An Essay on the Foundations of Geometry
{"WorkMasterId":5220,"WpPageId":252979,"ParentWpPageId":189742,"Slug":"an-essay-on-the-foundations-of-geometry","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/an-essay-on-the-foundations-of-geometry/","RelativeUrl":"theos/humanities/philosophy/philosophers/bertrand-russell/an-essay-on-the-foundations-of-geometry/","HasFullText":true,"RawHtmlLength":691339,"CleanHtmlLength":635229,"Kicker":"Philosophy Work","Title":"An Essay on the Foundations of Geometry","Deck":"Russell examines the logical and philosophical basis of geometry at the edge of neo-Kantian, mathematical, and analytic approaches to space.","BackLink":{"Text":"Back to Bertrand Russell","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/"},"AuthorCard":{"Label":"Author","Title":"Bertrand Russell","Url":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/","MediaHref":"","ImageSrc":"https://chrisdeasy.com/wp-content/uploads/bertrand-russell-01-1954-portrait-2.jpg","ImageAlt":"Bertrand Russell Portrait, 1954","FilterTerra":"Western Europe","ClickText":"Bertrand Russell","ClickHref":"https://chrisdeasy.com/theos/humanities/philosophy/philosophers/bertrand-russell/","Copies":["1872 CE – 1970 CE","Trellech, Monmouthshire","British analytic philosopher, logician, mathematician, social critic, and Nobel laureate from Trellech whose logicism, theory of descriptions, logical atomism, epistemology, philosophy of language, ethics, pacifism, secular critique, and political writing shaped analytic philosophy and twentieth-century public reason."]},"ContextCards":[{"Label":"Period","Key":"Period:5","Title":"Contemporary History","DateText":"1945 CE – 2065 CE","Url":"https://chrisdeasy.com/theos/humanities/philosophy/eras-of-thought/philosophers-of-contemporary-history/"},{"Label":"Era","Key":"Era:12","Title":"World War Era","DateText":"1914 CE – 1944 CE","Url":"https://chrisdeasy.com/theos/humanities/philosophy/eras-of-thought/philosophers-of-modern-history/philosophers-of-the-world-war-era/"},{"Label":"Composition","Title":"1897 CE","Url":"","DateText":""}],"DateNote":"Displayed year is the publication year of Russell\u0027s early geometry book.","GeoCards":[{"Label":"Region","Key":"Region:1"},{"Label":"Terra Avita","Key":"TerraAvita:1"},{"Label":"Terra Avita Region","Key":"TerraAvitaRegion:2"},{"Label":"Modern Country","Key":"Country:GBR:1"}],"OriginalTitle":"","Language":"English","DisciplineCards":[{"Label":"Primary Discipline","Key":"Discipline:philosophy-of-science"},{"Label":"Secondary Discipline","Key":"Discipline:logic"}],"Tradition":"Analytic philosophy, logicism, British empiricism, social criticism, secular humanism, and twentieth-century public reason","FullText":{"Title":"Full Text","Copy":"Public-domain full text from Project Gutenberg eBook #52091 .","Url":"","Label":"","Kicker":"","Cards":[]},"CoreThesis":["Russell examines the logical and philosophical basis of geometry at the edge of neo-Kantian, mathematical, and analytic approaches to space."],"Classification":{"AlternateTitles":"","KeyConcepts":"An Essay on the Foundations of Geometry; Bertrand Russell; logicism; descriptions; logical atomism; knowledge; language; science; ethics; politics; religion; public reason","Methodology":"Logical analysis, formal argument, empiricist reconstruction, linguistic analysis, public criticism, historical explanation, and social-philosophical argument.","Structure":"Accepted work page for Russell under the Core Major scope; minor journalism, duplicate anthologies, individual letters, source/testimony pages, and works merely about Russell are excluded."},"Arguments":["Connects Russell\u0027s technical work in logic and language with his epistemology, philosophy of science, ethics, politics, secular criticism, and public writing."],"Influence":{"InfluencedBy":"Frege, Peano, Leibniz, Hume, Mill, Moore, Whitehead, Cantor, Cambridge mathematics, British empiricism, and anti-idealism.","InfluenceOn":""},"Significance":["Part of the Core Major Russell corpus that made him central to analytic philosophy, mathematical logic, public ethics, secular critique, and twentieth-century intellectual life.","Used in debates about reference, logic, mathematics, science, knowledge, mind, language, liberalism, religion, education, power, and public responsibility."],"EvidenceNote":["Accepted as Russell\u0027s first major philosophical book and a starting point for his work on mathematical foundations."],"MainSections":[{"Kind":"RawSection","Title":"Full Versions","BodyHtml":"\u003cdiv class=\"dz-philo__full-version-grid\"\u003e\n \u003carticle class=\"dz-philo__full-version-card\"\u003e\n \u003cp class=\"dz-philo__full-version-provider\"\u003eProject Gutenberg\u003c/p\u003e\n \u003ch3 class=\"dz-philo__full-version-title\"\u003eProject Gutenberg eBook #52091\u003c/h3\u003e\n \u003cp class=\"dz-philo__full-version-meta\"\u003eHtmlText · Imported\u003c/p\u003e\n \u003ca class=\"dz-philo__full-version-link\" href=\"https://www.gutenberg.org/ebooks/52091\"\u003eOpen full version\u003c/a\u003e\n \u003c/article\u003e\n \u003c/div\u003e"},{"Kind":"TextSection","Title":"Core Thesis","Paragraphs":["Russell examines the logical and philosophical basis of geometry at the edge of neo-Kantian, mathematical, and analytic approaches to space."]},{"Kind":"FieldSection","Title":"Classification","Fields":[{"Label":"Alternate Titles","Value":""},{"Label":"Key Concepts","Value":"An Essay on the Foundations of Geometry; Bertrand Russell; logicism; descriptions; logical atomism; knowledge; language; science; ethics; politics; religion; public reason"},{"Label":"Methodology","Value":"Logical analysis, formal argument, empiricist reconstruction, linguistic analysis, public criticism, historical explanation, and social-philosophical argument."},{"Label":"Structure","Value":"Accepted work page for Russell under the Core Major scope; minor journalism, duplicate anthologies, individual letters, source/testimony pages, and works merely about Russell are excluded."}]},{"Kind":"TextSection","Title":"Arguments","Paragraphs":["Connects Russell\u0027s technical work in logic and language with his epistemology, philosophy of science, ethics, politics, secular criticism, and public writing."]},{"Kind":"FieldSection","Title":"Influence","Fields":[{"Label":"Influenced By","Value":"Frege, Peano, Leibniz, Hume, Mill, Moore, Whitehead, Cantor, Cambridge mathematics, British empiricism, and anti-idealism."},{"Label":"Influence On","Value":"Analytic philosophy, mathematical logic, philosophy of language, logical atomism, logical positivism, secular humanism, public philosophy, peace activism, and twentieth-century liberal thought."}]},{"Kind":"TextSection","Title":"Significance","Paragraphs":["Part of the Core Major Russell corpus that made him central to analytic philosophy, mathematical logic, public ethics, secular critique, and twentieth-century intellectual life.","Used in debates about reference, logic, mathematics, science, knowledge, mind, language, liberalism, religion, education, power, and public responsibility."]},{"Kind":"TextSection","Title":"Evidence Note","Paragraphs":["Accepted as Russell\u0027s first major philosophical book and a starting point for his work on mathematical foundations."]},{"Kind":"RawSection","Title":"Full Text","BodyHtml":"\u003cp class=\"dz-philo__section-copy dz-philo__full-text-source\"\u003ePublic-domain full text from \u003ca href=\"https://www.gutenberg.org/ebooks/52091\"\u003eProject Gutenberg eBook #52091\u003c/a\u003e.\u003c/p\u003e\n \u003carticle class=\"dz-philo__full-text-body\"\u003e\r\n\u003cdiv class=\"transnote\"\u003e\r\n\u003cp\u003e\u003cstrong\u003eTRANSCRIBER\u0027S NOTE\u003c/strong\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"customcover\"\u003eThe cover image was created by the transcriber\r\nand is placed in the public domain.\u003c/p\u003e\r\n\r\n\u003cp\u003eObvious typographical errors and punctuation errors have been\r\ncorrected after careful comparison with other occurrences within\r\nthe text and consultation of external sources.\u003c/p\u003e\r\n\r\n\u003cp\u003eMore detail can be found at the \u003ca href=\"#TN\"\u003eend of the book\u003c/a\u003e.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\r\n\u003cp class=\"p6\" /\u003e\r\n\u003cp class=\"pfs120\"\u003eTHE\u003c/p\u003e\r\n\r\n\u003cp class=\"p1 pfs150\"\u003eFOUNDATIONS OF GEOMETRY.\u003c/p\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\u003cp class=\"pfs100\"\u003e\r\n\u003cspan class=\"antiqua\"\u003eLondon:\u003c/span\u003e C. J. CLAY \u003cspan class=\"fs70\"\u003eAND\u003c/span\u003e SONS,\u003c/p\u003e\r\n\r\n\u003cp class=\"pfs80\"\u003eCAMBRIDGE UNIVERSITY PRESS WAREHOUSE,\u003cbr /\u003e\r\nAVE MARIA LANE.\u003c/p\u003e\r\n\u003cp class=\"pfs70\"\u003e\u003cspan class=\"antiqua\"\u003eGlasgow:\u003c/span\u003e 263, ARGYLE STREET.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-deco-75.jpg\" width=\"75\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp class=\"pfs70\"\u003e\r\n\u003cspan class=\"antiqua\"\u003eLeipzig:\u003c/span\u003e F. A. BROCKHAUS.\u003cbr /\u003e\r\n\u003cspan class=\"antiqua\"\u003eNew York:\u003c/span\u003e THE MACMILLAN COMPANY.\u003cbr /\u003e\r\n\u003cspan class=\"antiqua\"\u003eBombay:\u003c/span\u003e GEORGE BELL AND SONS.\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\r\n\u003ch1\u003e\r\n\u003cspan class=\"large\"\u003eAN ESSAY\u003c/span\u003e\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"xs\"\u003eON THE\u003c/span\u003e\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"xl\"\u003eFOUNDATIONS OF GEOMETRY\u003c/span\u003e\u003c/h1\u003e\r\n\r\n\u003cp class=\"p6\" /\u003e\r\n\u003cp class=\"pfs70\"\u003eBY\u003c/p\u003e\r\n\r\n\u003cp class=\"pfs100\"\u003eBERTRAND A. W. RUSSELL. M.A.\u003c/p\u003e\r\n\u003cp class=\"pfs60\"\u003eFELLOW OF TRINITY COLLEGE, CAMBRIDGE.\u003c/p\u003e\r\n\r\n\u003cp class=\"p6\" /\u003e\r\n\u003cp class=\"pfs90\"\u003eCAMBRIDGE:\u003c/p\u003e\r\n\u003cp class=\"pfs80\"\u003eAT THE UNIVERSITY PRESS.\u003c/p\u003e\r\n\u003cp class=\"pfs90\"\u003e1897\u003c/p\u003e\r\n\r\n\u003cp class=\"pfs70\"\u003e[\u003cem\u003eAll Rights reserved.\u003c/em\u003e]\u003c/p\u003e\r\n\u003cp class=\"p2\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\r\n\u003cp class=\"p6\" /\u003e\r\n\u003cp class=\"pfs60\"\u003e\r\n\u003cspan class=\"antiqua\"\u003eCambridge:\u003c/span\u003e\u003cbr /\u003e\u003cbr /\u003e\r\nPRINTED BY J. AND C. F. CLAY,\u003cbr /\u003e\u003cbr /\u003e\r\nAT THE UNIVERSITY PRESS.\u003c/p\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap\" /\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003ePREFACE.\u003c/h2\u003e\r\n\r\n\u003cp class=\"drop-capx\"\u003eThe present work is based on a dissertation submitted at\r\nthe Fellowship Examination of Trinity College, Cambridge,\r\nin the year 1895. Section B of the third chapter is in\r\nthe main a reprint, with some serious alterations, of an article\r\nin \u003ccite\u003eMind\u003c/cite\u003e (New Series, No. 17). The substance of the book has\r\nbeen given in the form of lectures at the Johns Hopkins\r\nUniversity, Baltimore, and at Bryn Mawr College, Pennsylvania.\u003c/p\u003e\r\n\r\n\u003cp\u003eMy chief obligation is to Professor Klein. Throughout the\r\nfirst chapter, I have found his \"Lectures on non-Euclidean\r\nGeometry\" an invaluable guide; I have accepted from him the\r\ndivision of Metageometry into three periods, and have found\r\nmy historical work much lightened by his references to previous\r\nwriters. In Logic, I have learnt most from Mr Bradley, and\r\nnext to him, from Sigwart and Dr Bosanquet. On several\r\nimportant points, I have derived useful suggestions from\r\nProfessor James\u0027s \"Principles of Psychology.\"\u003c/p\u003e\r\n\r\n\u003cp\u003eMy thanks are due to Mr G. F. Stout and Mr A. N.\r\nWhitehead for kindly reading my proofs, and helping me by\r\nmany useful criticisms. To Mr Whitehead I owe, also, the\r\ninestimable assistance of constant criticism and suggestion\r\nthroughout the course of construction, especially as regards\r\nthe philosophical importance of projective Geometry.\u003c/p\u003e\r\n\r\n\u003cp class=\"p2 smcap\"\u003eHaslemere.\u003c/p\u003e\r\n\u003cp class=\"pad2\"\u003e\u003cem\u003eMay, 1897.\u003c/em\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003cp class=\"pfs80\"\u003eTO\u003c/p\u003e\r\n\r\n\u003cp class=\"p1 pfs100\"\u003eJOHN McTAGGART ELLIS McTAGGART\u003c/p\u003e\r\n\r\n\u003cp class=\"p2 pfs80\"\u003eTO WHOSE DISCOURSE AND FRIENDSHIP IS OWING\u003c/p\u003e\r\n\u003cp class=\"pfs80\"\u003eTHE EXISTENCE OF THIS BOOK.\u003c/p\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap\" /\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_vii\" id=\"Page_vii\"\u003e[Pg vii]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003ch2\u003e\u003ca name=\"CONTENTS\" id=\"CONTENTS\"\u003eTABLE OF CONTENTS.\u003c/a\u003e\u003c/h2\u003e\r\n\r\n\u003cdiv class=\"p1 center fs90\"\u003e\r\n\u003ctable border=\"0\" cellpadding=\"4\" cellspacing=\"0\" width=\"95%\" summary=\"\"\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large\"\u003eINTRODUCTION.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc small tdpp\"\u003eOUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,\r\nPSYCHOLOGY AND MATHEMATICS.\u003c/td\u003e\u003ctd class=\"tdr\"\u003e\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdr xs\"\u003ePAGE\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N1\"\u003e1.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl wd90\"\u003eThe problem first received a modern form through Kant, who connected the \u003cem\u003eà priori\u003c/em\u003e with the subjective\u003c/td\u003e\u003ctd class=\"tdr\"\u003e1\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N2\"\u003e2.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eA mental state is subjective, for Psychology, when its immediate cause does not lie in the outer world\u003c/td\u003e\u003ctd class=\"tdr\"\u003e2\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N3\"\u003e3.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eA piece of knowledge is \u003cem\u003eà priori\u003c/em\u003e, for Epistemology, when without it knowledge would be impossible\u003c/td\u003e\u003ctd class=\"tdr\"\u003e2\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N4\"\u003e4.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe subjective and the \u003cem\u003eà priori\u003c/em\u003e belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay\u003c/td\u003e\u003ctd class=\"tdr\"\u003e3\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N5\"\u003e5.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMy test of the \u003cem\u003eà priori\u003c/em\u003e will be purely logical: what knowledge is necessary for experience?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e3\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N6\"\u003e6.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut since the necessary is hypothetical, we must include, in the \u003cem\u003eà priori\u003c/em\u003e, the ground of necessity\u003c/td\u003e\u003ctd class=\"tdr\"\u003e4\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N7\"\u003e7.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThis may be the essential postulate of our science, or the element, in the subject-matter, which is necessary to experience;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e4\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N8\"\u003e8.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich, however, are both at bottom the same ground\u003c/td\u003e\u003ctd class=\"tdr\"\u003e5\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N9\"\u003e9.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eForecast of the work\u003c/td\u003e\u003ctd class=\"tdr\"\u003e5\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large\"\u003eCHAPTER I.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc small tdpp\"\u003eA SHORT HISTORY OF METAGEOMETRY.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N10\"\u003e10.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMetageometry began by rejecting the axiom of parallels\u003c/td\u003e\u003ctd class=\"tdr\"\u003e7\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N11\"\u003e11.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIts history may be divided into three periods: the synthetic, the metrical and the projective\u003c/td\u003e\u003ctd class=\"tdr\"\u003e7\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N12\"\u003e12.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe first period was inaugurated by Gauss,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e10\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_viii\" id=\"Page_viii\"\u003e[viii]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N13\"\u003e13.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhose suggestions were developed independently by Lobatchewsky\u003c/td\u003e\u003ctd class=\"tdr\"\u003e10\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N14\"\u003e14.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd Bolyai\u003c/td\u003e\u003ctd class=\"tdr\"\u003e11\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N15\"\u003e15.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe purpose of all three was to show that the axiom of parallels could not be deduced from the others, since its denial did not lead to contradictions\u003c/td\u003e\u003ctd class=\"tdr\"\u003e12\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N16\"\u003e16.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe second period had a more philosophical aim, and was inspired chiefly by Gauss and Herbart\u003c/td\u003e\u003ctd class=\"tdr\"\u003e13\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N17\"\u003e17.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe first work of this period, that of Riemann, invented two new conceptions:\u003c/td\u003e\u003ctd class=\"tdr\"\u003e14\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N18\"\u003e18.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe first, that of a manifold, is a class-conception, containing space as a species,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e14\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N19\"\u003e19.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd defined as such that its determinations form a collection of magnitudes\u003c/td\u003e\u003ctd class=\"tdr\"\u003e15\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N20\"\u003e20.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe second, the measure of curvature of a manifold, grew out of curvature in curves and surfaces\u003c/td\u003e\u003ctd class=\"tdr\"\u003e16\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N21\"\u003e21.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBy means of Gauss\u0027s analytical formula for the curvature of surfaces,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e19\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N22\"\u003e22.