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I want to learn Euclidean geometry But I always find books about it that use coordinates and even some linear algebra and other stuff that was discovered thousands of years after geometry but I want to learn how many ideas of geometry has been discovered and evolved for thousands of years without any modern techniques like analytical geometry or linear algebra or integration ...etc.

I also want 3 types of books

1-and the more important books that cover many of theorems of Euclid geometry before Desecrates without coordinates or any modern math like How Archimedes found out the volume of solids like sphere and area of parabolas without integration etc.. .

2-books that are more focus on problem solving and hard and practise questions like IMO training books

3- books that cover more advanced topics in Euclidean geometry after Desecrates that use modern methods.

pie
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    What about reading Euclid's The Elements?!! – Ted Shifrin Jun 30 '23 at 22:46
  • @Ted Shifrin interesting,is this book really written by Euclid ? but I am worried because this book is too old that id doesn't has many theorems that has been discovered hundreds of years after Euclid – pie Jun 30 '23 at 22:53
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    Yes, Euclid really wrote it. You don't have to read it in Greek; there are well-done English translations with lots of commentary and elaboration. The standard one is by Sir Thomas Heath. I would start there and not worry so much about what came between Euclid and Descartes. – Ted Shifrin Jun 30 '23 at 22:58
  • @Ted Shifrin I just checked the book and it seems great and I will read it thank you for your suggestion but I am still worried about "what came between Euclid and Descartes" but for now I will read the entire book and maybe in the future if I want to learn more about geometry I will re-ask this question – pie Jun 30 '23 at 23:30
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    In all seriousness, I had a much better than average high school geometry course (back in 1968) and we never got past Euclid (except for a tiny bit of analytic geometry using cartesian coordinates); and we certainly did far less 3D geometry than Euclid does. – Ted Shifrin Jun 30 '23 at 23:37
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    @TedShifrin It seems to me that OP is interested in “advanced” Euclidean geometry, similar to what I do, which is why they ask about results and theorems derived after Euclid – 冥王 Hades Jul 01 '23 at 13:00
  • @冥王 Hades is there a book that you would recommend to me? – pie Jul 01 '23 at 17:31
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    @pie Honestly it depends entirely on your current level and how far you wish to go in Euclidean geometry. You can definitely read The Elements as Ted recommended and in fact I also recommend that you start with it. But beyond that, it just depends on how far you wish to go. You can certainly find books on advanced Euclidean geometry used for IMO preparation containing difficult problems. If you’d like, I can link several threads that answer that question with specific recommendations – 冥王 Hades Jul 01 '23 at 22:17
  • @冥王Hades that would be great and very helpful , but I am worried that IMO books would focus more on the hard problems and problem solving skills rather than the the theorems itself but anyway still i need a source for problems – pie Jul 02 '23 at 01:38

5 Answers5

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The book series you are looking for is Kiselev's Geometry. These are small yet beautifully bound hardcover books that cover planimetry and stereometry.

Kiselev's page can be found on Amazonwhich contains both books. I do not recommend buying them there because their price is way off the mark. I have both copies and I don't believe I paid more than $\$30$ for each book. In fact, I may have paid less.

John Douma
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  • These are great books! I found them online but they seem very small and don't contain much geometry – pie Jul 03 '23 at 20:51
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Perhaps you can have a look to the book of Akopyan. It is free on his webpage, but you can also buy it via Amazon.. This book contains no coordinates, but no text neither. You have plenty of theorems explained by a picture, and you have to prove it by yourself..

https://users.mccme.ru/akopyan/papers/EnGeoFigures.pdf

Thomas
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If you want to study Euclidean geometry, why not get hold of Euclid’s Elements? Old and still good, inexpensive and easy to find. He doesn’t tell you how to find things out, but rather gives you an orderly, step-by-step, and dramatic demonstration of results.

Edward Porcella
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  • Great suggestion! I will begin with his book but what about all the geometry between Euclid and Descartes? – pie Jul 04 '23 at 07:44
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I'm fond of Lachlan, Modern Pure Geometry, (1893).

Lots more where that came from - geometry textbooks from the period 1825-1925 - most of them on archive.org. Some are mentioned in Lachlan's introduction, so you can search for them in archive.org.

For something completely different, take a look at the opinionated and idiosyncratic Coolidge, A History of the Conic Sections and Quadric Surfaces (1945). Since you ask about Archimedes, the book mentions J.I. Heath, The Works Of Archimedes (1817) which contains translations of Archimedes' works On The Sphere and Cylinder and Quadrature of the Parabola. See also T.L. Heath's 1896 translation of Apollonius, Conic Sections, pgs lviii-lxvii

brainjam
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  • that would be great if you can recommend more books , but why nobody cares about geometry after 1925 it seems that we forgot about it – pie Jul 05 '23 at 01:11
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    @pie, Lots of people cared after 1925, but it became less mainstream and by the 1970's had disappeared from most curricula. But there has been a revival in recent decades. I've added a few more recommendations. – brainjam Jul 07 '23 at 16:05
  • how do you know about these books? and how do you search for good books – pie Jul 08 '23 at 04:16
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    @pie Good question, no easy answer. Usually I don't look for "good books". Rather I'm searching for a specific answer to a problem or topic that I'm interested in. 30 years ago it would be in a math library, browsing the books, periodicals and card catalogs. Now it's the internet. For me, specific questions that catch my curiosity are the keys to learning a larger topic. And along the way you find books and papers that are worth reading all the way through. Many disciplines are like this, but especially math, both learning and researching: https://news.ycombinator.com/item?id=10548173 – brainjam Jul 08 '23 at 17:21
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I guess I'll recommend the book reviewed in this video https://youtu.be/1B2BtvWsz-M

زكريا حسناوي
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  • شكله بسيط جدا و لكن اللي انا عايزة كتب في المواضيع المتقدمة في الهندسة – pie Jul 03 '23 at 20:35
  • لربما يجب عليك الاطلاع على هذا الكتاب https://www.academia.edu/86234543/PROBLEMS_IN_PLANE_AND_SOLID_GEOMETRY_v_1_Plane_Geometry_Viktor_Prasolov_translated_and_edited_by_Dimitry_Leites – زكريا حسناوي Jul 03 '23 at 23:54