Lessons in Geometry, Vol. 1: Plane Geometry (English and French Edition) by Jacques Hadamard (PDF)

Description

This is a book in the tradition of Euclidean synthetic geometry written by one of the twentieth century’s great mathematicians. The original audience was pre-college teachers, but it is useful as well to gifted high school students and college students, in particular, to mathematics majors interested in geometry from a more advanced standpoint. The text starts where Euclid starts, and covers all the basics of plane Euclidean geometry. But this text does much more. It is at once pleasingly classic and surprisingly modern. The problems (more than 450 of them) are well-suited to exploration using the modern tools of dynamic geometry software. For this reason, the present edition includes a CD of dynamic solutions to select problems, created using Texas Instruments’ TI-NspireTM Learning Software. The TI-NspireTM documents demonstrate connections among problems and–through the free trial software included on the CD–will allow the reader to explore and interact with Hadamard’s Geometry in new ways. The material also includes introductions to several advanced topics. The exposition is spare, giving only the minimal background needed for a student to explore these topics. Much of the value of the book lies in the problems, whose solutions open worlds to the engaged reader. And so this book is in the Socratic tradition, as well as the Euclidean, in that it demands of the reader both engagement and interaction. A forthcoming companion volume that includes solutions, extensions, and classroom activities related to the problems can only begin to open the treasures offered by this work. We are just fortunate that one of the greatest mathematical minds of recent times has made this effort to show to readers some of the opportunities that the intellectual tradition of Euclidean geometry has to offer.

User’s Reviews

Editorial Reviews: Review Hadamard’s prose is clean and clear, focused and without frills. –MAA Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is the translation of a work firstly published in 1898. Hadamard was a first rate mathematician, but – as explained in the preface to the translation – cared a lot about classical geometry, having revised his work fourteen times in his life. What this book has that is so different from others? I believe that, first of all, it blends classical theorems of geometry with advanced ones (for instance, inversive geometry). Probably most of us are not exposed to some advanced techniques in high school mathematics, even though more than a century passed since the first edition. But the book is written for high school teachers and students. It is a superb introduction to a subject that is not very much treated even in college. But beware – you must work through the exercises to fully apreciate this book, because exercises complement the text. Even if you do not succeed to solve them all (very difficult task), the simple act of trying to solve makes you advance your knowledge. This is the second difference of this book, the interaction of text and exercises, much stronger than today’s books, many of whom are just recipes for problem-solving. Thirdly, and difficult to apreciate on first reading, is its complete rigour. It proves everything, but with a method that is uncommon today: using rigid motion. Sometimes it is difficult to see the rigour and we think that it is lacking somewhere. But the interesting about this method, very rarely used today, is that it makes geometry much more dynamic and alive. Even for those who follow the now standard method developped by Birkhoff (sometimes called the “metric” approach), where the rigour is more easily seen, it is useful to know the rigid motion method, that in my opinion is better to make new discoveries, although more difficult to make the derivations strictly rigorous. It is, however, the method that emerges from Klein’s Erlanger Program, and has its origins in the same place that the synthetic “static” method: in Euclid’s. For the derivation of Proposition I.4 in Euclid is by rigid motion, although Euclid seems to avoid this method.

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