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich enables us to define a \u003cem\u003econstant\u003c/em\u003e measure of curvature of a three-dimensional space without reference to a fourth dimension\u003c/td\u003e\u003ctd class=\"tdr\"\u003e20\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N23\"\u003e23.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe main result of Riemann\u0027s mathematical work was to show that, if magnitudes are independent of place, the measure of curvature of space must be constant\u003c/td\u003e\u003ctd class=\"tdr\"\u003e21\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N24\"\u003e24.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHelmholtz, who was more of a philosopher than a mathematician,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e22\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N25\"\u003e25.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eGave a new but incorrect formulation of the essential axioms,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e23\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N26\"\u003e26.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd deduced the quadratic formula for the infinitesimal arc, which Riemann had assumed\u003c/td\u003e\u003ctd class=\"tdr\"\u003e24\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N27\"\u003e27.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBeltrami gave Lobatchewsky\u0027s planimetry a Euclidean interpretation,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e25\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N28\"\u003e28.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich is analogous to Cayley\u0027s theory of distance;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e26\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N29\"\u003e29.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd dealt with \u003cem\u003en\u003c/em\u003e-dimensional spaces of constant negative curvature\u003c/td\u003e\u003ctd class=\"tdr\"\u003e27\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N30\"\u003e30.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity\u003c/td\u003e\u003ctd class=\"tdr\"\u003e27\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N31\"\u003e31.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eCayley reduced metrical properties to projective properties, relative to a certain conic or quadric, the Absolute;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e28\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N32\"\u003e32.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd Klein showed that the Euclidean or non-Euclidean systems result, according to the nature of the Absolute;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e29\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N33\"\u003e33.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHence Euclidean \u003cem\u003espace\u003c/em\u003e appeared to give rise to all the kinds of Geometry, and the question, which is true, appeared reduced to one of convention\u003c/td\u003e\u003ctd class=\"tdr\"\u003e30\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N34\"\u003e34.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut this view is due to a confusion as to the nature of the coordinates employed\u003c/td\u003e\u003ctd class=\"tdr\"\u003e30\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_ix\" id=\"Page_ix\"\u003e[ix]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N35\"\u003e35.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eProjective coordinates have been regarded as dependent on distance, and thus really metrical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e31\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N36\"\u003e36.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut this is not the case, since anharmonic ratio can be projectively defined\u003c/td\u003e\u003ctd class=\"tdr\"\u003e32\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N37\"\u003e37.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eProjective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e33\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N38\"\u003e38.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe true connection of Cayley\u0027s measure of distance with non-Euclidean Geometry is that suggested by Beltrami\u0027s Saggio, and worked out by Sir R. Ball,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e36\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N39\"\u003e39.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich provides a Euclidean equivalent for every non-Euclidean proposition, and so removes the possibility of contradictions in Metageometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e38\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N40\"\u003e40.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eKlein\u0027s elliptic Geometry has not been proved to have a corresponding variety of space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e39\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N41\"\u003e41.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe geometrical use of imaginaries, of which Cayley demanded a philosophical discussion,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e41\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N42\"\u003e42.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHas a merely technical validity,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e42\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N43\"\u003e43.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd is capable of giving geometrical results only when it begins and ends with real points and figures\u003c/td\u003e\u003ctd class=\"tdr\"\u003e45\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N44\"\u003e44.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWe have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it\u003c/td\u003e\u003ctd class=\"tdr\"\u003e46\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N45\"\u003e45.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSophus Lie has applied projective methods to Helmholtz\u0027s formulation of the axioms, and has shown the axiom of Monodromy to be superfluous\u003c/td\u003e\u003ctd class=\"tdr\"\u003e46\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N46\"\u003e46.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMetageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy\u003c/td\u003e\u003ctd class=\"tdr\"\u003e50\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N47\"\u003e47.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMetrical Geometry has three indispensable axioms,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e50\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N48\"\u003e48.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich we shall find to be not results, but conditions, of measurement,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e51\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N49\"\u003e49.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd which are nearly equivalent to the three axioms of projective Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e52\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N50\"\u003e50.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBoth sets of axioms are necessitated, not by facts, but by logic\u003c/td\u003e\u003ctd class=\"tdr\"\u003e52\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large\"\u003eCHAPTER II.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc small tdpp\"\u003eCRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N51\"\u003e51.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eA criticism of representative modern theories need not begin before Kant\u003c/td\u003e\u003ctd class=\"tdr\"\u003e54\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N52\"\u003e52.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eKant\u0027s doctrine must be taken, in an argument about Geometry, on its purely logical side\u003c/td\u003e\u003ctd class=\"tdr\"\u003e55\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_x\" id=\"Page_x\"\u003e[x]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N53\"\u003e53.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eKant contends that since Geometry is apodeictic, space must be \u003cem\u003eà priori\u003c/em\u003e and subjective, while since space is \u003cem\u003eà priori\u003c/em\u003e and subjective, Geometry must be apodeictic\u003c/td\u003e\u003ctd class=\"tdr\"\u003e55\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N54\"\u003e54.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMetageometry has upset the first line of argument, not the second\u003c/td\u003e\u003ctd class=\"tdr\"\u003e56\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N55\"\u003e55.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe second may be attacked by criticizing either the distinction of synthetic and analytic judgments, or the first two arguments of the metaphysical deduction of space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e57\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N56\"\u003e56.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eModern Logic regards every judgment as both synthetic and analytic,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e57\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N57\"\u003e57.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut leaves the \u003cem\u003eà priori\u003c/em\u003e, as that which is presupposed in the possibility of experience\u003c/td\u003e\u003ctd class=\"tdr\"\u003e59\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N58\"\u003e58.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eKant\u0027s first two arguments as to space suffice to prove \u003cem\u003esome\u003c/em\u003e form of externality, but not necessarily Euclidean space, a necessary condition of experience\u003c/td\u003e\u003ctd class=\"tdr\"\u003e60\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N59\"\u003e59.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAmong the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann\u003c/td\u003e\u003ctd class=\"tdr\"\u003e62\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N60\"\u003e60.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eRiemann regarded space as a particular kind of manifold, i.e. wholly quantitatively\u003c/td\u003e\u003ctd class=\"tdr\"\u003e63\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N61\"\u003e61.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHe therefore unduly neglected the qualitative adjectives of space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e64\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N62\"\u003e62.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis philosophy rests on a vicious disjunction\u003c/td\u003e\u003ctd class=\"tdr\"\u003e65\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N63\"\u003e63.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis definition of a manifold is obscure,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e66\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N64\"\u003e64.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd his definition of measurement applies only to space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e67\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N65\"\u003e65.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThough mathematically invaluable, his view of space as a manifold is philosophically misleading\u003c/td\u003e\u003ctd class=\"tdr\"\u003e69\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N66\"\u003e66.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHelmholtz attacked Kant both on the mathematical and on the psychological side;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e70\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N67\"\u003e67.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut his criterion of apriority is changeable and often invalid;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e71\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N68\"\u003e68.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis proof that non-Euclidean spaces are imaginable is inconclusive;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e72\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N69\"\u003e69.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd his assertion of the dependence of measurement on rigid bodies, which may be taken in three senses,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e74\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N70\"\u003e70.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIs wholly false if it means that the axiom of Congruence actually asserts the existence of rigid bodies,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e75\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N71\"\u003e71.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIs untrue if it means that the necessary reference of geometrical propositions to matter renders pure Geometry empirical,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e76\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N72\"\u003e72.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd is inadequate to his conclusion if it means, what is true, that \u003cem\u003eactual\u003c/em\u003e measurement involves approximately rigid bodies\u003c/td\u003e\u003ctd class=\"tdr\"\u003e78\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N73\"\u003e73.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eGeometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e80\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N74\"\u003e74.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eErdmann accepted the conclusions of Riemann and Helmholtz,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e81\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xi\" id=\"Page_xi\"\u003e[xi]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N75\"\u003e75.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd regarded the axioms as necessarily successive steps in classifying space as a species of manifold\u003c/td\u003e\u003ctd class=\"tdr\"\u003e82\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N76\"\u003e76.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis deduction involves four fallacious assumptions, namely:\u003c/td\u003e\u003ctd class=\"tdr\"\u003e82\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N77\"\u003e77.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThat conceptions must be abstracted from a series of instances;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e83\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N78\"\u003e78.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThat all definition is classification;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e83\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N79\"\u003e79.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThat conceptions of magnitude can be applied to space as a whole;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e84\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N80\"\u003e80.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd that if conceptions of magnitude could be so applied, all the adjectives of space would result from their application\u003c/td\u003e\u003ctd class=\"tdr\"\u003e86\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N81\"\u003e81.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eErdmann regards Geometry alone as incapable of deciding on the truth of the axiom of Congruence,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e86\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N82\"\u003e82.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich he affirms to be empirically proved by Mechanics.\u003c/td\u003e\u003ctd class=\"tdr\"\u003e88\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N83\"\u003e83.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe variety and inadequacy of Erdmann\u0027s tests of apriority\u003c/td\u003e\u003ctd class=\"tdr\"\u003e89\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N84\"\u003e84.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eInvalidate his final conclusions on the theory of Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e90\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N85\"\u003e85.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eLotze has discussed two questions in the theory of Geometry:\u003c/td\u003e\u003ctd class=\"tdr\"\u003e93\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N86\"\u003e86.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(1) He regards the possibility of non-Euclidean spaces as suggested by the subjectivity of space,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e93\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N87\"\u003e87.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd rejects it owing to a mathematical misunderstanding,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e96\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N88\"\u003e88.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHaving missed the most important sense of their possibility,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e96\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N89\"\u003e89.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich is that they fulfil the logical conditions to which any form of externality must conform\u003c/td\u003e\u003ctd class=\"tdr\"\u003e97\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N90\"\u003e90.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(2) He attacks the mathematical procedure of Metageometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e98\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N91\"\u003e91.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe attack begins with a question-begging definition of parallels\u003c/td\u003e\u003ctd class=\"tdr\"\u003e99\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N92\"\u003e92.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eLotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e99\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N93\"\u003e93.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis criticism of Helmholtz\u0027s analogies rests wholly on mathematical mistakes\u003c/td\u003e\u003ctd class=\"tdr\"\u003e101\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N94\"\u003e94.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHis proof that space must have three dimensions rests on neglect of different orders of infinity\u003c/td\u003e\u003ctd class=\"tdr\"\u003e104\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N95\"\u003e95.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHe attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous\u003c/td\u003e\u003ctd class=\"tdr\"\u003e107\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N96\"\u003e96.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eLotze\u0027s objections fall under four heads\u003c/td\u003e\u003ctd class=\"tdr\"\u003e108\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N97\"\u003e97.