The author of this article: Liu Ruixiang, [Meet] Here thank you Teacher Liu for your support!
Because I am a fan of "Geometric Origins", when I received the "Mathematical Circle" series of books from Hunan Science and Technology Publishing House, the first book I couldn't wait to read was this "Window of Euclid". What's more, the author of the book, Leonard Monlordino, together with Hawking, wrote a "Brief History of Time< popular edition of >" and "Big Design".
When you think of Primitive Geometry or Euclid, what comes to mind first? I guess most people think of elementary geometry, or the fifth postulate. But this book tells us that this is not enough. The author named this book "The Window of Euclid" because he wanted to show us a wider world through this window. The book deals with five revolutions: the birth of abstract and probative ideas, the unification of algebra and geometry, the discovery of curved spaces, relativity, and string theory. Some people may wonder: Why are the last two parts physics and not mathematics? In fact, mathematics—especially geometry—has always been part of physics: the ancients studied the structures of space that they could experience directly, and modern humans studied the structures of spaces of various scales. The difference between the two is only that the ancients believed that the spatial structure they could directly experience was the only form of our universe, while modern people had more available options, and the ancients did not develop handy research tools, which in this case refers to coordinate systems and analytic geometry. From the above brief introduction, we can see that what has developed from the Primitive Geometry is far more than non-Euclidean geometry.
Even when it comes to non-European geometry, this book has its own unique features. We know that the so-called non-Euclidean geometry is closely related to the fifth public set. This research has run through almost the entire history of elementary mathematics (in the West). Regarding this relay, the book reads:
For more than 2,000 years, as humanity has continued from generation to generation, national borders have changed, political systems have flourished and declined, and the earth has orbited the sun for about 1 trillion miles, but thinkers are still committed to the study of Euclid geometry, and there is nothing wrong with their idol's theory, only a tiny point of doubt: Can't the pesky parallel postulate be proven?
From this passage and the specific content introduced in the book, we can better appreciate the famous words of the mathematician Xiang Wuyi: "If we take understanding nature as our interest, and can inherit and strive for excellence from generation to generation, then the perfection of the basic structure of nature is within reach." ”
The general popular science works that introduce non-European geometry only talk about this postulate, and only very professional books will talk about the relationship between the second postulate and the parallel line - if there is no second postulate as a guarantee, the parallel line does not exist. But the book also mentions two other axioms (nominals): what is close and coincident is equal, and that any two points must be connected in a straight line. Once spherical geometry is introduced, the concept of "middle" fails. I think this may have inspired Hilbert to set up a specific axiom of order in his axiom system.
Even for the study of the fifth postulate, the book takes a different perspective, for example, today we may be more about Bower, Lobachevsky rather than Gauss, although the latter was one of the pioneers of non-Euclidean geometry, but after all, he did not publish works on non-Euclidean geometry during his lifetime. But the book gives Gauss a detailed account.
Throughout the book, this is not only a very good popular reading of the history of science, but also a very good popular science work, which can help the general reader understand many scientific concepts, both more classic and very modern, and the difficulty is moderate and the content is extensive. Finally, it is worth mentioning that the author and translator's writing is very good, there is no sense of obscurity in reading, and it is so smooth that it is impossible to release the volume.