Thales: The first person in the childhood of science丨Expand the scroll

Thales, an ancient Greek mathematician and philosopher, is the first mathematician in history to have his name recorded. In later documents, many of his feats are recorded, such as calculating the height of the pyramid, proving Thales' theorem, introducing many geometric propositions, etc. Thales was a learned wise man, the founder of the Miletus School, and the "first of the seven wise men" in the Socratic era. It is said that he was the first person to ask the question "What is the origin of the world?" More importantly, Thales used rational thinking to get rid of mythological prophecies and seek truth through natural phenomena, which ultimately led human civilization to the path of science.

Written by | Cai Tianxin

If you don't know geometry, please don't enter.

—Plato

1Thales of Miletus

There are many coincidences in the history of human civilization. For example, on April 23, 1616, Shakespeare, the greatest writer in the English-speaking world, and Cervantes, the greatest writer in the Spanish-speaking world, died on the same day. This day later became "World Book Day". In addition, Leonardo da Vinci, an Italian painter and "the most perfect representative of the Renaissance", was born on April 15, 1452 in the Julian calendar, which is also April 23 in the Gregorian calendar. On January 8, 1642, Galileo, the greatest Italian scientist, died, and within a year, Newton, the greatest British scientist, was born. Earlier, mathematicians and philosophers in ancient Greece emerged in large numbers, just as writers and artists emerged in large numbers in Italy during the Renaissance.

In 1266, the second year after the great poet Dante was born in Florence, the city gave birth to Giotto, the most outstanding artist of that century. Italians generally believe that the greatest era in the history of art began with him. According to the British art historian Sir Gombrich, before Giotto, people looked at artists like an excellent carpenter or tailor, and they often did not even sign their works; after Giotto, art history became the history of artists.

In contrast, mathematicians are much luckier. The first mathematician whose name was recorded was Thales of ancient Greece (about 625-547 BC), who lived nearly 19 centuries earlier than Giotto. Thales was born in Miletus, Asia Minor, on the east coast of the Aegean Sea (today, the west coast of the Asian part of Turkey), which was one of the 12 city-states of Ionia. Ionia was originally a scattered tribe in Greece. It was not until they immigrated to Asia Minor that they formed a community. That region was therefore called Ionia. It became rich and powerful through trade and then became an alliance.

Thales is known as the leader of the "Seven Sages of Greece" in the pre-Socratic era. The other six are Solon of Athens, Chilon of Sparta, Cleobulus of Rhodes, Periander of Corinth, Pittacus of Lesbos, and Bias of Asia Minor. Due to the long time and the fact that people's thoughts could only be spread orally at that time, except for Thales and Solon, the life stories of the other five sages are untraceable. We only know that they were all politicians and rulers, and each of them only had one or two mottos passed down.

For example, the famous saying of Bias is: "Too many people make things chaotic", the famous saying of Periander is: "Think twice before you act", the famous saying of Pidacus is: "Seize the opportunity", and the famous saying of Cleobulus is: "Be moderate in everything". "Be moderate in everything" is similar to Chinese Confucianism, just like Solon's famous saying "Avoid extremes"; and the famous saying of Periander is close to the Chinese idiom "Think twice before you act", which was originally said by Ji Wenzi, a minister of the State of Lu. However, even Confucius did not approve of his fear of gain and loss. Many maxims of Thales and Solon have been passed down to later generations. I particularly appreciate Thales' "Know thyself" and Solon's "Words are the mirror of actions".

Miletus was the largest city in the East of Greece at that time. Miletus was originally the name of a region in Crete. It can be seen that most of the residents were immigrants from Crete. In Miletus, merchant rule replaced clan aristocracy, so the thinking was relatively free and open, and many famous figures in the fields of literature, art, science and philosophy were produced. The blind poet Homer and the later historian Herodotus came from Ionia. It is said that Thales also engaged in business in his early years, traveled to Egypt and Babylon, learned and mastered mathematics and astronomy, and his later research involved physics, engineering and philosophy in addition to these two fields.

