CINXE.COM

<!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/_ff77d0f0508c7e0bad470e7d6b7f28fa/kl_viewport_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/v2_ff77d0f0508c7e0bad470e7d6b7f28fa__en.js"></script><script>_OC_Hooks = ["_OC_Page", "_OC_SearchReload", "_OC_TocReload", "_OC_EmptyFunc", "_OC_SearchPage", "_OC_QuotePage" ];for (var _OC_i = 0; _OC_i < _OC_Hooks.length; _OC_i++) {eval("var " + _OC_Hooks[_OC_i] + ";");}function _OC_InitHooks () {for (var i = 0; i < _OC_Hooks.length; i++) {var func = arguments[i];eval( _OC_Hooks[i] + " = func;");}}</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=AFLRE71ZWkG5987UZcrvC8T9a01hR8MzUqEeYXon8KPr-jKOZjC1zcbHUG-pjcORGy8Nf0zb0h-3gMaXYRp8FhoTKQsPPVPklTrxj7BQGgrhmd2eevAhDB0YS9UvJa6B9Hphpga-pr0H"/><link rel="image_src" href="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE71ZWkG5987UZcrvC8T9a01hR8MzUqEeYXon8KPr-jKOZjC1zcbHUG-pjcORGy8Nf0zb0h-3gMaXYRp8FhoTKQsPPVPklTrxj7BQGgrhmd2eevAhDB0YS9UvJa6B9Hphpga-pr0H"/><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%26q%3Dlength%26source%3Dgbs_word_cloud_r%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&q=length&source=gbs_word_cloud_r&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. Accessible mode has the same essential features but works better with your reader."><img border="0" src="//www.google.com/images/cleardot.gif"alt="Screen reader users: click this link for accessible mode. Accessible mode has the same essential features but works better with your reader."></a></div><div class="kd-appbar"><h2 class="kd-appname"><a href="/books">Books</a></h2><div class="kd-buttonbar left" id="left-toolbar-buttons"><a id="appbar-view-print-sample-link" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&printsec=frontcover&vq=length&source=gbs_vpt_read"></a><a id="appbar-view-ebook-sample-link" href="https://play.google.com/books/reader?id=FyToBwAAQBAJ&source=gbs_vpt_read"></a><a id="appbar-patents-prior-art-finder-link" href=""></a><a id="appbar-patents-discuss-this-link" href="" data-is-grant=""></a><a id="appbar-read-patent-link" href=""></a><a id="appbar-download-pdf-link" href=""></a></div><div class="kd-buttonbar right" id="right-toolbar-buttons"></div></div><div style="display: none"><ol id="ofe-gear-menu-contents" class="gbmcc"><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="https://www.google.com/accounts/Login?service=print&continue=https://books.google.com.sg/books%3Fop%3Dlibrary&hl=en">My library</a></li><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="http://books.google.com.sg/support/topic/4359341?hl=en-SG">Help</a></li><li class="gbe gbmtc"><a class="gbmt goog-menuitem-content" id="" href="https://books.google.com.sg/advanced_book_search">Advanced Book Search</a></li></ol></div><div id="volume-main"><div id="volume-left"><div id=menu_container ><div id="menu_scroll_wrapper"><div id="menu_scroll" role="navigation"><div id="gb-get-book-container"><a href="https://play.google.com/store/books/details?id=FyToBwAAQBAJ&rdid=book-FyToBwAAQBAJ&rdot=1&source=gbs_vpt_read&pcampaignid=books_booksearch_viewport" id="gb-get-book-content">Buy eBook - SGD 86.32</a></div><p id="gb-buy-options-trigger" class="gb-buy-options-link">Get this book in print</p><h3 class=about_title><a name="buy_anchor"></a></h3><div id=buy class=about_content><div id=buy_v><ul style="list-style-type: none; padding-left: 0; margin: 0;"><li><a style="white-space:normal" href="http://www.springer.com/978-1-4612-6135-3?utm_medium=referral&utm_source=google_books&utm_campaign=3_pier05_buy_print&utm_content=en_08082017" dir=ltr onMouseOver="this.href='http://www.springer.com/978-1-4612-6135-3?utm_medium\x3dreferral\x26utm_source\x3dgoogle_books\x26utm_campaign\x3d3_pier05_buy_print\x26utm_content\x3den_08082017';return false" onMouseDown="this.href='/url?client\x3dca-google-print\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3dFyToBwAAQBAJ\x26q\x3dhttp://www.springer.com/978-1-4612-6135-3%3Futm_medium%3Dreferral%26utm_source%3Dgoogle_books%26utm_campaign%3D3_pier05_buy_print%26utm_content%3Den_08082017\x26usg\x3dAOvVaw2cy90lzm47WMSaDu_pDh8y\x26source\x3dgbs_buy_r';return true"><span dir=ltr>Springer Shop</span></a></li><li><a style="white-space:normal" href="http://www.amazon.com/gp/search?index=books&linkCode=qs&keywords=9781461261353" dir=ltr onMouseOver="this.href='http://www.amazon.com/gp/search?index\x3dbooks\x26linkCode\x3dqs\x26keywords\x3d9781461261353';return false" onMouseDown="this.href='/url?client\x3dca-google-print\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3dFyToBwAAQBAJ\x26q\x3dhttp://www.amazon.com/gp/search%3Findex%3Dbooks%26linkCode%3Dqs%26keywords%3D9781461261353\x26usg\x3dAOvVaw0EFotrn3m5lqiGrV8plgFC\x26source\x3dgbs_buy_r';return true"><span dir=ltr>Amazon.com</span></a></li><li><a style="white-space:normal" href="http://www.mphonline.com/books/nsearch.aspx?do=detail&pcode=9781461261353" dir=ltr onMouseOver="this.href='http://www.mphonline.com/books/nsearch.aspx?do\x3ddetail\x26pcode\x3d9781461261353';return false" onMouseDown="this.href='/url?client\x3dca-google-print\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3dFyToBwAAQBAJ\x26q\x3dhttp://www.mphonline.com/books/nsearch.aspx%3Fdo%3Ddetail%26pcode%3D9781461261353\x26usg\x3dAOvVaw2MukSnMva-sC4uHKipqMWB\x26source\x3dgbs_buy_r';return true"><span dir=ltr>MPH</span></a></li><li><a style="white-space:normal" href="https://www.popular.com.sg/catalogsearch/result/?q=9781461261353" dir=ltr onMouseOver="this.href='https://www.popular.com.sg/catalogsearch/result/?q\x3d9781461261353';return false" onMouseDown="this.href='/url?client\x3dca-google-print\x26format\x3dgoogleprint\x26num\x3d0\x26id\x3dFyToBwAAQBAJ\x26q\x3dhttps://www.popular.com.sg/catalogsearch/result/%3Fq%3D9781461261353\x26usg\x3dAOvVaw1_5J_KN43d1isvAFP6u95r\x26source\x3dgbs_buy_r';return true"><span dir=ltr>Popular</span></a></li><li><hr style="margin-right: 20%; color: #666;"></li><li><a style="white-space:normal" href="https://books.google.com.sg/url?id=FyToBwAAQBAJ&pg=PA165&q=http://www.worldcat.org/oclc/1157340479&clientid=librarylink&usg=AOvVaw3MK7UXZcdolvWHpcH6rPIT&source=gbs_buy_r"><span dir=ltr>Find in a library</span></a></li><li><a class="secondary" style="white-space:normal" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&sitesec=buy&source=gbs_buy_r" id="get-all-sellers-link"><span dir=ltr>All sellers</span> »</a></li></ul></div></div><div class=menu id=menu><div class="menu_content" style="margin-bottom:6px"><div style="margin-bottom:4px"><div class="sidebarnav"><table border="0" cellpadding="0" cellspacing="0"><tr><td><div class="sidebarcover"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&printsec=frontcover&vq=length" onclick="_OC_Page('PP1',this.href); return false;"><img src="https://books.google.com.sg/books/publisher/content?id=FyToBwAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl&imgtk=AFLRE73g8qY_VONntkbiKTmek8q_CO01SOG2e0tJ0nG47V5hzsoxZ6d4UHUPHctP4FVt3e76GFfM6DM07P_4SI8ZIsqlDXaeXZZOwNr8Eo_ZqJqKhx7v1ImN4EF5_67FaGJIhcGiG3gl" alt="Front Cover" title="Front Cover" height=80 border=1 id=summary-frontcover ></a></div></td><td></td></tr></table></div><div style="clear:both"></div></div><div id="volume-info-sidebar"><h1 class="gb-volume-title" dir=ltr>A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary ...</h1><span class="addmd">By I.M. Yaglom</span></div><div style="margin-bottom:3px"><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="Go" accesskey=i></span></td><td class="swv-td-space"><div> </div></td><td><input type=submit value="Go"></td></tr></table><script type="text/javascript">if (window['_OC_autoDir']) {_OC_autoDir('search_form_input');}</script></form></div><div><p><a id="sidebar-atb-link" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&vq=length&source=gbs_navlinks_s"><span dir=ltr>About this book</span></a></p></div></div></div><div><div id="navbarContainer" class="gb-navbar"></div><script>_OC_InitNavbar({"child_node":[{"title":"My library","url":"https://books.google.com.sg/books?uid=114584440181414684107\u0026source=gbs_lp_bookshelf_list","id":"my_library","collapsed":true},{"title":"My History","url":"","id":"my_history","collapsed":true}],"highlighted_node_id":""});</script><a href="/intl/en/googlebooks/tos.html" target="_blank">Terms of Service</a><h3 class=about_title><a name="pub_info_anchor"></a></h3><div id=pub_info class=about_content><div id=pub_info_v><table cellspacing=0><tr><td><a href="https://books.google.com.sg/url?id=FyToBwAAQBAJ&pg=PA165&q=http://www.springer.com/shop&linkid=1&usg=AOvVaw3qjA2jyxFK2Ek-6onqUbwf&source=gbs_pub_info_r" style="text-decoration:none"><img width=135 height=35 border=0 src="https://pagead2.googlesyndication.com/pagead/imgad?id=CJPF-5HA9_PCvwEQvgEYMTIIXq6w7W8OyhQ" alt="Springer Science & Business Media"></a><tr><td style="font-size:84.6%;color:#666666">Pages displayed by permission of <a class=link_aux href="https://books.google.com.sg/url?id=FyToBwAAQBAJ&pg=PA165&q=http://www.springer.com/shop&linkid=1&usg=AOvVaw3qjA2jyxFK2Ek-6onqUbwf&source=gbs_pub_info_r">Springer Science & Business Media</a>. <a style="color:#7777cc;white-space:normal" href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&printsec=copyright&vq=length&source=gbs_pub_info_r">Copyright</a>. </table></div></div></div></div></div></div></div><div id="volume-center"><div id="scroll_atb" role="main"><div id="toolbar_container"><div style="float:left;white-space:nowrap"><table cellpadding=0 cellspacing=0><tr><td id="l_toolbar"></td><td class=toolbar-pc-cell><table cellpadding=0 cellspacing=0><tr><td class=no-jump-cell align=right><span id=page_label style="margin-right:.5em">Page 165</span></td><td class=arrow style="padding-right:2px"><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA164&lpg=PA165&focus=viewport&vq=length" onclick="_OC_EmptyFunc(this.href); return false;"><div class=pagination><div id=prev_btn alt="Previous Page" title="Previous Page" class="SPRITE_pagination_v2_left"></div></div></a></td><td class=arrow><a href="https://books.google.com.sg/books?id=FyToBwAAQBAJ&pg=PA166&lpg=PA165&focus=viewport&vq=length" onclick="_OC_EmptyFunc(this.href); return false;"><div class=pagination><div id=next_btn alt="Next Page" title="Next Page" class="SPRITE_pagination_v2_right"></div></div></a></td></tr></table></td><td>  </td><td id=view_toolbar></td><td id=view_new></td></tr></table></div><div style="float:right"><table cellpadding=0 cellspacing=0><tr><td></td><td id="r_toolbar" style="white-space:nowrap"></td></tr></table></div><div style="clear:both"></div></div><div id="search_bar"></div><div class="gback"><div id="viewport" class="viewport" tabindex="0"><a name="page" accesskey="c"></a><table class="viewport-table" id="container" align="center" cellpadding="0" cellspacing="0"><tr><td valign="top" align="center"><noscript><style type=text/css>.imgg { width:575px;height:872px;background:#eee;padding-bottom:25px}</style><div class=imgg><div align=center><table border=0 cellpadding=0 cellspacing=0 width=500 align=center style="margin-top:2em"><tr><td rowspan=2 valign=top style="width:9px;background:#fff url('/googlebooks/bbl_l.gif') top left repeat-y"><img src="/googlebooks/bbl_tl.gif" width=9 height=7 alt=""></td><td style="background:url('/googlebooks/bbl_t.gif') top left repeat-x"><img width=1 height=7 alt=""></td><td rowspan=2 valign=top style="width:10px;background:#fff url('/googlebooks/bbl_r.gif') top right repeat-y"><img src="/googlebooks/bbl_tr.gif" width=10 height=7 alt=""></td></tr><tr><td align=center style="background:#ff9;text-align:center;line-height:1.2em"><div style="margin:1em"><img width=60 height=60 align=absmiddle src="/googlebooks/restricted_logo.gif" alt=""> <span style="font-weight:bold;font-size:1.2em"><br>Restricted Page</span></div><div style="margin:1em" align=left>You have reached your viewing limit for this book (<a href=https://books.google.com.sg/support/answer/43729?topic=9259&hl=en>why?</a>).</div></td></tr><tr><td><img src="/googlebooks/bbl_bl.gif" width=9 height=9 alt=""></td><td style="background:url('/googlebooks/bbl_b.gif') bottom left repeat-x"><img width=1 height=9 alt=""></td><td><img src="/googlebooks/bbl_br.gif" width=10 height=9 alt=""></td></tr></table></div></div></noscript></td></tr></table></div></div><script>_OC_addFlags({Host:"https://books.google.com.sg/", IsBooksUnifiedLeftNavEnabled:1, IsBooksRentalEnabled:1, IsBrowsingHistoryEnabled:1, IsZipitFolderCollectionEnabled:1});_OC_Run({"page":[{"pid":"PP1","flags":32,"order":0},{"pid":"PR3","order":3,"title":"iii"},{"pid":"PR4","order":4,"title":"iv"},{"pid":"PR17","order":17,"title":"xvii"},{"pid":"PR18","order":18,"title":"xviii"},{"pid":"PA1","order":19,"title":"1"},{"pid":"PA2","order":20,"title":"2"},{"pid":"PA3","order":21,"title":"3"},{"pid":"PA4","order":22,"title":"4"},{"pid":"PA5","order":23,"title":"5"},{"pid":"PA6","order":24,"title":"6"},{"pid":"PA7","order":25,"title":"7"},{"pid":"PA8","order":26,"title":"8"},{"pid":"PA9","order":27,"title":"9"},{"pid":"PA10","order":28,"title":"10"},{"pid":"PA11","order":29,"title":"11"},{"pid":"PA12","order":30,"title":"12"},{"pid":"PA13","order":31,"title":"13"},{"pid":"PA14","order":32,"title":"14"},{"pid":"PA15","order":33,"title":"15"},{"pid":"PA16","order":34,"title":"16"},{"pid":"PA17","order":35,"title":"17"},{"pid":"PA18","order":36,"title":"18"},{"pid":"PA19","order":37,"title":"19"},{"pid":"PA20","order":38,"title":"20"},{"pid":"PA21","order":39,"title":"21"},{"pid":"PA22","order":40,"title":"22"},{"pid":"PA23","order":41,"title":"23"},{"pid":"PA24","order":42,"title":"24"},{"pid":"PA25","order":43,"title":"25"},{"pid":"PA26","order":44,"title":"26"},{"pid":"PA27","order":45,"title":"27"},{"pid":"PA28","order":46,"title":"28"},{"pid":"PA29","order":47,"title":"29"},{"pid":"PA30","order":48,"title":"30"},{"pid":"PA31","order":49,"title":"31"},{"pid":"PA32","order":50,"title":"32"},{"pid":"PA33","order":51,"title":"33"},{"pid":"PA34","order":52,"title":"34"},{"pid":"PA35","order":53,"title":"35"},{"pid":"PA36","order":54,"title":"36"},{"pid":"PA37","order":55,"title":"37"},{"pid":"PA38","order":56,"title":"38"},{"pid":"PA39","order":57,"title":"39"},{"pid":"PA40","order":58,"title":"40"},{"pid":"PA41","order":