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Lessons in Geometry

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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.