Among the "Seven Sages of Greece", only Thales was a knowledgeable scholar. Thales founded the Milesian School, attempting to get rid of religion and seek truth through natural phenomena. He believed that there was life and movement everywhere, and that water was the origin of all things. Here we tell an anecdote related to water. When he was young, Thales took advantage of his business to have extensive contact with society. Once, he used mules to transport salt. One mule slipped into a stream, and part of the salt was dissolved, which immediately reduced the burden a lot. So the mule rolled every time it crossed the stream. In order to change the bad habits of this animal, Thales let it carry a sponge. As a result, after absorbing water, the weight doubled. From then on, the mule never dared to repeat the same trick again.

Statue of Thales at Miletus

It is said that when Thales was in Egypt, he had mastered mathematical skills. He used the proportional relationship between the sun's shadow and the height of the pole to calculate the height of the pyramid. There is a widely circulated story that on a sunny day, Thales inserted a pole vertically into the ground. When the shadow of the pole was equal to the height of the pole (some records say it was the height and the shadow of a person), he measured the length of the pyramid's shadow, and this length was the height of the pyramid. However, since the base of the pyramid is large and not a point, it can only be accurately measured at a special sunlight angle. The upgraded version of this story is that Thales erected a pole at the end point of the pyramid's shadow, and with the help of the projection of sunlight, formed two similar triangles, and the ratio of the tower height to the pole height was equal to the ratio of the length of the two shadows.

2Thales in the Eyes of Philosophers

Although Thales has left his name in history, we still know very little about his life. Fortunately, several later philosophers and writers mentioned some anecdotes about him in their works. From them, we can understand his personality and temperament. Perhaps this is the earliest mathematical story. Unfortunately, although there were some famous mathematicians in ancient China, such a cultural atmosphere was not formed, and humanities scholars rarely cared about the work of scientists. A few exceptions are that Zhuangzi recorded the concept of infinity expounded by the famous scholar Huizi in "The World", and "Zhoubi Suanjing" also wrote about Zhou Gong and the official Shang Gao discussing the Pythagorean number.

Plato was both a philosopher and a mathematician. It is said that the gate of the academy he founded had a sign saying, "No entry for those who do not know geometry", and at the back door of the academy it was written, "Only those who know philosophy can govern the country". Theaetetus is an important work of Plato. Theaetetus (ca. 417-369 BC) was a student of Socrates and Plato's senior brother. He was present with Plato when his teacher died. He was a mathematician and philosopher, the founder of solid geometry, and the main interlocutor of Theaetetus (and Plato's other work, The Sophist).

Plato explored the nature of "knowledge" in this article, which can be seen as a commemoration of his teacher and senior. Regarding "what is knowledge", Socrates told the young Theaetetus three answers, namely "feeling", "belief" and "belief plus reasoning", but he finally denied them all, because for those who seek knowledge, the most important thing is the process. The book records an anecdote about Thales. Once he looked up at the sky and accidentally fell into the ditch next to him. A beautiful Thracian maid laughed at him and said, "You can't see what's right at your feet, how can you know what's in the sky?" Thales did not respond to this, but the question from Solon, the archon of Athens, stung him.

Thales may be the first of many wise men who lived alone for life. According to Plutarch, a Roman biographer in the first century (more than six centuries after Thales' death), Solon, who was fourteen years older than Thales, once visited Thales in Miletus. Solon was a politician, reformer, and legislator. He served as the first archon of the city-state of Athens in 594 BC. In addition, he was also a successful businessman who liked to travel to famous mountains and rivers, investigate social customs, and even achieved great success in poetry creation, earning him the reputation of "the first poet of Athens." Although Solon's poems mainly praised the city-state of Athens and its laws, he also criticized and condemned the greed, tyranny and cruelty of the nobles, and firmly believed that morality was better than wealth.