59,"title":"41"},{"pid":"PA42","order":60,"title":"42"},{"pid":"PA43","order":61,"title":"43"},{"pid":"PA44","order":62,"title":"44"},{"pid":"PA45","order":63,"title":"45"},{"pid":"PA46","order":64,"title":"46"},{"pid":"PA47","order":65,"title":"47"},{"pid":"PA48","order":66,"title":"48"},{"pid":"PA49","order":67,"title":"49"},{"pid":"PA50","order":68,"title":"50"},{"pid":"PA51","order":69,"title":"51"},{"pid":"PA52","order":70,"title":"52"},{"pid":"PA53","order":71,"title":"53"},{"pid":"PA54","order":72,"title":"54"},{"pid":"PA55","order":73,"title":"55"},{"pid":"PA56","order":74,"title":"56"},{"pid":"PA57","order":75,"title":"57"},{"pid":"PA58","order":76,"title":"58"},{"pid":"PA59","order":77,"title":"59"},{"pid":"PA60","order":78,"title":"60"},{"pid":"PA61","order":79,"title":"61"},{"pid":"PA62","order":80,"title":"62"},{"pid":"PA63","order":81,"title":"63"},{"pid":"PA64","order":82,"title":"64"},{"pid":"PA65","order":83,"title":"65"},{"pid":"PA66","order":84,"title":"66"},{"pid":"PA67","order":85,"title":"67"},{"pid":"PA68","order":86,"title":"68"},{"pid":"PA69","order":87,"title":"69"},{"pid":"PA70","order":88,"title":"70"},{"pid":"PA71","order":89,"title":"71"},{"pid":"PA72","order":90,"title":"72"},{"pid":"PA73","order":91,"title":"73"},{"pid":"PA74","order":92,"title":"74"},{"pid":"PA75","order":93,"title":"75"},{"pid":"PA76","order":94,"title":"76"},{"pid":"PA77","order":95,"title":"77"},{"pid":"PA78","order":96,"title":"78"},{"pid":"PA79","order":97,"title":"79"},{"pid":"PA80","order":98,"title":"80"},{"pid":"PA81","order":99,"title":"81"},{"pid":"PA82","order":100,"title":"82"},{"pid":"PA83","order":101,"title":"83"},{"pid":"PA84","order":102,"title":"84"},{"pid":"PA85","order":103,"title":"85"},{"pid":"PA86","order":104,"title":"86"},{"pid":"PA87","order":105,"title":"87"},{"pid":"PA88","order":106,"title":"88"},{"pid":"PA89","order":107,"title":"89"},{"pid":"PA90","order":108,"title":"90"},{"pid":"PA91","order":109,"title":"91"},{"pid":"PA92","order":110,"title":"92"},{"pid":"PA93","order":111,"title":"93"},{"pid":"PA94","order":112,"title":"94"},{"pid":"PA95","order":113,"title":"95"},{"pid":"PA96","order":114,"title":"96"},{"pid":"PA97","order":115,"title":"97"},{"pid":"PA98","order":116,"title":"98"},{"pid":"PA99","order":117,"title":"99"},{"pid":"PA100","order":118,"title":"100"},{"pid":"PA101","order":119,"title":"101"},{"pid":"PA102","order":120,"title":"102"},{"pid":"PA103","order":121,"title":"103"},{"pid":"PA104","order":122,"title":"104"},{"pid":"PA105","order":123,"title":"105"},{"pid":"PA106","order":124,"title":"106"},{"pid":"PA107","order":125,"title":"107"},{"pid":"PA108","order":126,"title":"108"},{"pid":"PA109","order":127,"title":"109"},{"pid":"PA110","order":128,"title":"110"},{"pid":"PA111","order":129,"title":"111"},{"pid":"PA112","order":130,"title":"112"},{"pid":"PA113","order":131,"title":"113"},{"pid":"PA114","order":132,"title":"114"},{"pid":"PA115","order":133,"title":"115"},{"pid":"PA116","order":134,"title":"116"},{"pid":"PA117","order":135,"title":"117"},{"pid":"PA118","order":136,"title":"118"},{"pid":"PA119","order":137,"title":"119"},{"pid":"PA120","order":138,"title":"120"},{"pid":"PA121","order":139,"title":"121"},{"pid":"PA122","order":140,"title":"122"},{"pid":"PA123","order":141,"title":"123"},{"pid":"PA124","order":142,"title":"124"},{"pid":"PA125","order":143,"title":"125"},{"pid":"PA126","order":144,"title":"126"},{"pid":"PA127","order":145,"title":"127"},{"pid":"PA128","order":146,"title":"128"},{"pid":"PA129","order":147,"title":"129"},{"pid":"PA130","order":148,"title":"130"},{"pid":"PA131","order":149,"title":"131"},{"pid":"PA132","order":150,"title":"132"},{"pid":"PA133","order":151,"title":"133"},{"pid":"PA134","order":152,"title":"134"},{"pid":"PA135","order":153,"title":"135"},{"pid":"PA136","order":154,"title":"136"},{"pid":"PA137","order":155,"title":"137"},{"pid":"PA138","order":156,"title":"138"},{"pid":"PA139","order":157,"title":"139"},{"pid":"PA140","order":158,"title":"140"},{"pid":"PA141","order":159,"title":"141"},{"pid":"PA142","order":160,"title":"142"},{"pid":"PA143","order":161,"title":"143"},{"pid":"PA144","order":162,"title":"144"},{"pid":"PA146","order":164,"title":"146"},{"pid":"PA147","order":165,"title":"147"},{"pid":"PA148","order":166,"title":"148"},{"pid":"PA150","order":168,"title":"150"},{"pid":"PA151","order":169,"title":"151"},{"pid":"PA154","order":172,"title":"154"},{"pid":"PA155","order":173,"title":"155"},{"pid":"PA156","order":174,"title":"156"},{"pid":"PA157","order":175,"title":"157"},{"pid":"PA158","order":176,"title":"158"},{"pid":"PA159","order":177,"title":"159"},{"pid":"PA161","order":179,"title":"161"},{"pid":"PA162","order":180,"title":"162"},{"pid":"PA163","order":181,"title":"163"},{"pid":"PA164","flags":8,"order":182,"title":"164"},{"pid":"PA165","flags":8,"order":183,"title":"165"},{"pid":"PA166","flags":8,"order":184,"title":"166"},{"pid":"PA168","flags":8,"order":186,"title":"168"},{"pid":"PA169","order":187,"title":"169"},{"pid":"PA170","order":188,"title":"170"},{"pid":"PA174","order":192,"title":"174"},{"pid":"PA175","order":193,"title":"175"},{"pid":"PA176","order":194,"title":"176"},{"pid":"PA177","order":195,"title":"177"},{"pid":"PA178","order":196,"title":"178"},{"pid":"PA179","order":197,"title":"179"},{"pid":"PA182","order":200,"title":"182"},{"pid":"PA183","order":201,"title":"183"},{"pid":"PA184","order":202,"title":"184"},{"pid":"PA185","order":203,"title":"185"},{"pid":"PA187","order":205,"title":"187"},{"pid":"PA188","order":206,"title":"188"},{"pid":"PA189","order":207,"title":"189"},{"pid":"PA190","order":208,"title":"190"},{"pid":"PA192","order":210,"title":"192"},{"pid":"PA193","order":211,"title":"193"},{"pid":"PA194","order":212,"title":"194"},{"pid":"PA195","order":213,"title":"195"},{"pid":"PA196","order":214,"title":"196"},{"pid":"PA197","order":215,"title":"197"},{"pid":"PA198","order":216,"title":"198"},{"pid":"PA199","order":217,"title":"199"},{"pid":"PA200","order":218,"title":"200"},{"pid":"PA201","order":219,"title":"201"},{"pid":"PA202","order":220,"title":"202"},{"pid":"PA205","order":223,"title":"205"},{"pid":"PA208","order":226,"title":"208"},{"pid":"PA209","order":227,"title":"209"},{"pid":"PA210","order":228,"title":"210"},{"pid":"PA211","order":229,"title":"211"},{"pid":"PA212","order":230,"title":"212"},{"pid":"PA213","order":231,"title":"213"},{"pid":"PA214","order":232,"title":"214"},{"pid":"PA215","order":233,"title":"215"},{"pid":"PA216","order":234,"title":"216"},{"pid":"PA217","order":235,"title":"217"},{"pid":"PA218","order":236,"title":"218"},{"pid":"PA219","order":237,"title":"219"},{"pid":"PA220","order":238,"title":"220"},{"pid":"PA222","order":240,"title":"222"},{"pid":"PA223","order":241,"title":"223"},{"pid":"PA224","order":242,"title":"224"},{"pid":"PA226","order":244,"title":"226"},{"pid":"PA227","order":245,"title":"227"},{"pid":"PA228","order":246,"title":"228"},{"pid":"PA230","order":248,"title":"230"},{"pid":"PA231","order":249,"title":"231"},{"pid":"PA232","order":250,"title":"232"},{"pid":"PA233","order":251,"title":"233"},{"pid":"PA234","order":252,"title":"234"},{"pid":"PA235","order":253,"title":"235"},{"pid":"PA236","order":254,"title":"236"},{"pid":"PA237","order":255,"title":"237"},{"pid":"PA238","order":256,"title":"238"},{"pid":"PA239","order":257,"title":"239"},{"pid":"PA240","order":258,"title":"240"},{"pid":"PA242","order":260,"title":"242"},{"pid":"PA243","order":261,"title":"243"},{"pid":"PA245","order":263,"title":"245"},{"pid":"PA246","order":264,"title":"246"},{"pid":"PA247","order":265,"title":"247"},{"pid":"PA249","order":267,"title":"249"},{"pid":"PA250","order":268,"title":"250"},{"pid":"PA252","order":270,"title":"252"},{"pid":"PA253","order":271,"title":"253"},{"pid":"PA254","order":272,"title":"254"},{"pid":"PA256","order":274,"title":"256"},{"pid":"PA257","order":275,"title":"257"},{"pid":"PA259","order":277,"title":"259"},{"pid":"PA260","order":278,"title":"260"},{"pid":"PA263","order":281,"title":"263"},{"pid":"PA264","order":282,"title":"264"},{"pid":"PA265","order":283,"title":"265"},{"pid":"PA266","order":284,"title":"266"},{"pid":"PA268","order":286,"title":"268"},{"pid":"PA270","order":288,"title":"270"},{"pid":"PA271","order":289,"title":"271"},{"pid":"PA272","order":290,"title":"272"},{"pid":"PA273","order":291,"title":"273"},{"pid":"PA274","order":292,"title":"274"},{"pid":"PA278","order":296,"title":"278"},{"pid":"PA279","order":297,"title":"279"},{"pid":"PA282","order":300,"title":"282"},{"pid":"PA283","order":301,"title":"283"},{"pid":"PA285","order":303,"title":"285"},{"pid":"PA288","order":306,"title":"288"},{"pid":"PA289","order":307,"title":"289"},{"pid":"PA291","order":309,"title":"291"},{"pid":"PA295","order":313,"title":"295"},{"pid":"PA299","order":317,"title":"299"},{"pid":"PA301","order":319,"title":"301"},{"pid":"PA302","order":320,"title":"302"},{"pid":"PA303","order":321,"title":"303"},{"pid":"PA306","order":322,"title":"306"},{"pid":"PA307","order":323,"title":"307"},{"pid":"PA309","order":325,"title":"309"}],"prefix":"https://books.google.com.sg/books?id=FyToBwAAQBAJ\u0026lpg=PA165\u0026vq=length"},{"fullview":false,"page_width":575,"page_height":872,"font_height":0,"first_content_page":19,"disable_twopage":true,"initial_zoom_width_override":0,"show_print_pages_button":false,"title":"A Simple Non-Euclidean Geometry and Its Physical Basis","subtitle":"An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity","attribution":"By I.M. Yaglom","additional_info":{"[JsonBookInfo]":{"BuyLinks":[{"Seller":"Springer Shop","Url":"http://www.springer.com/978-1-4612-6135-3?utm_medium=referral\u0026utm_source=google_books\u0026utm_campaign=3_pier05_buy_print\u0026utm_content=en_08082017","TrackingUrl":"/url?client=ca-google-print\u0026format=googleprint\u0026num=0\u0026id=FyToBwAAQBAJ\u0026q=http://www.springer.com/978-1-4612-6135-3%3Futm_medium%3Dreferral%26utm_source%3Dgoogle_books%26utm_campaign%3D3_pier05_buy_print%26utm_content%3Den_08082017\u0026usg=AOvVaw2cy90lzm47WMSaDu_pDh8y","IsPublisher":true},{"Seller":"Amazon.com","Url":"http://www.amazon.com/gp/search?index=books\u0026linkCode=qs\u0026keywords=9781461261353","TrackingUrl":"/url?client=ca-google-print\u0026format=googleprint\u0026num=0\u0026id=FyToBwAAQBAJ\u0026q=http://www.amazon.com/gp/search%3Findex%3Dbooks%26linkCode%3Dqs%26keywords%3D9781461261353\u0026usg=AOvVaw0EFotrn3m5lqiGrV8plgFC"},{"Seller":"MPH","Url":"http://www.mphonline.com/books/nsearch.aspx?do=detail\u0026pcode=9781461261353","TrackingUrl":"/url?client=ca-google-print\u0026format=googleprint\u0026num=0\u0026id=FyToBwAAQBAJ\u0026q=http://www.mphonline.com/books/nsearch.aspx%3Fdo%3Ddetail%26pcode%3D9781461261353\u0026usg=AOvVaw2MukSnMva-sC4uHKipqMWB"},{"Seller":"Popular","Url":"https://www.popular.com.sg/catalogsearch/result/?q=9781461261353","TrackingUrl":"/url?client=ca-google-print\u0026format=googleprint\u0026num=0\u0026id=FyToBwAAQBAJ\u0026q=https://www.popular.com.sg/catalogsearch/result/%3Fq%3D9781461261353\u0026usg=AOvVaw1_5J_KN43d1isvAFP6u95r"}],"AboutUrl":"https://books.google.com.sg/books?id=FyToBwAAQBAJ","PreviewUrl":"https://books.google.com.sg/books?id=FyToBwAAQBAJ","allowed_syndication_flags":{"allow_disabling_chrome":true},"online_access":{"url":"https://play.google.com/store/books/details?id=FyToBwAAQBAJ\u0026rdid=book-FyToBwAAQBAJ\u0026rdot=1","price":"SGD 86.32","list_price":"SGD 123.32"},"TocLine":[{"Title":"Introduction ","Pid":"PA1","PgNum":"1","Order":19},{"Title":"Principle of duality coparallelograms and cotrapezoids ","Pid":"PA55","PgNum":"55","Order":73},{"Title":"Proofs of the principle of duality ","Pid":"PA64","PgNum":"64","Order":82},{"Title":"Circles and Cycles ","Pid":"PA77","PgNum":"77","Order":95},{"Title":"The circumcycle and incycle of a triangle ","Pid":"PA104","PgNum":"104","Order":122},{"Title":"Power of a point with respect to a circle or cycle inversion ","Pid":"PA117","PgNum":"117","Order":135},{"Title":"Conclusion ","Pid":"PA158","PgNum":"158","Order":176},{"Title":"Supplement A Nine plane geometries ","Pid":"PA214","PgNum":"214","Order":232},{"Title":"Supplement B Axiomatic characterization ","Pid":"PA242","PgNum":"242","Order":260},{"Title":"Analytic models of ","Pid":"PA258","PgNum":"258","Order":276},{"Title":"Bibliography ","Pid":"PA289","PgNum":"289","Order":307},{"Title":"Index of Names ","Pid":"PA302","PgNum":"302","Order":320}]}},"table_of_contents_page_id":"PR17","max_resolution_image_width":1280,"max_resolution_image_height":1941,"num_toc_pages":2,"quality_info":"We have no quality information about this book.","volume_id":"FyToBwAAQBAJ","permission_info":"Pages displayed by permission of \u003ca class=link_aux href=\"https://books.google.com.sg/url?id=FyToBwAAQBAJ\u0026pg=PA165\u0026q=http://www.springer.com/shop\u0026linkid=1\u0026usg=AOvVaw3qjA2jyxFK2Ek-6onqUbwf\u0026source=gbs_pub_info_r\"\u003eSpringer Science \u0026amp; Business Media\u003c/a\u003e","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},"publisher":"Springer Science \u0026 Business Media","publication_date":"2012.12.06","subject":"Mathematics","num_pages":307,"sample_url":"https://play.google.com/books/reader?id=FyToBwAAQBAJ\u0026source=gbs_vpt_hover","synposis":"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.","my_library_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fop%3Dlibrary\u0026hl=en","is_magazine":false,"is_public_domain":false,"last_page":{"pid":"PA309","order":325,"title":"309"}},{"enableUserFeedbackUI":true,"pseudocontinuous":true,"is_cobrand":false,"sign_in_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fid%3DFyToBwAAQBAJ%26q%3Dlength%26source%3Dgbs_word_cloud_r%26hl%3Den\u0026hl=en","isEntityPageViewport":false,"showViewportOnboarding":false,"showViewportPlainTextOnboarding":false},{"page":[{"pid":"PA165","flags":8,"order":183,"vq":"length"}]},null,{"number_of_results":80,"search_results":[{"page_id":"PR14","page_number":"xiv","snippet_text":"... \u003cb\u003elength\u003c/b\u003e is measured \u0026quot; the same way \u0026quot; but angles are not . By measuring angles as well as \u003cb\u003elengths\u003c/b\u003e in three different ways we are led to nine plane geometries . In Supplement A some of these geometries are realized as the intrinsic\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=FyToBwAAQBAJ\u0026pg=PR14\u0026vq=length"},{"page_id":"PR15","page_number":"xv","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the directed segment AA1 . ( 6 ) ( p . 