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eTwo other semi-philosophical objections may be urged,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e109\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N98\"\u003e98.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eOne of which, the absence of similarity, has been made the basis of attack by Delbœuf,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e110\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N99\"\u003e99.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut does not form a valid ground of objection\u003c/td\u003e\u003ctd class=\"tdr\"\u003e111\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N100\"\u003e100.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eRecent French speculation on the foundations of Geometry has suggested few new views\u003c/td\u003e\u003ctd class=\"tdr\"\u003e112\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N101\"\u003e101.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAll homogeneous spaces are \u003cem\u003eà priori\u003c/em\u003e possible, and the decision between them is empirical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e114\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u0026nbsp;\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xii\" id=\"Page_xii\"\u003e[xii]\u003c/a\u003e\u003c/span\u003e\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large\"\u003eCHAPTER III.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large tdpp smcap\"\u003eSection A. the axioms of projective geometry.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N102\"\u003e102.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eProjective Geometry does not deal with magnitude, and applies to all spaces alike\u003c/td\u003e\u003ctd class=\"tdr\"\u003e117\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N103\"\u003e103.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIt will be found wholly \u003cem\u003eà priori\u003c/em\u003e\u003c/td\u003e\u003ctd class=\"tdr\"\u003e117\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N104\"\u003e104.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIts axioms have not yet been formulated philosophically\u003c/td\u003e\u003ctd class=\"tdr\"\u003e118\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N105\"\u003e105.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eCoordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points\u003c/td\u003e\u003ctd class=\"tdr\"\u003e118\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N106\"\u003e106.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe possibility of distinguishing various points is an axiom\u003c/td\u003e\u003ctd class=\"tdr\"\u003e119\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N107\"\u003e107.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment\u003c/td\u003e\u003ctd class=\"tdr\"\u003e119\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N108\"\u003e108.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar\u003c/td\u003e\u003ctd class=\"tdr\"\u003e120\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N109\"\u003e109.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHence follows, by extension, the principle of projective transformation\u003c/td\u003e\u003ctd class=\"tdr\"\u003e121\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N110\"\u003e110.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBy which figures qualitatively indistinguishable from a given figure are obtained\u003c/td\u003e\u003ctd class=\"tdr\"\u003e122\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N111\"\u003e111.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnharmonic ratio may and must be descriptively defined\u003c/td\u003e\u003ctd class=\"tdr\"\u003e122\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N112\"\u003e112.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe quadrilateral construction is essential to the projective definition of points,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e123\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N113\"\u003e113.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd can be projectively defined,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e124\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N114\"\u003e114.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBy the general principle of projective transformation\u003c/td\u003e\u003ctd class=\"tdr\"\u003e126\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N115\"\u003e115.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe principle of duality is the mathematical form of a philosophical circle,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e127\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N116\"\u003e116.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory\u003c/td\u003e\u003ctd class=\"tdr\"\u003e128\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N117\"\u003e117.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWe define the point as that which is spatial, but contains no space, whence other definitions follow\u003c/td\u003e\u003ctd class=\"tdr\"\u003e128\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N118\"\u003e118.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhat is meant by qualitative equivalence in Geometry?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e129\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N119\"\u003e119.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eTwo pairs of points on one straight line, or two pairs of straight lines through one point, are qualitatively equivalent\u003c/td\u003e\u003ctd class=\"tdr\"\u003e129\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N120\"\u003e120.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThis explains why \u003cem\u003efour\u003c/em\u003e collinear points are needed, to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given\u003c/td\u003e\u003ctd class=\"tdr\"\u003e130\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N121\"\u003e121.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAny two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property\u003c/td\u003e\u003ctd class=\"tdr\"\u003e131\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N122\"\u003e122.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThree axioms are used by projective Geometry,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e132\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xiii\" id=\"Page_xiii\"\u003e[xiii]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N123\"\u003e123.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd are required for qualitative spatial comparison,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e132\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N124\"\u003e124.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich involves the homogeneity, relativity and passivity of space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e133\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N125\"\u003e125.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe conception of a form of externality,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e134\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N126\"\u003e126.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBeing a creature of the intellect, can be dealt with by pure mathematics\u003c/td\u003e\u003ctd class=\"tdr\"\u003e134\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N127\"\u003e127.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe resulting doctrine of extension will be, for the moment, hypothetical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e135\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N128\"\u003e128.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut is rendered assertorical by the necessity, for experience, of some form of externality\u003c/td\u003e\u003ctd class=\"tdr\"\u003e136\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N129\"\u003e129.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAny such form must be relational\u003c/td\u003e\u003ctd class=\"tdr\"\u003e136\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N130\"\u003e130.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd homogeneous\u003c/td\u003e\u003ctd class=\"tdr\"\u003e137\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N131\"\u003e131.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd the relations constituting it must appear infinitely divisible\u003c/td\u003e\u003ctd class=\"tdr\"\u003e137\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N132\"\u003e132.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIt must have a finite integral number of dimensions,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e139\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N133\"\u003e133.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eOwing to its passivity and homogeneity\u003c/td\u003e\u003ctd class=\"tdr\"\u003e140\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N134\"\u003e134.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd to the systematic unity of the world\u003c/td\u003e\u003ctd class=\"tdr\"\u003e140\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N135\"\u003e135.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eA one-dimensional form alone would not suffice for experience\u003c/td\u003e\u003ctd class=\"tdr\"\u003e141\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N136\"\u003e136.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSince its elements would be immovably fixed in a series\u003c/td\u003e\u003ctd class=\"tdr\"\u003e142\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N137\"\u003e137.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eTwo positions have a relation independent of other positions,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e143\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N138\"\u003e138.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSince positions are wholly defined by mutually independent relations\u003c/td\u003e\u003ctd class=\"tdr\"\u003e143\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N139\"\u003e139.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHence projective Geometry is wholly \u003cem\u003eà priori\u003c/em\u003e,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e146\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N140\"\u003e140.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThough metrical Geometry contains an empirical element\u003c/td\u003e\u003ctd class=\"tdr\"\u003e146\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large tdpp smcap\"\u003eSection B. the axioms of metrical geometry.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N141\"\u003e141.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMetrical Geometry is distinct from projective, but has the same fundamental postulate\u003c/td\u003e\u003ctd class=\"tdr\"\u003e147\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N142\"\u003e142.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIt introduces the new idea of motion, and has three \u003cem\u003eà priori\u003c/em\u003e axioms\u003c/td\u003e\u003ctd class=\"tdr\"\u003e148\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc tdpp\"\u003eI. \u003cem\u003eThe Axiom of Free Mobility.\u003c/em\u003e\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N143\"\u003e143.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eMeasurement requires a criterion of spatial equality\u003c/td\u003e\u003ctd class=\"tdr\"\u003e149\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N144\"\u003e144.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich is given by superposition, and involves the axiom of Free Mobility\u003c/td\u003e\u003ctd class=\"tdr\"\u003e150\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N145\"\u003e145.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe denial of this axiom involves an action of empty space on things\u003c/td\u003e\u003ctd class=\"tdr\"\u003e151\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N146\"\u003e146.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThere is a mathematically possible alternative to the axiom,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e152\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N147\"\u003e147.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich, however, is logically and philosophically untenable\u003c/td\u003e\u003ctd class=\"tdr\"\u003e153\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N148\"\u003e148.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThough Free Mobility is \u003cem\u003eà priori\u003c/em\u003e, actual measurement is empirical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e154\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xiv\" id=\"Page_xiv\"\u003e[xiv]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N149\"\u003e149.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSome objections remain to be answered, concerning\u0026mdash;\u003c/td\u003e\u003ctd class=\"tdr\"\u003e154\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N150\"\u003e150.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(1) The comparison of volumes and of Kant\u0027s symmetrical objects\u003c/td\u003e\u003ctd class=\"tdr\"\u003e154\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N151\"\u003e151.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(2) The measurement of time, where congruence is impossible\u003c/td\u003e\u003ctd class=\"tdr\"\u003e156\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N152\"\u003e152.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(3) The immediate perception of spatial magnitude; and\u003c/td\u003e\u003ctd class=\"tdr\"\u003e157\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N153\"\u003e153.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(4) The Geometry of non-congruent surfaces\u003c/td\u003e\u003ctd class=\"tdr\"\u003e158\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N154\"\u003e154.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eFree Mobility includes Helmholtz\u0027s Monodromy\u003c/td\u003e\u003ctd class=\"tdr\"\u003e159\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N155\"\u003e155.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eFree Mobility involves the relativity of space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e159\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N156\"\u003e156.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eFrom which, reciprocally, it can be deduced\u003c/td\u003e\u003ctd class=\"tdr\"\u003e160\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N157\"\u003e157.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eOur axiom is therefore \u003cem\u003eà priori\u003c/em\u003e in a double sense\u003c/td\u003e\u003ctd class=\"tdr\"\u003e160\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc tdpp\"\u003eII. \u003cem\u003eThe Axiom of Dimensions.\u003c/em\u003e\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N158\"\u003e158.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSpace must have a finite integral number of dimensions\u003c/td\u003e\u003ctd class=\"tdr\"\u003e161\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N159\"\u003e159.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut the restriction to three is empirical\u003c/td\u003e\u003ctd class=\"tdr\"\u003e162\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N160\"\u003e160.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe general axiom follows from the relativity of position\u003c/td\u003e\u003ctd class=\"tdr\"\u003e162\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N161\"\u003e161.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe limitation to three dimensions, unlike most empirical knowledge, is accurate and certain\u003c/td\u003e\u003ctd class=\"tdr\"\u003e163\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc tdpp\"\u003eIII. \u003cem\u003eThe Axiom of Distance.\u003c/em\u003e\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N162\"\u003e162.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe axiom of distance corresponds, here, to that of the straight line in projective Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e164\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N163\"\u003e163.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe possibility of spatial measurement involves a magnitude uniquely determined by two points,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e164\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N164\"\u003e164.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSince two points must have some relation, and the passivity of space proves this to be independent of external reference\u003c/td\u003e\u003ctd class=\"tdr\"\u003e165\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N165\"\u003e165.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThere can be only one such relation\u003c/td\u003e\u003ctd class=\"tdr\"\u003e166\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N166\"\u003e166.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThis must be measured by a curve joining the two points,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e166\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N167\"\u003e167.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd the curve must be uniquely determined by the two points\u003c/td\u003e\u003ctd class=\"tdr\"\u003e167\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N168\"\u003e168.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSpherical Geometry contains an exception to this axiom,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e168\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N169\"\u003e169.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich, however, is not quite equivalent to Euclid\u0027s\u003c/td\u003e\u003ctd class=\"tdr\"\u003e168\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N170\"\u003e170.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe exception is due to the fact that two points, in spherical space, may have an external relation unaltered by motion,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e169\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N171\"\u003e171.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhich, however, being a relation of linear magnitude, presupposes the possibility of linear magnitude\u003c/td\u003e\u003ctd class=\"tdr\"\u003e170\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N172\"\u003e172.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eA relation between two points must be a line joining them\u003c/td\u003e\u003ctd class=\"tdr\"\u003e170\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N173\"\u003e173.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eConversely, the existence of a unique line between two points can be deduced from the nature of a form of externality,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e171\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N174\"\u003e174.