Guide
Professor Cai Tianxin of the School of Mathematics of Zhejiang University recently published "Mathematics and Art" (Jiangsu People's Publishing House, 2021.7), which was selected into the "Guangming Book List" of Guangming Daily, interviewed and reported by the reporter of Jiefang Daily, excerpted from the full page of China Reading Daily, and will also be promoted by China Science News... "Mr. Sai" invited authors to excerpt a section from the chapter "Greek Mathematics and Greek Art" for the benefit of readers. This article has been slightly revised.
Written by | Cai Tianxin
Aristotle's Poetics and Euclid's (Geometric) Primitives, respectively, are the highest theoretical crystallizations of literary theory and natural science in the classical era, each influencing the future generation for more than 2,000 years. Aristotle and Euclid were both students of Plato's Academy, though the two did not know each other and may not even belong to the same era. In essence, their works are imitations of three-dimensional space, except that the former is an imitation of the image space and the latter is an imitation of the abstract world.
Wang Ruonan painted
<h1 class="pgc-h-arrow-right" data-track="13" >01 Plato Academy</h1>
In 500 BC, Pythagoras was killed at Metapontum near Taranto. The following year, the First Large-Scale War between Eurasian Continents in World History, the Persian-Greek War, broke out. The war was caused by the invasion of Greece by the powerful Persian Achaemenid dynasty for expansion. Unexpectedly, the war lasted for nearly half a century.
The opposite turned out, and after the Battle of Marathon, the Battle of Thermos Pass, and the Battle of Salami Bay, the Greek city-state and institutions survived, while the Persian Empire was in a state of collapse. This war was an unprecedented cultural integration in human history, and its influence far exceeded that of Persia and Greece, promoting the exchange and development of Eastern and Western civilizations, and promoting the development and progress of science, art and human society. Greek civilization was preserved and flourished, becoming the foundation of future Western civilization.
The year the Persian-Greek War ended, socrates (470-399 BC) had just turned 20 years old, and Athens was in the golden age of Pericles (495-429 BC). Socrates came from a poor background, his father was a carver, his mother was a midwife, he himself worked as a carver and stonemason, and is said to have participated in the construction of the Acropolis. Socrates had a flat nose, thick lips, bulging eyes, and a short stature. Although his appearance is ordinary, he speaks plainly and has a sacred thought in his head. At that time the wise men gathered in Athens from all over the country, bringing to democracy many new trends in knowledge and free debate, and Socrates learned from the wise men, and was also influenced by the Ophis and the Pythagorean school.
Socrates was a very learner in his youth, and later became familiar with Homer's epic poems and became a very learned man by self-study. He lived a hard life all his life, no matter the cold and heat, he wore an ordinary single coat, often did not wear shoes, and ate and drank even more exquisitely. Socrates devoted himself to learning and made a living by imparting knowledge, without setting up a museum or receiving remuneration, and his biography and ideological achievements were recorded by his disciples, the two most outstanding of whom were Plato and Xenophon. Socrates' doctrine was mystical, and he opposed the study of nature as blasphemous. He liked to raise objections, loved to satirize people, and his main weapon was rebuttal arguments, also called cross-examination, to point out the potential confusion and absurdity of other people's views, which was called the Socratic method by later generations.
In 429 BC, After being re-elected as chief general, Pericles was killed by the plague and Athens flourished. By this time Socrates had become a well-known figure, and many children of rich and poor families gathered around him to ask him for advice. Socrates often said, "I know only that I know nothing." "Only God is wise." He prided himself on his ignorance and believed that everyone should admit their ignorance. However, in 399 BC, Socrates was charged with "blasphemy" and sentenced to the traditional capital punishment of drinking ducks. He was charged with corrupting youth, flouting the gods worshipped in the city-state, and engaging in bizarre religious practices.
Because Socrates approached the allegation with disdain, while at the same time making a "defense" equivalent to admitting that it was true, he was found guilty by the court by 280 votes to 220. At that time, if Socrates had agreed to pay a fine that did not have to be too large, he could not have been sentenced to death. Then he took a tough stance, saying that he was actually the great benefactor of society and should enjoy the treatment of state support as an outstanding person. As a result, the claim angered the court, and the death sentence was passed by more votes. For more than a month, friends came to see him in prison every day, and some tried to help him escape, but he refused. In the end he chose to drink the poisonous viola juice and died slowly and painfully.
Socrates' death, especially his fearlessness before his death, gave his disciple Plato a deep stimulus. Although he came from a prominent background, he gave up the idea of politics and devoted his life to philosophical research, calling his mentor Socrates "the wisest, most just, most outstanding man ever seen." After Socrates' death, Plato left Athens and began a long journey, visiting Asia Minor, Egypt, Cyrene (present-day Libya), southern Italy, and Sicily. Around that time, Plato began to write dialogues, mostly with Socrates as the central figures, and other characters mostly real. But whether his writing began before or after Socrates' death is uncertain.
