation to generation. There has never been such a perfect textbook for more than 2000 years. It has hundreds of editions, second only to the Bible. Archimedes: Archimedes is called one of the three great mathematicians. Archimedes (287-2 12 BC) was born in a small village near Syracuse, Greece. He was born into a noble family, and was related to King Hilong of Syracuse, and his family was very rich. Archimedes's father was an astronomer and mathematician, knowledgeable and humble. Influenced by his family, Archimedes became interested in mathematics, astronomy, especially ancient Greek geometry. When he was just eleven years old, with the help of his relationship with the royal family, he was sent to study in Alexandria, Egypt. Alexandria, located at the mouth of the Nile, was one of the centers of cultural trade at that time. There are magnificent museums, libraries and talented people here, which are praised as "the capital of wisdom" by the world. Archimedes studied and lived here for many years and had close contacts with many scholars. He absorbed the excellent cultural heritage of the East and ancient Greece and made great contributions to his later scientific career. In 2 12 BC, Roman troops invaded Syracuse, and Archimedes was killed by Roman soldiers at the age of 75. Archimedes' body is buried in Sicily, and the tombstone is engraved with a figure of a cylinder engraved with a ball to commemorate his outstanding contribution to geometry. However, with the publication of Research on Conic Curves, it marks the end of an era, and mathematics has entered a dark period since then. When the church kept controlling people's minds, the darkness of the Middle Ages still hung over mathematics.