Statue of Solon in the U.S. House of Representatives

For example, Solon once wrote in a poem: "Evil men often become rich, while good men often become poor; but we are unwilling to exchange morality for wealth; because morality is eternal, while wealth changes owners every day." This shows that Solon is a man of personality. Thales has a motto, "Too much prudence will only bring disaster", which is quite contradictory to Solon's motto "Avoid extremes". Sure enough, as they talked, a small wave arose between the two. Solon asked Thales why he had not married yet, and Thales was obviously unhappy and did not answer.

A few days later, Solon, who was emotional and loved writing poetry and traveling, got news that a young man who died in Athens might be his son, which made him heartbroken. At this time, Thales appeared with a smile. After telling him that the news was purely fictitious, he stated that the reason why he did not want to marry and have children was that he was afraid of facing the pain of losing his relatives. It is said that when Thales was middle-aged, his mother urged him to get married, but he replied, "It's not time yet." Later, when he entered his old age, his old mother urged him to get married again, but he replied, "It's past that time now."

As mentioned above, Brutarch’s works were very popular during the Renaissance. He was highly praised by the French writer Montaigne, and many of Shakespeare’s plays were based on his works. After each record, he also made comments. For example, regarding Thales’ view on marriage, he wrote, “It is neither reasonable nor valuable to not obtain what is necessary because of fear of losing it… In any case, we must not use poverty to prevent the loss of property, to avoid the loss of friends by living in isolation, or to avoid the death of children by childlessness; we should deal with all misfortunes with reason.”

Aristotle, a disciple of Plato and a philosopher who lived nearly three centuries later than Thales, also told a story about Thales in his book Metaphysics. Once, based on his agricultural knowledge and meteorological data, Thales predicted that the olives would have a bumper harvest the following year, so he raised funds in advance to rent all the oil presses in the area at a low price. Later, as he expected, the oil presses were in short supply. So he rented them out at a high price and ended up making a huge fortune. He did this not to become a millionaire, but to answer some people's ridicule of him: If you are really smart, why don't you get rich? At the same time, he also warned people that knowledge is better than wealth.

The interaction between Thales and Solon may be the earliest friendship between mathematicians and politicians, or mathematicians and poets in history.

3. Versatile Thales

The first historian of mathematics to be recorded was Eudemus of Rhodes (about the second half of the 4th century BC), who was a favorite student of Aristotle. He wrote books on the history of arithmetic, geometry and astronomy, but unfortunately they have all been lost. He also co-edited the complete works of his mentor Aristotle. Fortunately, Proclus (410-485) wrote the "Eudemus Synopsis" when he was annotating "The Elements", which was mainly based on Eudemus' "History of Geometry". Eudemus wrote in the book, "... (Thales) introduced the study of geometry (from Egypt) to Greece. He himself discovered many propositions and guided his students to study the basic principles that could deduce other propositions."

Proclus was the tutor and the last director of Plato's Academy. He had a famous saying, "Where there is number, there is beauty." It is because of his record that we know that Thales proved five theorems in internal plane geometry, including Thales' theorem, which are all included in the "Mathematics" textbook of middle schools today. Thales proved that "the diameter of a circle divides the circle into two equal parts; the two base angles of an isosceles triangle are equal; the vertical angles formed by two intersecting straight lines are equal; if two triangles have two angles and one side that are equal, then the two triangles are congruent."

Of course, Thales' most valuable mathematical work is the proposition now known as the "Thales Theorem": "The indirect angle on a semicircle is a right angle." As we all know, the "Thales Theorem" is the first theorem in the history of mathematics named after a mathematician. Euclid listed it as Proposition 31 in Volume 3 of "Elements of Geometry". The so-called indirect angle refers to the vertex on the circle, and the intersection of the two sides with the circle is at the two ends of a straight line. Its proof uses the equivalent form of the parallel postulate, that is, the sum of the three internal angles of a triangle is 180 degrees. The details are as follows:

Suppose the diameter AC divides the circle equally, and the vertex of the circumference angle is B. To prove that angle ABC is a right angle, connect the center of the circle 0 and point B. Since the two base angles of an isosceles triangle are equal, we know that angle A equals angle ABO, angle C equals angle CBO, so angle A plus angle C equals angle ABC. Since the sum of the interior angles of a triangle is 180 degrees, angle ABC is 90 degrees, which is a right angle.