Plato's head, now in the Capitolini Museum in Rome
It was during his journey that Plato came into contact with several mathematicians, including a descendant of the Pythagorean school in Sicily, and studied mathematics himself. After returning to Athens, Plato collaborated with people to create a school in the northeast of the city that resembled a modern private university (the term is used to commemorate a hero of the battle, which now means academy of sciences or institutions of higher learning). The school had classrooms, dining rooms, auditoriums, gardens and dormitories, and Plato was the head of the school, and he and his assistants taught various courses. In addition to being invited to return to Sicily twice to lecture, he spent the last 40 years of his life in the academy, which itself miraculously existed for nine hundred years, like the legendary Pythagorean school.
As a philosopher, Plato had a profound influence on the philosophy of Europe and the development of culture and society as a whole. He wrote a total of 36 books during his lifetime, most of which were written in the form of dialogues. The content is mainly about political and moral issues, but also about metaphysics and theology. For example, in the "State Chapter" he proposes that all people, men and women, should have the opportunity to show their talents and enter the governing body. In the Book of Drinking, the unmarried wise man of his life also speaks of lust, "Desire is from the soul, to attain the goodness of longing, and the object is eternal beauty." "In the most popular words, to love a beauty is actually to seek the immortality of life through the body and heirs of the beauty.
Although Plato himself did not make particularly prominent contributions to the study of mathematics (some attribute analytical (1) and reductionarith (2) to him), his academy was the center of Greek mathematical activity of his time, and most of the important mathematical achievements were achieved by his disciples. For example, the irrational study of the square root or the high power root of general integers (including the generation and resolution of the first mathematical crisis caused by the discovery of irrational numbers), the construction of the regular octahedron and the regular twenty-sided body, the invention of the conic curve and the exhaustion method (the invention of the former was to solve the multicube problem (3)), and so on.
The inquiry into the philosophy of mathematics also began with Plato. In his view, the object of mathematical study should be the eternal and unchanging relationships in the world of ideas, not the vagaries of the material world of sensations. He distinguishes not only mathematical concepts from the corresponding entities in reality, but also strictly from the geometry used in the discussion to represent them. For example, the idea of a triangle is unique, but there are many triangles, and there are various imperfect facsimiles corresponding to these triangles, i.e., real objects with various triangle shapes. In this way, the abstract definition of mathematical concepts, which began in Pythagoras, takes another step forward.
Of all of Plato's writings, the most influential is undoubtedly the Republic. The book consists of 10 dialogues, the core of which outlines the philosophy of metaphysics and science. Part 6 deals with mathematical hypotheses and proofs. He wrote, "Those who study geometry, arithmetic, and so on, first assume that odd numbers, even numbers, three types of angles, and things like that are known. ...... Starting from the known assumptions, push back in a consistent manner until the desired conclusion is reached. It can be seen that deductive reasoning has become popular in the school. Plato also strictly limited mathematical cartographic tools to rulers and compasses, which played an important role in the formation of the later Euclidean system of geometric axioms.
The 5 regular polyhedra known as Plato's polyhedra
When it comes to geometry, we all know that it was Plato's vehement, an important part of the precision science he conceived of to take 10 years to study. Plato considered the God who created the world to be a "great geometrist" who himself systematically elaborated on the (only) five regular polyhedra features and diagrams, so much so that they were called "Platonic bodies" (illustrations) by later generations. A widely circulated story from the 6th century AD says that at the entrance of Plato's Academy, "Those who do not understand geometry are not allowed to enter." In any case, Plato was fully aware of the importance of mathematics in the search for human ideals, and in his posthumous book, the Laws, he even described those who ignored this importance as "pigs."
Regrettably, Plato praised "God as a geometrician" on the one hand, and expelled the poet from the "Republic" on the other. He once counted the two major crimes of the artist, "Art is not true, it cannot give people the truth; art is bad and bad, and it confuses people's hearts." But Plato was not opposed to poetry, he wrote in the State: "The pastimed, pleasant poetry proves that it has a reason for existence in a well-managed city-state, and we are more than happy to accept it, because we ourselves can feel its charm." But it is always ungodly to turn our backs on what we believe to be the truth and to cling to poetry, because poetry and poets interfere with our peaceful souls and rational judgments about the world. In other words, Plato proceeded from reason and stood in the position of the government.
Plato's Academy produced countless outstanding students, including the all-rounder Aristotle. In the field of mathematics, there are Thasted, the founder of stereo geometry (who was also co-founder of Plato's Academy), Odoxos, the founder of the theory of proportions (he proposed that numbers and quantities are two different concepts, the former limited to rational numbers, and he was the first to build geometric models of the universe), and Menaechmus (a student of Eudoxos), the discoverer of conical curves. Even the great mathematician Euclid came to Akademi in his early years to study, all of which earned Plato and his academy the reputation of "the founder of mathematicians". Here I would like to introduce Eudoxos first.
