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A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary ... - I.M. Yaglom - Google Books
<!DOCTYPE html><html><head><title>A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary ... - I.M. Yaglom - Google Books</title><link rel="stylesheet" href="/books/css/_a33f2a89320471e58c940b9287b9d4eb/kl_about_this_book_kennedy_full_bundle.css" type="text/css" /><link rel="stylesheet"href="https://fonts.googleapis.com/css2?family=Product+Sans:wght@400"><script src="/books/javascript/atb_a33f2a89320471e58c940b9287b9d4eb__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/A_Simple_Non_Euclidean_Geometry_and_Its.html?id=FyToBwAAQBAJ"/><meta property="og:url" content="https://books.google.com/books/about/A_Simple_Non_Euclidean_Geometry_and_Its.html?id=FyToBwAAQBAJ"/><meta name="title" content="A Simple Non-Euclidean Geometry and Its Physical Basis"/><meta name="description" content="There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the Bibliography. This geometry, also called hyperbolic geometry, is part of the required subject matter of many mathematics departments in universities and teachers' colleges-a reflec tion of the view that familiarity with the elements of hyperbolic geometry is a useful part of the background of future high school teachers. Much attention is paid to hyperbolic geometry by school mathematics clubs. Some mathematicians and educators concerned with reform of the high school curriculum believe that the required part of the curriculum should include elements of hyperbolic geometry, and that the optional part of the curriculum should include a topic related to hyperbolic geometry. I The broad interest in hyperbolic geometry is not surprising. This interest has little to do with mathematical and scientific applications of hyperbolic geometry, since the applications (for instance, in the theory of automorphic functions) are rather specialized, and are likely to be encountered by very few of the many students who conscientiously study (and then present to examiners) the definition of parallels in hyperbolic geometry and the special features of configurations of lines in the hyperbolic plane. The principal reason for the interest in hyperbolic geometry is the important fact of "non-uniqueness" of geometry; of the existence of many geometric systems."/><meta property="og:title" content="A Simple Non-Euclidean Geometry and Its Physical Basis"/><meta property="og:type" content="book"/><meta property="og:site_name" content="Google Books"/><meta property="og:image" content="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71GQWNoqHqUP3Mdw85KuraSnzECdP51VXmT32Ro08gG7aylMZuhAkoMM7LCgyYVawcbO5cy1w4N5234VFyIlU9TdgJJ5sIlWe6wBOdQuE6iuvjM_46JMn30Hw5R2kucK2xW9EXt"/><link rel="image_src" href="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71GQWNoqHqUP3Mdw85KuraSnzECdP51VXmT32Ro08gG7aylMZuhAkoMM7LCgyYVawcbO5cy1w4N5234VFyIlU9TdgJJ5sIlWe6wBOdQuE6iuvjM_46JMn30Hw5R2kucK2xW9EXt"/><script></script><style>#gbar,#guser{font-size:13px;padding-top:1px !important;}#gbar{height:22px}#guser{padding-bottom:7px !important;text-align:right}.gbh,.gbd{border-top:1px solid #c9d7f1;font-size:1px}.gbh{height:0;position:absolute;top:24px;width:100%}@media all{.gb1{height:22px;margin-right:.5em;vertical-align:top}#gbar{float:left}}a.gb1,a.gb4{text-decoration:underline !important}a.gb1,a.gb4{color:#00c !important}.gbi .gb4{color:#dd8e27 !important}.gbf .gb4{color:#900 !important} #gbar { padding:.3em .6em !important;}</style></head><body class=""><div id=gbar><nobr><a target=_blank class=gb1 href="https://www.google.com.sg/search?tab=pw">Search</a> <a target=_blank class=gb1 href="https://www.google.com.sg/imghp?hl=en&tab=pi">Images</a> <a target=_blank class=gb1 href="https://maps.google.com.sg/maps?hl=en&tab=pl">Maps</a> <a target=_blank class=gb1 href="https://play.google.com/?hl=en&tab=p8">Play</a> <a target=_blank class=gb1 href="https://www.youtube.com/?tab=p1">YouTube</a> <a target=_blank class=gb1 href="https://news.google.com/?tab=pn">News</a> <a target=_blank class=gb1 href="https://mail.google.com/mail/?tab=pm">Gmail</a> <a target=_blank class=gb1 href="https://drive.google.com/?tab=po">Drive</a> <a target=_blank class=gb1 style="text-decoration:none" href="https://www.google.com.sg/intl/en/about/products?tab=ph"><u>More</u> »</a></nobr></div><div id=guser width=100%><nobr><span id=gbn class=gbi></span><span id=gbf class=gbf></span><span id=gbe></span><a target=_top id=gb_70 href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fid%3DFyToBwAAQBAJ%26source%3Dgbs_navlinks_s%26hl%3Den&hl=en&ec=GAZACg" class=gb4>Sign in</a></nobr></div><div class=gbh style=left:0></div><div class=gbh style=right:0></div><div role="alert" style="position: absolute; left: 0; right: 0;"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&source=gbs_navlinks_s&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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Yaglom</span></a></div><div><span dir=ltr>Springer Science & Business Media</span>, <span dir=ltr>6 Dec 2012</span> - <a class="secondary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22&source=gbs_ge_summary_r&cad=0"><span dir=ltr>Mathematics</span></a> - <span dir=ltr>307 pages</span></div></div><div id=synopsis><div id=synopsis-window><div id=synopsistext dir=ltr class="sa">There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the Bibliography. This geometry, also called hyperbolic geometry, is part of the required subject matter of many mathematics departments in universities and teachers' colleges-a reflec tion of the view that familiarity with the elements of hyperbolic geometry is a useful part of the background of future high school teachers. Much attention is paid to hyperbolic geometry by school mathematics clubs. Some mathematicians and educators concerned with reform of the high school curriculum believe that the required part of the curriculum should include elements of hyperbolic geometry, and that the optional part of the curriculum should include a topic related to hyperbolic geometry. I The broad interest in hyperbolic geometry is not surprising. This interest has little to do with mathematical and scientific applications of hyperbolic geometry, since the applications (for instance, in the theory of automorphic functions) are rather specialized, and are likely to be encountered by very few of the many students who conscientiously study (and then present to examiners) the definition of parallels in hyperbolic geometry and the special features of configurations of lines in the hyperbolic plane. The principal reason for the interest in hyperbolic geometry is the important fact of "non-uniqueness" of geometry; of the existence of many geometric systems.