100 must-read books for college students

Chapter 12 The World's Oldest Mathematics Tome
Chapter 11 The oldest mathematical masterpiece in the world: "The Elements of Geometry"

Mathematics works written more than 2000 years ago are still taught in secondary schools around the world as a model, so their status is firm and enduring
, only Euclid's "Elements of Geometry".Since this book came out in the 3rd century BC, it has long been regarded as the first book of geometry.

As a textbook, thousands of editions of it have been published in various languages ​​around the world, and there are countless explanatory articles.

Its scope and influence are no less than that of Christianity's "Bible".

Masterpieces of the School of Mathematics

Euclid was born about 365 BC, and his life story is difficult to verify. He once studied under Plato, and then lived and worked
Made in Alexandria, Egypt.

The emergence of such a brilliant work "Elements of Geometry" is not accidental. The city of Alexandria in the 3rd century BC was the central

The center of economy, science and culture in the eastern part of the sea, there is a world-renowned library with a collection of 70 volumes, as well as a museum
, laboratories, observatories and other cultural and scientific facilities.A large number of mathematicians were working in Alexandria at that time, and some of their ingenuity

The writings are still shining today.Among them, at the invitation of Ptolemy I, he came to Alexandria to preside over the mathematics school.

Euclid is an outstanding mathematician with a "comprehensive" style.

Before that, many people made a lot of pioneer work for him.The first is that the ancient Egyptians remeasured floods year after year.

A wealth of land surveying knowledge was accumulated in the course of the overuse of the land, and then the ancient Greek mathematicians carried out a preliminary study on these geometric knowledge.

step by step.Thales, the earliest mathematician in ancient Greece, began to prove mathematical propositions around 600 BC, making geometry
The first step towards becoming a deductive science.Then Pythagoras used numbers to explain everything, turning mathematics from concrete things
Abstracted from objects, established some mathematical theories, discovered the Pythagorean theorem, incommensurable quantities, added to the early geometry
A lot of content.Subsequent Eudoxus school founded the theory of proportionality, established the theory with axiomatic method, so that the proportion is also applicable to different
Commensurable quantities expand the scope of application of geometry.Mathematics has also been valued by ancient Greek philosophers, Plato is particularly strong
In order to emphasize the role of mathematics in training intelligence, it was written on the gate of his academy: "Those who do not understand geometry are not allowed to enter."Plato
and Aristotle et al. also developed formal logic closely related to geometry.In Plato's conception, reasoning and judgment

On the basis of research, Aristotle established the deductive method of syllogism, and made a clear statement on the law of identity, contradiction and excluded middle.

stated.All this work paved the way for Euclid to write the Elements.

Euclid studied hard and worked hard. Between 330 BC and 320 BC, he wrote the great work of mathematics "The Elements of Geometry".

》, on the basis of predecessors, established a system of rigorous, incisive discussion of the geometry system.

The original manuscript of Euclid's "Elements of Geometry" has long been lost, and the various versions we see now are based on revisions, annotations

Reorganized from the original and translated versions.The most widely circulated version is the revision of "Elements of Geometry" made by Tyne at the end of the 4th century.
book.The closest version to the original is thought to be a Greek manuscript found in the Vatican Library in the early 19th century.According to 1842
According to the statistics from the year to the end of the 19th century, more than 1000 editions of "Elements of Geometry" have been published in various languages.end of some versions
Two volumes were added, so there are 15 volumes of "Elementary Geometry".

The first Chinese translation of "Elements of Geometry" was translated based on the annotations of the German mathematician Kravis, and was translated by the missionary Matteo Ricci.

Interpretation, written by Xu Guangqi, only translated the first 6 volumes, which were engraved and published in Beijing in 1607. More than 200 years later, Li Shanlan, a mathematician in the late Qing Dynasty
Cooperating with the Englishman William Williams, he translated all the last 9 volumes of "Elements of Geometry" and published them in 1858.

"Elements of Geometry" Mathematics Complete Book

欧几里得著的《几何原本》共13卷，内含119个定义、5条公设和5条公理，前后推出467个命
question.It focuses on geometry, and some chapters are devoted to issues such as number theory and irrational quantities. Looking at the whole book, there is no doubt that it is

The Encyclopedia of Mathematics at the time.

From Volume [-] to Volume [-] of "Elements of Geometry", it mainly discusses the propositions related to straight lines and circles in plane geometry.volume first gives some definitions
, postulates and axioms, and then discuss and prove the propositions one by one.Part of the content is an elaborate summary of previous experience, such as: "

Proposition 47. In a right triangle, the square on the side opposite the right angle is equal to the square between the two squares on the sides sandwiching the right angle.

and "." Proposition 48. In a triangle, if the square on one side is equal to the sum of the squares on the other two sides, then its

The angle between his two sides must be a right angle". This is the positive proposition and converse proposition of the Pythagorean theorem (Pythagorean theorem). Some content
The content is the creation of the author himself, such as the famous fifth postulate: "If a line intersects two lines, and if two lines intersect on the same side,

If the sum of the angles is less than two right angles, the two straight lines must intersect at a point on this side after infinitely extending.” Many propositions are still used in middle school until now.
He's theorem or rule, such as: "Proposition 4, if the two sides of a triangle and their included angles are correspondingly equal to the other triangle

If the two sides and the included angles of the triangle are equal, then the two triangles are congruent. "

Volume [-] is devoted to the theory of proportion, applicable to commensurable or incommensurable quantities.Put geometry on solid ground with the theory of proportion
Based on the outstanding achievements of ancient Greek mathematics, although the content of this volume is based on the work of the predecessor Eudocus, but

Euclid was the first to systematically arrange and prove the proportional theory, and his contribution was enormous.