Map of ancient Athens, Plato's Academy in the northern suburbs
Eudoxus (c. 400 BC – c. 347 BC) was born in Cnidus, Asia Minor, and worked as an assistant physician in his youth. In 368 BC, Eudoxos accompanied the doctor to Athens, where he studied for two months, and he lived in the outer port of Piraeus, during which he walked dozens of kilometers almost every day to the Plato's Academy in Athens to study mathematics, astronomy and philosophy, and at the same time established a personal friendship with Plato. Years later, Eudoxos established his own academy on the south shore of the Marmara Sea, and eventually led some of his students to relocate to Athens, most likely joining platonic academy.
<h1 class="pgc-h-arrow-right" data-track="212" >02 Aristotle and Poetics</h1>
It is well known that the most outstanding students of Plato's Academy were not the four mathematicians mentioned at the end of the previous verse, but the all-round aristotle (384-322 BC). Aristotle was born in Stagira, a small town on the Halkidiki Peninsula in northern Greece. His father was the physician of the Macedonian king Amintas III, who was the father of Philip II and the grandfather of Alexander the Great. As the son of a doctor, Aristotle came into contact with hippocrates (460-377 BC), the "father of medicine", who understood the concepts and practices of medicine and biology at an early age, so that his philosophical thought later had a distinctly biological tendency.
Aristotle's head
Aristotle died when he was young, and he was entrusted with the custody of a relative. At the age of 17, Aristotle was sent to the Plato Academy in Athens, where he spent 20 years. He was taught by Plato and other teachers to be energetic, active in thought, and experienced rhetoric and debate. In 367 BC, Plato suddenly felt unwell at a wedding feast, then retreated to a corner and died quietly. The academy was inherited by Plato's nephew, and Aristotle left Athens and began roaming like his teacher. As for the reason for his departure, it may be because he did not manage the academy, which is different from the philosophy of the new headmaster, or it may be because of his Macedonian background.
Aristotle was in the company of his classmate and friend Xenocratis, who first went to the port city of Assus on the northwest coast of Asia Minor, where they began to study anatomy and to write Political Science. Aristotle was active and Xenocratic was taciturn, and was likened by his teacher Plato to "one needs spurs, the other needs a cage..." However, when they worked together at Assos's new school, the two different personalities made a match made in heaven. The patron was a wealthy soldier who, when he visited Plato's Academy, had the ambition to build one in Asia Minor, extending Greek institutions and Greek philosophy to the lands of Asia.
In Assos, Aristotle married the patron's niece, Pisias, and they had a daughter. "Political Science" is the pioneering work of Western political science research, based on his and his students' investigation of the political and legal systems of the Greek city-states, its core content is the problem of the city-state, based on the premise that "man is a natural political animal", analyzing the formation and foundation of the city-state, as well as the education of citizens, etc., and putting forward the ideal city-state. The book also depicts his ideal marriage, with her husband 37 and his wife 18. The author was 37 years old, so it has been speculated that Pisias was 18 at the time. Aristotle had expressed his wish that the two would be buried together after death, but unfortunately his wife died too early, and he later found a partner (it is impossible to determine whether the wife or lover), and she bore him a son.
After three years in Assos, Aristotle came to the island of Lesvos, across the water, home to Sappho, greece's first female poetess, who is now synonymous with lesbianism. Aristotle co-founded a school with a friend in the capital, Mytilene, where his interests turned to biology and he conducted pioneering research. He stressed the importance of observation for biology, saying that "theories must be confirmed by what is observed in order to enjoy real honor." Plato believed that the soul was an independent entity, temporarily dwelling in the body, while Aristotle believed that the soul was a life force that essentially constituted an individual together with the body.
At the turn of 343 and 342 BC, the 42-year-old Aristotle was invited by Philip II to the Macedonian capital of Pella to work as a tutor for Alexander, 13. He tried to train the prince according to the heroic image of Homer's epic, who later became one of the greatest kings and warriors in human history, but the relationship between the two is said to be not harmonious. Aristotle believed that the Greeks were above all else and taught his disciples that their subjects should be forbidden to intermarry with Gentiles, but instead he married a Persian noblewoman and even executed Aristotle's nephew for treason. However, Alexander later gave his teacher a gift to rebuild his hometown of Stuckira, which had been destroyed by his father.