</div></div></div><div class="search_box_wrapper"><form action=/books id=search_form style="margin:0px;padding:0px;" method=get> <input type=hidden name="id" value="FyToBwAAQBAJ"><table cellpadding=0 cellspacing=0 class="swv-table"><tr><td class="swv-td-search"><span><input id=search_form_input type=text maxlength=1024 class="text_flat swv-input-search" aria-label="Search in this book" name=q value="" title="Search inside" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Search inside"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form><div id="preview-link"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&printsec=frontcover" class="primary"><span dir=ltr>Preview this book</span> »</a></div></div></td> </tr></table><div id="summary-second-column"></div></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="selected_pages_anchor"></a>Selected pages</h3><div id=selected_pages class=about_content><div id=selected_pages_v><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PR3&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&pg=PR3&img=1&zoom=1&sig=ACfU3U2v_ZiOyUxXeJZGRp3MlkztWx4y6A" alt="Title Page" title="Title Page" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PR3&source=gbs_selected_pages&cad=1" >Title Page</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PR17&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&pg=PR17&img=1&zoom=1&sig=ACfU3U1qznK3ObUG2TsDAV7HM4hsB0YV8g" alt="Table of Contents" title="Table of Contents" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PR17&source=gbs_selected_pages&cad=1" >Table of Contents</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA302&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&pg=PA302&img=1&zoom=1&sig=ACfU3U2BXj8WunHAuy-cg7zaSjgrP5pAOg" alt="Index" title="Index" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA302&source=gbs_selected_pages&cad=1" >Index</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA290&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&pg=PA290&img=1&zoom=1&sig=ACfU3U35I_u2Uy8KifvPWj806EGdsPQFJg" alt="References" title="References" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA290&source=gbs_selected_pages&cad=1" >References</a></div><div style="clear:both;"></div></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="toc_anchor"></a>Contents</h3><div id=toc class=about_content><div id=toc_v><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA1&source=gbs_toc_r&cad=2" ><span title="Introduction " style="white-space:nowrap"><span dir=ltr>Introduction </span></span></a></div></td><td class="toc_number" align=right>1</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Principle of duality coparallelograms and cotrapezoids </span></span></div></td><td class="toc_number" align=right>55</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Proofs of the principle of duality </span></span></div></td><td class="toc_number" align=right>64</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Circles and Cycles </span></span></div></td><td class="toc_number" align=right>77</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>The circumcycle and incycle of a triangle </span></span></div></td><td class="toc_number" align=right>104</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Power of a point with respect to a circle or cycle inversion </span></span></div></td><td class="toc_number" align=right>117</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Conclusion </span></span></div></td><td class="toc_number" align=right>158</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><div class="second_toc_column"><div class="second_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Supplement A Nine plane geometries </span></span></div></td><td class="toc_number" align=right>214</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Supplement B Axiomatic characterization </span></span></div></td><td class="toc_number" align=right>242</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Analytic models of </span></span></div></td><td class="toc_number" align=right>258</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Bibliography </span></span></div></td><td class="toc_number" align=right>289</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA302&source=gbs_toc_r&cad=2" ><span title="Index of Names " style="white-space:nowrap"><span dir=ltr>Index of Names </span></span></a></div></td><td class="toc_number" align=right>302</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr><tr><td class="toc_entry"><div class="toc_entry"><a class="primary" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&printsec=copyright" ><span title="Copyright" style="white-space:nowrap"><span dir=ltr>Copyright</span></span></a></div></td><td class="toc_number" align=right></td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><div style="clear:left; height:1em"></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN146126135X&id=FyToBwAAQBAJ'>View all</a></h3><div id=book_other_versions class=about_content><div id=book_other_versions_v><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=llnvAAAAMAAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/publisher/content?id=llnvAAAAMAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=llnvAAAAMAAJ&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary ...</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22I.M.