Old and New Topics in Geometry: Volume I: Projective, Neutral and Basic Euclidean Geometry

by Franz Rothe

The present first volume begins with Hilbert's axioms from the emph{Foundations of Geometry}. After some discussion of logic and axioms in general, incidence geometries, especially the finite ones, and affine and projective geometry in two and three dimensions are treated. As in Hilbert's system, there follow sections about the axioms of order, and congruence in neutral geometry, the axioms of measurement and of completeness, and deviating from Hilbert, about circles.
The insight of independence of the parallel axiom leaves many open roads to pursue, but the desire to develop a natural as well as completely axiomatic system remains. In this context, the classification of Hilbert planes into three types,---as semi-euclidean, semi-elliptic or semi-hyperbolic, known as the uniformity theorem is a important step beyond Euclid. A further step is the introduction of the axiom of the unbounded opening of an angle. In the spirit of Hilbert, a completely axiomatic theory of area in neutral geometry is set up, before branching into the cases of Euclidean, or semi-hyperbolic, or semi-elliptic geometry.
Finally, the attention has to focus of Euclidean geometry. The text gets a more educational and even more elementary flavor. Here I start with Thales theorem about the angle in a semicircle, and continue with Euclid's related theorems about angles in a circle. The most simple parts of the Euclidean geometry are given in detail, as well as the later parts about similarity, and area in Euclidean geometry with the theorem of Pythagoras and trigonometry, and finally the measurement of the circle.

550 pages, Kindle Edition

Published October 18, 2022

Reviews

[Lucia Maranghello · Rating 4 out of 5]

The subject of geometry has always been one of great fascination and intrigue for mathematicians and scholars alike. In Franz Rothe's book, Old and New Topics in Geometry, readers are presented with a comprehensive exploration of this fascinating subject. As an accomplished author and holder of a doctorate in mathematics from the esteemed University of Tuebingen in Germany, Rothe brings a wealth of knowledge and experience to this book. From Hilbert's axioms to Pythagoras's theorem, Rothe delves into the foundations of geometry, providing acute and insightful details on the subject's principles, theorems, and hypotheses.

While this book may be quite complex for those without a solid foundation in mathematics, it can be an excellent reference material for students of geometry. The author explains the material with proper introductions, comprehensive vocabulary, and detailed solutions to complex numeric problems, making it a valuable resource for those wanting to enhance their knowledge in the field of mathematics. One aspect of this book that stands out is its well-structured content page, which offers a basic understanding of what readers can expect to learn in each chapter. This feature is particularly helpful, especially for those who find math books uninviting to read. Moreover, Rothe's writing style is engaging and provides readers with an understanding of Euclidean geometry that is both interesting and accessible. The author's ability to explain complex mathematical concepts with clarity is a testament to his expertise in the subject matter.

If you have an affinity for logic and geometry and are looking for a well-structured, comprehensive guide to this subject, Old and New Topics in Geometry is highly recommended. Whether you are a student or a professional mathematician, this book offers valuable insights into the foundations of geometry that will enhance your understanding of this fascinating subject.

[Mayra · Rating 5 out of 5]

For students and enthusiasts of Geometry, this book is an invaluable resource. Although I am not in a profession that necessitates the use of Geometry, I have always found the subject incredibly fascinating, and this book has deepened my understanding of several complex concepts. The author's lucid and concise explanation of the subject matter, coupled with a formal approach and the use of specific vocabulary, makes it comprehensible even for an amateur like me.

Furthermore, the book's organization is exemplary, with its contents arranged in a coherent and systematic manner, making it easy to follow. The topics gradually progress in complexity, allowing for an enjoyable read while learning. Commencing with Projective, Neutral and Basic Euclidean Geometry, a comprehensive and distinct volume that delves into the history of geometry from its inception. The book commences with an elucidation of Hilbert's axioms, which form the bedrock of Geometry's foundations. It also discusses logic and axioms in general and covers affine and projective geometry in two and three dimensions, among other topics. The book culminates with sections about the axioms of order and congruence in neutral geometry, the axioms of measurement and completeness, and deviates from Hilbert regarding circles.

I consider this book to be an excellent educational resource that both students and enthusiasts of Geometry will find extremely informative. I appreciated that the book is set up in a way that you can jump around the book to different topics if needed.

I am eager to share my passion for Geometry with my friends and intend to introduce them to this wonderful book. I plan on revisiting the book with them to review some basic topics and then delve into the more advanced ones. I am confident they will enjoy it as much as I did.

[Huda · Rating 5 out of 5]

Author Franz Rothe takes readers on a deep dive into the foundations of Geometry. In keeping with Hilbert's system, the book delves into sections about the axioms of order and congruence in neutral geometry, the axioms of measurement and completeness, and introduces the concept of circles, which deviates from Hilbert's original work. The insight into the independence of the parallel axiom leads to numerous avenues for further exploration while maintaining the goal of developing a natural and fully axiomatic system.