Thales also made extraordinary achievements in mathematics. He has a famous saying, "Water is the best." He believed that sunlight evaporates water, mist rises from the water surface to form clouds, and clouds turn into rain, so he asserted that water is the essence of all things. It is worth mentioning that Thales also classified metals as "water", probably because they can melt. He also believed that the earth is a disk floating on the water, and earthquakes are vibrations caused by the drift of the earth through the water. Although this view was later proved to be wrong, he dared to reveal the true face of nature and established his own ideological system (he also believed that the earth is a disk floating on the water), so he is recognized as the ancestor of Greek philosophy.

In fact, Thales was the first person to raise the philosophical question of "What is the origin of all things?" and try to answer it. Thales pioneered the spirit of rationalism, the tradition of materialism and the principle of universality. In theology, he was a polytheist and believed that the world was full of various gods. Despite this, he still sought reasons from nature itself, rather than from the elusive personalities of the gods in human form, to explain the world. This is the importance of Thales. In his later years, Thales recruited students and founded the Milesian School.

In physics, the discovery that friction in amber produces static electricity is also attributed to Thales. He was well versed in astronomy, and had estimated the size of the sun and the moon, confirmed the Little Bear constellation, pointed out that it was helpful for navigation, and for the first time determined the length of a year to be 365 days. Herodotus, who died two years before Plato was born, is known as the "Father of History". His work "History" is the first prose work in the West to be handed down intact, so he is also considered the founder of Western literature and an outstanding representative of humanism.

Latin edition of Herodotus's Histories (Venice, 1494)

Herodotus's book records that Thales had accurately predicted a solar eclipse in 585 BC, which stopped the war. The two sides were Ariades of Lydia and Cyaxares of Media. The battle lasted for five consecutive years without a clear winner, resulting in the loss of lives and corpses everywhere. Thales predicted in advance that a solar eclipse would occur, and he claimed that God was against the war and would use a solar eclipse as a warning. As a result, when the two armies were fighting fiercely, the day suddenly turned into night, and the soldiers were very frightened. They remembered Thales' warning and stopped the war and made peace. That was the earliest solar eclipse clearly recorded in the West, and the exact date was May 28.

As for the method used by Thales, later scholars believe that it may be the Saros cycle discovered by the ancient Babylonians.* Some modern scholars also believe that Thales' time did not have the knowledge to accurately predict the location and time of a solar eclipse. Eudemus believed that Thales already knew that the four seasons divided by the spring equinox, summer solstice, autumnal equinox and winter solstice were of unequal length.

*Saros cycle is an astronomical term for the period of solar and lunar eclipses, which is about 6585.32 days long, equivalent to 18 years and 10.3 or 11.3 days (depending on whether there are 5 or 4 leap years). In each Saros cycle, there are about 43 solar eclipses and 28 lunar eclipses.

4Thales 's students and legacy

More important than the "Thales theorem" is that Thales introduced the idea of ​​proof of propositions. Thales pioneered the proof of mathematical propositions, which marked the rise of human understanding of objective things from sensibility to rationality. This was an extraordinary leap in the history of mathematics, which was carried forward by Euclid when he wrote "Elements" a few centuries later. Not only that, Thales' proof also used some axioms or propositions whose truth had been confirmed. Although there is no original document to prove that Thales achieved all these achievements, the above records have been passed down to this day, making him the first mathematician and the originator of argumentative geometry.