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd necessarily leads to distance, when quantity is applied to it\u003c/td\u003e\u003ctd class=\"tdr\"\u003e172\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xv\" id=\"Page_xv\"\u003e[xv]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N175\"\u003e175.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHence the axiom of distance, also, is \u003cem\u003eà priori\u003c/em\u003e in a double sense\u003c/td\u003e\u003ctd class=\"tdr\"\u003e172\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N176\"\u003e176.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eNo metrical coordinate system can be set up without the straight line\u003c/td\u003e\u003ctd class=\"tdr\"\u003e174\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N177\"\u003e177.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eNo axioms besides the above three are necessary to metrical Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e175\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N178\"\u003e178.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut these three are necessary to the direct measurement of any continuum\u003c/td\u003e\u003ctd class=\"tdr\"\u003e176\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N179\"\u003e179.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eTwo philosophical questions remain for a final chapter\u003c/td\u003e\u003ctd class=\"tdr\"\u003e177\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc large\"\u003eCHAPTER IV.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003c/td\u003e\u003ctd class=\"tdc small tdpp\"\u003ePHILOSOPHICAL CONSEQUENCES.\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N180\"\u003e180.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhat is the relation to experience of a form of externality in general?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e178\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N181\"\u003e181.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThis form is the class-conception, containing every possible intuition of externality; and some such intuition is necessary to experience\u003c/td\u003e\u003ctd class=\"tdr\"\u003e178\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N182\"\u003e182.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhat relation does this view bear to Kant\u0027s?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e179\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N183\"\u003e183.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eIt is less psychological, since it does not discuss whether space is given in sensation,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e180\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N184\"\u003e184.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd maintains that not only space, but any form of externality which renders experience possible, must be given in sense-perception\u003c/td\u003e\u003ctd class=\"tdr\"\u003e181\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N185\"\u003e185.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eExternality should mean, not externality to the Self, but the mutual externality of presented things\u003c/td\u003e\u003ctd class=\"tdr\"\u003e181\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N186\"\u003e186.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWould this be unknowable without a given form of externality?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e182\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N187\"\u003e187.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBradley has proved that space and time preclude the existence of mere particulars,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e182\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N188\"\u003e188.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd that knowledge requires the \u003cem\u003eThis\u003c/em\u003e to be neither simple nor self-subsistent\u003c/td\u003e\u003ctd class=\"tdr\"\u003e183\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N189\"\u003e189.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eTo prove that experience requires a form of externality, I assume that all knowledge requires the recognition of identity in difference\u003c/td\u003e\u003ctd class=\"tdr\"\u003e184\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N190\"\u003e190.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSuch recognition involves time\u003c/td\u003e\u003ctd class=\"tdr\"\u003e184\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N191\"\u003e191.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd some other form giving simultaneous diversity\u003c/td\u003e\u003ctd class=\"tdr\"\u003e185\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N192\"\u003e192.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe above argument has not deduced sense-perception from the categories, but has shown the former, unless it contains a certain element, to be unintelligible to the latter\u003c/td\u003e\u003ctd class=\"tdr\"\u003e186\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N193\"\u003e193.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eHow to account for the realization of this element, is a question for metaphysics\u003c/td\u003e\u003ctd class=\"tdr\"\u003e187\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_xvi\" id=\"Page_xvi\"\u003e[xvi]\u003c/a\u003e\u003c/span\u003e\r\n \u003ca href=\"#N194\"\u003e194.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eWhat are we to do with the contradictions in space?\u003c/td\u003e\u003ctd class=\"tdr\"\u003e188\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N195\"\u003e195.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThree contradictions will be discussed in what follows\u003c/td\u003e\u003ctd class=\"tdr\"\u003e188\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N196\"\u003e196.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(1) The antinomy of the Point proves the relativity of space,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e189\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N197\"\u003e197.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd shows that Geometry must have some reference to matter,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e190\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N198\"\u003e198.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBy which means it is made to refer to spatial order, not to empty space\u003c/td\u003e\u003ctd class=\"tdr\"\u003e191\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N199\"\u003e199.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe causal properties of matter are irrelevant to Geometry, which must regard it as composed of unextended atoms, by which points are replaced\u003c/td\u003e\u003ctd class=\"tdr\"\u003e191\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N200\"\u003e200.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(2) The circle in defining straight lines and planes is overcome by the same reference to matter\u003c/td\u003e\u003ctd class=\"tdr\"\u003e192\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N201\"\u003e201.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003e(3) The antinomy that space is relational and yet more than relational,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e193\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N202\"\u003e202.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eSeems to depend on the confusion of empty space with spatial order\u003c/td\u003e\u003ctd class=\"tdr\"\u003e193\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N203\"\u003e203.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eKant regarded empty space as the subject-matter of Geometry,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e194\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N204\"\u003e204.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eBut the arguments of the Aesthetic are inconclusive on this point,\u003c/td\u003e\u003ctd class=\"tdr\"\u003e195\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N205\"\u003e205.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eAnd are upset by the mathematical antinomies, which prove that spatial order should be the subject-matter of Geometry\u003c/td\u003e\u003ctd class=\"tdr\"\u003e196\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N206\"\u003e206.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe apparent thinghood of space is a psychological illusion, due to the fact that spatial relations are immediately given\u003c/td\u003e\u003ctd class=\"tdr\"\u003e196\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N207\"\u003e207.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eThe apparent divisibility of spatial relations is either an illusion, arising out of empty space, or the expression of the possibility of quantitatively different spatial relations\u003c/td\u003e\u003ctd class=\"tdr\"\u003e197\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N208\"\u003e208.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eExternality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation\u003c/td\u003e\u003ctd class=\"tdr\"\u003e198\u003c/td\u003e\u003c/tr\u003e\r\n\u003ctr\u003e\u003ctd class=\"tdl\"\u003e\u003ca href=\"#N209\"\u003e209.\u003c/a\u003e\u003c/td\u003e\u003ctd class=\"tdl\"\u003eConclusion\u003c/td\u003e\u003ctd class=\"tdr\"\u003e199\u003c/td\u003e\u003c/tr\u003e\r\n\u003c/table\u003e\u003c/div\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003chr class=\"chap\" /\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_1\" id=\"Page_1\"\u003e[Pg 1]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003eINTRODUCTION.\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"fs60\"\u003eOUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,\r\nPSYCHOLOGY AND MATHEMATICS.\u003c/span\u003e\u003c/h2\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e1.\u003c/b\u003e\u003ca name=\"N1\" id=\"N1\"\u003e\u003c/a\u003e\r\nGeometry, throughout the 17th and 18th centuries,\r\nremained, in the war against empiricism, an impregnable\r\nfortress of the idealists. Those who held\u0026mdash;as was generally\r\nheld on the Continent\u0026mdash;that certain knowledge, independent of\r\nexperience, was possible about the real world, had only to\r\npoint to Geometry: none but a madman, they said, would\r\nthrow doubt on its validity, and none but a fool would deny\r\nits objective reference. The English Empiricists, in this\r\nmatter, had, therefore, a somewhat difficult task; either they\r\nhad to ignore the problem, or if, like Hume and Mill, they\r\nventured on the assault, they were driven into the apparently\r\nparadoxical assertion that Geometry, at bottom, had no certainty\r\nof a different \u003cem\u003ekind\u003c/em\u003e from that of Mechanics\u0026mdash;only the\r\nperpetual presence of spatial impressions, they said, made\r\nour experience of the truth of the axioms so wide as to seem\r\nabsolute certainty.\u003c/p\u003e\r\n\r\n\u003cp\u003eHere, however, as in many other instances, merciless logic\r\ndrove these philosophers, whether they would or no, into\r\nglaring opposition to the common sense of their day. It was\r\nonly through Kant, the creator of modern Epistemology, that\r\nthe geometrical problem received a modern form. He reduced\r\nthe question to the following hypotheticals: If Geometry has\r\napodeictic certainty, its matter, \u003cem\u003ei.e.\u003c/em\u003e space, must be \u003cem\u003eà priori\u003c/em\u003e, and\r\nas such must be purely subjective; and conversely, if space is\r\npurely subjective, Geometry must have apodeictic certainty.\r\nThe latter hypothetical has more weight with Kant, indeed it\r\nis ineradicably bound up with his whole Epistemology; nevertheless\r\nit has, I think, much less force than the former. Let us\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_2\" id=\"Page_2\"\u003e[2]\u003c/a\u003e\u003c/span\u003e\r\naccept, however, for the moment, the Kantian formulation, and\r\nendeavour to give precision to the terms \u003cem\u003eà priori\u003c/em\u003e and \u003cem\u003esubjective\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e2.\u003c/b\u003e\u003ca name=\"N2\" id=\"N2\"\u003e\u003c/a\u003e\r\nOne of the great difficulties, throughout this controversy,\r\nis the extremely variable use to which these words, as\r\nwell as the word \u003cem\u003eempirical\u003c/em\u003e, are put by different authors. To\r\nKant, who was nothing of a psychologist, \u003cem\u003eà priori\u003c/em\u003e and \u003cem\u003esubjective\u003c/em\u003e\r\nwere almost interchangeable terms\u003ca name=\"FNanchor_1_1\" id=\"FNanchor_1_1\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_1_1\" class=\"fnanchor\"\u003e[1]\u003c/a\u003e; in modern usage there is,\r\non the whole, a tendency to confine the word \u003cem\u003esubjective\u003c/em\u003e to\r\nPsychology, leaving \u003cem\u003eà priori\u003c/em\u003e to do duty for Epistemology. If\r\nwe accept this differentiation, we may set up, corresponding\r\nto the problems of these two sciences, the following provisional\r\ndefinitions: \u003cem\u003eà priori\u003c/em\u003e applies to any piece of knowledge which,\r\nthough perhaps elicited by experience, is \u003cem\u003elogically\u003c/em\u003e presupposed\r\nin experience: \u003cem\u003esubjective\u003c/em\u003e applies to any mental state whose\r\nimmediate cause lies, not in the external world, but within\r\nthe limits of the subject. The latter definition, of course, is\r\nframed exclusively for Psychology: from the point of view\r\nof physical Science all mental states are subjective. But for\r\na Science whose matter, strictly speaking, is \u003cem\u003eonly\u003c/em\u003e mental states,\r\nwe require, if we are to use the word to any purpose, some\r\ndifferentia among mental states, as a mark of a more special\r\nsubjectivity on the part of those to which this term is applied.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow the only mental states whose immediate causes lie\r\nin the external world are \u003cem\u003esensations\u003c/em\u003e. A pure sensation is, of\r\ncourse, an impossible abstraction\u0026mdash;we are never wholly passive\r\nunder the action of an external stimulus\u0026mdash;but for the purposes\r\nof Psychology the abstraction is a useful one. Whatever, then,\r\nis not sensation, we shall, in Psychology, call subjective. It\r\nis in sensation alone that we are directly affected by the external\r\nworld, and only here does it give us direct information\r\nabout itself.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e3.\u003c/b\u003e\u003ca name=\"N3\" id=\"N3\"\u003e\u003c/a\u003e\r\nLet us now consider the epistemological question, as\r\nto the sort of knowledge which can be called \u003cem\u003eà priori\u003c/em\u003e. Here\r\nwe have nothing to do\u0026mdash;in the first instance, at any rate\u0026mdash;with\r\nthe cause or genesis of a piece of knowledge; we accept\r\nknowledge as a datum to be analysed and classified. Such\r\nanalysis will reveal a formal and a material element in\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_3\" id=\"Page_3\"\u003e[3]\u003c/a\u003e\u003c/span\u003e\r\nknowledge. The formal element will consist of postulates which\r\nare required to make knowledge possible at all, and of all\r\nthat can be deduced from these postulates; the material element,\r\non the other hand, will consist of all that comes to\r\nfill in the form given by the formal postulates\u0026mdash;all that is\r\ncontingent or dependent on experience, all that might have\r\nbeen otherwise without rendering knowledge impossible. We\r\nshall then call the formal element \u003cem\u003eà priori\u003c/em\u003e, the material element\r\nempirical.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e4.\u003c/b\u003e\u003ca name=\"N4\" id=\"N4\"\u003e\u003c/a\u003e\r\nNow what is the connection between the subjective\r\nand the \u003cem\u003eà priori\u003c/em\u003e? It is a connection, obviously\u0026mdash;if it exists\r\nat all\u0026mdash;from the outside, \u003cem\u003ei.e.\u003c/em\u003e not deducible directly from the\r\nnature of either, but provable\u0026mdash;if it can be proved\u0026mdash;only by\r\na general view of the conditions of both. The question, what\r\nknowledge is \u003cem\u003eà priori\u003c/em\u003e, must, on the above definition, depend\r\non a logical analysis of knowledge, by which the conditions\r\nof possible experience may be revealed; but the question, what\r\nelements of a cognitive state are subjective, is to be investigated\r\nby pure Psychology, which has to determine what, in\r\nour perceptions, belongs to sensation, and what is the work\r\nof thought or of association. Since, then, these two questions\r\nbelong to different sciences, and can be settled independently,\r\nwill it not be wise to conduct the two investigations separately?\r\nTo decree that the \u003cem\u003eà priori\u003c/em\u003e shall always be subjective, seems\r\ndangerous, when we reflect that such a view places our results,\r\nas to the \u003cem\u003eà priori\u003c/em\u003e, at the mercy of empirical psychology. How\r\nserious this danger is, the controversy as to Kant\u0027s pure intuition\r\nsufficiently shows.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e5.\u003c/b\u003e\u003ca name=\"N5\" id=\"N5\"\u003e\u003c/a\u003e\r\nI shall, therefore, throughout the present Essay, use\r\nthe word \u003cem\u003eà priori\u003c/em\u003e without any psychological implication. My\r\ntest of apriority will be purely logical: Would experience be\r\nimpossible, if a certain axiom or postulate were denied? Or,\r\nin a more restricted sense, which gives apriority only within\r\na particular science: Would experience as to the subject-matter\r\nof that science be impossible, without a certain axiom or postulate?\r\nMy results also, therefore, will be purely logical. If\r\nPsychology declares that some things, which I have declared\r\n\u003cem\u003eà priori\u003c/em\u003e, are not subjective, then, failing an error of detail in\r\nmy proofs, the connection of the \u003cem\u003eà priori\u003c/em\u003e and the subjective,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_4\" id=\"Page_4\"\u003e[4]\u003c/a\u003e\u003c/span\u003e\r\nso far as those things are concerned, must be given up. There\r\nwill be no discussion, accordingly, throughout this Essay, of\r\nthe relation of the \u003cem\u003eà priori\u003c/em\u003e to the subjective\u0026mdash;a relation which\r\ncannot determine what pieces of knowledge are \u003cem\u003eà priori\u003c/em\u003e, but\r\nrather depends on that determination, and belongs, in any\r\ncase, rather to Metaphysics than to Epistemology.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e6.\u003c/b\u003e\u003ca name=\"N6\" id=\"N6\"\u003e\u003c/a\u003e\r\nAs I have ventured to use the word \u003cem\u003eà priori\u003c/em\u003e in a\r\nslightly unconventional sense, I will give a few elucidatory\r\nremarks of a general nature.