In 335 BC, aristotle, who was nearly half a hundred years old, returned to Athens from Stuckyla, when plato's academy was in its prime under the helm of Xenokrati, and he founded his own school in the forest on the outskirts of the city, Lu Garden, named after the mythical wolf slayer Lyceum. Aristotle was informal, often walking with his disciples and teaching at the same time, so it was called the Proms, and LuYuan also became a school that integrated teaching and research. Twelve years later, when Alexander the Great died on the way to battle, there was a brief and fierce wave of anti-Macedonians in Athens, and Aristotle, feeling that he was in danger, left Athens for Chalcis, the capital of his mother's hometown, the island of Evia, north of Athens. The following year, Aristotle died on the island due to a stomach ailment.
Like Akademi, Lu Yuan educates students through argument and discussion, and does not require them to blindly accept the teacher's point of view. Aristotle's style is concise and clear, full of personality, and his ideas can be summarized as follows: experience is the source of knowledge, logic is its structure. The 1st-century BC Roman writer Cicero wrote in the School of The School that "Aristotle's gentle style is like a golden river." His writings are abundant, with hundreds of volumes surviving in the late antiquity period, and the ancient bibliography lists as many as 170 of his independent works. Even today, more than 30 copies and about 2,000 printed pages have survived. Aristotle mentioned in His Treatise on Philosophy the five stages of the development of civilization, namely: the production of the necessities of life, the creation of art, the art of politics, the glorification of life, and the philosophy from nature to God.
In general, Aristotle's writings fall into two categories. One is published before death, and some are lost later; the other is collected and preserved by others without being published by oneself or not intending to be published at all. Poetics may have belonged to the latter, and was written after his return to Athens at the end of his study tour, and can be described as a work of his mature period. The extant text, which are his lecture notes in Lü Yuan or the notes of his protégés, are rigorous but obscure, may not have been collated and processed, and are prone to different interpretations. The Poetics was divided into two volumes, but unfortunately the second volume was lost. In the first volume, it is said that comedy will be discussed later, and it may be in the second volume. In any case, "Poetics" is the crystallization of Aristotle's aesthetic thought, the pioneering work of Western aesthetics.
Arabic translation of the 10th century Poetics
It is said that the Poetics and other works of Aristotle slept in the cellar for more than a hundred years, and were later circulated after being collated and revised by the Promsist philosopher Andronicus of Rhodios (1st century BC). The Poetics was translated into Syriac in the 6th century and into Arabic in the 10th century, and the earliest surviving version is an 11th-century Byzantine manuscript. At the beginning of the Renaissance, poetics was translated from Arabic into Latin. Since the late 15th century, poetics has had an increasing influence on European literary and aesthetic thought, and classical literature and aesthetics have enshrined it as a guigao. Even the construction of modern contemporary art and aesthetic theory is inseparable from the study of "Poetics", and people draw ideological nourishment from different angles and ways.
There are twenty-six chapters in the existing Poetics, which can be roughly divided into three parts: one to five chapters, which deal with the nature of art as imitation, distinguishing different art forms in turn, and tracing the origin and historical development of art; chapters 6 to twenty-four and twenty-six, discussing the characteristics and constituent elements of tragedy, and comparing epics and tragedies; and twenty-five chapters, the principles and methods of criticism and rebuttal. The book mainly discusses three philosophical issues of art, namely the nature of art, the meaning of tragedy, and the role of art, and its aesthetic ideas can also be summarized into three points: imitation, tragedy, and purification. Among them, the author believes that the most essential thing is imitation.
Aristotle began by pointing out that the nature of art is imitation. He argues that epics, tragedies, comedies, Dionysus, and various other arts "are imitations ... imitates human character, emotions and activities". But what Aristotle called "imitation" is not the same as the imitation and imitation that we usually understand, but also has the broad meaning of the imitation of nature by skill. The "poem" in Poetics refers to the art of creation (artistic creation), which expresses people and human life. In fact, Greek art was originally characterized by harmony, solemnity, tranquility, etc., artistically reproducing life.
Before Aristotle, Greek philosophers had spoken of "imitation," but given it a different meaning. Pythagoras believed that beauty was a logarithmic imitation; Heraclitus advocated that art imitated the harmony of nature; Socrates said that paintings, statues and other arts not only imitated the image of beauty, but also could imitate human emotions and personality through images. Plato, in the State Chapter, discussed in detail that poetry is imitation. He believes that the carpenter imitates the "form" of the bed to make the wooden bed, and the bed painted by the painter is only a mirror image of the wooden bed, not a real existence. Art is far from ideas and reason, not only does not contribute to the governance of the city-state and civic morality, but also rebels against truth and goodness, resulting in a conflict between reason and lust. It is for this reason that Plato advocated the expulsion of the poet from the Republic.
Aristotle's philosophical ideas differed from those of his teachers, and he had very different understandings of the "imitative" nature of art. In his view, real things, including human activities, are real and have various meanings; poetry imitates human activities and creates the true existence of art in works; "imitation" is not only an external image, but also a manifestation of human nature and activity, showing this "existence" meaning of man. Moreover, "imitation" is an intellectual curiosity activity, obtaining truth in a figurative way and forming creative knowledge about people; the "imitation" of art is not only driven by feelings and desires, it relies on "practical wisdom" to perceive life and grasp the true meaning of life. Therefore, the art of imitation is a noble intellectual activity.