Volume 1 talks about similar figures.Such as: "Proposition 19, the ratio of the areas of equal height triangles is equal to the ratio of their base lengths". "Proposition [-], relative
The ratio of the areas of triangles is equal to the square of the ratio of their corresponding sides." "Proposition 31. In a right triangle, the side opposite the right angle
The area of ​​the above figure is equal to the product of the areas of the similar figures on the two sides sandwiching the right angle". Proposition 31 is actually the Pythagorean theorem
expansion.

Volumes [-] to [-] discuss the properties of integers and the ratio of integers, which belong to the content of elementary number theory.In this part, Euclid uses the line segment
Representative numbers are discussed by using the area of ​​a rectangle to represent the product of two numbers.Such as: "Definition 16, the number obtained by multiplying two numbers is called the average
surface, its side length is the original two numbers”; “Definition 17, the number obtained by multiplying three numbers is called a solid, and its side length is the original
Three numbers". Many propositions in number theory are quite rigorous, such as "Proposition 14 in Volume [-], if a number is divisible by several prime numbers
If it is the smallest number that can be completely divided, then there is no other prime number that can divide that number except these few prime numbers.”

Volume 4 studies the relationship between commensurable and incommensurable measures involving straight lines, which are incommensurable.The main research can be expressed by a±b (a, b are two commensurable straight lines) to represent various straight lines.Such as: "Definition [-], it is known that squares on a straight line are comparable, and any area commensurable with it is comparable; any area incommensurable with it is incomparable, and the degree of their side lengths is also Incomparable."

Volume [-] to Volume [-] discuss solid geometry, including straight lines, planes, the relationship between lines and planes, and the relationship between planes
Relations, polygonal angles, similar solids, columns, cones, tables, spheres, and regular polyhedra.

The definitive work on the axioms of geometry

Euclid sorted out and summarized the geometric knowledge accumulated in the production practice and scientific research of the predecessors for a long time, forming a deductive body
Department, wrote the first mathematical work "Elements of Geometry" with rigorous theory and complete system in history.

The "Elements" has been circulated in the form of manuscripts in the past, and it has been extensively studied by many mathematicians over the centuries.
notes and comments.Although Euclid was influenced by the philosophical thoughts that emphasized ideas and neglected practice at that time, all of the "Originals" are

The definitions, axioms and theorems of images do not have content to solve practical problems, but because it has a rigorous theoretical system, it is
People still pay attention to mathematics education and mathematics research. After the 12th century, "Elements of Geometry" was adopted as a university textbook.

After the advent of printing around 1500 AD, this work was quickly reprinted in large numbers, and more than 1000 editions appeared.
It has become the most reprinted and researched book in the history of the Western world, second only to the "Bible".In the 17th and 18th centuries,

The writings of Euclid are the foundation of mathematics teaching in the West. "Elements of Geometry" once "firmly" controlled the teaching of geometry for a long time,
So much so that the Swedish poet G. M. Behrman wrote: "Even now thinking of Euclid... I have to wipe my sweat
forehead. "

So much admiration for the Elements has historically given such authority to the Euclidean system that

Some of the shortcomings of Euclidean geometry are either ignored or believed in.For example, for Euclid's fifth postulate, many

Mathematicians have long considered it inappropriate, and successive dynasties have tried to make a more definite proof of it, but after failing one by one, they still dare not doubt its reliability.
reliability.Until the beginning of the 19th century, Gauss, known as the "Prince of Mathematics", had secretly explored several completely different fifth postulates.

However, he still dared not publish it, for fear of being scolded as a "lunatic" who blasphemed the classics.

In the 19th century, the sublation of Euclid's fifth postulate finally caused a revolution in geometry.Russian mathematician Lobachevsky
Stand up bravely and announce the replacement of the fifth postulate with the new parallel postulate, thus creating a new geometry - Roche Geometry.

Then there was the birth of Riemannian geometry, which is commonly known as non-Euclidean geometry.

The emergence of non-Euclidean geometry aroused people's greater enthusiasm for geometry research.After many studies, mathematicians have discovered that non-Euclidean geometry
It does not completely negate Euclidean geometry, non-Euclidean geometry is more applicable to the large-scale material world of the universe, but in the small-scale space of daily life
Under these conditions, Euclidean geometry is sufficiently accurate. At the end of the 19th century, the German mathematician Hilbert elaborated the "Elements of Geometry".
The refinement of the heart makes the axiom system of Euclidean geometry more perfect.Until today, part of the essence of "Elements of Geometry" is still
It is an indispensable and unique content for cultivating students' logical reasoning ability in mathematics teaching; people are building houses, building bridges, repairing
In the production practice of Lu and others, the results that need to be calculated can be obtained more accurately by using Euclidean geometry.In our country, the most
The earliest Chinese translation of "Elements of Geometry" was translated by Xu Guangqi, a politician and scientist in the late Ming Dynasty. Xu Guangqi believed that this book "does not need to be

Doubt, don't guess, don't try, don't have to change", "If you want to get rid of it, you can't get it, if you want to refute it, you can't get it, if you want to reduce it, you can't get it, if you want to go forward, you can't get it."

It is impossible to get it after the post”, and gave a high evaluation. Of course, with the development of production and science, people’s research on mathematics
It is undeniable that the ancient "Elements of Geometry", which was produced 2300 years ago, still has
It is extremely vigorous.

"Elements of Geometry" was still a middle school mathematics textbook in all countries in the world until the beginning of the 20th century.
The value, status and role of masterpieces in their field and education are discussed.

(End of this chapter)