+Yaglom%22" class="secondary"><span dir=ltr>I.M. Yaglom</span></a></span><br/><span><span style="color:#999">Snippet view</span> - 1979</span><br/></span></div></td><td align=right></td></tr></table></div></div><div class="one-third-column"><div class="crsiwrapper"><table class="rsi" cellspacing=0 cellpadding=0 border=0><tr><td class="coverdstd" align="center"><a href="https://books.google.com.sg/books?id=46upoAEACAAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/publisher/content?id=46upoAEACAAJ&printsec=frontcover&img=1&zoom=5" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=46upoAEACAAJ&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>A Simple Non-Euclidean Geometry and Its Physical Basis</span></a><br><span style="line-height: 1.3em; font-size:-1;"><span><a href="" class="secondary"><span dir=ltr>B Gordon</span></a>,<a href="" class="secondary"><span dir=ltr>I M Yaglom</span></a>,<a href="" class="secondary"><span dir=ltr>A Shenitzer</span></a></span><br/><span><span style="color:#999">No preview available</span> - 1979</span><br/></span></div></td><td align=right></td></tr></table></div></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"A Simple Non-Euclidean Geometry and Its Physical Basis","authors":"I.M. Yaglom","bib_key":"UOM:39015017136782","pub_date":"28 Feb 1979","snippet":"There are many technical and popular accounts, both in Russian and in other languages, of the non-Euclidean geometry of Lobachevsky and Bolyai, a few of which are listed in the ...","subject":"Gardening","info_url":"https://books.google.com.sg/books?id=llnvAAAAMAAJ\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=llnvAAAAMAAJ\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/publisher/content?id=llnvAAAAMAAJ\u0026printsec=frontcover\u0026img=1\u0026zoom=1","num_pages":338,"viewability":1,"preview":"noview","embeddable":false,"my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false,"subtitle":"An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity"},{"title":"A Simple Non-Euclidean Geometry and Its Physical Basis","authors":"B Gordon, I M Yaglom, A Shenitzer","bib_key":"ISBN:1461261368","pub_date":"28 Feb 1979","info_url":"https://books.google.com.sg/books?id=46upoAEACAAJ\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=46upoAEACAAJ\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/publisher/content?id=46upoAEACAAJ\u0026printsec=frontcover\u0026img=1\u0026zoom=1","num_pages":332,"viewability":4,"preview":"noview","embeddable":false,"my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false}]);})();</script></div></div></div><div class=vertical_module_list_row><h3 class=about_title><a name="word_cloud_anchor"></a>Common terms and phrases</h3><div id=word_cloud class=about_content><div id=word_cloud_v><style type="text/css">.cloud9 {color: #7777cc;font-size: 10px;}.cloud8 {color: #6963CC;font-size: 10.5px;}.cloud7 {color: #6057CC;font-size: 11px;}.cloud6 {color: #574BCC;font-size: 11.5px;}.cloud5 {color: #4E3DCC;font-size: 12px;}.cloud4 {color: #4632CC;font-size: 14px;}.cloud3 {color: #3D26CC;font-size: 16px;}.cloud2 {color: #341ACC;font-size: 18px;}.cloud1 {color: #2B0DCC;font-size: 20px;}.cloud0 {color: #2200CC;font-size: 22px;}.cloud {margin-top: 4px;line-height: 24px;}.cloud a {margin-right: 6px;text-decoration: none;}.cloud a:hover {text-decoration: underline;}</style><div class=cloud><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=A%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>A₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=angle+between+lines&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>angle between lines</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=angle+bisectors&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>angle bisectors</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=axioms&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>axioms</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=B%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>B₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Cayley-Klein+geometries&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Cayley-Klein geometries</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=center+Q&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>center Q</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=chord&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>chord</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=circle&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>circle</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=coefficient&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>coefficient</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=complex+numbers&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>complex numbers</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=concept&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>concept</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=congruent&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>congruent</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=coordinate+system&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>coordinate system</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=corresponding&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>corresponding</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=curvature&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>curvature</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=curve&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>curve</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=cyclic+rotation&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>cyclic rotation</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=defined&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>defined</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=definition&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>definition</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=denote&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>denote</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=direction&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>direction</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=distance&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>distance</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=double+numbers&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>double numbers</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=doublet&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>doublet</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=dual&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>dual</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=elliptic&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>elliptic</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=elliptic+geometry&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>elliptic geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=equal&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>equal</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=equation&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>equation</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Euclidean+geometry&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Euclidean geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Euclidean+plane&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Euclidean plane</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=example&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>example</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=fact&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>fact</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Figure&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Figure</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=follows&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>follows</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=formulas&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>formulas</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Galilean+geometry&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Galilean geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Galilean+plane&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Galilean plane</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Galilean+transformations&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Galilean transformations</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=given&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>given</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Hence&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Hence</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=hyperbolic+geometry&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>hyperbolic geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=hyperbolic+plane&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>hyperbolic plane</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=implies&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>implies</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=inertial+reference+frame&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>inertial reference frame</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=inscribed+angle&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>inscribed angle</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=intersection+point&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>intersection point</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=k%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>k₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=l%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>l₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=length&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>length</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=maps&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>maps</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=mechanical&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>mechanical</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=midpoints&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>midpoints</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Minkowskian+geometry&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Minkowskian geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Minkowskian+plane&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Minkowskian plane</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=non-Euclidean&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>non-Euclidean</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=Non-Euclidean+Geometry&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Non-Euclidean Geometry</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=pair&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>pair</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=perpendicular&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>perpendicular</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=point+at+infinity&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>point at infinity</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=principle+of+duality&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>principle of duality</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=principle+of+relativity&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>principle of relativity</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=proof&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>proof</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=properties&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>properties</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=radical+axis&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>radical axis</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=radius&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>radius</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=relations&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>relations</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=respect&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>respect</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=S%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>S₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=scalar+product&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>scalar product</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=second+kind&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>second kind</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=segment&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>segment</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=set+of+points&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>set of points</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=sides&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>sides</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=sinh&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>sinh</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=space&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>space</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=special+line&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>special