Aristotle's imitation can be summarized in the following three aspects. First, all art arises from imitation. Second, imitation is human nature, and art evolves and perfects in the realization of human nature. He believes that man is born with the gift of imitation; man has an innate ability to perceive beauty in beautiful things. In his view, from childhood we have had the instinct to imitate; man is the most imitative animal, through which man acquires his first experience and knowledge; imitation is essentially a curiosity, and it is precisely because of this that man is distinguished from other animals.
Third, imitation should express necessity, probability and type. Aristotle believed that the duty of the poet was not to record what had happened, but to describe what might happen out of necessity, probabilism, of a particular kind of person and thing. The difference between history and poetry is not only whether it is written in rhyme, but Herodotus's History, if rewritten as rhyme, is still a work of history. He analyzes the difference between poetry and history: history describes facts that have already occurred, while poetry describes things that may have happened in the past, now, and in the future out of necessity and probability; history records only individual events and people, and poetry shows "universal nature" in possible events and people, and is therefore "more philosophical and serious" than history.
"Poetics" and "Poetic Art" published by the People's Literature Publishing House.
Poetics is the first and most important document in the history of European aesthetics, and Aristotle took the lead in using scientific views and methods to elucidate aesthetic concepts and study literary and artistic issues. Unfortunately, in the surviving 26 chapters of Poetics, Aristotle deals primarily with epics and tragedies, without dealing with the plastic arts or even lyric poetry. Probably because lyric poetry had no layout, the ancient Greeks considered it to be music. However, the ancient Greeks left behind a few but exquisite works in carving and architecture, although some of them are only remnants or imitations.
Myron (480–440 BC) was an ancient Greek sculptor whose original bronze work Discus Thrower (c. 450 BC) has been lost, and we now see imitations of marble from the Roman period. Phidias (490-430 BC), a contemporary of Miron, was a great sculptor, but unfortunately his masterpiece The Three Goddesses of Destiny in the Parthenon Temple in Athens had a severed limb and no head. The Venus de loser in the Louvre in Paris is a marble work from about 150 BC, but it was already in decline in Greek art, so some art historians believe that it is an imitation of the work of the 4th century BC. Venus was discovered in 1820 on the island of Milos in the Aegean Sea by a farmer, and the human body dynamics match the natural folds of the shu roll, with a unique charm, and was acquired by the French after several twists and turns.
The closest complete carving is Laocon, the work of the sons and sons of rhode island sculptors in the 1st century BC, based on the story of the Trojan War in Homer's epic poem. Laocon, a Trojan priest, saw the Trojan horse and warned his countrymen not to drag them into the city (which contained Athenian soldiers). Athena, the patron saint of Athens, punished Laocon for "leaking the chance" by releasing two giant pythons to entangle him and a pair of twin sons. The sculpture depicts this shocking scene, reflecting God's deterrent power against the masses, including the resentment of society that "those who tell the truth do not have a good end". In 1506, the group of sculptures was discovered at the site of the Roman Baths of Titus and acquired by Pope Julius II, now in the Vatican Museums. Due to its early discovery, it had a huge influence on renaissance art. Michelangelo praised: It's incredible. The German aesthetician Gothold Lessing (1729-1781) wrote the book of the same name accordingly, which is one of the most important aesthetic documents in Europe.
Architecturally, around the 6th century BC, ancient Greek columns underwent a petrification process, changing from wood, mud bricks or clay to stone columns. As a result, some of the buildings have survived, the most famous of which is undoubtedly the Pathenon Temple in the Acropolis. According to the 1st-century BC Roman architect Vitruvius's Ten Books of Architecture, it was designed by the 5th-century BC architects Ictinos and Callicarats. Like other buildings of that era, the Parthenon used a golden section ratio, i.e. a 0.618:1 ratio of width to height. At the same time, the column style draws on the specialties of various city-states, such as the Ionian style on the coast of Eastern Asia Minor, which is rich in soft, lively feminine beauty; the Doric style of the Southern Peninsula, which is rich in strong tone and solid male beauty; and the Collins style is more slender and decorative. The earlier Temple of Athens is best preserved, as it was converted into a Christian church in the Middle Ages.
