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<!DOCTYPE html><html><head><title>Geometry: Euclid and Beyond - Robin Hartshorne - 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/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta property="og:url" content="https://books.google.com/books/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C"/><meta name="title" content="Geometry: Euclid and Beyond"/><meta name="description" content="In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks."/><meta property="og:title" content="Geometry: Euclid and Beyond"/><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/content?id=EJCSL9S6la0C&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE73WeVK6M0yCVl8isnKFRsJc-M2gE4t9Y1sYpZSsILlv1YExDbDQA1wnqoE00fN1IGuLQA7L7urStOD328VrpsSpOZZaYOnqDAc0YeZej1fpYxvjzCUlDFa1zeHzTQO_V_n5EvP9"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&printsec=frontcover&img=1&zoom=1&edge=curl&imgtk=AFLRE73WeVK6M0yCVl8isnKFRsJc-M2gE4t9Y1sYpZSsILlv1YExDbDQA1wnqoE00fN1IGuLQA7L7urStOD328VrpsSpOZZaYOnqDAc0YeZej1fpYxvjzCUlDFa1zeHzTQO_V_n5EvP9"/><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%3DEJCSL9S6la0C%26redir_esc%3Dy%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/about/Geometry_Euclid_and_Beyond.html?id=EJCSL9S6la0C&redir_esc=y&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.</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="redir_esc" value="y"><input type=hidden name="id" value="EJCSL9S6la0C"><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=EJCSL9S6la0C&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=EJCSL9S6la0C&pg=PR9&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&pg=PR9&img=1&zoom=1&sig=ACfU3U2pYkV8eovtnQAoiY9UoHVNPYMEIA" 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=EJCSL9S6la0C&pg=PR9&source=gbs_selected_pages&cad=1" >Table of Contents</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&pg=PA503&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&pg=PA503&img=1&zoom=1&sig=ACfU3U3MTWbRP8qJ5FWAJvdEWDesKrpVNA" alt="Index" title="Index" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=EJCSL9S6la0C&pg=PA503&source=gbs_selected_pages&cad=1" >Index</a></div><div class="selectedpagesthumbnail"><a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&pg=PA496&source=gbs_selected_pages&cad=1" ><img src="https://books.google.com.sg/books/content?id=EJCSL9S6la0C&pg=PA496&img=1&zoom=1&sig=ACfU3U2y0WZJH-rPv4gExoejKKlsf1gHEw" alt="References" title="References" height="160" border="1"></a><br/><a class="primary" href="https://books.google.com.sg/books?id=EJCSL9S6la0C&pg=PA496&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"><span style="white-space:nowrap"><span dir=ltr>Introduction </span></span></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>Euclids Geometry </span></span></div></td><td class="toc_number" align=right>7</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>Hilberts Axioms </span></span></div></td><td class="toc_number" align=right>65</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>Geometry over Fields </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>Congruence of Segments and Angles </span></span></div></td><td class="toc_number" align=right>140</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>Rigid Motions and SAS </span></span></div></td><td class="toc_number" align=right>148</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>NonArchimedean Geometry </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><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Segment Arithmetic </span></span></div></td><td class="toc_number" align=right>165</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>Cubic and Quartic Equations </span></span></div></td><td class="toc_number" align=right>270</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>Finite Field Extensions </span></span></div></td><td class="toc_number" align=right>280</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>NonEuclidean Geometry </span></span></div></td><td class="toc_number" align=right>295</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>History of the Parallel Postulate </span></span></div></td><td class="toc_number" align=right>296</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>Neutral Geometry </span></span></div></td><td class="toc_number" align=right>304</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>Archimedean Neutral Geometry </span></span></div></td><td class="toc_number" align=right>319</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>NonEuclidean Area </span></span></div></td><td class="toc_number" align=right>326</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>Circular Inversion </span></span></div></td><td class="toc_number" align=right>334</td></tr><tr><td class="toc_border"> </td><td class="toc_border"></td></tr></table></div></div><br style="clear:both;"/></div><span onclick="_OC_setListSectionVisible('toc_h', 1)" class=morelesslink id=toc_hc0 style="display:none"><br>More</span><div id=toc_hd1><div class="first_toc_column"><div class="first_toc_pad"><table><tr><td class="toc_entry"><div class="toc_entry"><span style="white-space:nowrap"><span dir=ltr>Similar Triangles </span></span></div></td><td class="toc_number" align=right>175</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>Introduction of Coordinates </span></span></div></td><td class="toc_number" align=right>186</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>Area </span></span></div></td><td class="toc_number" align=right>195</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>Area in Euclids Geometry </span></span></div></td><td class="toc_number" align=right>196</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>Measure of Area Functions </span></span></div></td><td class="toc_number" align=right>205</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>Dissection </span></span></div></td><td class="toc_number" align=right>212</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>Quadratura Circuli </span></span></div></td><td class="toc_number" align=right>221</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>Euclids Theory of Volume </span></span></div></td><td class="toc_number" align=right>226</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>Hilberts Third Problem </span></span></div></td><td class="toc_number" align=right>231</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>Construction Problems and Field Extensions </span></span></div></td><td class="toc_number" align=right>241</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>Three Famous Problems </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>The Regular 17Sided Polygon </span></span></div></td><td class="toc_number" align=right>250</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>Constructions with Compass and Marked Ruler </span></span></div></td><td class="toc_number" align=right>259</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>Circles Determined by Three Conditions </span></span></div></td><td class="toc_number" align=right>346</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 Poincar茅 Model </span></span></div></td><td class="toc_number" align=right>355</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>Hyperbolic Geometry </span></span></div></td><td class="toc_number" align=right>373</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>Hilberts Arithmetic of Ends </span></span></div></td><td class="toc_number" align=right>388</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>Hyperbolic Trigonometry </span></span></div></td><td class="toc_number" align=right>403</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>Characterization of Hilbert Planes </span></span></div></td><td