Teach by example
Euclid was born around 325 BC, an ancient Greek mathematician whose name was inextricably linked to geometry, who was famous for compiling the Primitive Geometry, but little was known about his life, and he was the founder of the Alexandrian school. Probably taught by Plato in his early years, he was an apprentice in Alexandria at the invitation of king Ptolemaic, who consulted Euclid and asked him if he could make the proof a little simpler and easier to understand, and Euclid confronted the king and said, "In geometry there is no flat way for the emperor to walk." "He was a gentle educator.
Once, a student had just finished learning the first proposition and asked, "What will you get after learning geometry?" Euclid then called for three coins and said, "He wants to make a real profit in his studies." It is evident that Euclid is rigorous in his studies and opposes the ideological style of not being willing to study opportunistically assiduously.
In the 6th century BC, the geometric knowledge of ancient Egypt and ancient Babylon was introduced to ancient Greece, and the combination of developed philosophical ideas in ancient Greece, especially formal logic, greatly promoted the development of geometry. Between the 6th century BC and the 3rd century BC, the ancient Greeks were very eager to use the laws of logic to organize a large amount of empirical, scattered geometric knowledge into a rigorous and complete system, and by the 3rd century BC, "classical geometry" had been basically formed, thus making mathematics enter the "golden age". Plato once wrote a large banner on the gate of his school, "No one who does not understand geometry can enter." Euclid's "Geometric Primitives" was in such a period that it inherited and carried forward the research results of previous generations, and collected the essence of them.
Primitive Geometry
Euclidean Primitive Geometry deduces a series of axioms and postulates as the starting point of the book. There are 13 volumes, and most of the current middle school geometry textbooks are the contents of Euclidean's "Geometric Primitives". The Pythagorean theorem is prominent in Euclidean's Primitive Geometry. In the West, the Pythagorean theorem is called the Pythagorean theorem, but the time of its discovery is that in China and ancient Babylon and ancient India are hundreds of years earlier than Pythagoras, so we call it the Pythagorean theorem or quotient gaudorem. In Euclidean's Primitive Geometry, the proof method of the Pythagorean theorem is to take the three sides of the right triangle as the edges, make squares to the outside, and then use the area method to prove it, which is very much agreed with this ingenious idea, so this method is still generally retained in middle school textbooks.
It is said that the British philosopher Hobbes once accidentally flipped through Euclidean's "Geometric Origins", saw the proof of the Pythagorean theorem, did not believe such an inference, and was very surprised after reading it, and could not help but shout: "God, this is impossible." So he carefully read the proofs of each proposition from the back to the front, until the axioms and postulates, and was finally impressed by the rigor and clarity of his proof process.
Part of Euclidean Primitive Geometry is related to the early Homo sapiens school studying three well-known geometric drawing problems, particularly the drawing method of connecting regular polygons within circles. Euclidean's "Geometric Primitives" only draws straight lines with a straight ruler without a scale, and draws circles with compasses as axioms, limiting the "ruler" to drawing. So the geometry of the drawing of the situation appears "possible" and "impossible". Here Euclid gives only the practice of positive three, four, five, six, and fifteen-sided shapes, which, together with continuous dithalectonic arcs, can be extended to positive 2n, 3 (2n), 5 (2n), 15 (2n) edges. Thus, we can imagine that Euclid must have tried other methods of drawing regular polygons, but did not do so. Therefore, after the advent of Euclidean's "Geometric Origins", regular polygonal drawing aroused great interest.
The theory of proportion in Euclidean's Primitive Geometry is the highest achievement of the book. Prior to this, the Pythagoreans also had the theory of proportions, but did not apply to the ratio of immutability of quantities, and Euclid described Eudochsos's theory of proportions here in order to get out of this dilemma. The definition of two proportional equalities, that is, the definition of proportions, applies to all metrics of metric and non-metric, and it saves the Beech school of similar forms and other theories, which is a very important achievement.
It is said that a Czechoslovak priest, Bolzano, was on vacation in Prague when he suddenly fell ill, chills and pain. In order to distract himself, he picked up Euclidean's Primitive Geometry, and when he read about proportionality, he was so struck by this clever treatment that he was so excited that he completely forgot his pain. Afterwards, whenever his friend fell ill, he recommended reading Euclidean's Geometric Primitives on scale.
Euclidean's "Geometric Origins" draws on the deductive proof and deductive reasoning of Thales and Plato, completely embodies Aristotle's mathematical logic ideas, becomes the earliest example of axiomatic methods to establish a deductive system, and is the best teaching material for the training of mathematical logical thinking. However, it was also logically flawed in some respects, and once gave rise to the famous "fifth postulate test" activity in the history of mathematics, which gave birth to Lobachevsky geometry in the early 19th century. The birth of Roche geometry broke the concept of a unified space in Euclidean geometry and promoted further exploration of the broad field of geometry. Subsequently, large-scale logical tinkering with the axiom system of Euclidean Primitive Geometry was initiated. The German mathematician Hilbert, using the essence of modern views to set repairs, published "Geometric Foundations" in 1879, proposing a complete and concise axiomatic system of Euclidean geometry, which made Euclidean geometry reach a high degree of abstraction, logic, and mathematics, pushed the axiomatic methods to modernization, and established a unified axiomatic system.
This is also a major contribution to the development of geometry by Euclidean's Primitive Geometry. Euclidean's Primitive Geometry quickly and completely replaced all the works of the same type that preceded it, and even made them disappear.
The earliest Chinese translation was published in 1607 by the Italian missionaries Matteo Ricci and Xu Guangqi, and only the first 6 volumes of the 15 volumes were translated, which was the first mathematical translation work in China. It was named "Primitive Geometry", and it was from here that the name "Geometry" Chinese began. The introduction of the last 9 volumes was added two and a half centuries later in 1857 by the Qing dynasty scholar Li Shanlan and the Englishman Veleyali.
Source: The Story of a Foreign Mathematician by Wang Weiguo