The book establishes a completely axiomatic theory of area in neutral geometry before branching into the cases of Euclidean, semi-hyperbolic, or semi-elliptic geometry. Rothe’s writing style adopts a more educational and elementary tone when delving into Euclidean geometry, beginning with Thales' theorem about the angle in a semicircle and continuing with Euclid's related theorems about angles in a circle. The author’s background in mathematics shines through in his writing, and he shares a wealth of knowledge with readers. The book provides a detailed exposition of the simpler aspects of Euclidean geometry, including topics such as similarity, area using the theorem of Pythagoras and trigonometry, and finally, the measurement of the circle. The approach is in line with Hilbert's spirit of rigor and axiomatic foundations, making it an invaluable resource for those interested in the foundations and development of geometry.

Old and New Topics in Geometry is an excellent resource for those interested in the foundations and development of geometry. The book provides a thorough treatment of the subject matter, maintaining Hilbert's spirit of rigor and axiomatic foundations while remaining engaging and accessible to readers at all levels of expertise.

[Celeste · Rating 5 out of 5]

As an experienced professor, mathematician, and talented musician, author Franz Rothe delivers a series of comprehensive books that explain various aspects of Mathematics. In this volume, theoretical axioms and theorems are described and illustrated with clear graphics and figures. The book commences with a concise but rigorous overview of the basic notions of geometry, using straightforward and modern language.

The book also contains problems that challenge readers, and the solutions provided are detailed, allowing the reader to master basic notions and techniques and further their knowledge of the subject matter. The structure of the text is well-organized and coherent, allowing readers to follow the different propositions stated in the book and guiding them to discover more complex concepts. It is important to note that this book requires the reader to have a solid foundation in basic and advanced notions of geometry.

As a reader who has always found mathematics and geometry challenging, I found this book to be incredibly helpful. Rothe's ability to explain this subject matter is impressive and his years of experience teaching at the university level are evident on every page. I was able to read the book quickly, and now I refer back to it whenever I need to solve a specific problem or work with a particular theorem. The ideas are presented clearly and simply, making it easy to follow.

Old and New Topics in Geometry is a textbook that I highly recommend for anyone looking to expand their knowledge of this fascinating subject matter. It is a valuable resource that should be kept within reach for anyone interested in furthering their understanding of geometry.

[Juan · Rating 5 out of 5]

Old and New Topics in Geometry is an impressive work that demonstrates the author Franz Rothe's vast experience and extensive career in the field of Mathematics. This first volume provides a comprehensive overview of essential topics such as Hilbert's axioms, incidence geometries, projective geometry, and Euclidean geometry, among others.
It should be noted that the book uses highly specific terminology and vocabulary, which may not be suitable for everyone. However, the author does an excellent job of guiding the reader through these complex concepts and making them accessible to all. The book's structure is well-planned, and the concepts are explained with clear examples and helpful graphics, making it an invaluable resource for both students and parents. As a reader, I found that I was able to learn many things that I had not previously learned during my high school education. While the book is highly informative, my only critique is that the index is somewhat rudimentary, and some explanations could benefit from more detail. However, it's impossible to include everything in a single book, especially considering the amount of math condensed into these pages.
Despite this, Old and New Topics in Geometry is a must-have resource for anyone interested in learning or teaching geometry. It serves as an excellent "go-to" book for explanations of various concepts and theorems, but it should not be the only resource utilized. Overall, Rothe's writing style and the book's contents are both brilliant and engaging, making it a highly recommended read for anyone interested in geometry.

[Marina Lujan · Rating 5 out of 5]

This captivating book provides a comprehensive and rigorous treatment of fundamental geometry topics. The book focuses on Hilbert's axioms, Geometries of incidence, affine and projective geometry in two and three dimensions. The author's approach to presenting Hilbert's formal approach to Euclidean geometry is highly instructive, surprisingly simple to follow, and is a reflection of David Hilbert's monumental contribution to mathematics. The author skillfully investigates the properties of geometric figures in relation to their incidence, including intersection and connection, and provides concrete examples to help develop a better understanding of its structure and applications.

Throughout the book, author Franz Rothe presents the contents in an engaging and informative way. The author has taken special care to ensure that the contents are comprehensive, and the numerous examples and illustrations serve to reinforce the concepts and apply them in concrete situations. Whether you are a geometry student or a professional, this book is an essential reference. It is a valuable contribution to the field of mathematics and is an excellent resource for anyone interested in exploring this fascinating branch of mathematics. Rothe uses his background in Mathematics to provide readers with an exceptionally well-written book that deserves nothing less than five stars. Among the many books available on the subject, this book stands out for its clarity, comprehensive coverage, and engaging presentation. I highly recommend it to anyone seeking to deepen their understanding of geometry.