class="toc_number" align=right>415</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>Polyhedra </span></span></div></td><td class="toc_number" align=right>435</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 Five Regular Solids </span></span></div></td><td class="toc_number" align=right>436</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>Eulers and Cauchys Theorems </span></span></div></td><td class="toc_number" align=right>448</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>Brief Euclid </span></span></div></td><td class="toc_number" align=right>481</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>References </span></span></div></td><td class="toc_number" align=right>495</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=EJCSL9S6la0C&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;"/><span onclick="_OC_setListSectionVisible('toc_h', 0)" class=morelesslink id=toc_hc1 style="display:none"><br>Less</span></div><script type="text/javascript">if (window['_OC_setListSectionVisible']) {_OC_setListSectionVisible('toc_h', 0);}</script></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:ISBN0387986502&id=EJCSL9S6la0C'>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=C5fSBwAAQBAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=C5fSBwAAQBAJ&printsec=frontcover&img=1&zoom=5&edge=curl" border="0" height="80"></a></td><td valign=top><div class=resbdy><a class="primary cresbdy" href="https://books.google.com.sg/books?id=C5fSBwAAQBAJ&printsec=frontcover&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Geometry: Euclid and Beyond</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:%22Robin+Hartshorne%22" class="secondary"><span dir=ltr>Robin Hartshorne</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2013</span><br/></span></div></td><td 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2010</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=bZkTswEACAAJ&source=gbs_book_other_versions_r&cad=3" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=bZkTswEACAAJ&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=bZkTswEACAAJ&source=gbs_book_other_versions_r&cad=3"><span dir=ltr>Geometry: Euclid and Beyond</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:%22Robin+Hartshorne%22" class="secondary"><span dir=ltr>Robin Hartshorne</span></a></span><br/><span><span style="color:#999">No preview 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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=EJCSL9S6la0C&q=algebraic&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>algebraic</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=altitudes&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>altitudes</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=angle+bisectors&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>angle bisectors</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Archimedes&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Archimedes</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Cartesian+plane&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Cartesian plane</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=circle+with+center&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>circle with center</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=circular+inversion&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>circular inversion</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=congruent&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>congruent</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=convex&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>convex</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=cross-ratio&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>cross-ratio</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=cube&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>cube</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=define&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>define</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=definition&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>definition</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=dihedral+angles&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>dihedral angles</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=dissection&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>dissection</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=draw&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>draw</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=equal+content&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>equal content</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=equation&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>equation</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=equidecomposable&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>equidecomposable</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=equilateral+triangle&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>equilateral triangle</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=equivalent&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>equivalent</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Euclid%27s+Elements&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Euclid's Elements</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Euclidean+plane&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Euclidean plane</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&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=EJCSL9S6la0C&q=Exercise&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>Exercise</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=exists&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>exists</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=field+extension&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>field extension</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=field+F&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>field F</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=figure&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>figure</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=finite+number&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>finite number</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=follows&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>follows</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Galois+group&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Galois group</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=given+line&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>given line</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&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=EJCSL9S6la0C&q=Hilbert+plane&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>Hilbert plane</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Hilbert%27s+axioms&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>Hilbert's axioms</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=hyperbolic+plane&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>hyperbolic plane</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=icosahedron&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>icosahedron</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=inscribed&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>inscribed</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=intersection&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>intersection</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=isomorphic&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>isomorphic</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=isosceles&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>isosceles</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=lemma&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>lemma</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Let+ABC&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>Let