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe \u003cem\u003eà priori\u003c/em\u003e, since Kant at any rate, has generally stood\r\nfor the necessary or apodeictic element in knowledge. But\r\nmodern logic has shown that necessary propositions are always,\r\nin one aspect at least, hypothetical. There may be, and usually\r\nis, an implication that the connection, of which necessity is\r\npredicated, has some existence, but still, necessity always points\r\nbeyond itself to a \u003cem\u003eground\u003c/em\u003e of necessity, and asserts this ground\r\nrather than the actual connection. As Bradley points out,\r\n\"arsenic poisons\" remains true, even if it is poisoning no one.\r\nIf, therefore, the \u003cem\u003eà priori\u003c/em\u003e in knowledge be primarily the necessary,\r\nit must be the necessary on some hypothesis, and the\r\n\u003cem\u003eground\u003c/em\u003e of necessity must be included as \u003cem\u003eà priori\u003c/em\u003e. But the\r\nground of necessity is, so far as the necessary connection in\r\nquestion can show, a mere fact, a merely categorical judgment.\r\nHence necessity alone is an insufficient criterion of apriority.\u003c/p\u003e\r\n\r\n\u003cp\u003eTo supplement this criterion, we must supply the hypothesis\r\nor ground, on which alone the necessity holds, and this ground\r\nwill vary from one science to another, and even, with the progress\r\nof knowledge, in the same science at different times.\r\nFor as knowledge becomes more developed and articulate, more\r\nand more necessary connections are perceived, and the merely\r\ncategorical truths, though they remain the foundation of apodeictic\r\njudgments, diminish in relative number. Nevertheless,\r\nin a fairly advanced science such as Geometry, we can, I think,\r\npretty completely supply the appropriate ground, and establish,\r\nwithin the limits of the isolated science, the distinction between\r\nthe necessary and the merely assertorical.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e7.\u003c/b\u003e\u003ca name=\"N7\" id=\"N7\"\u003e\u003c/a\u003e\r\nThere are two grounds, I think, on which necessity\r\nmay be sought within any science. These may be (very\r\nroughly) distinguished as the ground which Kant seeks in the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_5\" id=\"Page_5\"\u003e[5]\u003c/a\u003e\u003c/span\u003e\r\n\u003ccite\u003eProlegomena\u003c/cite\u003e, and that which he seeks in the \u003ccite\u003ePure Reason\u003c/cite\u003e.\r\nWe may start from the existence of our science as a fact, and\r\nanalyse the reasoning employed with a view to discovering\r\nthe fundamental postulate on which its logical possibility depends;\r\nin this case, the postulate, and all which follows from\r\nit alone, will be \u003cem\u003eà priori\u003c/em\u003e. Or we may accept the existence of\r\nthe subject-matter of our science as our basis of fact, and\r\ndeduce dogmatically whatever principles we can from the\r\nessential nature of this subject-matter. In this latter case,\r\nhowever, it is not the whole empirical nature of the subject-matter,\r\nas revealed by the subsequent researches of our science,\r\nwhich forms our ground; for if it were, the whole science\r\nwould, of course, be \u003cem\u003eà priori\u003c/em\u003e. Rather it is that element, in the\r\nsubject-matter, which makes \u003cem\u003epossible\u003c/em\u003e the branch of experience\r\ndealt with by the science in question\u003ca name=\"FNanchor_2_2\" id=\"FNanchor_2_2\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_2_2\" class=\"fnanchor\"\u003e[2]\u003c/a\u003e. The importance of this\r\ndistinction will appear more clearly as we proceed\u003ca name=\"FNanchor_3_3\" id=\"FNanchor_3_3\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_3_3\" class=\"fnanchor\"\u003e[3]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e8.\u003c/b\u003e\u003ca name=\"N8\" id=\"N8\"\u003e\u003c/a\u003e\r\nThese two grounds of necessity, in ultimate analysis, fall\r\ntogether. The \u003cem\u003emethods\u003c/em\u003e of investigation in the two cases differ\r\nwidely, but the \u003cem\u003eresults\u003c/em\u003e cannot differ. For in the first case, by\r\nanalysis of the science, we discover the postulate on which alone\r\nits reasonings are possible. Now if reasoning in the science\r\nis impossible without some postulate, this postulate must be\r\nessential to experience of the subject-matter of the science,\r\nand thus we get the second ground. Nevertheless, the two\r\nmethods are useful as supplementing one another, and the\r\nfirst, as starting from the actual science, is the safest and\r\neasiest method of investigation, though the second seems the\r\nmore convincing for exposition.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e9.\u003c/b\u003e\u003ca name=\"N9\" id=\"N9\"\u003e\u003c/a\u003e\r\nThe course of my argument, therefore, will be as follows:\r\nIn the first chapter, as a preliminary to the logical analysis of\r\nGeometry, I shall give a brief history of the rise and development\r\nof non-Euclidean systems. The second chapter will prepare the\r\nground for a constructive theory of Geometry, by a criticism\r\nof some previous philosophical views; in this chapter, I shall\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_6\" id=\"Page_6\"\u003e[6]\u003c/a\u003e\u003c/span\u003e\r\nendeavour to exhibit such views as partly true, partly false,\r\nand so to establish, by preliminary polemics, the truth of such\r\nparts of my own theory as are to be found in former writers.\r\nA large part of this theory, however, cannot be so introduced,\r\nsince the whole field of projective Geometry, so far as I am\r\naware, has been hitherto unknown to philosophers. Passing,\r\nin the third chapter, from criticism to construction, I shall\r\ndeal first with projective Geometry. This, I shall maintain,\r\nis necessarily true of any form of externality, and is, since\r\nsome such form is necessary to experience, completely \u003cem\u003eà priori\u003c/em\u003e.\r\nIn metrical Geometry, however, which I shall next consider,\r\nthe axioms will fall into two classes: (1) Those common to\r\nEuclidean and non-Euclidean spaces. These will be found,\r\non the one hand, essential to the possibility of measurement\r\nin any continuum, and on the other hand, necessary properties\r\nof any form of externality with more than one dimension.\r\nThey will, therefore, be declared \u003cem\u003eà priori\u003c/em\u003e. (2) Those axioms\r\nwhich distinguish Euclidean from non-Euclidean spaces. These\r\nwill be regarded as wholly empirical. The axiom that the\r\nnumber of dimensions is three, however, though empirical,\r\nwill be declared, since small errors are here impossible, exactly\r\nand certainly true of our actual world; while the two remaining\r\naxioms, which determine the value of the space-constant, will\r\nbe regarded as only approximately known, and certain only\r\nwithin the errors of observation\u003ca name=\"FNanchor_4_4\" id=\"FNanchor_4_4\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_4_4\" class=\"fnanchor\"\u003e[4]\u003c/a\u003e. The fourth chapter, finally,\r\nwill endeavour to prove, what was assumed in Chapter III.,\r\nthat some form of externality is necessary to experience, and\r\nwill conclude by exhibiting the logical impossibility, if knowledge\r\nof such a form is to be freed from contradictions, of\r\nwholly abstracting this knowledge from all reference to the\r\nmatter contained in the form.\u003c/p\u003e\r\n\r\n\u003cp\u003eI shall hope to have touched, with this discussion, on all\r\nthe main points relating to the Foundations of Geometry.\u003c/p\u003e\r\n\r\n\r\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_1_1\" id=\"Footnote_1_1\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_1_1\"\u003e\u003cspan class=\"label\"\u003e[1]\u003c/span\u003e\u003c/a\u003e Cf. Erdmann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAxiome der Geometrie\u003c/cite\u003e, p. 111: \u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"Für Kant sind Apriorität\r\nund ausschliessliche Subjectivität allerdings Wechselbegriffe.\"\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_2_2\" id=\"Footnote_2_2\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_2_2\"\u003e\u003cspan class=\"label\"\u003e[2]\u003c/span\u003e\u003c/a\u003e I use \"experience\" here in the widest possible sense, the sense in which\r\nthe word is used by Bradley.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_3_3\" id=\"Footnote_3_3\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_3_3\"\u003e\u003cspan class=\"label\"\u003e[3]\u003c/span\u003e\u003c/a\u003e Where the branch of experience in question is essential to all experience,\r\nthe resulting apriority may be regarded as absolute; where it is necessary only\r\nto some special science, as relative to that science.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_4_4\" id=\"Footnote_4_4\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_4_4\"\u003e\u003cspan class=\"label\"\u003e[4]\u003c/span\u003e\u003c/a\u003e I have given no account of these empirical proofs, as they seem to be constituted\r\nby the whole body of physical science. Everything in physical science,\r\nfrom the law of gravitation to the building of bridges, from the spectroscope to\r\nthe art of navigation, would be profoundly modified by any considerable inaccuracy\r\nin the hypothesis that our actual space is Euclidean. The observed\r\ntruth of physical science, therefore, constitutes overwhelming empirical evidence\r\nthat this hypothesis is very approximately correct, even if not rigidly true.\u003c/p\u003e\u003c/div\u003e\r\n\u003c/div\u003e\r\n\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_7\" id=\"Page_7\"\u003e[7]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003e\u003ca name=\"CHAP_I\" id=\"CHAP_I\"\u003e\u003c/a\u003eCHAPTER I.\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"fs60\"\u003eA SHORT HISTORY OF METAGEOMETRY.\u003c/span\u003e\u003c/h2\u003e\r\n\r\n\r\n\u003cp\u003e\u003cb\u003e10.\u003c/b\u003e\u003ca name=\"N10\" id=\"N10\"\u003e\u003c/a\u003e\r\nWhen a long established system is attacked, it usually\r\nhappens that the attack begins only at a single point, where\r\nthe weakness of the established doctrine is peculiarly evident.\r\nBut criticism, when once invited, is apt to extend much further\r\nthan the most daring, at first, would have wished.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"poetry-container\"\u003e\u003cdiv class=\"poetry\"\u003e\r\n\u003cp class=\"verseq\"\u003e\"First cut the liquefaction, what comes last,\u003c/p\u003e\r\n\u003cp class=\"verse\"\u003eBut Fichte\u0027s clever cut at God himself?\"\u003c/p\u003e\r\n\u003c/div\u003e\u003c/div\u003e\r\n\r\n\u003cp class=\"noindent\"\u003eSo it has been with Geometry. The liquefaction of Euclidean\r\northodoxy is the axiom of parallels, and it was by the refusal\r\nto admit this axiom without proof that Metageometry began.\r\nThe first effort in this direction, that of Legendre\u003ca name=\"FNanchor_5_5\" id=\"FNanchor_5_5\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_5_5\" class=\"fnanchor\"\u003e[5]\u003c/a\u003e, was inspired\r\nby the hope of deducing this axiom from the others\u0026mdash;a hope\r\nwhich, as we now know, was doomed to inevitable failure.\r\nParallels are defined by Legendre as lines in the same plane,\r\nsuch that, if a third line cut them, it makes the sum of the\r\ninterior and opposite angles equal to two right angles. He\r\nproves without difficulty that such lines would not meet, but\r\nis unable to prove that non-parallel lines in a plane must meet.\r\nSimilarly he can prove that the sum of the angles of a triangle\r\ncannot exceed two right angles, and that if any one triangle has\r\na sum equal to two right angles, all triangles have the same\r\nsum; but he is unable to prove the existence of this one\r\ntriangle.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e11.\u003c/b\u003e\u003ca name=\"N11\" id=\"N11\"\u003e\u003c/a\u003e\r\nThus Legendre\u0027s attempt broke down; but mere failure\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_8\" id=\"Page_8\"\u003e[8]\u003c/a\u003e\u003c/span\u003e\r\ncould prove nothing. A bolder method, suggested by Gauss,\r\nwas carried out by Lobatchewsky and Bolyai\u003ca name=\"FNanchor_6_6\" id=\"FNanchor_6_6\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_6_6\" class=\"fnanchor\"\u003e[6]\u003c/a\u003e. If the axiom\r\nof parallels is logically deducible from the others, we shall, by\r\ndenying it and maintaining the rest, be led to contradictions.\r\nThese three mathematicians, accordingly, attacked the problem\r\nindirectly: they denied the axiom of parallels, and yet obtained\r\na logically consistent Geometry. They inferred that the axiom\r\nwas logically independent of the others, and essential to the\r\nEuclidean system. Their works, being all inspired by this\r\nmotive, may be distinguished as forming the first period in\r\nthe development of Metageometry.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe second period, inaugurated by Riemann, had a much\r\ndeeper import: it was largely philosophical in its aims and\r\nconstructive in its methods. It aimed at no less than a logical\r\nanalysis of all the essential axioms of Geometry, and regarded\r\nspace as a particular case of the more general conception of\r\na \u003cem\u003emanifold\u003c/em\u003e. Taking its stand on the methods of analytical\r\nmetrical Geometry, it established two non-Euclidean systems,\r\nthe first that of Lobatchewsky, the second\u0026mdash;in which the axiom\r\nof the straight line, in Euclid\u0027s form, was also denied\u0026mdash;a new\r\nvariety, by analogy called spherical. The leading conception in\r\nthis period is the \u003cem\u003emeasure of curvature\u003c/em\u003e, a term invented by\r\nGauss, but applied by him only to surfaces. Gauss had shown\r\nthat free mobility on surfaces was only possible when the\r\nmeasure of curvature was constant; Riemann and Helmholtz\r\nextended this proposition to \u003cem\u003en\u003c/em\u003e dimensions, and made it the\r\nfundamental property of space.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the third period, which begins with Cayley, the philosophical\r\nmotive, which had moved the first pioneers, is less\r\napparent, and is replaced by a more technical and mathematical\r\nspirit. This period is chiefly distinguished from the second, in\r\na mathematical point of view, by its method, which is projective\r\ninstead of metrical. The leading mathematical conception here\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_9\" id=\"Page_9\"\u003e[9]\u003c/a\u003e\u003c/span\u003e\r\nis the Absolute (\u003ci lang=\"de\" xml:lang=\"de\"\u003eGrundgebild\u003c/i\u003e), a figure by relation to which all\r\nmetrical properties become projective. Cayley\u0027s work, which\r\nwas very brief, and attracted little attention, has been perfected\r\nand elaborated by F. Klein, and through him has found general\r\nacceptance. Klein has added to the two kinds of non-Euclidean\r\nGeometry already known, a third, which he calls elliptic; this\r\nthird kind closely resembles Helmholtz\u0027s spherical Geometry,\r\nbut is distinguished by the important difference that, in it,\r\ntwo straight lines meet in only one point\u003ca name=\"FNanchor_7_7\" id=\"FNanchor_7_7\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_7_7\" class=\"fnanchor\"\u003e[7]\u003c/a\u003e. The distinctive\r\nmark of the spaces represented by both is that, like the surface\r\nof a sphere, they are finite but unbounded. The reduction of\r\nmetrical to projective properties, as will be proved hereafter,\r\nhas only a technical importance; at the same time, projective\r\nGeometry is able to deal directly with those purely descriptive\r\nor qualitative properties of space which are common to Euclid\r\nand Metageometry alike. The third period has, therefore, great\r\nphilosophical importance, while its method has, mathematically,\r\nmuch greater beauty and unity than that of the second; it is\r\nable to treat all kinds of space at once, so that every symbolic\r\nproposition is, according to the meaning given to the symbols,\r\na proposition in whichever Geometry we choose. This has the\r\nadvantage of proving that further research cannot lead to contradictions\r\nin non-Euclidean systems, unless it at the same\r\nmoment reveals contradictions in Euclid. These systems, therefore,\r\nare logically as sound as that of Euclid himself.\u003c/p\u003e\r\n\r\n\u003cp\u003eAfter this brief sketch of the characteristics of the three\r\nperiods, I will proceed to a more detailed account. It will be\r\nmy aim to avoid, as far as possible, all technical mathematics,\r\nand bring into relief only those fundamental points in the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_10\" id=\"Page_10\"\u003e[10]\u003c/a\u003e\u003c/span\u003e\r\nmathematical development, which seem of logical or philosophical\r\nimportance.\u003c/p\u003e\r\n\r\n\r\n\u003ch3\u003eFirst Period.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e12.\u003c/b\u003e\u003ca name=\"N12\" id=\"N12\"\u003e\u003c/a\u003e\r\nThe originator of the whole system, \u003cem\u003eGauss\u003c/em\u003e, does not\r\nappear, as regards strictly non-Euclidean Geometry, in any of\r\nhis hitherto published papers, to have given more than results;\r\nhis proofs remain unknown to us. Nevertheless he was the\r\nfirst to investigate the consequences of denying the axiom of\r\nparallels\u003ca name=\"FNanchor_8_8\" id=\"FNanchor_8_8\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_8_8\" class=\"fnanchor\"\u003e[8]\u003c/a\u003e, and in his letters he communicated these consequences\r\nto some of his friends, among whom was Wolfgang Bolyai. The\r\nfirst mention of the subject in his letters occurs when he was\r\nonly 18; four years later, in 1799, writing to W. Bolyai, he\r\nenunciates the important theorem that, in hyperbolic Geometry,\r\nthere is a maximum to the area of a triangle. From later\r\nwritings it appears that he had worked out a system nearly, if\r\nnot quite, as complete as those of Lobatchewsky and Bolyai\u003ca name=\"FNanchor_9_9\" id=\"FNanchor_9_9\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_9_9\" class=\"fnanchor\"\u003e[9]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eIt is important to remember, however, that Gauss\u0027s work on\r\ncurvature, which \u003cem\u003ewas\u003c/em\u003e published, laid the foundation for the\r\nwhole method of the second period, and was undertaken,\r\naccording to Riemann and Helmholtz\u003ca name=\"FNanchor_10_10\" id=\"FNanchor_10_10\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_10_10\" class=\"fnanchor\"\u003e[10]\u003c/a\u003e, with a view to an\r\n(unpublished) investigation of the foundations of Geometry.\r\nHis work in this direction will, owing to its method, be better\r\ntreated of under the second period, but it is interesting to\r\nobserve that he stood, like many pioneers, at the head of two\r\ntendencies which afterwards diverged.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e13.