Finally, we would like to touch upon the different views of Aristotle and Plato on private property. Historically, the concept of property has changed frequently, generally referring only to land and slaves in the early days, and later expanding to intangible assets such as patents and copyrights, as well as anything that was considered necessary for life and freedom. Inspired by the Spartans, Plato opposed all kinds of private property, which he believed corrupted the soul and fueled greed, jealousy, and violence. Aristotle's view was the opposite, he believed that it was easier for owners of common property to fight among themselves, and that private property was essential to human progress because people would be motivated to work hard and thus become moral people. The Polish-born American historian Richard Pipes (1923-) put it bluntly: "Humans must have it in order to survive." ”
Speaking of private property, it has to do with today's hot Bitcoin. In the age of the Internet, people are eager to digitize personal property, which requires solving two problems. One is scarcity, and digital products are wary of being copied and redistributed. The second is transferability, only in this way can assets be traded, otherwise ownership will lose power. These two points are also attributes of money, and once solved, digital products will become the new numismatic form. Bitcoin satisfies these two points, first, it reinforces consensus rules and protects them through software, creating digital scarcity; second, transferability can be guaranteed by public key cryptography, that is, non-forgeable digital signatures. The most famous public key cryptography is the RSA (three professors' surnames at the Massachusetts Institute of Technology) system, which is constructed using Euler's theorem in number theory and the Great Derivation Ofe Technique of the Southern Song Dynasty mathematician Qin Jiushao (1202-1261), which says that if the integers a and m are mutually elemental, then there is a unique solution to the congruent ax≡1 (mod m).
<h1 class="pgc-h-arrow-right" data-track="213" >03 Euclid and the Original</h1>
If Aristotle's Poetics identifies the nature of art as imitation, it is an imitation of the image of the three-dimensional space of human life. Well, we can say that Euclid's Original, a little later than him, is an abstract imitation of the same three-dimensional space. Regarding the mathematician Euclid, no one knows the exact year of birth and death, the place where he was born, grew up, died, and was buried (4), and he became famous for his "Original". Before the 19th century, Euclid was synonymous with geometry, and there was a popular saying that the total print of the Original was second only to the Bible, and Euclid was regarded as the most influential of all pure mathematicians on the course of world history.
A 19th-century statue of Euclid. It is now in the collection of the University Museum of Oxford
We now know that around 300 BC, Euclid was active as a mathematician in Alexandria, the cultural and educational center of ancient Greece on the Mediterranean side, when it was the capital of the Ptolemaic dynasty and the second largest city in Egypt. This suggests that Euclid lived later than Aristotle, but earlier than Archimedes, another great mathematician. It is mainly based on two works, one is the Outline of the Development of Geometry (c. 450, abbreviated as the Compendium of Geometry), the tutor of the late Plato School, which is his commentary on the Original. The other is the 4th-century mathematician Papos's "Compilation of Mathematics" (abbreviated as "Compilation").
The Compendium states that Euclid was a man of the time of Ptolemy I (c. 367-282 BC, reigned 323-285 BC), who studied in Athens in his early years and was well versed in Plato's teachings. His Originals quote the work of several Platonic figures, such as Eudoxos, who himself was a member of the Platonic School. It is also mentioned in the book that Archimedes' writings also quote propositions from the Original. The Original establishes postulates and axioms, which are clearly influenced by Aristotle's logical ideas. The Repertory records that Apollonius of Perga (c. 262–190 BCE), the master of the conical curve, lived in Alexandria for a long time and became acquainted with Euclid's students.
The Compendium recounts an anecdote: King Ptolemy asked Euclid if there were any shortcuts to learning geometry besides the Original, and Euclid replied: "There is no king's way in geometry." This phrase was later popularized as "there is no smooth way to seek knowledge". The 5th-century editor Stobias also recorded an anecdote about Euclid: When a new student asked what he would gain from studying geometry in the future, Euclid did not answer positively, but asked his slave to give him a penny, and then said, "Because he always wants to gain from his studies." Thus, Euclid demanded that students be gradual and oppose opportunistic and skillful pragmatism.
Euclid's "Original" is an epoch-making work that has been passed down from the time of its birth to the present day and has had a sustained and significant impact on the progress of human civilization. The historical significance of the Original is that it is the first example of a logical deduction established by axiomatic methods, and the mathematical knowledge accumulated before is fragmentary, like bricks and tiles and wood, and Euclid has to use logical methods to organize and combine this knowledge so that it is in a strict system and built into a towering edifice. It can be said that Euclid was the most outstanding architect in the field of mathematics.
From Thales to Euclid, Greek mathematics has gone through three centuries of development. From the Ionian school's opening of geometric arguments to the Pythagoreans' extraction of abstract numbers from concrete things, it provided the material and cornerstone for the birth of the Original. After the Persian-Greek War, Athens became a cultural center, where the Homo sapiens school proposed the three major problems of geometric drawing, the rules of using rulers and compasses, and became the golden rule of Euclidean geometry. Antiphon's "exhaustion method", which tried to solve the problem of "turning circles into squares", gave birth to modern limit theory ideas. Democritus deduced from the atomic method that the volume of the cone is one-third of the same bottom isometric cylinder, which has become an important calculation method of the Original.