[Ivan M · Rating 5 out of 5]

Thoughtfully divided into six main sections, each dedicated to specific areas of geometry, author Franz Rothe has written a scholarly and extensive compilation of vital topics in geometry, spanning both classical and contemporary perspectives. Rothe adeptly covers a wide range of topics in geometry, including elucidating fundamental principles of Euclidean geometry, such as the Pythagorean theorem. The author presents classical theorems alongside their proofs, expertly narrating the history of non-Euclidean geometry from the inception of hyperbolic geometry to more recent work on elliptic geometry.  

In Old and New Topics in Geometry, Franz Rothe excels in explaining complex concepts, making them understandable to even those without a strong mathematical background. The topics discussed in the book are complex, so I would recommend using this book in a study group or in a classroom setting to ask questions. I highly recommend this book to readers who possess a basic understanding of geometry and seek to expand their knowledge of the subject. Dr. Rothe's profound knowledge of the subject is evident in his writing, making the reader genuinely interested in a complex topic such as Euclidean geometry.  

I am eager to continue reading more of this talented author's works, and I am confident that his years of experience and expertise will continue to enlighten and inspire readers. 

[Almiria · Rating 4 out of 5]

Old and New Topics in Geometry is an impressive first volume in a series of geometry books that delves into the subject matter with great detail. Written by Franz Rothe, a seasoned educator with over a decade of experience teaching Fundamentals of Geometry, this book showcases the author's extensive knowledge in the field. Encompassing a broad range of mathematical equations and theorems, this volume includes prominent topics such as Hilbert's Axioms of Geometry, Desargues' Theorem, the Theorem of Pappus, Dedekind's axiom, Proclus' Theorem, and many more.

However, it is important to note that this book is not for the faint of heart or for beginners in the subject. After a rather abrupt preface, readers are presented with a plethora of equations and theorems accompanied by diagrams that lack introductory explanations. The book assumes a strong foundation in mathematics and geometry, making it challenging for readers who are not already well-versed in these areas. Nevertheless, this book is undoubtedly an excellent resource for scholars and mathematicians seeking detailed and in-depth content on geometrical equations and debates. The book is not ideal for beginners, but its comprehensive and extensive knowledge make it an excellent choice for those with a firm grasp of the subject matter.

I highly recommend this book for its sheer amount of information and Rothe's expertise.

[Chico's Mom · Rating 4 out of 5]

This book is an informative reference that is perfect for those who are wanting to learn more about Geometry and its history. Covering a range of topics, including Affine and projective Geometry, neutral geometry, and Euclidean Geometry. The author provides clear explanations of the principles, formulas, and examples with solutions for each topic. It is especially helpful for educators who are seeking a comprehensive guide to teaching geometry in the classroom.

While the book is text-heavy, the explanations are easy to understand and follow. The book provides a deep understanding of geometry beyond what is typically taught in high school, and I feel like the author takes the basics of Geometry to a deeper level. The author's presentation of the law of sines and cosines is particularly noteworthy as it is explained more simply than in high school curriculums. There are also some gaps in the book that could be improved to enhance the overall reading experience. The author includes examples and images along with each discussion, making it easier for readers who are visual learners to follow along.

This highly detailed read is an excellent reference for professors to use in their classrooms and for students who are looking for a deeper understanding of geometry. It offers a unique perspective on the subject matter, making it a valuable addition to anyone’s library.

[Liz · Rating 5 out of 5]

Geometry is often taught in schools as a collection of memorized formulas and definitions, leaving many students struggling to understand its relevance and application. However, the study of geometry can be a powerful tool for problem-solving and gaining insight into the world around us. In Old and New Topics in Geometry by Franz Rothe offers a refreshing approach to the subject that emphasizes its practical applications and historical context.

Rothe's book begins with a detailed explanation of Hilbert's axioms, which serve as the foundation for modern treatments of Euclidean geometry. Throughout the book's approximately 500 pages, Rothe expertly guides readers through a range of topics, using clear graphics and illustrations to help explain abstract concepts.

While this book may be best suited for advanced students in the field of mathematics, it offers valuable insights for anyone looking to deepen their understanding of geometry. The index at the end of the book is a useful tool for quickly locating specific topics, and the comprehensive approach taken by Rothe makes it an indispensable reference for mathematicians and students alike.

I enjoyed this well-written and well-organized book and I feel it deserves a place on the bookshelf of any serious student of mathematics. Franz Rothe has done an excellent job collecting and presenting this material.