\u003c/b\u003e\u003ca name=\"N13\" id=\"N13\"\u003e\u003c/a\u003e\r\n\u003cem\u003eLobatchewsky\u003c/em\u003e, a professor in the University of Kasan,\r\nfirst published his results, in their native Russian, in the\r\nproceedings of that learned body for the years 1829\u0026ndash;1830.\r\nOwing to this double obscurity of language and place, they\r\nattracted little attention, until he translated them into French\u003ca name=\"FNanchor_11_11\" id=\"FNanchor_11_11\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_11_11\" class=\"fnanchor\"\u003e[11]\u003c/a\u003e\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_11\" id=\"Page_11\"\u003e[11]\u003c/a\u003e\u003c/span\u003e\r\nand German\u003ca name=\"FNanchor_12_12\" id=\"FNanchor_12_12\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_12_12\" class=\"fnanchor\"\u003e[12]\u003c/a\u003e: even then, they do not appear to have obtained\r\nthe notice they deserved, until, in 1868, Beltrami unearthed\r\nthe article in Crelle, and made it the theme of a brilliant\r\ninterpretation.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the introduction to his little German book, Lobatchewsky\r\nlaments the slight interest shown in his writings by his compatriots,\r\nand the inattention of mathematicians, since Legendre\u0027s\r\nabortive attempt, to the difficulties in the theory of parallels.\r\nThe body of the work begins with the enunciation of several\r\nimportant propositions which hold good in the system proposed\r\nas well as in Euclid: of these, some are in any case independent\r\nof the axiom of parallels, while others are rendered so by\r\nsubstituting, for the word \"parallel,\" the phrase \"not intersecting,\r\nhowever far produced.\" Then follows a definition,\r\nintentionally framed so as to contradict Euclid\u0027s: With respect\r\nto a given straight line, all others in the same plane may be\r\ndivided into two classes, those which cut the given straight line,\r\nand those which do not cut it; a line which is the limit between\r\nthe two classes is called \u003cem\u003eparallel\u003c/em\u003e to the given straight line. It\r\nfollows that, from any external point, two parallels can be\r\ndrawn, one in each direction. From this starting-point, by\r\nthe Euclidean synthetic method, a series of propositions are\r\ndeduced; the most important of these is, that in a triangle the\r\nsum of the angles is always less than, or always equal to two\r\nright angles, while in the latter case the whole system becomes\r\northodox. A certain analogy with spherical Geometry\u0026mdash;whose\r\nmeaning and extent will appear later\u0026mdash;is also proved, consisting\r\nroughly in the substitution of hyperbolic for circular functions.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e14.\u003c/b\u003e\u003ca name=\"N14\" id=\"N14\"\u003e\u003c/a\u003e\r\nVery similar is the system of \u003cem\u003eJohann Bolyai\u003c/em\u003e, so similar,\r\nindeed, as to make the independence of the two works, though\r\na well-authenticated fact, seem all but incredible. Johann\r\nBolyai first published his results in 1832, in an appendix to\r\na work by his father Wolfgang, entitled; \u003cspan lang=\"la\" xml:lang=\"la\"\u003e\"Appendix, scientiam\r\nspatii absolute veram exhibens: a veritate aut falsitate\r\nAxiomatis XI. Euclidei (a priori haud unquam decidenda)\r\nindependentem; adjecta ad casum falsitatis, quadratura circuli\r\ngeometrica.\"\u003c/span\u003e Gauss, whose bosom friend he became at college\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_12\" id=\"Page_12\"\u003e[12]\u003c/a\u003e\u003c/span\u003e\r\nand remained through life, was, as we have seen, the inspirer of\r\nWolfgang Bolyai, and used to say that the latter was the only\r\nman who appreciated his philosophical speculations on the\r\naxioms of Geometry; nevertheless, Wolfgang appears to have\r\nleft to his son Johann the detailed working out of the hyperbolic\r\nsystem. The works of both the Bolyai are very rare, and\r\ntheir method and results are known to me only through the\r\nrenderings of Frischauf and Halsted\u003ca name=\"FNanchor_13_13\" id=\"FNanchor_13_13\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_13_13\" class=\"fnanchor\"\u003e[13]\u003c/a\u003e. Both as to method and\r\nas to results, the system is very similar to Lobatchewsky\u0027s, so\r\nthat neither need detain us here. Only the initial postulates,\r\nwhich are more explicit than Lobatchewsky\u0027s, demand a brief\r\nattention. Frischauf\u0027s introduction, which has a philosophical\r\nand Newtonian air, begins by setting forth that Geometry deals\r\nwith absolute (empty) space, obtained by abstracting from the\r\nbodies in it, that two figures are called congruent when they\r\ndiffer only in position, and that the axiom of Congruence is\r\nindispensable in all determination of spatial magnitudes. Congruence\r\nwas to refer to geometrical bodies, with none of the\r\nproperties of ordinary bodies except impenetrability (Erdmann,\r\n\u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAxiome der Geometrie\u003c/cite\u003e, p. 26). A straight line is defined as\r\ndetermined by two of its points\u003ca name=\"FNanchor_14_14\" id=\"FNanchor_14_14\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_14_14\" class=\"fnanchor\"\u003e[14]\u003c/a\u003e, and a plane as determined by\r\nthree. These premisses, with a slight exception as to the straight\r\nline, we shall hereafter find essential to every Geometry. I have\r\ndrawn attention to them, as it is often supposed that non-Euclideans\r\ndeny the axiom of Congruence, which, here and\r\nelsewhere, is never the case. The stress laid on this axiom by\r\nBolyai is probably due to the influence of Gauss, whose work on\r\nthe curvature of surfaces laid the foundation for the use made\r\nof congruence by Helmholtz.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e15.\u003c/b\u003e\u003ca name=\"N15\" id=\"N15\"\u003e\u003c/a\u003e\r\nIt is important to remember that, throughout the\r\nperiod we have just reviewed, the purpose of hyperbolic\r\nGeometry is indirect: not the truth of the latter, but the\r\nlogical independence of the axiom of parallels from the rest, is\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_13\" id=\"Page_13\"\u003e[13]\u003c/a\u003e\u003c/span\u003e\r\nthe guiding motive of the work. If, by denying the axiom of\r\nparallels while retaining the rest, we can obtain a system free\r\nfrom logical contradictions, it follows that the axiom of parallels\r\ncannot be implicitly contained in the others. If this be so,\r\nattempts to dispense with the axiom, like Legendre\u0027s, cannot be\r\nsuccessful; Euclid must stand or fall with the suspected axiom.\r\nOf course, it remained possible that, by further development,\r\nlatent contradictions might have been revealed in these systems.\r\nThis possibility, however, was removed by the more direct and\r\nconstructive work of the second period, to which we must now\r\nturn our attention.\u003c/p\u003e\r\n\r\n\r\n\u003ch3\u003eSecond Period.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e16.\u003c/b\u003e\u003ca name=\"N16\" id=\"N16\"\u003e\u003c/a\u003e\r\nThe work of Lobatchewsky and Bolyai remained, for\r\nnearly a quarter of a century, without issue\u0026mdash;indeed, the\r\ninvestigations of Riemann and Helmholtz, when they came,\r\nappear to have been inspired, not by these men, but rather by\r\nGauss\u003ca name=\"FNanchor_15_15\" id=\"FNanchor_15_15\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_15_15\" class=\"fnanchor\"\u003e[15]\u003c/a\u003e and Herbart. We find, accordingly, very great difference,\r\nboth of aim and method, between the first period and the second.\r\nThe former, beginning with a criticism of one point in Euclid\u0027s\r\nsystem, preserved his synthetic method, while it threw over one\r\nof his axioms. The latter, on the contrary, being guided by a\r\nphilosophical rather than a mathematical spirit, endeavoured to\r\nclassify the conception of space as a species of a more general\r\nconception: it treated space algebraically, and the properties it\r\ngave to space were expressed in terms, not of intuition, but of\r\nalgebra. The aim of Riemann and Helmholtz was to show, by\r\nthe exhibition of logically possible alternatives, the empirical\r\nnature of the received axioms. For this purpose, they conceived\r\nspace as a particular case of a manifold, and showed that various\r\nrelations of magnitude (\u003ci lang=\"de\" xml:lang=\"de\"\u003eMassverhältnisse\u003c/i\u003e) were mathematically\r\npossible in an extended manifold. Their philosophy, which\r\nseems to me not always irreproachable, will be discussed in\r\nChapter II.; here, while it is important to remember the\r\nphilosophical motive of Riemann and Helmholtz, we shall\r\nconfine our attention to the mathematical side of their work.\r\nIn so doing, while we shall, I fear, somewhat maim the system\r\nof their thoughts, we shall secure a closer unity of subject, and\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_14\" id=\"Page_14\"\u003e[14]\u003c/a\u003e\u003c/span\u003e\r\na more compact account of the purely mathematical development.\r\nBut there is, in my opinion, a further reason for\r\nseparating their philosophy from their mathematics. While\r\ntheir philosophical purpose was, to prove that all the axioms\r\nof Geometry are empirical, and that a different content of our\r\nexperience might have changed them all, the unintended result\r\nof their mathematical work was, if I am not mistaken, to afford\r\nmaterial for an \u003cem\u003eà priori\u003c/em\u003e proof of certain axioms. These axioms,\r\nthough they believed them to be unnecessary, were always\r\nintroduced in their mathematical works, before laying the\r\nfoundations of non-Euclidean systems. I shall contend, in\r\nChapter III., that this retention was logically inevitable, and\r\nwas not merely due, as they supposed, to a desire for conformity\r\nwith experience. If I am right in this, there is a divergence\r\nbetween Riemann and Helmholtz the philosophers, and Riemann\r\nand Helmholtz the mathematicians. This divergence makes it\r\nthe more desirable to trace the mathematical development\r\napart from the accompanying philosophy.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e17.\u003c/b\u003e\u003ca name=\"N17\" id=\"N17\"\u003e\u003c/a\u003e\r\n\u003cem\u003eRiemann\u0027s\u003c/em\u003e epoch-making work, \"\u003ccite lang=\"de\" xml:lang=\"de\"\u003eUeber die Hypothesen,\r\nwelche der Geometrie zu Grande liegen\u003c/cite\u003e\u003ca name=\"FNanchor_16_16\" id=\"FNanchor_16_16\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_16_16\" class=\"fnanchor\"\u003e[16]\u003c/a\u003e\", was written, and read\r\nto a small circle, in 1854; owing, however, to some changes\r\nwhich he desired to make in it, it remained unpublished till\r\n1867, when it was published by his executors. The two\r\nfundamental conceptions, on whose invention rests the historic\r\nimportance of this dissertation, are that of a \u003cem\u003emanifold\u003c/em\u003e, and\r\nthat of the \u003cem\u003emeasure of curvature\u003c/em\u003e of a manifold. The former\r\nconception serves a mainly philosophical purpose, and is designed,\r\nprincipally, to exhibit space as an instance of a more\r\ngeneral conception. On this aspect of the manifold, I shall\r\nhave much to say in Chapter II.; its mathematical aspect,\r\nwhich alone concerns us here, is less complicated and less\r\nfruitful of controversy. The latter conception also serves a\r\ndouble purpose, but its mathematical use is the more prominent.\r\nWe will consider these two conceptions successively.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e18.\u003c/b\u003e\u003ca name=\"N18\" id=\"N18\"\u003e\u003c/a\u003e\r\n(1) \u003cem\u003eConception of a manifold\u003ca name=\"FNanchor_17_17\" id=\"FNanchor_17_17\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_17_17\" class=\"fnanchor\"\u003e[17]\u003c/a\u003e.\u003c/em\u003e The general purpose\r\nof Riemann\u0027s dissertation is, to exhibit the axioms as successive\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_15\" id=\"Page_15\"\u003e[15]\u003c/a\u003e\u003c/span\u003e\r\nsteps in the classification of the species space. The axioms of\r\nGeometry, like the marks of a scholastic definition, appear as\r\nsuccessive determinations of class-conceptions, ending with\r\nEuclidean space. We have thus, from the analytical point of\r\nview, about as logical and precise a formulation as can be\r\ndesired\u0026mdash;a formulation in which, from its classificatory character,\r\nwe seem certain of having nothing superfluous or redundant, and\r\nobtain the axioms explicitly in the most desirable form, namely\r\nas adjectives of the conception of space. At the same time, it\r\nis a pity that Riemann, in accordance with the metrical bias\r\nof his time, regarded space as primarily a magnitude\u003ca name=\"FNanchor_18_18\" id=\"FNanchor_18_18\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_18_18\" class=\"fnanchor\"\u003e[18]\u003c/a\u003e, or\r\nassemblage of magnitudes, in which the main problem consists\r\nin assigning quantities to the different elements or points,\r\nwithout regard to the qualitative nature of the quantities\r\nassigned. Considerable obscurity thus arises as to the whole\r\nnature of magnitude\u003ca name=\"FNanchor_19_19\" id=\"FNanchor_19_19\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_19_19\" class=\"fnanchor\"\u003e[19]\u003c/a\u003e. This view of Geometry underlies the\r\ndefinition of the manifold, as the general conception of which\r\nspace forms a special case. This definition, which is not very\r\nclear, may be rendered as follows.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e19.\u003c/b\u003e\u003ca name=\"N19\" id=\"N19\"\u003e\u003c/a\u003e\r\nConceptions of magnitude, according to Riemann, are\r\npossible there only, where we have a general conception,\r\ncapable of various determinations (\u003ci lang=\"de\" xml:lang=\"de\"\u003eBestimmungsweisen\u003c/i\u003e). The\r\nvarious determinations of such a conception together form a\r\n\u003cem\u003emanifold\u003c/em\u003e, which is continuous or discrete, according as the passage\r\nfrom one determination to another is continuous or discrete.\r\nParticular bits of a manifold, or quanta, can be compared by\r\ncounting when discrete, and by measurement when continuous.\r\n\"Measurement consists in a superposition of the magnitudes to\r\nbe compared. If this be absent, magnitudes can only be\r\ncompared when one is part of another, and then only the more\r\nor less, not the how much, can be decided\" (p. 256). We thus\r\nreach the general conception of a manifold of several dimensions,\r\nof which space and colours are mentioned as special cases.\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_16\" id=\"Page_16\"\u003e[16]\u003c/a\u003e\u003c/span\u003e\r\nTo the absence of this conception Riemann attributes the\r\n\"obscurity\" which, on the subject of the axioms, \"lasted from\r\nEuclid to Legendre\" (p. 254). And Riemann certainly has\r\nsucceeded, from an algebraic point of view, in exhibiting, far\r\nmore clearly than any of his predecessors, the axioms which\r\ndistinguish spatial quantity from other quantities with which\r\nmathematics is conversant. But by the assumption, from the\r\nstart, that space can be regarded as a quantity, he has been led\r\nto state the problem as: What sort of magnitude is space?\r\nrather than: What must space be in order that we may be able\r\nto regard it as a magnitude at all? He does not realise,\r\neither\u0026mdash;indeed in his day there were few who realized\u0026mdash;that\r\nan elaborate Geometry is possible which does not deal with\r\nspace as a quantity at all. His definition of space as a species\r\nof manifold, therefore, though for analytical purposes it defines,\r\nmost satisfactorily, the nature of spatial magnitudes, leaves\r\nobscure the true ground for this nature, which lies in the\r\nnature of space as a system of relations, and is anterior to the\r\npossibility of regarding it as a system of magnitudes at all.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut to proceed with the mathematical development of\r\nRiemann\u0027s ideas. We have seen that he declared measurement\r\nto consist in a superposition of the magnitudes to be compared.\r\nBut in order that this may be a possible means of determining\r\nmagnitudes, he continues, these magnitudes must be independent\r\nof their position in the manifold (p. 259). This can\r\noccur, he says, in several ways, as the simplest of which, he\r\nassumes that the lengths of lines are independent of their\r\nposition. One would be glad to know what other ways are\r\npossible: for my part, I am unable to imagine any other\r\nhypothesis on which magnitude would be independent of place.\r\nSetting this aside, however, the problem, owing to the fact that\r\nmeasurement consists in superposition, becomes identical with\r\nthe determination of the most general manifold in which\r\nmagnitudes are independent of place. This brings us to\r\nRiemann\u0027s other fundamental conception, which seems to me\r\neven more fruitful than that of a manifold.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e20.\u003c/b\u003e\u003ca name=\"N20\" id=\"N20\"\u003e\u003c/a\u003e\r\n(2) \u003cem\u003eMeasure of curvature.\u003c/em\u003e This conception is due to\r\nGauss, but was applied by him only to surfaces; the novelty in\r\nRiemann\u0027s dissertation was its extension to a manifold of \u003cem\u003en\u003c/em\u003e\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_17\" id=\"Page_17\"\u003e[17]\u003c/a\u003e\u003c/span\u003e\r\ndimensions. This extension, however, is rather briefly and\r\nobscurely expressed, and has been further obscured by Helmholtz\u0027s\r\nattempts at popular exposition. The term \u003cem\u003ecurvature\u003c/em\u003e,\r\nalso, is misleading, so that the phrase has been the source of\r\nmore misunderstanding, even among mathematicians, than any\r\nother in Pangeometry. It is often forgotten, in spite of\r\nHelmholtz\u0027s explicit statement\u003ca name=\"FNanchor_20_20\" id=\"FNanchor_20_20\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_20_20\" class=\"fnanchor\"\u003e[20]\u003c/a\u003e, that the \"measure of curvature\"\r\nof an \u003cem\u003en\u003c/em\u003e-dimensional manifold is a purely analytical\r\nexpression, which has only a symbolic affinity to ordinary\r\ncurvature. As applied to three-dimensional space, the implication\r\nof a four-dimensional \"plane\" space is wholly misleading;\r\nI shall, therefore, generally use the term space-constant instead\u003ca name=\"FNanchor_21_21\" id=\"FNanchor_21_21\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_21_21\" class=\"fnanchor\"\u003e[21]\u003c/a\u003e.\r\nNevertheless, as the conception grew, historically, out of that\r\nof curvature, I will give a very brief exposition of the historical\r\ndevelopment of theories of curvature.