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<!DOCTYPE html><html><head><title>Elementary Mathematics from an Advanced Standpoint: Geometry - Felix Klein - Google Books</title><link rel="stylesheet" href="/books/css/_ff77d0f0508c7e0bad470e7d6b7f28fa/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_ff77d0f0508c7e0bad470e7d6b7f28fa__en.js"></script><link rel="canonical" href="https://books.google.com/books/about/Elementary_Mathematics_from_an_Advanced.html?id=fj-ryrSBuxAC"/><meta property="og:url" content="https://books.google.com/books/about/Elementary_Mathematics_from_an_Advanced.html?id=fj-ryrSBuxAC"/><meta name="title" content="Elementary Mathematics from an Advanced Standpoint"/><meta name="description" content="&quot;Nothing comparable to it.&quot; — Mathematics Teacher. This comprehensive three-part treatment begins with a consideration of the simplest geometric manifolds: line-segment, area, and volume as relative magnitudes; the Grassmann determinant principle for the plane and the Grassmann principle for space; classification of the elementary configurations of space according to their behavior under transformation of rectangular coordinates; and derivative manifolds. The second section, on geometric transformations, examines affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. The text concludes with a systematic discussion of geometry and its foundations. 1939 edition. 141 figures."/><meta property="og:title" content="Elementary Mathematics from an Advanced Standpoint"/><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=fj-ryrSBuxAC&printsec=frontcover&img=1&zoom=1&imgtk=AFLRE72Gjz-AxLI8_CkckQvncNeT6uzKCydXoWEXnpBBd49xF0zzoK2ypKsqTXBAHDqZKkHDrFjdaKboOxj2cWtAAUrB2849sj975bVvjz-oPZ8odBKL2iw7Zoji-D52nSm_4gChSr2t"/><link rel="image_src" href="https://books.google.com.sg/books/content?id=fj-ryrSBuxAC&printsec=frontcover&img=1&zoom=1&imgtk=AFLRE72Gjz-AxLI8_CkckQvncNeT6uzKCydXoWEXnpBBd49xF0zzoK2ypKsqTXBAHDqZKkHDrFjdaKboOxj2cWtAAUrB2849sj975bVvjz-oPZ8odBKL2iw7Zoji-D52nSm_4gChSr2t"/><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%3Dfj-ryrSBuxAC%26pg%3DPA167%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?id=fj-ryrSBuxAC&pg=PA167&redir_esc=y&hl=en&output=html_text" title="Screen reader users: click this link for accessible mode. 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the Grassmann determinant principle for the plane and the Grassmann principle for space; classification of the elementary configurations of space according to their behavior under transformation of rectangular coordinates; and derivative manifolds. The second section, on geometric transformations, examines affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. The text concludes with a systematic discussion of geometry and its foundations. 1939 edition. 141 figures.</div></div></div><div class="search_box_wrapper"></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="book_other_versions_anchor"></a>Other editions - <a href='https://books.google.com.sg/books?q=editions:ISBN0486434818&id=fj-ryrSBuxAC'>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=6k8x5Vul__IC&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=6k8x5Vul__IC&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=6k8x5Vul__IC&printsec=frontcover&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>Elementary Mathematics from an Advanced Standpoint: Geometry</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:%22Felix+Klein%22" class="secondary"><span dir=ltr>Felix Klein</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2004</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=JazDAgAAQBAJ&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=JazDAgAAQBAJ&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=JazDAgAAQBAJ&printsec=frontcover&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>Elementary Mathematics from an Advanced Standpoint: Geometry</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:%22Felix+Klein%22" class="secondary"><span dir=ltr>Felix Klein</span></a></span><br/><span><span style="color:#99522e">Limited preview</span> - 2012</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=N-WuzgEACAAJ&source=gbs_book_other_versions_r&cad=1" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=N-WuzgEACAAJ&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=N-WuzgEACAAJ&source=gbs_book_other_versions_r&cad=1"><span dir=ltr>Elementary Mathematics From an Advanced Standpoint: Geometry; 2</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:%22Felix+1849-1925+Klein%22" class="secondary"><span dir=ltr>Felix 1849-1925 Klein</span></a></span><br/><span><span style="color:#999">No preview available</span> - 2021</span><br/></span></div></td><td align=right></td></tr></table></div></div><script>(function () {var fn = window['_OC_WSBookList'] || window['_OC_BookList'];fn && fn('book_other_versions', [{"title":"Elementary Mathematics from an Advanced Standpoint","authors":"Felix Klein","bib_key":"ISBN:9780486434810","pub_date":"1 Jan 2004","snippet":"This comprehensive treatment features analytic formulas, enabling precise formulation of geometric facts, and it covers geometric manifolds and transformations, concluding with ...","subject":"Mathematics","info_url":"https://books.google.com.sg/books?id=6k8x5Vul__IC\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=6k8x5Vul__IC\u0026printsec=frontcover\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/content?id=6k8x5Vul__IC\u0026printsec=frontcover\u0026img=1\u0026zoom=1\u0026edge=curl","num_pages":226,"viewability":2,"preview":"partial","embeddable":true,"my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false,"subtitle":"Geometry"},{"title":"Elementary Mathematics from an Advanced Standpoint","authors":"Felix Klein","bib_key":"ISBN:9780486167411","pub_date":"10 Feb 2012","snippet":"This comprehensive treatment features analytic formulas, enabling precise formulation of geometric facts, and it covers geometric manifolds and transformations, concluding with ...","subject":"Mathematics","info_url":"https://books.google.com.sg/books?id=JazDAgAAQBAJ\u0026source=gbs_book_other_versions","preview_url":"https://books.google.com.sg/books?id=JazDAgAAQBAJ\u0026printsec=frontcover\u0026source=gbs_book_other_versions","thumbnail_url":"https://books.google.com.sg/books/content?id=JazDAgAAQBAJ\u0026printsec=frontcover\u0026img=1\u0026zoom=1\u0026edge=curl","num_pages":226,"viewability":2,"preview":"partial","embeddable":true,"list_price":"SGD 22.26","sale_price":"SGD 15.14","buy_url":"https://play.google.com/store/books/details?id=JazDAgAAQBAJ\u0026rdid=book-JazDAgAAQBAJ\u0026rdot=1","is_ebook":true,"preview_ebook_url":"https://play.google.com/books/reader?id=JazDAgAAQBAJ","add_to_my_ebooks_url":"https://books.google.com.sg/books?op=add\u0026sig=ACfU3U3oi7CjQfB1nUrPg9BnLEEtzGbB3g\u0026uid=114584440181414684107\u0026as_coll=7\u0026id=JazDAgAAQBAJ","my_ebooks_url":"https://www.google.com/accounts/Login?service=print\u0026continue=https://books.google.com.sg/books%3Fas_coll%3D7\u0026hl=en","sale_price_better":true,"has_flowing_text":true,"has_scanned_text":true,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false,"subtitle":"Geometry"},{"title":"Elementary Mathematics From an Advanced Standpoint","authors":"Felix 1849-1925 Klein","bib_key":"ISBN:1014468647","pub_date":"9 Sept 2021","snippet":"This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. 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Arbib</span></a></span><br/><span><span style="color:#999">Snippet view</span> - 1972</span><br/></span></div></td><td align=right></td></tr></table></div><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=-33hBOyyqygC&source=gbs_citations_module_r&cad=2" ><img alt="" class="coverthumb hover-card-attach-point" src="https://books.google.com.sg/books/content?id=-33hBOyyqygC&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=-33hBOyyqygC&printsec=references&vq=%22Elementary+Mathematics+from+an+Advanced+Standpoint%22&source=gbs_citations_module_r&cad=2"><span dir=ltr>Philosophy of Geometry from Riemann to Poincaré</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:%22R.+Torretti%22" class="secondary"><span dir=ltr>R. Torretti</span></a></span><br/><span><span style="color:#999">No preview available</span> - 1984</span><br/></span></div></td><td align=right></td></tr></table></div><a class="secondary" href="https://books.google.com.sg/books?q=cites:ISBN0486434818&id=fj-ryrSBuxAC&source=gbs_citations_module_r&cad=2"><span dir=ltr>All Book Search results &raquo;</span></a></div></div><div style="clear:both"></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>Elementary Mathematics from an Advanced Standpoint: Geometry</span><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Dover+Books+on+Mathematics%22&source=gbs_metadata_r&cad=3"><i><span dir=ltr>Volume 2 of Dover Books on Mathematics</span></i></a><br><a class="primary" href="https://www.google.com.sg/search?tbo=p&tbm=bks&q=bibliogroup:%22Elementary+Mathematics+from+an+Advanced+Standpoint%22&source=gbs_metadata_r&cad=3"><i><span dir=ltr>Volume 1 of Elementary Mathematics from an Advanced Standpoint</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:%22Felix+Klein%22&source=gbs_metadata_r&cad=3"><span dir=ltr>Felix Klein</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, reprint</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Publisher</span></td><td class="metadata_value"><span dir=ltr>Courier Corporation, 2004</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>ISBN</span></td><td class="metadata_value"><span dir=ltr>0486434818, 9780486434810</span></td></tr><tr class="metadata_row"><td class="metadata_label"><span dir=ltr>Length</span></td><td class="metadata_value"><span dir=ltr>214 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=3"><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/Elementary_Mathematics_from_an_Advanced.bibtex?id=fj-ryrSBuxAC&output=bibtex"><span dir=ltr>BiBTeX</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Elementary_Mathematics_from_an_Advanced.enw?id=fj-ryrSBuxAC&output=enw"><span dir=ltr>EndNote</span></a> <a class="gb-button " href="https://books.google.com.sg/books/download/Elementary_Mathematics_from_an_Advanced.ris?id=fj-ryrSBuxAC&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, IsBooksRentalEnabled:1, IsBrowsingHistoryEnabled: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%3Dfj-ryrSBuxAC%26pg%3DPA167%26redir_esc%3Dy%26hl%3Den\u0026hl=en"}, {"volume_id":"fj-ryrSBuxAC","is_ebook":false,"volumeresult":{"has_flowing_text":false,"has_scanned_text":false,"can_download_pdf":false,"can_download_epub":false,"is_pdf_drm_enabled":false,"is_epub_drm_enabled":false},"sample_url":"https://play.google.com/books/reader?id=fj-ryrSBuxAC\u0026source=gbs_atb_hover","is_browsable":false,"is_public_domain":false}, 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