line</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=sphere&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>sphere</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=stereographic+projection&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>stereographic projection</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=tangent&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>tangent</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=theorem&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>theorem</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=theory+of+relativity&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>theory of relativity</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=triangle+ABC&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>triangle ABC</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=uniform+motions&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>uniform motions</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=vector&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>vector</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=velocity&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>velocity</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=y-axis&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>y-axis</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=y%E2%82%81&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>y₁</span></a> <a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&q=zero&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>zero</span></a></div></div></div></div><div class=vertical_module_list_row><h3 class="about_title">Bibliographic information</h3><div class="about_content" id="metadata_content" style="padding-bottom:.3em"><div class="metadata_sectionwrap"><table id="metadata_content_table"><tr class="metadata_row"><td class="metadata_label">Title</td><td class="metadata_value"><span dir=ltr>A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Heidelberg+Science+Library%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Heidelberg Science Library</span></i></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Author</span></td><td class="metadata_value"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=inauthor:%22I.M.+Yaglom%22&source=gbs_metadata_r&cad=5"><span dir=ltr>I.M. Yaglom</span></a></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Translated by</span></td><td class="metadata_value"><span dir=ltr>A. Shenitzer</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Contributor</span></td><td class="metadata_value"><span dir=ltr>B. Gordon</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Edition</span></td><td class="metadata_value"><span dir=ltr>illustrated</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Springer Science & Business Media, 2012</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>146126135X, 9781461261353</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>307 pages</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Subjects</span></td><td class="metadata_value"><div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics%22" itemprop="url" dir=ltr><span itemprop="title">Mathematics</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry%22" itemprop="url" dir=ltr><span itemprop="title">Geometry</span></a></div> › <div style="display:inline" itemscope itemtype="http://data-vocabulary.org/Breadcrumb"><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+Geometry+General%22" itemprop="url" dir=ltr><span itemprop="title">General</span></a></div><br><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=subject:%22Mathematics+/+Geometry+/+General%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Mathematics / Geometry / General</span></a></td></tr><tr class="metadata_row"><td> </td><td> </td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Export Citation</span></td><td class="metadata_value"><a class="gb-button " href="https://books.google.com.sg/books/download/A_Simple_Non_Euclidean_Geometry_and_Its.bibtex?id=FyToBwAAQBAJ&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/A_Simple_Non_Euclidean_Geometry_and_Its.enw?id=FyToBwAAQBAJ&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/A_Simple_Non_Euclidean_Geometry_and_Its.ris?id=FyToBwAAQBAJ&output=ris"><span dir=ltr>RefMan</span></a></td></tr></table></div><div style="clear:both"></div></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBooksUnifiedLeftNavEnabled:1, IsBrowsingHistoryEnabled:1, IsBooksRentalEnabled:1, IsZipitFolderCollectionEnabled:1});_OC_Run({"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3DFyToBwAAQBAJ%26source%3Dgbs_navlinks_s%26hl%3Den\u0026hl=en"}, {"volume_id":"FyToBwAAQBAJ","is_ebook":true,"volumeresult":{"has_flowing_text":false,"has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"sample_url":"https://play.google.com/books/reader?id=FyToBwAAQBAJ\u0026source=gbs_atb_hover","is_browsable":true,"is_public_domain":false}, {});</script><div id="footer_table" style="font-size:83%;text-align:center;position:relative;top:20px;height:4.5em;margin-top:2em"><div style="margin-bottom:8px"><a href="/intl/en/googlebooks/about.html"><nobr><nobr>About Google Books</nobr></nobr></a> - <a href="/intl/en/googlebooks/privacy.html"><nobr><nobr>Privacy Policy</nobr></nobr></a> - <a href="/intl/en/googlebooks/tos.html"><nobr><nobr>Terms of Service</nobr></nobr></a> - <a href="http://books.google.com.sg/support/partner/?hl=en-SG"><nobr><nobr>Information for Publishers</nobr></nobr></a> - <a href="http://books.google.com.sg/support/answer/180577?hl=en-SG&url=https://books.google.com.sg/books?id=FyToBwAAQBAJ&source=gbs_navlinks_s&hl=en&v=FyToBwAAQBAJ&is=atb"><nobr><nobr>Report an issue</nobr></nobr></a> - <a href="http://books.google.com.sg/support/topic/4359341?hl=en-SG"><nobr><nobr>Help</nobr></nobr></a> - <a href="https://www.google.com.sg/"><nobr><nobr>Google Home</nobr></nobr></a></div></div></div></div></div><script>(function() {var href = window.location.href;if (href.indexOf('?') !== -1) {var parameters = href.split('?')[1].split('&');for (var i = 0; i < parameters.length; i++) {var param = parameters[i].split('=');if (param[0] == 'focus') {var elem = document.getElementById(param[1]);if (elem) {elem.focus();}}}}})();</script>