\u003c/p\u003e\r\n\r\n\u003cp\u003eJust as the notion of \u003cem\u003elength\u003c/em\u003e was originally derived from the\r\nstraight line, and extended to other curves by dividing them\r\ninto infinitesimal straight lines, so the notion of \u003cem\u003ecurvature\u003c/em\u003e was\r\nderived from the circle, and extended to other curves by\r\ndividing them into infinitesimal circular arcs. Curvature may\r\nbe regarded, originally, as a measure of the amount by which a\r\ncurve departs from a straight line; in a circle, which is similar\r\nthroughout, this amount is evidently constant, and is measured\r\nby the reciprocal of the radius. But in all other curves, the\r\namount of curvature varies from point to point, so that it\r\ncannot be measured without infinitesimals. The measure\r\nwhich at once suggests itself is, the curvature of the circle most\r\nnearly coinciding with the curve at the point considered.\r\nSince a circle is determined by three points, this circle will\r\npass through three consecutive points of the curve. We have\r\nthus defined the curvature of any curve, plane or tortuous; for,\r\nsince any three points lie in a plane, such a circle can always\r\nbe described.\u003c/p\u003e\r\n\r\n\u003cp\u003eIf we now pass to a surface, what we want is, by analogy,\r\na measure of its departure from a plane. The curvature, as\r\nabove defined, has become indeterminate, for through any point\r\nof the surface we can draw an infinite number of arcs, which\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_18\" id=\"Page_18\"\u003e[18]\u003c/a\u003e\u003c/span\u003e\r\nwill not, in general, all have the same curvature. Let us, then,\r\ndraw all the geodesics joining the point in question to neighbouring\r\npoints of the surface in all directions. Since these\r\narcs form a singly infinite manifold, there will be among\r\nthem, if they have not all the same curvature, one arc of\r\nmaximum, and one of minimum curvature\u003ca name=\"FNanchor_22_22\" id=\"FNanchor_22_22\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_22_22\" class=\"fnanchor\"\u003e[22]\u003c/a\u003e. The product of\r\nthese maximum and minimum curvatures is called the \u003cem\u003emeasure\r\nof curvature\u003c/em\u003e of the surface at the point under consideration.\r\nTo illustrate by a few simple examples: on a sphere, the\r\ncurvatures of all such lines are equal to the reciprocal of the\r\nradius of the sphere, hence the measure of curvature everywhere\r\nis the square of the reciprocal of the radius of the sphere.\r\nOn any surface, such as a cone or a cylinder, on which straight\r\nlines can be drawn, these have no curvature, so that the\r\nmeasure of curvature is everywhere zero\u0026mdash;this is the case, in\r\nparticular, with the plane. In general, however, the measure\r\nof curvature of a surface varies from point to point.\u003c/p\u003e\r\n\r\n\u003cp\u003eGauss, the inventor of this conception\u003ca name=\"FNanchor_23_23\" id=\"FNanchor_23_23\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_23_23\" class=\"fnanchor\"\u003e[23]\u003c/a\u003e, proved that, in\r\norder that two surfaces may be developable upon each other\u0026mdash;\u003cem\u003ei.e.\u003c/em\u003e\r\nmay be such that one can be bent into the shape of the\r\nother without stretching or tearing\u0026mdash;it is necessary that\r\nthe two surfaces should have equal measures of curvature at\r\ncorresponding points. When this is the case, every figure\r\nwhich is possible on the one is, in general, possible on the\r\nother, and the two have practically the same Geometry\u003ca name=\"FNanchor_24_24\" id=\"FNanchor_24_24\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_24_24\" class=\"fnanchor\"\u003e[24]\u003c/a\u003e. As\r\na corollary, it follows that a necessary condition, for the free\r\nmobility of figures on any surface, is the constancy of the\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_19\" id=\"Page_19\"\u003e[19]\u003c/a\u003e\u003c/span\u003e\r\nmeasure of curvature\u003ca name=\"FNanchor_25_25\" id=\"FNanchor_25_25\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_25_25\" class=\"fnanchor\"\u003e[25]\u003c/a\u003e. This condition was proved to be\r\nsufficient, as well as necessary, by Minding\u003ca name=\"FNanchor_26_26\" id=\"FNanchor_26_26\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_26_26\" class=\"fnanchor\"\u003e[26]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e21.\u003c/b\u003e\u003ca name=\"N21\" id=\"N21\"\u003e\u003c/a\u003e\r\nSo far, all has been plain sailing\u0026mdash;we have been dealing\r\nwith purely geometrical ideas in a purely geometrical manner\u0026mdash;but\r\nwe have not, as yet, found any sense of the measure of\r\ncurvature, in which it can be extended to space, still less to\r\nan \u003cem\u003en\u003c/em\u003e-dimensional manifold. For this purpose, we must examine\r\nGauss\u0027s method, which enables us to determine the measure\r\nof curvature of a surface at any point as an inherent property,\r\nquite independent of any reference to the third dimension.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe method of determining the measure of curvature from\r\nwithin is, briefly, as follows: If any point on the surface be\r\ndetermined by two coordinates, \u003cem\u003eu\u003c/em\u003e, \u003cem\u003ev\u003c/em\u003e, then small arcs of the\r\nsurface are given by the formula\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003eds\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e = \u003cem\u003eEdu\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e + \u003cem\u003e2Fdu dv\u003c/em\u003e + \u003cem\u003eGdv\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e,\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"noindent\"\u003ewhere \u003cem\u003eE\u003c/em\u003e, \u003cem\u003eF\u003c/em\u003e, \u003cem\u003eG\u003c/em\u003e are, in general, functions of \u003cem\u003eu\u003c/em\u003e, \u003cem\u003ev\u003c/em\u003e.\u003ca name=\"FNanchor_27_27\" id=\"FNanchor_27_27\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_27_27\" class=\"fnanchor\"\u003e[27]\u003c/a\u003e From this\r\nformula alone, without reference to any space outside the surface,\r\nwe can determine the measure of curvature at the point\r\n\u003cem\u003eu\u003c/em\u003e, \u003cem\u003ev\u003c/em\u003e, as a function of \u003cem\u003eE\u003c/em\u003e, \u003cem\u003eF\u003c/em\u003e, \u003cem\u003eG\u003c/em\u003e and their differentials with respect\r\nto \u003cem\u003eu\u003c/em\u003e and \u003cem\u003ev\u003c/em\u003e. Thus we may regard the measure of curvature of\r\na surface as an inherent property, and the above geometrical\r\ndefinition, which involved a reference to the third dimension,\r\nmay be dropped. But at this point a caution is necessary. It\r\nwill appear in \u003ca href=\"#N176\"\u003eChap. III. (§ 176)\u003c/a\u003e, that it is logically impossible\r\nto set up a precise coordinate system, in which the coordinates\r\nrepresent spatial magnitudes, without the axiom of Free\r\nMobility, and this axiom, as we have just seen, holds on surfaces\r\nonly when the measure of curvature is constant. Hence\r\nour definition of the measure of curvature will only be \u003cem\u003ereally\u003c/em\u003e\r\nfree from reference to the third dimension, when we are dealing\r\nwith a surface of constant measure of curvature\u0026mdash;a point which\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_20\" id=\"Page_20\"\u003e[20]\u003c/a\u003e\u003c/span\u003eRiemann entirely overlooks. This caution, however, applies\r\nonly in space, and if we take the coordinate system as presupposed\r\nin the conception of a manifold, we may neglect the\r\ncaution altogether\u0026mdash;while remembering that the possibility of\r\na coordinate system in space involves axioms to be investigated\r\nlater. We can thus see how a meaning might be found,\r\nwithout reference to any higher dimension, for a constant\r\nmeasure of curvature of three-dimensional space, or for any\r\nmeasure of curvature of an \u003cem\u003en\u003c/em\u003e-dimensional manifold in general.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e22.\u003c/b\u003e\u003ca name=\"N22\" id=\"N22\"\u003e\u003c/a\u003e\r\nSuch a meaning is supplied by Riemann\u0027s dissertation,\r\nto which, after this long digression, we can now return. We\r\nmay define a continuous manifold as any continuum of elements,\r\nsuch that a single element is defined by \u003cem\u003en\u003c/em\u003e continuously variable\r\nmagnitudes. This definition does not really include space, for\r\ncoordinates in space do not define a point, but its relations to\r\nthe origin, which is itself arbitrary. It includes, however, the\r\nanalytical conception of space with which Riemann deals, and\r\nmay, therefore, be allowed to stand for the moment. Riemann\r\nthen assumes that the difference\u0026mdash;or distance, as it may be\r\nloosely called\u0026mdash;between any two elements is comparable, as\r\nregards magnitude, to the difference between any other two.\r\nHe assumes further, what it is Helmholtz\u0027s merit to have\r\nproved, that the difference \u003cem\u003eds\u003c/em\u003e between two consecutive elements\r\ncan be expressed as the square root of a quadratic function of\r\nthe differences of the coordinates: \u003cem\u003ei.e.\u003c/em\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003eds\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e = Σ\u003csub\u003e1\u003c/sub\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\r\n Σ\u003csub\u003e1\u003c/sub\u003e\u003csup\u003e\u003cem\u003en\u003c/em\u003e\u003c/sup\u003e\r\n \u003cem\u003ea\u003csub\u003eik\u003c/sub\u003e dx\u003csub\u003ei\u003c/sub\u003e.dx\u003csub\u003ek\u003c/sub\u003e\u003c/em\u003e ,\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"noindent\"\u003ewhere the coefficients \u003cem\u003ea\u003csub\u003eik\u003c/sub\u003e\u003c/em\u003e are, in general, functions of the coordinates\r\n\u003cem\u003ex\u003csub\u003e1\u003c/sub\u003e x\u003csub\u003e2\u003c/sub\u003e … x\u003csub\u003e\u003cem\u003en\u003c/em\u003e\u003c/sub\u003e\u003c/em\u003e.\r\n\u003ca name=\"FNanchor_28_28\" id=\"FNanchor_28_28\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_28_28\" class=\"fnanchor\"\u003e[28]\u003c/a\u003e\r\nThe question is: How are we to obtain a\r\ndefinition of the measure of curvature out of this formula? It is\r\nnoticeable, in the first place, that, just as in a surface we found\r\nan infinite number of \u003cem\u003eradii\u003c/em\u003e of curvature at a point, so in a\r\nmanifold of three or more dimensions we must find an infinite\r\nnumber of \u003cem\u003emeasures\u003c/em\u003e of curvature at a point, one for every two-dimensional\r\nmanifold passing through the point, and contained\r\nin the higher manifold. What we have first to do, therefore, is\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_21\" id=\"Page_21\"\u003e[21]\u003c/a\u003e\u003c/span\u003e\r\nto define such two-dimensional manifolds. They must consist,\r\nas we saw on the surface, of a singly infinite series of geodesics\r\nthrough the point. Now a geodesic is completely determined\r\nby one point and its direction at that point, or by one point\r\nand the next consecutive point. Hence a geodesic through\r\nthe point considered is determined by the ratios of the increments\r\nof coordinates, \u003cem\u003edx\u003csub\u003e1\u003c/sub\u003e dx\u003csub\u003e2\u003c/sub\u003e … dx\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e. Suppose we have two\r\nsuch geodesics, in which the \u003cem\u003ei\u003c/em\u003e′th increments are respectively\r\n\u003cem\u003ed′x\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e and \u003cem\u003ed″x\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e. Then all the geodesics given by\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003edx\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e = λ′\u003cem\u003ed′x\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e + λ″\u003cem\u003ed″x\u003csub\u003ei\u003c/sub\u003e\u003c/em\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figright\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep021.jpg\" width=\"200\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp class=\"noindent\"\u003eform a singly infinite series, since they contain one parameter,\r\nnamely λ′: λ″. Such a series of geodesics, therefore,\r\nmust form a two-dimensional manifold,\r\nwith a measure of curvature\r\nin the ordinary Gaussian sense.\r\nThis measure of curvature can be\r\ndetermined from the above formula\r\nfor the elementary arc, by\r\nthe help of Gauss\u0027s general formula\r\nalluded to above. We thus obtain an infinite number of\r\nmeasures of curvature at a point, but from\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003en\u003c/em\u003e.(\u003cem\u003en\u003c/em\u003e\u0026nbsp;\u0026ndash;\u0026nbsp;1)\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e2\u003c/span\u003e\r\n\u003c/span\u003e\r\nof these,\r\nthe rest can be deduced (Riemann, Gesammelte Werke,\r\np. 262). When all the measures of curvature at a point are\r\nconstant, and equal to all the measures of curvature at any\r\nother point, we get what Riemann calls a manifold of constant\r\ncurvature. In such a manifold free mobility is possible, and\r\npositions do not differ intrinsically from one another. If \u003cem\u003ea\u003c/em\u003e\r\nbe the measure of curvature, the formula for the arc becomes,\r\nin this case,\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003eds\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e =\r\nΣ\u003cem\u003edx\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e \u003csub\u003e\u003csub\u003e\u003cspan class=\"xxl\"\u003e/\u003c/span\u003e\u003c/sub\u003e\u003c/sub\u003e\r\n\u003cspan class=\"xxl\"\u003e(\u003c/span\u003e1 +\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003ea\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e4\u003c/span\u003e\r\n\u003c/span\u003e\r\nΣ\u003cem\u003ex\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e\u003cspan class=\"xxl\"\u003e)\u003c/span\u003e\u003csup\u003e\u003csup\u003e\u003csup\u003e\u003cspan class=\"xs\"\u003e2\u003c/span\u003e\u003c/sup\u003e\u003c/sup\u003e\u003c/sup\u003e.\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"noindent\"\u003eIn this case only, as I pointed out above, can the term \"measure\r\nof curvature\" be properly applied to space without reference\r\nto a higher dimension, since free mobility is logically indispensable\r\nto the existence of quantitative or metrical Geometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e23.\u003c/b\u003e\u003ca name=\"N23\" id=\"N23\"\u003e\u003c/a\u003e\r\nThe mathematical result of Riemann\u0027s dissertation\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_22\" id=\"Page_22\"\u003e[22]\u003c/a\u003e\u003c/span\u003e\r\nmay be summed up as follows. Assuming it possible to apply\r\nmagnitude to space, \u003cem\u003ei.e.\u003c/em\u003e to determine its elements and figures\r\nby means of algebraical quantities, it follows that space can be\r\nbrought under the conception of a manifold, as a system of\r\nquantitatively determinable elements. Owing, however, to the\r\npeculiar nature of spatial measurement, the quantitative determination\r\nof space demands that magnitudes shall be independent\r\nof place\u0026mdash;in so far as this is not the case, our measurement will\r\nbe necessarily inaccurate. If we now assume, as the quantitative\r\nrelation of distance between two elements, the square root of a\r\nquadratic function of the coordinates\u0026mdash;a formula subsequently\r\nproved by Helmholtz and Lie\u0026mdash;then it follows, since magnitudes\r\nare to be independent of place, that space must, within the\r\nlimits of observation, have a constant measure of curvature, or\r\nmust, in other words, be homogeneous in all its parts. In the\r\ninfinitesimal, Riemann says (p. 267), observation could not\r\ndetect a departure from constancy on the part of the measure\r\nof curvature; but he makes no attempt to show how Geometry\r\ncould remain possible under such circumstances, and the only\r\nGeometry he has constructed is based entirely on Free Mobility.\r\nI shall endeavour to prove, in Chapter III., that any metrical\r\nGeometry, which should endeavour to dispense with this axiom,\r\nwould be logically impossible. At present I will only point out\r\nthat Riemann, in spite of his desire to prove that all the axioms\r\ncan be dispensed with, has nevertheless, in his mathematical\r\nwork, retained three fundamental axioms, namely, Free Mobility,\r\nthe finite integral number of dimensions, and the axiom that\r\ntwo points have a unique relation, namely distance. These, as\r\nwe shall see hereafter, are retained, in actual mathematical\r\nwork, by all metrical Metageometers, even when they believe,\r\nlike Riemann and Helmholtz, that no axioms are philosophically\r\nindispensable.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e24.\u003c/b\u003e\u003ca name=\"N24\" id=\"N24\"\u003e\u003c/a\u003e\r\n\u003cem\u003eHelmholtz\u003c/em\u003e, the historically nearest follower of Riemann,\r\nwas guided by a similar empirical philosophy, and arrived\r\nindependently at a very similar method of formulating the\r\naxioms. Although Helmholtz published nothing on the subject\r\nuntil after Riemann\u0027s death, he had then only just seen\r\nRiemann\u0027s dissertation (which was published posthumously),\r\nand had worked out his results, so far as they were then\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_23\" id=\"Page_23\"\u003e[23]\u003c/a\u003e\u003c/span\u003e\r\ncompleted, in entire independence both of Riemann and of\r\nLobatchewsky. Helmholtz is by far the most widely read of\r\nall writers on Metageometry, and his writings, almost alone,\r\nrepresent to philosophers the modern mathematical standpoint\r\non this subject. But his importance is much greater, in this\r\ndomain, as a philosopher than as a mathematician; almost his\r\nonly original mathematical result, as regards Geometry, is his\r\nproof of Riemann\u0027s formula for the infinitesimal arc, and even\r\nthis proof was far from rigid, until Lie reformed it by his\r\nmethod of continuous groups. In this chapter, therefore, only\r\ntwo of his writings need occupy us, namely the two articles\r\nin the \u003ccite lang=\"de\" xml:lang=\"de\"\u003eWissenschaftliche Abhandlungen\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, entitled respectively\r\n\u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"Ueber die thatsächlichen Grundlagen der Geometrie,\"\u003c/span\u003e\r\n1866 (p. 610 ff.), and \u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"Ueber die Thatsachen, die der Geometrie\r\nzum Grunde liegen,\"\u003c/span\u003e 1868 (p. 618 ff.).\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e25.