39 ) . SAA , = y1y is the signed special \u003cb\u003elength\u003c/b\u003e of the directed special segment AA ,. ( 7 ) ( p . 41 ) . 8 , denotes the signed Galilean magnitude of the directed Galilean angle from 1 to\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=FyToBwAAQBAJ\u0026pg=PR15\u0026vq=length"},{"page_id":"PA14","page_number":"14","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of a segment , magnitude of an angle , area ? [ Hint : In Euclidean geometry , a trapezoid is a quadrilateral with a pair of parallel opposite sides , and a circle is the locus of points equidistant from a fixed point Q. Could\u0026nbsp;..."},{"page_id":"PA16","page_number":"16","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of this segment if its direction ( from 0 to A ) agrees with the positive direction on o , and the negative of that \u003cb\u003elength\u003c/b\u003e otherwise . [ We use the notations OA and OA for segments and directed segments , respectively . ] The\u0026nbsp;..."},{"page_id":"PA35","page_number":"35","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the vertical segments AA1 , BB1 , CC1 , ... between 1 and 11 . The restriction of the shear ( la ) to a special line m is a translation along m ( by an amount depending on m . ) . Hence the segments AA , d are mapped onto the\u0026nbsp;..."},{"page_id":"PA36","page_number":"36","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of parallel segments is meaningful in Galilean geometry . However , if AB || MN , then a shear may map ... \u003cb\u003elength\u003c/b\u003e e , and the perpendicular distance between them is e . Hence area ( S \u0026#39; ) = e2 = area ( S ) . It is plausible\u0026nbsp;..."},{"page_id":"PA38","page_number":"38","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the segment on the y - axis between I and 1 , and Vk2 + 1 = √tan2q + 1 = 1 / cosy , where o is the angle formed by each of the lines / and / , and the x - axis ( cf. p . 10 above ; in particular , Fig . 11a ) . We call the\u0026nbsp;..."},{"page_id":"PA40","page_number":"40","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the circular arc NN , cut off by 7 and 7 , from the unit circle S centered at Q ( cf. the Euclidean configuration 32a and the Galilean configuration 32b ; naturally , by the \u0026quot; \u003cb\u003elength\u003c/b\u003e of the arc \u0026quot; NN1 of the Galilean circle\u0026nbsp;..."},{"page_id":"PA41","page_number":"41","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of NN1 . If the equations of I and l1 are y = kx + s and y = k1x + s1 , and Q has coordinates ( x , y ) , then the equation of m is x = xo + 1 . Hence N and N , have coordinates ( x 。 + 1 , k ( x + 1 ) + s ) and ( x + 1 , k1 ( x\u0026nbsp;..."},{"page_id":"PA42","page_number":"42","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the directed segment MM , between 1 and l1 belonging to a special line ( the special line is arbitrary ; cf. Fig . 35 ) . This definition makes sense because the motions ( 1 ) map special lines onto special lines . If the\u0026nbsp;..."},{"page_id":"PA47","page_number":"47","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of the sides of the triangle , i.e. , the ( positive ) distances | dBc | = a , | dc | = b , | dAB | = c , and the letters A , B , C stand not only for the vertices of the triangle but also for the ( positive ) magnitudes of its\u0026nbsp;..."},{"page_id":"PA48","page_number":"48","snippet_text":"... \u003cb\u003elength\u003c/b\u003e dB of a segment AB by the same letters AB . ( Such notation will appear , for example , in some equations involving the \u003cb\u003elengths\u003c/b\u003e of the sides of a triangle . ) The \u003cb\u003elengths\u003c/b\u003e a , b , and c of the sides of a triangle in the Galilean\u0026nbsp;..."},{"page_id":"PA49","page_number":"49","snippet_text":"... Fig . 37 justify calling AP , BQ , and CR the altitudes , in Galilean geometry , of AABC . ] We denote the \u003cb\u003elengths\u003c/b\u003e of the Ул Ул The R C B A \u0026#39; ha C \u0026#39; 0 01 Figure 42a Figure 42b altitudes by ha , hò , and h¿ : ha. 4. The triangle 49."},{"page_id":"PA50","page_number":"50","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of B\u0026#39;C \u0026#39; and of the altitude ha coincide with their Euclidean \u003cb\u003elengths\u003c/b\u003e . Hence S \u0026#39; = a\u0026#39;h\u0026#39;a where S \u0026#39; is the area of △ A\u0026#39;B\u0026#39;C \u0026#39; ( note that S \u0026#39; is the Euclidean as well as the Galilean area of A\u0026#39;B\u0026#39;C \u0026#39; ; the area of a figure is\u0026nbsp;..."},{"page_id":"PA52","page_number":"52","snippet_text":"... \u003cb\u003elength\u003c/b\u003e a , and the congruent sides CA , CA1 of \u003cb\u003elength\u003c/b\u003e b . However , AB = a + b , while A1B = a − b , so triangles ABC and ABC are not congruent . Similarly , triangles ABC and A , BC in Figure 46b have a common side BC enclosed by\u0026nbsp;..."},{"page_id":"PA53","page_number":"53","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of ) segments by a fixed number k . An example of such a map is a ( positive ) central dilatation with center O , which we define as a transformation of the plane , taking every point A onto the point A \u0026#39; of the ray OA such that\u0026nbsp;..."},{"page_id":"PA54","page_number":"54","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of its diagonals , the angles between the diagonals and between opposite sides , the \u003cb\u003elengths\u003c/b\u003e of its midlines , the \u003cb\u003elength\u003c/b\u003e of the segment joining the midpoints of its diagonals , etc. ( b ) Find relations involv- ing the\u0026nbsp;..."},{"page_id":"PA59","page_number":"59","snippet_text":"... \u003cb\u003elength\u003c/b\u003e is half the sum of their \u003cb\u003elengths\u003c/b\u003e.13 We call a quadrilateral ABCD in the Galilean plane with sides AB = a , BC = b , CD = c , DA = d and parallel vertices B and D ( this , we recall , means that BD is a special line ; see Fig\u0026nbsp;..."},{"page_id":"PA64","page_number":"64","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e AB = c , BC = a , CA = 5 , then AW : WB = ɓ : ã , BU : UC = c : b , CV : VA = a : c , and AW BU CV Б с a ... \u003cb\u003elengths\u003c/b\u003e of two collinear segments is the same as the ratio of their Euclidean \u003cb\u003elengths\u003c/b\u003e . Since Ceva\u0026#39;s theorem only\u0026nbsp;..."},{"page_id":"PA71","page_number":"71","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e are the same : dAB = dcp or , in the case of \u0026quot; special vectors \u0026quot; whose carriers are special lines , AB = 8CD ( Fig . 62a ) . [ This rule is equivalent to the parallelogram law : If AB = CD and their carriers are distinct , then\u0026nbsp;..."