Thales usually spoke humorously and philosophically. His answer to the question “How to live an upright life?” was: “No,

Do what you hate others to do. This is similar to Confucius' "Do not do to others what you do not want others to do to you" in "The Analects of Confucius Yan Yuan". When someone asked, "What is the strangest thing you have ever seen?" Thales replied, "Long-lived tyrants." And "When you make a discovery, what do you want to get?" He, who has never received any rewards, replied, "When you tell others, don't say it's your discovery, but say it's my discovery. This is the highest reward for you."

Thales's theories and ideas have a huge influence. He pioneered a rational exploration of the world and became the first true mathematician and scientist. Under his leadership, the Greeks broke free from the shackles of God and began to consciously explore the mysteries of numbers, shapes and even the universe, revealing the original secrets of nature. After centuries of efforts, mathematics transitioned from a concrete, experimental stage to an abstract, theoretical stage, gradually forming an independent, deductive doctrine, which ultimately led to the prosperity of Greek science, art and philosophy, and then influenced Europe and the entire world.

Among Thales' students, Anaximander and Anaximenes were the most accomplished. Anaximander (c. 610-545 BC) was born in Miletus and served as a leader in a colony near the Black Sea. He believed that the world was not made of water, but of a special basic form that was not familiar to us (a certain entity that constituted the four elements of earth, air, water, and fire). This entity split into opposites such as cold and hot, dry and wet during movement, thus producing all things. The world was created from it and returned to it. He was the first philosopher to write down his ideas in prose, rather than in poetry like Homer or Hesiod.

Anaximander created a reductio ad absurdum from which he deduced that humans evolved from sea fish and that higher animals evolved from lower animals. He proposed an important cosmological view, in which the Earth was the center of the universe (a free-floating cylinder) and the Sun, Moon, and stars were arranged in a ring around the Earth. This cosmological view lasted for more than 2,000 years before the emergence of Copernicus's theory. Anaximander was said to be a good actor, with dramatic clothing and speech. He led an embassy to Sparta, where he demonstrated two of his great inventions - the sundial and the world map (the first drawn by humans). Unfortunately, his book "On Nature" has been lost.

Anaximenes (586-526 BC) had a different view. He believed that the world was made up of air, and the condensation and dispersion of air produced various forms of matter. Like his two predecessors of the Milesian School, his philosophy was also monistic. The Egyptians and Babylonians used gods to explain the formation and nature of the world, while they made a naturalistic interpretation. Anaximenes made such an assumption: gas is a universal thing, it can spontaneously enter our souls, thereby achieving the purpose of controlling the body.

It is said that Anaximenes had thousands of disciples. Once when he was teaching, he asked students to put down their notebooks and listen carefully. He promised to give everyone handouts after class. As a result, he only gave a blank piece of paper and asked everyone to write down what they heard. Only Pythagoras, who was passing by and listening to the class, wrote down everything. This story is hard to tell whether it is true or not, but it is full of philosophy. It involves the art of teachers or managers, which is to let students or employees take responsibility for themselves and learn and master important practical skills. Anaximander once said: Endless suffering is a raw material of nature. Anaximenes gave an example to support it-wool is compressed because it is to be made into a rug.

In addition, some scholars have classified the historian, writer and traveler Hecataeus (c. 550-476 BC) as a student of Thales. Hecataeus not only wrote the earliest travelogue in a beautiful and concise style (he traveled extensively in the Persian Empire), but was also a pioneer in geography and anthropology. He has a famous saying that has been passed down to later generations: "Egypt is the gift of the Nile." But judging from his birth year and the year of Thales' death, he could not have been the latter's student. In any case, Hecataeus was a Milesian under Persian rule, and he was the first recorded historian in ancient Greece. Herodotus, who was born shortly before his death, also imitated him and said: "Geometry is the gift of the Nile."

2022, Hangzhou Xixi

This article is authorized to be excerpted from Cai Tianxin: Mathematical Legend (The Commercial Press, November 2022)