It is worth mentioning that about a century before Euclid, Hippocrates of Chios (about 460 BC) wrote a work on geometric principles, which may have been the prototype of the Original, but unfortunately did not survive. This Hippocrates predates the eponymous "Father of Medicine" and was born on the island of Chios off the coast of Asia Minor, northwest of Samos. Legend has it that he was a merchant who went to Athens to complain because his goods were robbed by pirates and failed to recover his belongings. As a result, he stayed in Athens, listened to mathematical lectures, and finally became able to live by teaching geometry. Aristotle, on the other hand, had a different theory, believing that he had been deceived by the Byzantine tax collectors. Hippocrates discovered and proved the "crescent theorem" using Thales' theorem and Pythagorean theorem: the sum of the areas of a right-angled triangle on a semicircle, the area of the bow formed by the two right-angled edges (lune AC and lune CB) is equal to the area of the triangle. ⑤
Proof of the crescent theorem
The most influential on the Original is the Platonic School. Plato attached great importance to mathematics, with particular emphasis on the abstract nature of ultimate reality and the importance of mathematics in the training of philosophical thinking, and his disciple Eudoxos used axioms to establish the theory of proportions, mostly taken from his work in Book 5 of the Original. Eudoxos also improved Antiphon's exhaustion method for the first time in mathematical proof, making it an important method of argument in the Original. The formal logic created by Aristotle created the necessary conditions for the rigorous system of Euclidean geometry. It is worth mentioning that the "circle cutting technique" founded by Liu Hui, a mathematician in the Wei and Jin dynasties in China, is also an exhaustion method, but Anti feng starts from the square, while Liu Hui starts from the regular hexagon.
Of course, "Original" also has the limitations of the times. For example, as early as the 5th century BC, Zero of Elea (c. 490-c. 425 BC, of which Pericles was a student) of the Ilia School in southern Italy proposed several famous paradoxes, the proof of which used the law of attribution, forcing mathematicians and philosophers of that time and later to begin to think about infinite problems. This is reflected in "Original", but it is more or less avoided. For example, in the first volume of the public set III, it is clear that the line segments can be "infinitely" extended, but it is said that the line segments can be "arbitrarily" extended; for example, the fourth book proposition XX, which obviously proves that "there are infinite numbers of prime numbers", but it is written that "the number of prime numbers is more than any given number".
The Original consists of fifteen volumes. The first six volumes deal with geometry, the next four volumes are about number theory, but are described in geometric terms, and the last five volumes are still about geometry. The first volume begins with 23 definitions, two of which are extraordinary: A point is that which has no part; A line is breadthless length. This is followed by definitions of planes, right angles, vertical angles, sharp angles, obtuse angles, parallel lines, etc. After the definition are five public settings, the first four are:
A straight line can be made from any point to another;
Line segments can be arbitrarily extended;
A circle can be made with any center and any radius (the above is euclidean diagramming);
All right angles are equal.
If a straight line intersects two straight lines, and the same side inner angle is less than the two right angles, then the two straight lines must be extended to intersect on one side of the two inner angles.
The narrative of the fifth postulate is somewhat cumbersome and puzzling, but it is most famous until more than two thousand years later, when the 18th-century Scottish mathematician Playfair (1748-1819) gave the concise form (Pleifer postulate) that we know today:
A little outside the line can be made and can only be made as a single parallel line.
The postulate is followed by the axioms, also five, the first two of which are:
Quantities equal to the same amount are equal to each other;
The whole is larger than the part.
After that, there are no postulates or axioms in each volume, but only propositions, and each volume has dozens. Volume I has 48 propositions and now looks relatively elementary. Proposition 5, for example, is a common topic of the ancient Greeks: the two bottom angles of the isosceles triangle are equal, and their outer angles are equal. At that time, the level of mathematics in Europe was indeed very low, and Proposition 5 was actually considered to be "Donkey Bridge", which means "stupid difficulty".
Volume 2 is a little deeper, where proposition 5 is the solution of the unary quadratic equation, which basically gives the formula solution in today's middle school, and its equivalent form can be derived from the following Pythagorean array:
where n can be any positive integer. According to Proclus, this array was given by Pythagoras. He also pointed out that Plato gave another array, i.e. that
Proposition 13 is the cosine theorem known in today's secondary schools
However , Euclid described the cosine theorem geometrically , and did not have trigonometric functions. Book VII begins with number theory, with 22 definitions at the beginning, the last of which is the definition of a perfect number: a number equals its own part (the true factor).
The sum of these numbers is called the perfect number. It was only in Book IX that Proposition 36 proved the famous result of perfect numbers: if
is a prime number, then
is the perfect number. It is worth mentioning that the inverse theorem of this conclusion was not proved by the Swiss mathematician Euler until more than two thousand years later. So we have: an even number is perfect if and only if it is of the following form
Here n and
are prime numbers, wherein
Now known as The Mersenne Prime, named after the 17th-century French mathematician and priest Mason, 51 Mersenne primes and 51 even perfect numbers have been found so far, and not a single odd perfect number has been found. Are there infinitely more than one even perfect number? And are there odd perfect numbers? It is the oldest problem in the history of mathematics, and it is also one of the most difficult problems.