},{"page_id":"PA80","page_number":"80","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of an arbitrary arc s = arc AB of S to twice the inscribed angle a subtended by s : S r = . 2α ( 3a ) If we wish to adapt the above definition of radius to a cycle Z , then we must show that for a given cycle Z and any arc AB of\u0026nbsp;..."},{"page_id":"PA81","page_number":"81","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of an arc AB of a curve г is the limit of the \u003cb\u003elength\u003c/b\u003e AA , + A1A2 + + An - 1A + AnВ of a polygonal line AA A2 ... AB ( Fig . 71a ) inscribed in Ã as the longest link in the polygonal line tends to zero . Since the Galilean\u0026nbsp;..."},{"page_id":"PA85","page_number":"85","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of arc AB is s and the angle between the tangents AT and BT , at A and B is o , then we define the average curvature Pav of T on the arc AB as Pav S where s = arc AB and y = LTAT \u0026#39; ( AT \u0026#39; || BT1 ) ( 6 ) ( Fig . 76 ) .9\u0026nbsp;..."},{"page_id":"PA86","page_number":"86","snippet_text":"... \u003cb\u003elength\u003c/b\u003e s and angle o ( in particular , the \u003cb\u003elength\u003c/b\u003e s of the arc AB is equal to the \u003cb\u003elength\u003c/b\u003e dB of the chord AB ) . In the special case where I is a cycle Z ( see Fig . 77b ) , the average curvature Pay on the arc AB is q / s , where s\u0026nbsp;..."},{"page_id":"PA87","page_number":"87","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the arc AA1 ; our problem is to compute p = lim11 → ( △ ❤ / As ) . = In the Galilean case , this is quite easy . Let A1 = ( x + △ x , y + Ay ) . Since the Galilean angle between two lines is equal to the difference of the\u0026nbsp;..."},{"page_id":"PA89","page_number":"89","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the arc QA of I measured from some arbitrary but fixed \u0026quot; origin \u0026quot; Q on T to A. This statement is the so - called fundamental theorem on plane curves . Equation ( 9 ) is called the natural equation of the curve г. 13 It is\u0026nbsp;..."},{"page_id":"PA90","page_number":"90","snippet_text":"... \u003cb\u003elength\u003c/b\u003e , and the radius of curvature r = 1 / p = limo ( As / Aq ) involves the ratio of arc \u003cb\u003elength\u003c/b\u003e angle , it follows that every transformation of the Euclidean plane which preserves angles and multiplies distances by a factor k\u0026nbsp;..."},{"page_id":"PA91","page_number":"91","snippet_text":"... \u003cb\u003elength\u003c/b\u003e . In this sense , no point on a line is different from another . However , this property of a line does 15Today these definitions are viewed as the least fortunate aspect of the otherwise remarkable work of Euclid . The author of\u0026nbsp;..."},{"page_id":"PA92","page_number":"92","snippet_text":"... \u003cb\u003elength\u003c/b\u003e are congruent , so that nothing distinguishes one of its points from another . The homogeneity of a line and of a circle manifests itself in the existence of a \u0026quot; linear glide \u0026quot; and a \u0026quot; circular glide , \u0026quot; defined respectively as a\u0026nbsp;..."},{"page_id":"PA95","page_number":"95","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of segments unchanged . Hence dAp = dA\u0026#39;p = dA\u0026#39;p \u0026#39; = dр\u0026#39;B \u0026#39; = dрB . \u0026quot; 18 We give another example of a similar kind . It is well known that the midpoints of a family of parallel chords of a circle S lie on a line — a diameter d of\u0026nbsp;..."},{"page_id":"PA96","page_number":"96","snippet_text":"... \u003cb\u003elength\u003c/b\u003e decreases . Then , in the limit , the points A , B and the midpoint C ( on d ) of the chord AB coincide . We conclude that the tangent t to a cycle Z at the end of a diameter d of Z is parallel to the chords bisected by d\u0026nbsp;..."},{"page_id":"PA101","page_number":"101","snippet_text":"... \u003cb\u003elength\u003c/b\u003e AB to be always positive , i.e. , we work with nondirected distances , then the condition AB = const ... \u003cb\u003elength\u003c/b\u003e AB ( = d ) of the segment of the tangent m to the cycle Z contained between the sides of the angle aLb is equal\u0026nbsp;..."},{"page_id":"PA104","page_number":"104","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of the corresponding segments . It is clear that there is a unique cycle through the points A , B , C ; it can be defined as , say , the set of points from each of which the segment AB is seen at a Galilean angle equal to C\u0026nbsp;..."},{"page_id":"PA106","page_number":"106","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of its sides are duals . Finally , the_radius r = lim ( As / Aq ) of a cycle and its curvature p = lim ( Aq / As ) are Δφ = 0 duals . It follows that the dual of the formulas a b с = 2R A B C = = for the radius of the\u0026nbsp;..."},{"page_id":"PA115","page_number":"115","snippet_text":"... \u003cb\u003elength\u003c/b\u003e x of the segment C1R . Since x - 2 = C1R - C , B = BR and we have and thus i.e. , x + 2 = C1R + AC¡ = AR , a2 — ( x − £ ) 2 = b2 — ( x + ; ) 2 ( = CR2 ) , - - - ( x + 2 ) 2 - ( x − 2 ) 2 = b2 — a2 , 2cx = b2 - a2 . Consequently\u0026nbsp;..."},{"page_id":"PA118","page_number":"118","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of one of the segments MA 。 and MB 。 is d + r and the \u003cb\u003elength\u003c/b\u003e of the other segment is ❘d - r ] . The sign of the product MA MB is the same as the sign of the difference d - r . Thus in all cases the power of the point M with\u0026nbsp;..."},{"page_id":"PA120","page_number":"120","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e x1 - x 。 and x2 - xo , and therefore the power of M with respect to S is ( x1 − xo ) ( x2 − xo ) = ( xo - x1 ) ( xo − x2 ) . Since , in this case , the equation of S is or equivalently , - ( x − x1 ) ( x − x2 ) = 0 , x2 +\u0026nbsp;..."},{"page_id":"PA130","page_number":"130","snippet_text":"... \u003cb\u003elength\u003c/b\u003e . In other words , the special line e through M is equidistant from the special line d and from the special line ƒ through K. In turn , this implies that if N is the intersection point of e and KL , then PA \u0026#39; = MN = AA \u0026#39; ( since\u0026nbsp;..."},{"page_id":"PA135","page_number":"135","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the segment A\u0026#39;N \u0026#39; = AN = PR ) and use the fact ( implied by the definition of angle in Galilean geometry ) that and N\u0026#39;Y = LN\u0026#39;A\u0026#39;M \u0026#39; , A\u0026#39;N \u0026#39; y XN = L MAN AN UR = LQPR , PR M M 12 Figure 123 then we obtain the equalities LN\u0026#39;A\u0026#39;M\u0026nbsp;..."},{"page_id":"PA136","page_number":"136","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the arcs MN , M\u0026#39;N \u0026#39; , and QR , i.e. , the \u003cb\u003elength\u003c/b\u003e of MN ) . This fact the last equation above yield the relation ( 38 ) . To prove that an inversion of the second kind is a conformal , i.e. , angle - pre- serving , mapping , we\u0026nbsp;..."},{"page_id":"PA138","page_number":"138","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of the sides of triangle ABC ( cf. p . 