Volume 10
Volume 11
Volume 12
Volume 13

Translator's Note
References

The pinnacle of ancient mathematics and a golden tower in the history of human intellect
The first domestic translation of the Greek original of Euclid's Elements

A textbook that builds the foundation of mathematics and learns the framework of thinking.
The foundation for the development of human reason for over two thousand years

'Euclid's Elements' is a mathematical classic that was born in ancient Greece around 300 BC.
It is called the pinnacle of ancient mathematics, as it incorporates the basic problems of geometry, number theory, algebra, and analysis that had been developed before its birth into a single system.
This work, which synthesized mathematics from earlier periods, including ancient Babylonia, Egypt, and Greece, had a profound influence on the intellectual history of humanity for 2,300 years from its creation until the 20th century.
This is evidenced by the fact that it was printed earlier than any other book in human history, excluding the Bible, and that it has an overwhelmingly greater number of editions than any other secular book.
The National Research Foundation of Korea's Academic Masterpiece Translation 『Elements』, the first complete translation of the original Greek text in Korea, was translated as closely as possible to the original text with the intention of defining and explaining mathematical phenomena in the language of the time when mathematics was born.
This also stems from the intention to view the modern era through the lens of the classics, rather than viewing the classics through the lens of the modern era.


The 『Principles』 clearly presents the following steps: creatively presenting universal and abstract concepts related to shapes and numbers, abstracting phenomena and clearly setting them as problems, proving the set problems—that is, eliminating all possible doubts—and then synthesizing the facts into a single theory.
It is also important to note that all of this unfolds without any prior knowledge.
Therefore, 『Principles』 was not only a basic mathematics textbook, but also a textbook for learning the framework of thinking, which is discovery, argument, and synthesis.
The phrase “Let no one ignorant of geometry enter” written at the entrance to Plato’s Academy can be understood in that sense.
Thus, 『Principles』 has been with the great intellect of mankind for a long time.
So, as the center of civilization changed from medieval Islam to modern Europe, it was translated first, then translated again, annotated, and reinterpreted, becoming the foundation for the development of human reason. This can be found in classics of science and philosophy, including Newton's Principia and Spinoza's Ethics, as well as the American Declaration of Independence and constitutional system.

“There is no book like Euclid’s Elements in mathematics in our metaphysics.

If you want to know what mathematics is, read Euclid's Elements.”
- Immanuel Kant

An abstract structure that laid the foundation for ancient mathematics
The aesthetics of rigorous systems and simple proof methods

『Principles』 is divided into 13 volumes, organized by topic, maintaining a strict format of definitions, propositions, and proofs without any unnecessary details.
Topics covered include basic plane figures, the existence and interrelationships of solid figures, the theoretical basis and application of proportion and similarity, the composition of the world of natural numbers, the classification and interrelationships of incommensurable sizes, and the characteristics of regular polyhedra inscribed in a sphere.
All of this begins with the definition, "A point is something that has no parts," and ends with the corollary, "There are only five regular polyhedra." In the process, the problem is clearly set up with about 500 propositions, auxiliary propositions, and corollaries, each of which is rigorously proven.
Each volume consists of between 14 and 115 propositions, and when a new topic begins, a definition appears to develop the topic, followed by propositions related to that definition.
It can be seen as a beautiful abstract construction that builds on the foundations of ancient mathematics.

Besides the beauty of the abstract system, another beauty of the Elements lies in its method of proof.
The fact is that all of these propositions can be proven logically based on just five axioms.
From a few simple facts, we can 'prove' basic mathematical phenomena that no one would suspect, and these proven mathematical facts, when combined with each other, become a tool for explaining higher-dimensional phenomena.
In this process, no observation, experience, authority, or compromise other than the agreement of definitions and postulates intervenes.
In other words, 『Principles』 is a vast deductive system.
Therefore, it is also an example of developing reasoning correctly.

“The long chain of very simple and easy proofs which geometers often use to prove very difficult things made me think of the following:
That is, everything that humans can perceive is interconnected in that way, and if we do not regard anything as true that is not true, and always observe the necessary order when deducing one thing from another, we can eventually reach it no matter how far it is, and discover it no matter how hidden it is.”
- René Descartes