ABC</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=limiting+parallel&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>limiting parallel</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=line+segments&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>line segments</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=marked+ruler&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>marked ruler</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=meet&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>meet</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=midpoint&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>midpoint</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&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=EJCSL9S6la0C&q=notion&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>notion</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=ordered+field&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>ordered field</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=parallel+axiom&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>parallel axiom</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=parallel+postulate&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>parallel postulate</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=perpendicular&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>perpendicular</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Poincar%C3%A9+model&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>Poincar茅 model</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=polygon&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>polygon</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=polyhedra&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>polyhedra</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=polyhedron&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>polyhedron</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=polynomial&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>polynomial</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=problem&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>problem</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=proof&source=gbs_word_cloud_r&cad=4" class="cloud0"><span dir=ltr>proof</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Proposition&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>Proposition</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=prove&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>prove</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&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=EJCSL9S6la0C&q=real+Cartesian+plane&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>real Cartesian plane</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=real+numbers&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>real numbers</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=rectangle&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>rectangle</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=result&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>result</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=right+angles&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>right angles</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=right+triangle&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>right triangle</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=rigid+motion&source=gbs_word_cloud_r&cad=4" class="cloud1"><span dir=ltr>rigid motion</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=rotation&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>rotation</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=ruler+and+compass&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>ruler and compass</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Saccheri+quadrilateral&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>Saccheri quadrilateral</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=satisfying&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>satisfying</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=Section&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>Section</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=sides&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>sides</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=splitting+field&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>splitting field</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=square+roots&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>square roots</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=steps&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>steps</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=subgroup&source=gbs_word_cloud_r&cad=4" class="cloud7"><span dir=ltr>subgroup</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=suppose&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>suppose</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=symmetry&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>symmetry</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=tangent&source=gbs_word_cloud_r&cad=4" class="cloud4"><span dir=ltr>tangent</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=tetrahedron&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>tetrahedron</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=theorem&source=gbs_word_cloud_r&cad=4" class="cloud2"><span dir=ltr>theorem</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=theory+of+area&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>theory of area</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=triangle+ABC&source=gbs_word_cloud_r&cad=4" class="cloud5"><span dir=ltr>triangle ABC</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=unique&source=gbs_word_cloud_r&cad=4" class="cloud8"><span dir=ltr>unique</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=vertex&source=gbs_word_cloud_r&cad=4" class="cloud3"><span dir=ltr>vertex</span></a> <a href="https://books.google.com.sg/books?id=EJCSL9S6la0C&q=vertices&source=gbs_word_cloud_r&cad=4" class="cloud6"><span dir=ltr>vertices</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>Geometry: Euclid and Beyond</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Undergraduate+Texts+in+Mathematics%22&source=gbs_metadata_r&cad=5"><i><span dir=ltr>Undergraduate Texts in Mathematics</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:%22Robin+Hartshorne%22&source=gbs_metadata_r&cad=5"><span dir=ltr>Robin Hartshorne</span></a></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, 2005</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0387986502, 9780387986500</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>528 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>&nbsp</td><td>&nbsp</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/Geometry_Euclid_and_Beyond.bibtex?id=EJCSL9S6la0C&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Geometry_Euclid_and_Beyond.enw?id=EJCSL9S6la0C&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Geometry_Euclid_and_Beyond.ris?id=EJCSL9S6la0C&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%3DEJCSL9S6la0C%26redir_esc%3Dy%26hl%3Den\u0026hl=en"}, 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