114 ) , and that in the Galilean case we also have FC1 = | b - al / 2 ( cf. p . 114 ) . We now apply an inversion ( of the first kind ) with center C1 and coefficient k = ( C1F ) 2 = ( C1F1 )\u0026nbsp;..."},{"page_id":"PA140","page_number":"140","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e a , b , c of our triangle are positive . Also , the computations below reflect the situation in Figures 127a and 127b , where b \u0026gt; 2a . As a result , all the expressions below ( \u003cb\u003elengths\u003c/b\u003e of various segments ) are positive\u0026nbsp;..."},{"page_id":"PA146","page_number":"146","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the tangent K1A \u0026#39; from K1 to σ is independent of the choice of A on s . But then K1 must be the same for all A on s ; indeed , if K1 moves along the line QK , then the \u003cb\u003elength\u003c/b\u003e of the tangent from K , to σ changes regardless of\u0026nbsp;..."},{"page_id":"PA162","page_number":"162","snippet_text":"... \u003cb\u003elength\u003c/b\u003e and time so that the speed of light c is 1. This means that if the unit of time is one second , then the unit of \u003cb\u003elength\u003c/b\u003e is about 300,000 kilometers ( since the approximate speed of light is 300,000 km / sec ) . The line in Figure\u0026nbsp;..."},{"page_id":"PA163","page_number":"163","snippet_text":"... \u003cb\u003elength\u003c/b\u003e , then the world line of a ray of light is given by the same line 7. This implies that the axes Ot \u0026#39; and Ox \u0026#39; must form equal ( Euclidean ) angles with the axes Ot and Ox , i.e. , ≤ tot \u0026#39; = Lx\u0026#39;Ox = q . The latter conclusion\u0026nbsp;..."},{"page_id":"PA164","page_number":"164","snippet_text":"... \u003cb\u003elength\u003c/b\u003e OX 。 on Ox . The coordinate x \u0026#39; is equal to the new \u003cb\u003elength\u003c/b\u003e d \u0026#39; ( M\u0026#39;S ) of the segment M\u0026#39;S measured in terms of the unit of \u003cb\u003elength\u003c/b\u003e OX \u0026#39; on Ox \u0026#39; . If we recall that the equation of the line Ot \u0026#39; in the coordinate system { x , t }\u0026nbsp;..."},{"page_id":"PA165","page_number":"165","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of our rod in the reference frame { x \u0026#39; , t \u0026#39; } relative to which it moves with speed v ( with velocity —v if we take into consideration the direction of motion ) . To determine the new \u003cb\u003elength\u003c/b\u003e of the rod we must determine the x\u0026nbsp;..."},{"page_id":"PA166","page_number":"166","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of an object moving with speed v is known as the Fitzgerald contraction . - We now invert Eqs . ( 2a ) and ... \u003cb\u003elength\u003c/b\u003e of a rod R at rest with respect to { x \u0026#39; , t \u0026#39; } is multiplied by k \u0026#39; ( 1 - v2 ) ( 3 \u0026#39; ) with respect to\u0026nbsp;..."},{"page_id":"PA168","page_number":"168","snippet_text":"... \u003cb\u003elength\u003c/b\u003e T for H , has the greater \u003cb\u003elength\u003c/b\u003e T = 1 V1 - v2 ( 6b ) The seemingly paradoxical relation ( 6b ) can be deduced in another way . Consider a thin rod AB of \u003cb\u003elength\u003c/b\u003e 7 moving with velocity v in the direction of the line o\u0026nbsp;..."},{"page_id":"PA169","page_number":"169","snippet_text":"... \u003cb\u003elength\u003c/b\u003e is the same for both observers ; cf. ( 10 ) , p . 173. ] Since the speed of light has the same value 1 for the observers H and H \u0026#39; , the time ( as measured by the clock of H \u0026#39; ) taken by the ray of light to traverse the path AB\u0026nbsp;..."},{"page_id":"PA179","page_number":"179","snippet_text":"... \u003cb\u003elength\u003c/b\u003e and area . We shall find it convenient to put dAB = √2S ( AKBL ) . It is clear that if C is a point of the segment AB , then dAB = dAc + dCB ( 14 ) ( for the areas of the similar rectangles AKBL , AK1CL1 , and CK1⁄2BL1⁄2 in Fig\u0026nbsp;..."},{"page_id":"PA182","page_number":"182","snippet_text":"... Q , then it is natural to define the angle d , between I and I , as the Minkowskian \u003cb\u003elength\u003c/b\u003e of the arc NN , between 1 and 7 , belonging to the \u0026quot; unit circle \u0026quot; S with center Figure 158 S X y S 区 Q N y S N N Q. 182 Conclusion 158."},{"page_id":"PA183","page_number":"183","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the arc NN1 can be defined as the limit of the \u003cb\u003elengths\u003c/b\u003e of polygonal lines NM , M2 ... M , N1 inscribed in NN , whose longest edges approach zero . In this connection , it is of interest to point out that all edges of a\u0026nbsp;..."},{"page_id":"PA188","page_number":"188","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the segment AN cut off by the variable radius OM on the tangent AT to the hyperbola S at its vertex A. P Figure 166a shows that if o increases indefinitely , then cosho and sinh increae indefinitely and tanho tends to 1. Also\u0026nbsp;..."},{"page_id":"PA190","page_number":"190","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of the segments BC , CA , and AB can be compared . ( The \u003cb\u003elengths\u003c/b\u003e of these segments will be denoted by the same letters a , b , c as the segments themselves . ) Then if A , B , and C are not collinear , the largest side of the\u0026nbsp;..."},{"page_id":"PA192","page_number":"192","snippet_text":"... \u003cb\u003elength\u003c/b\u003e dap of the altitude AP of ABC dropped from A to the line BC ( Fig . 171 ) . The theorem on the concurrence of the medians of a triangle is valid in Minkowskian geometry , and is proved just as in Euclidean geometry ( cf. Sec . 4\u0026nbsp;..."},{"page_id":"PA195","page_number":"195","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e of segments of the second kind as imaginary . The sign of the product ( 28 ) is the same as in Euclidean or Galilean geometry . ] We shall refer to this signed product as the power of the point M with respect to the circle S. It\u0026nbsp;..."},{"page_id":"PA205","page_number":"205","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of a segment is expressed solely in terms of its projection to the x - axis . We now replace the segment AA , by its projection AB parallel to the y - axis . Similarly , we replace AÃ , by the vertical projection AB . Thus the\u0026nbsp;..."},{"page_id":"PA208","page_number":"208","snippet_text":"... \u003cb\u003elength\u003c/b\u003e , which in conjunction with the limiting process c → ∞o reduces Euclidean ( or Minkowskian ) geometry to Galilean geometry , is accompanied by a decrease in all angles . Indeed , in order to obtain finite expressions for the\u0026nbsp;..."},{"page_id":"PA211","page_number":"211","snippet_text":"... \u003cb\u003elength\u003c/b\u003e which is c times larger than the unit employed in Section 11. With this choice of unit of \u003cb\u003elength\u003c/b\u003e the following replacements must be made in all the formulas in Section 11 : t → t , x , 음 ( 49 ) [ see ( 38 ) , this section ]\u0026nbsp;..."