As early as the end of the 8th century, the Original was translated into Arabic by scholars of the time of Rashid, the fifth caliph of the Abbasid dynasty in Baghdad, and its first full Latin version was translated from Arabic around 1120 by the English empirical philosopher Adelard of Bath (c. 1116-1142), and the first complete English translation was by the Scottish merchant Henry Billingsley, - 1606) translated in 1570 by the original Greek, he served as mayor of London.
In 1808, Napoleon, a European and mathematician, found some Manuscripts of Mathematics in Greek in the Vatican Library and brought them back to Paris. Among them are manuscripts of euclid's two works, including the Original. A few years later, the Greek, Latin, and French editions of the Original were published simultaneously in France, known as the Vatican, and are recognized as the closest to the original.
Matteo Ricci and Xu Guangqi
The earliest Chinese edition of the Original was published in Beijing in 1607 and was co-translated by the Italian missionary Matteo Ricci (1552-1610) and the Ming dynasty scholar Xu Guangqi (1562-1633), with the words "Tessili Matteo Interpretation" and "Wu Song Xu Guangqi". This is also the beginning of the translation of Western mathematical classics in modern China, which has since opened the door to academic exchanges between China and the West.
They relied on a Revised Latin edition by the German Christoph Clavius (1537-1612), who translated only the first six volumes and gave it the title Geometric Primitives. Shortly after publication, Xu Guangqi's father died, and he helped the coffin to return to the south and worry about Ding for three years. In 1610, Xu Guangqi returned to Beijing to resume his post, and Matteo Ricci had unfortunately died of illness. It was not until 250 that the English missionary Alexander Wylie (1815-1887) and the Qing Dynasty mathematician Li Shanlan (1811-1882) collaborated to translate the Original and publish it in Shanghai.
Until the birth of non-European geometry in the 19th century, the Original was the main source of reasoning, theorems, and methods of geometry. It is also an important factor in the emergence of modern science, and its complete deductive reasoning structure even inspires thinkers. It can be said that "Original" is not only an imitation of the real world, but also provides the necessary tools for such imitation.
In a sense, this kind of imitation is the imitation that Aristotle spoke of in The Poetics. It can be said that around the same time, ancient Greece gave birth in a similar way to the Original and Poetics, which are the crystallization of the highest theories of mathematics and art. By the Time of the Renaissance, the methods and practices of painting were inseparable from Euclid geometry, the most prominent of which were the principle of perspective and the absence of shadows.
Mr. Wu Wenjun wrote a letter of recommendation (partial) for the book "Mathematics and Art" in his early years.
annotations:
(1) The analytical method is a method of decomposing complex things or phenomena into several simple components and conducting research separately. The analytical method is the symmetry of the comprehensive method, which is the method of linking the various components, aspects and factors to understand and grasp things or phenomena as a whole.
(2) The reductive fallacy method is a form of the method of counter-evidence. When using the counter-evidence, if there is only one situation in the proposition, it is enough to refute it, which is called the "reductive fallacy method". If there are multiple cases, they must be refuted one by one to prove that the proposition is true, and this method of counter-evidence is called the "exhaustive method".
(3) The times cube is one of the three major geometric problems of the so-called ancient Greece, and the other two problems are the transformation of circles into squares and third equal angles, all of which require only the use of rulers and compasses for drawing. It was not until the 19th century, with the advent of Galois theory and Lindemann's demonstration of transcendent numbers, that mathematicians figured out that these three problems were practically unsolvable.
(4) There is confusion that the mathematician and philosopher Eucleides of Megara, a century before Euclid, was a student of Socrates and one of the interlocutors of Plato's Theaethème, a city west of Athens.
(5) First, by Thales' theorem, the circumferential angle on the semicircle is a right angle; secondly, by the Pythagorean theorem; and then by the formula for calculating the area of the circle or semicircle, the sum of the areas of the semicircle with AC and CB as the diameter is equal to the area of the semicircle with AB as the diameter; and then subtracting the sum of the two common arc areas respectively, that is, the crescent theorem is realized.
About the Author
Cai Tianxin
Professor, doctoral supervisor, and Qiushi Distinguished Scholar of the School of Mathematics of Zhejiang University, his recent works include the revised edition of "Little Memories", "My University", "26 Cities", "Mathematics and Art", "Modern Introduction to Classical Number Theory" (Chinese and English editions), "Perfect Numbers and Chibonacci Sequence" (that is, out), editor-in-chief of "Poetry of the Subway" and "Poetry of High-speed Rail".
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