Pangeometry, Edited and Translated by Athanase Papadopoulos (Heritage of European Mathematics)

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But they can also try to go deeply into Lobachevskysconstructions and computations. This will need some investment of time and energy,but it is worth doing.

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Indeed, although it is relatively easy to follow the main stream ofLobachevskys ideas, some of the details in the Pangeometry are rather obscure, andsome of the computations are difcult to follow. Let me recall Gausss letter to the mathematician and astronomer C. Gerling,written on 8 February , in which he compared Lobachevskys writings to a junglethrough which it is difcult to nd a passage and perspective, without rst becomingacquainted with each tree that composes it.

Schumacher, dated 28 November In developing the subject,the author followed a road different from the one I took myself; Lobachevsky carriedout the task in a masterly fashion and in a truly geometric spirit. I consider it a duty tocall your attention to this book, since I have no doubt that it will give you a tremendouspleasure They show the respectthat Gauss had for Lobachevskys work. Concerning the Pangeometry itself, let mequote G. Halsted , a mathematician whose life was entirely dedicatedto the defence and popularisation of non-Euclidean geometry, and in particular ofLobachevskys writings, some of which he translated into English.

In his biographi-cal article Lobachevsky [62], Halsted wrote: Though Lobachevskys Geometric Re-searches on the Theory of Parallels, published in , of which my English transla-tion is now in its fourth edition and has been beautifully reproduced in Japan, remainseven today the simplest introduction to the subject which has ever appeared; yet in itLobachevsky has not reached that nal breadth of view given rst in John Bolyais2[die mehr einemverworrenenWalde gleichen, durch den es, ohne alle Bume erst einzeln kennen gelerntzu haben, schwer ist, einen Durchgang und bersicht zu nden.

Ich glaube, Sie auf das Buch aufmerksam machenzu mssen, welches Ihnen gewiss ganz exquisiten Genuss gewhren wird. Finally, let me quote J. Hoel , another famous defender and translator into French of non-Euclidean geometry texts, who wrote in his Notice sur la vie etles travaux de N. Lobatchefsky [76] that the Pangeometry is one of Lobachevskysmost remarkable works regarding the clarity of exposition.

Reading the major texts of our mathematical ancestors is an expression of theesteem and gratitude that we owe them. This alone is a sufcient reason for publishingthis English translation of the Pangeometry, and making it easily accessible to themathematical public. While I was working on this book, I not only thought about mathematics, but Ialso tried to enter into Lobachevskys mysterious intimate world, and to understandhis tragic life. This led me to include in this book, besides the mathematical material,historical and biographical elements.

Beyond the particular case of Lobachevsky, Ithink that including historical comments in mathematical texts can be very helpful inmaking the reader feel the charm of the subject. I would like to thank Manfred Karbe and Ernest B. Vinberg for their kind interest inthis translation and the commentary, and for the invaluable information they providedon various editions of Lobachevskys works. I am also grateful to Dmitry V. Million-schikov and Grigory M. Polotovsky for help in the references, to A.


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Mishchenkowho provided copies of the original versions of the Pangeometry, and to Evgenyi N. Sosov, Daniar H. Mushtari andYi Zhang who kindly transmitted pictures from Kazan.

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The pictures on p. Lobachevsky,in the series Outstanding scientists and graduates of Kazan State University. Frenkin who translated into Russian part of mycommentary, see []. At the occasion of that translation, I discussed with Frenkinseveral points in that commentary.


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  • I am also grateful to Sergei S. Demidov who readmy notes on Lobachevskys biography, corrected inaccuracies, and sent informationwhich was unknown to me. I amespecially grateful to Jeremy Gray for an illuminatinglong conversation about this book. Gray also read an early version of the manuscriptand suggested several valuable improvements. Finally, I would like to take this opportunity to thank Norbert ACampo fromwhomI learned a lot of Lobachevsky model-free hyperbolic geometry, Marie-Pascale forher care and patience, and Irene Zimmermann for her excellent work in the nal editing.

    A commentary on Lobachevskys Pangeometry Introduction On the content of Lobachevskys Pangeometry On hyperbolic geometry and its reception On models, and on model-free hyperbolic geometry A short list of references Some milestones for Lobachevskys works on geometry Bibliography On the present editionThe present volume contains my translation of the Pangeometry, followed by someadditional material.

    The translation itself is accompanied by footnotes. These footnotes mostly concernthe rst part of the Pangeometry, in which Lobachevsky briey reviewed his previousworks. The notes are intended to help the reader who does not know the bases ofLobachevsky geometry to nd his way more easily in the subject.

    After this translation, we have reproduced the original French and Russian texts ofthe Pangeometry. This will allow the readers of the French or Russian languages toaccess Lobachevskys original writings directly. After the translation and the original texts, I have included a translation of a Frenchbiography of Lobachevsky, which was originally written as a preface to the editionof his Collected geometric works, published in Kazan.


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    • Navigation menu.
    • About this book.
    • Pangeometry, Edited and Translated by Athanase Papadopoulos (Heritage of European Mathematics).
    • I wrote an introduction to this biography, and I accompanied the translation by footnotes. The introduction andthe footnotes are intended to update the biography. After this biography, I have included a commentary on the Pangeometry.

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      It is in-tended to help the interested reader to get a quick overviewon the whole memoir, and toput this memoir in a larger perspective. The commentary concerns some mathematicalpoints, as well as historical material and references. My translation of the Pangeometry is not completely literal.

      The reason is that,as I have already said, I think the Pangeometry will be read not only for its historicalvalue, but also by geometers and students in geometry who want to learn more aboutthe subject. Therefore I made a few minor formal changes that will make its readingeasier. Apart from the formal changes that are listed below, I tried to be most faithfulto the original text. Here is the list of changes that I made: I highlighted certain denitions angle of parallelism, side of parallelism, limitcircle, etc.

      I corrected a few misprints, and I included some small corrections that were donein the German edition by Liebmann. I added gures to the text. We have read issues to any of our locations.

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