},{"page_id":"PA214","page_number":"214","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of segments . According to this rule we choose a unit of \u003cb\u003elength\u003c/b\u003e OE on o and define the distance between points A and B on o by the formula3 d = AB OE ( 1 ) This distance is called parabolic ( hence the letter P in d ) . If C is a\u0026nbsp;..."},{"page_id":"PA215","page_number":"215","snippet_text":"... ( \u003cb\u003elengths\u003c/b\u003e of segments ) . It is convenient to assume that the segments on the right - hand side of the cross ratio ( A , B ; I , J ) = AI / AJ BI / BJ ( 2 ) of A , B , I , J are directed . In view of the fact that a real - valued\u0026nbsp;..."},{"page_id":"PA217","page_number":"217","snippet_text":"... \u003cb\u003elength\u003c/b\u003e that give rise to three geometries on a line o , so , too , are there three ways of measuring angles that give rise to three geometries in a pencil of lines with center O. The usual measure of angles in the pencil with center O\u0026nbsp;..."},{"page_id":"PA218","page_number":"218","snippet_text":"... \u003cb\u003elength\u003c/b\u003e on a line and one of three ways of measuring angles in a pencil with center O. This gives nine ways of measuring \u003cb\u003elengths\u003c/b\u003e and angles and thus the nine plane geometries listed in Table I. We are familiar with the geometries of\u0026nbsp;..."},{"page_id":"PA219","page_number":"219","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of arc AM ( ❤ and arc AM appear in Fig . 166a ) . Now x + Vx2 − 1 = x + y = = 1+ ( y / x ) 1+ ( y / x ) 1 / x V1- ( x2 / x2 ) 1+ ( y / x ) 1- ( y / x ) 1 + m = - m where m is the slope of OM . But then SOA , OM = Minkowskian\u0026nbsp;..."},{"page_id":"PA220","page_number":"220","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the arc AB . This measure of \u003cb\u003elength\u003c/b\u003e is clearly elliptic . By the angle Sab between the lines a and b of the elliptic 7See footnote 4 of the Preface ; cf. [ 79 ] . 0 Σ A B Σ A Figure 189a Σ \u0026#39;. 220 Supplement A. Nine plane\u0026nbsp;..."},{"page_id":"PA222","page_number":"222","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of a polygonal line is the sum of the \u003cb\u003elengths\u003c/b\u003e of its segments , and the \u003cb\u003elength\u003c/b\u003e of a segment with endpoints ( x , y , z ) and ( x1 , y1,71 ) is given by the formula - d2 = | ( x , −x ) 2 + ( y1 − y ) 2 — ( z1 − 2 ) 2 . In\u0026nbsp;..."},{"page_id":"PA223","page_number":"223","snippet_text":"... \u003cb\u003elength\u003c/b\u003e.13 A pencil of lines with center Q of our geometry is determined by the planes of the pencil with axis OQ = q which intersect the cone K given by x2 + y2 - z2 = 0 ( 5a ) ( see Figs . 191a and 191b ) . The structure of such a\u0026nbsp;..."},{"page_id":"PA224","page_number":"224","snippet_text":"... \u003cb\u003elength\u003c/b\u003e and angle is the doubly hyperbolic plane , and its motions are again rotations of the \u0026quot; sphere \u0026quot; Σ2 ( regarded as a set of pairs of antipodal points ) of Minkowskian space about its center O. We note that in hyperbolic and in\u0026nbsp;..."},{"page_id":"PA226","page_number":"226","snippet_text":"... \u003cb\u003elengths\u003c/b\u003e Measure of angles E Р E = = Р = = H = = A sinha second , then Figure 193 X Σ ( xí -. cos a = cos b cos c + sin b sin c cos A sin A sin B sin C sin a sin b sin c cos A = cos B cos C + sin B sin Ccos a A a = b + c B C sin a sin b\u0026nbsp;..."},{"page_id":"PA230","page_number":"230","snippet_text":"... sections of o \u0026#39; and of by planes passing through O ( 0,0,0 ) ; the distance between points A and B is the X Z 110 Figure 194a Figure 194b 316 16 Figure 195a 18 Figure 195b \u003cb\u003elength\u003c/b\u003e of the. 230 Supplement A. Nine plane geometries."},{"page_id":"PA231","page_number":"231","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the segment AB of the \u0026quot; line \u0026quot; joining A and B , measured in terms of semi - Euclidean or semi - Minkowskian geometry , and the angle between lines a and b is the angle between the corresponding planes measured in terms of the\u0026nbsp;..."},{"page_id":"PA240","page_number":"240","snippet_text":"... \u003cb\u003elength\u003c/b\u003e of the segment AB cut off by these planes on a fixed line q ( not parallel to the edge of the angle ) is called parabolic . Finally , the measure defined by the formula Sapk log L ( α , i ) / 4 ( α , ε ) Z ( B , ) / Z ( B , E )\u0026nbsp;..."},{"page_id":"PA247","page_number":"247","snippet_text":"... \u003cb\u003elength\u003c/b\u003e | a | of the vector a are defined by || a || = x2 + y2 , | a | = √x2 + y • ( 20❜e ) ( 20 ′′ e ) Finally , the ( undefined ) concept of a point is linked to the concept of a vector by a relation which associates to each\u0026nbsp;..."},{"page_id":"PA250","page_number":"250","snippet_text":"... , m } is an orthonormal basis of the Minkowskian plane , then the analogue of ( 20e ) is ab = xx1 — yy 1 . ( 20m ) If , as before , we denote the norm a2 of a vector a by || a || and its \u003cb\u003elength\u003c/b\u003e Va2 by lal , then in place of ( 20."},{"page_id":"PA252","page_number":"252","snippet_text":"... \u003cb\u003elength\u003c/b\u003e p1 = √ ( p2 ) , for such vectors ( in the coordinate system adopted above we can put [ p ( 0 , y ) | 1 = y ) . This special \u003cb\u003elength\u003c/b\u003e allows us to introduce a measure on special lines which is closely related to the measure of\u0026nbsp;..."},{"page_id":"PA256","page_number":"256","snippet_text":"... \u003cb\u003elength\u003c/b\u003e | a | of a vector a in Minkowskian three - space are given by the formulas || a || = x2 + y2 — z2 and | a | = √x2 + y2 − z2 ; ( 202m ) here it is necessary to distinguish spacelike , isotropic , and timelike vectors a with\u0026nbsp;..."},{"page_id":"PA271","page_number":"271","snippet_text":"... \u003cb\u003elength\u003c/b\u003e r = OM = dom = x and the Galilean angle y = ≤ xOM = Sox , Oм are called the polar coordinates of M. We associate to M the dual number z = x + \u0026amp; y = r ( 1 + ɛy ) = ( 30 \u0026#39; ) ( Fig . 214a ) . Now let M be a point in the Minkowskian\u0026nbsp;..."},{"page_id":"PA302","page_number":"302","snippet_text":"... , 13 HILBERT , David , 1862-1943 , 243 STUDY , Eduard , 1862-1930 , 265 KLEIN , Felix , 1849–1925 , vii WEYL , Hermann , 1885-1955 , vii Index of Subjects Acceleration 17 Arc \u003cb\u003elength\u003c/b\u003e Galilean 81 Minkowskian Index of Names.","page_url":"https://books.google.com.sg/books?id=FyToBwAAQBAJ\u0026pg=PA302\u0026vq=length"},{"page_id":"PA303","page_number":"303","snippet_text":"... \u003cb\u003elength\u003c/b\u003e Galilean 81 Minkowskian 183 Angle between two curves 128 elliptic measure of 217 Galilean 41 hyperbolic measure of 217 Minkowskian 182 parabolic measure of 217 Cayley - Klein geometries list of 218 plane analytic models of 258\u0026nbsp;...","page_url":"https://books.google.com.sg/books?id=FyToBwAAQBAJ\u0026pg=PA303\u0026vq=length"}],"search_query_escaped":"length"},{});</script></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>