\u003c/b\u003e\u003ca name=\"N25\" id=\"N25\"\u003e\u003c/a\u003e\r\nIn the first of these, which is chiefly philosophical,\r\nHelmholtz gives hints of his then uncompleted mathematical\r\nwork, but in the main contents himself with a statement of\r\nresults. He announces that he will prove Riemann\u0027s quadratic\r\nformula for the infinitesimal arc; but for this purpose, he says,\r\nwe have to \u003cem\u003estart\u003c/em\u003e with Congruence, since without it spatial\r\nmeasurement is impossible. Nevertheless, he maintains that\r\nCongruence is proved by experience. How we could, without\r\nthe help of measurement, discover lapses from Congruence, is a\r\npoint which he leaves undiscussed. He then enunciates the\r\nfour axioms which he considers essential to Geometry, as\r\nfollows:\u003c/p\u003e\r\n\r\n\u003cp\u003e(1) \u003cem\u003eAs regards continuity and dimensions.\u003c/em\u003e In a space of\r\n\u003cem\u003en\u003c/em\u003e dimensions, a point is uniquely determined by the measurement\r\nof \u003cem\u003en\u003c/em\u003e continuous variables (coordinates).\u003c/p\u003e\r\n\r\n\u003cp\u003e(2) \u003cem\u003eAs regards the existence of moveable rigid bodies.\u003c/em\u003e\r\nBetween the 2\u003cem\u003en\u003c/em\u003e coordinates of any point-pair of a rigid body,\r\nthere exists an equation which is the same for all congruent\r\npoint-pairs. By considering a sufficient number of point-pairs,\r\nwe get more equations than unknown quantities: this gives us\r\na method of determining the form of these equations, so as to\r\nmake it possible for them all to be satisfied.\u003c/p\u003e\r\n\r\n\u003cp\u003e(3) \u003cem\u003eAs regards free mobility.\u003c/em\u003e Every point can pass freely\r\nand continuously from one position to another. From (2) and\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_24\" id=\"Page_24\"\u003e[24]\u003c/a\u003e\u003c/span\u003e\r\n(3) it follows, that if two systems \u003cem\u003eA\u003c/em\u003e and \u003cem\u003eB\u003c/em\u003e can be brought into\r\ncongruence in any one position, this is also possible in every\r\nother position.\u003c/p\u003e\r\n\r\n\u003cp\u003e(4) \u003cem\u003eAs regards independence of rotation in rigid bodies\u003c/em\u003e\r\n(Monodromy). If (\u003cem\u003en\u003c/em\u003e\u0026nbsp;\u0026ndash;\u0026nbsp;1) points of a body remain fixed, so that\r\nevery other point can only describe a certain curve, then that\r\ncurve is closed.\u003c/p\u003e\r\n\r\n\u003cp\u003eThese axioms, says Helmholtz, suffice to give, with the\r\naxiom of three dimensions, the Euclidean and non-Euclidean\r\nsystems as the only alternatives. That they \u003cem\u003esuffice\u003c/em\u003e, mathematically,\r\ncannot be denied, but they seem, in some respects,\r\nto go too far. In the first place, there is no necessity to make\r\nthe axiom of Congruence apply to actual rigid bodies\u0026mdash;on this\r\nsubject I have enlarged in Chapter II.\u003ca name=\"FNanchor_29_29\" id=\"FNanchor_29_29\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_29_29\" class=\"fnanchor\"\u003e[29]\u003c/a\u003e Again, Free Mobility,\r\nas distinct from Congruence, hardly needs to be specially\r\nformulated: what barrier could empty space offer to a point\u0027s\r\nprogress? The axiom is involved in the homogeneity of\r\nspace, which is the same thing as the axiom of Congruence.\r\nMonodromy, also, has been severely criticized; not only is it\r\nevident that it might have been included in Congruence, but\r\neven from the purely analytical point of view, Sophus Lie has\r\nproved it to be superfluous\u003ca name=\"FNanchor_30_30\" id=\"FNanchor_30_30\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_30_30\" class=\"fnanchor\"\u003e[30]\u003c/a\u003e. Thus the axiom of Congruence,\r\nrightly formulated, includes Helmholtz\u0027s third and fourth\r\naxioms and part of his second axiom. All the four, or rather,\r\nas much of them as is relevant to Geometry, are consequences,\r\nas we shall see hereafter, of the one fundamental principle of\r\nthe relativity of position.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e26.\u003c/b\u003e\u003ca name=\"N26\" id=\"N26\"\u003e\u003c/a\u003e\r\nThe second article, which is mainly mathematical,\r\nsupplies the promised proof of the arc-formula, which is Helmholtz\u0027s\r\nmost important contribution to Geometry. Riemann\r\nhad \u003cem\u003eassumed\u003c/em\u003e this formula, as the simplest of a number of\r\nalternatives: Helmholtz proved it to be a necessary consequence\r\nof his axioms. The present paper begins with a short\r\nrepetition of the first, including the statement of the axioms, to\r\nwhich, at the end of the paper, two more are added, (5) that\r\nspace has three dimensions, and (6) that space is infinite. It\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_25\" id=\"Page_25\"\u003e[25]\u003c/a\u003e\u003c/span\u003eis supposed in the text, as also in the first paper, that the\r\nmeasure of curvature cannot be negative, and, consequently,\r\nthat an infinite space must be Euclidean. This error in both\r\npapers is corrected in notes, added after the appearance of\r\nBeltrami\u0027s paper on negative curvature. It is a sample of\r\nthe slightly unprofessional nature of Helmholtz\u0027s mathematical\r\nwork on this subject, which elicits from Klein the following\r\nremarks\u003ca name=\"FNanchor_31_31\" id=\"FNanchor_31_31\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_31_31\" class=\"fnanchor\"\u003e[31]\u003c/a\u003e: \"Helmholtz is not a mathematician by profession,\r\nbut a physicist and physiologist…. From this non-mathematical\r\nquality of Helmholtz, it follows naturally that he does not\r\ntreat the mathematical portion of his work with the thoroughness\r\nwhich one would demand of a mathematician by trade\r\n(\u003ccite\u003evon Fach\u003c/cite\u003e).\" He tells us himself that it was the physiological\r\nstudy of vision which led him to the question of the axioms,\r\nand it is as a physicist that he makes his axioms refer to actual\r\nrigid bodies. Accordingly, we find errors in his mathematics,\r\nsuch as the axiom of Monodromy, and the assumption that the\r\nmeasure of curvature must be positive. Nevertheless, the\r\nproof of Riemann\u0027s arc-formula is extremely able, and has, on\r\nthe whole, been substantiated by Lie\u0027s more thorough investigations.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e27.\u003c/b\u003e\u003ca name=\"N27\" id=\"N27\"\u003e\u003c/a\u003e\r\nHelmholtz\u0027s other writings on Geometry are almost\r\nwholly philosophical, and will be discussed at length in\r\nChapter II. For the present, we may pass to the only other\r\nimportant writer of the second period, \u003cem\u003eBeltrami\u003c/em\u003e. As his work is\r\npurely mathematical, and contains few controverted points, it\r\nneed not, despite its great importance, detain us long.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe \"\u003ccite class=\"fsnormal\" lang=\"it\" xml:lang=\"it\"\u003eSaggio di Interpretazione della Geometria non-Euclidea\u003c/cite\u003e\u003ca name=\"FNanchor_32_32\" id=\"FNanchor_32_32\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_32_32\" class=\"fnanchor\"\u003e[32]\u003c/a\u003e,\"\r\nwhich is principally confined to two dimensions,\r\ninterprets Lobatchewsky\u0027s results by the characteristic method\r\nof the second period. It shows, by a development of the work\r\nof Gauss and Minding\u003ca name=\"FNanchor_33_33\" id=\"FNanchor_33_33\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_33_33\" class=\"fnanchor\"\u003e[33]\u003c/a\u003e, that all the propositions in plane\r\nGeometry, which Lobatchewsky had set forth, hold, within\r\nordinary Euclidean space, on surfaces of constant negative\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_26\" id=\"Page_26\"\u003e[26]\u003c/a\u003e\u003c/span\u003ecurvature. It is strange, as Klein points out\u003ca name=\"FNanchor_34_34\" id=\"FNanchor_34_34\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_34_34\" class=\"fnanchor\"\u003e[34]\u003c/a\u003e, that this interpretation,\r\nwhich was known to Riemann and perhaps even to\r\nGauss, should have remained so long without explicit statement.\r\nThis is the more strange, as Lobatchewsky\u0027s \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003e\"Géométrie\r\nImaginaire\"\u003c/cite\u003e had appeared in Crelle, Vol. \u003cspan class=\"fs70\"\u003eXVII.\u003c/span\u003e\u003ca name=\"FNanchor_35_35\" id=\"FNanchor_35_35\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_35_35\" class=\"fnanchor\"\u003e[35]\u003c/a\u003e, and Minding\u0027s\r\narticle, from which the interpretation follows at once, had\r\nappeared in Crelle, Vol. \u003cspan class=\"fs70\"\u003eXIX.\u003c/span\u003e Minding had shewn that the\r\nGeometry of surfaces of constant negative curvature, in particular\r\nas regards geodesic triangles, could be deduced from\r\nthat of the sphere by giving the radius a purely imaginary\r\nvalue \u003cem\u003eia\u003c/em\u003e\u003ca name=\"FNanchor_36_36\" id=\"FNanchor_36_36\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_36_36\" class=\"fnanchor\"\u003e[36]\u003c/a\u003e. This result, as we have seen, had also been obtained\r\nby Lobatchewsky for his Geometry, and yet it took thirty years\r\nfor the connection to be brought to general notice.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e28.\u003c/b\u003e\u003ca name=\"N28\" id=\"N28\"\u003e\u003c/a\u003e\r\nIn Beltrami\u0027s Saggio, straight lines are, of course,\r\nreplaced by geodesics; his coordinates are obtained through\r\na point-by-point correspondence with an auxiliary plane, in\r\nwhich straight lines correspond to geodesics on the surface.\r\nThus geodesics have linear equations, and are always uniquely\r\ndetermined by two points. Distances on the surface, however,\r\nare not equal to distances on the plane; thus while the surface\r\nis infinite, the corresponding portion of the plane is contained\r\nwithin a certain finite circle. The distance of two points on\r\nthe surface is a certain function of the coordinates, not the\r\nordinary function of elementary Geometry. These relations\r\nof plane and surface are important in connection with Cayley\u0027s\r\ntheory of distance, which we shall have to consider next. If\r\nwe were to define distance on the plane as that function of\r\nthe coordinates which gives the corresponding distance on the\r\nsurface, we should obtain what Klein calls \"a plane with a\r\nhyperbolic system of measurement (\u003ci lang=\"de\" xml:lang=\"de\"\u003eMassbestimmung\u003c/i\u003e)\" in which\r\nCayley\u0027s theory of distance would hold. It is evident, however,\r\nthat the ordinary notion of distance has been presupposed in\r\nsetting up the coordinate system, so that we do not really\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_27\" id=\"Page_27\"\u003e[27]\u003c/a\u003e\u003c/span\u003eget alternative Geometries on one and the same plane. The\r\nbearing of these remarks will appear more fully when we come\r\nto consider Cayley and Klein.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e29.\u003c/b\u003e\u003ca name=\"N29\" id=\"N29\"\u003e\u003c/a\u003e\r\nThe value of Beltrami\u0027s Saggio, in his own eyes, lies in\r\nthe intelligible Euclidean sense which it gives to Lobatchewsky\u0027s\r\nplanimetry: the corresponding system of Solid Geometry, since\r\nit has no meaning for Euclidean space, is barely mentioned in\r\nthis work. In a second paper\u003ca name=\"FNanchor_37_37\" id=\"FNanchor_37_37\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_37_37\" class=\"fnanchor\"\u003e[37]\u003c/a\u003e, however, almost contemporaneous\r\nwith the first, he proceeds to consider the general theory of\r\n\u003cem\u003en\u003c/em\u003e-dimensional manifolds of constant negative curvature. This\r\npaper is greatly influenced by Riemann\u0027s dissertation; it begins\r\nwith the formula for the linear element, and proves from this\r\nfirst, that Congruence holds for such spaces, and next, that\r\nthey have, according to Riemann\u0027s definition, a constant negative\r\nmeasure of curvature. (It is instructive to observe, that both\r\nin this and in the former Essay, great stress is laid on the\r\nnecessity of the Axiom of Congruence.)\u003c/p\u003e\r\n\r\n\u003cp\u003eThis work has less philosophical interest than the former,\r\nsince it does little more than repeat, in a general form, the\r\nresults which the Saggio had obtained for two dimensions\u0026mdash;results\r\nwhich sink, when extended to \u003cem\u003en\u003c/em\u003e dimensions, to the\r\nlevel of mere mathematical constructions. Nevertheless, the\r\npaper is important, both as a restoration of negative curvature,\r\nwhich had been overlooked by Helmholtz, and as an analytical\r\ntreatment of Lobatchewsky\u0027s results\u0026mdash;a treatment which, together\r\nwith the Saggio, at last restored to them the prominence\r\nthey deserved.\u003c/p\u003e\r\n\r\n\r\n\u003ch3\u003eThird Period.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e30.\u003c/b\u003e\u003ca name=\"N30\" id=\"N30\"\u003e\u003c/a\u003e\r\nThe third period differs radically, alike in its methods\r\nand aims, and in the underlying philosophical ideas, from the\r\nperiod which it replaced. Whereas everything, in the second\r\nperiod, turned on measurement, with its apparatus of Congruence,\r\nFree Mobility, Rigid Bodies, and the rest, these\r\nvanish completely in the third period, which, swinging to the\r\nopposite extreme, regards quantity as a perfectly irrelevant\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_28\" id=\"Page_28\"\u003e[28]\u003c/a\u003e\u003c/span\u003e\r\ncategory in Geometry, and dispenses with congruence and the\r\nmethod of superposition. The ideas of this period, unfortunately,\r\nhave found no exponent so philosophical as Riemann\r\nor Helmholtz, but have been set forth only by technical\r\nmathematicians. Moreover the change of fundamental ideas,\r\nwhich is immense, has not brought about an equally great\r\nchange in actual procedure; for though spatial quantity is no\r\nlonger a part of projective Geometry, quantity is still employed,\r\nand we still have equations, algebraic transformations, and so\r\non. This is apt to give rise to confusion, especially in the\r\nmind of the student, who fails to realise that the quantities\r\nused, so far as the propositions are really projective, are mere\r\nnames for points, and not, as in metrical Geometry, actual\r\nspatial magnitudes.\u003c/p\u003e\r\n\r\n\u003cp\u003eNevertheless, the fundamental difference between this period\r\nand the former must strike any one at once. Whereas Riemann\r\nand Helmholtz dealt with metrical ideas, and took, as their\r\nfoundations, the measure of curvature and the formula for the\r\nlinear element\u0026mdash;both purely metrical\u0026mdash;the new method is\r\nerected on the formulae for transformation of coordinates required\r\nto express a given collineation. It begins by reducing\r\nall so-called metrical notions\u0026mdash;distance, angle, etc.\u0026mdash;to projective\r\nforms, and obtains, from this reduction, a methodological unity\r\nand simplicity before impossible. This reduction depends,\r\nhowever, except where the space-constant is negative, upon\r\nimaginary figures\u0026mdash;in Euclid, the circular points at infinity; it is\r\nmoreover purely symbolic and analytical, and must be regarded\r\nas philosophically irrelevant. As the question concerning the\r\nimport of this reduction is of fundamental importance to our\r\ntheory of Geometry, and as Cayley, in his Presidential Address\r\nto the British Association in 1883, formally challenged philosophers\r\nto discuss the use of imaginaries, on which it depends,\r\nI will treat this question at some length. But first let us see\r\nhow, as a matter of mathematics, the reduction is effected.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e31.\u003c/b\u003e\u003ca name=\"N31\" id=\"N31\"\u003e\u003c/a\u003e\r\nWe shall find, throughout this period, that almost\r\nevery important proposition, though misleading in its obvious\r\ninterpretation, has nevertheless, when rightly interpreted, a\r\nwide philosophical bearing. So it is with the work of \u003cem\u003eCayley\u003c/em\u003e,\r\nthe pioneer of the projective method.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_29\" id=\"Page_29\"\u003e[29]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eThe projective formula for angles, in Euclidean Geometry,\r\nwas first obtained by Laguerre, in 1853. This formula had,\r\nhowever, a perfectly Euclidean character, and it was left for\r\nCayley to generalize it so as to include both angles and\r\ndistances in Euclidean and non-Euclidean systems alike\u003ca name=\"FNanchor_38_38\" id=\"FNanchor_38_38\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_38_38\" class=\"fnanchor\"\u003e[38]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003eCayley\u003c/em\u003e was, to the last, a staunch supporter of Euclidean\r\n\u003cem\u003espace\u003c/em\u003e, though he believed that non-Euclidean \u003cem\u003eGeometries\u003c/em\u003e could\r\nbe applied, within Euclidean space, by a change in the definition\r\nof distance\u003ca name=\"FNanchor_39_39\" id=\"FNanchor_39_39\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_39_39\" class=\"fnanchor\"\u003e[39]\u003c/a\u003e. He has thus, in spite of his Euclidean orthodoxy,\r\nprovided the believers in the possibility of non-Euclidean spaces\r\nwith one of their most powerful weapons. In his \"Sixth\r\nMemoir upon Quantics\" (1859), he set himself the task of\r\n\"establishing the notion of distance upon purely descriptive\r\nprinciples.\" He showed that, with the ordinary notion of\r\ndistance, it can be rendered projective by reference to the\r\ncircular points and the line at infinity, and that the same is\r\ntrue of angles\u003ca name=\"FNanchor_40_40\" id=\"FNanchor_40_40\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_40_40\" class=\"fnanchor\"\u003e[40]\u003c/a\u003e. Not content with this, he suggested a new\r\ndefinition of distance, as the inverse sine or cosine of a certain\r\nfunction of the coordinates; with this definition, the properties\r\nusually known as metrical become projective properties, having\r\nreference to a certain conic, called by Cayley the Absolute.\r\n(The circular points are, analytically, a degenerate conic, so\r\nthat ordinary Geometry forms a particular case of the above.)\r\nHe proves that, when the Absolute is an \u003cem\u003eimaginary\u003c/em\u003e conic, the\r\nGeometry so obtained for two dimensions is spherical Geometry.\r\nThe correspondence with Lobatchewsky, in the case where\r\nthe Absolute is \u003cem\u003ereal\u003c/em\u003e, is not worked out: indeed there is,\r\nthroughout, no evidence of acquaintance with non-Euclidean\r\nsystems. The importance of the memoir, to Cayley, lies\r\nentirely in its proof that metrical is only a branch of descriptive\r\nGeometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e32.\u003c/b\u003e\u003ca name=\"N32\" id=\"N32\"\u003e\u003c/a\u003e\r\nThe connection of Cayley\u0027s Theory of Distance with\r\nMetageometry was first pointed out by Klein\u003ca name=\"FNanchor_41_41\" id=\"FNanchor_41_41\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_41_41\" class=\"fnanchor\"\u003e[41]\u003c/a\u003e. Klein showed\r\nin detail that, if the Absolute be real, we get Lobatchewsky\u0027s\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_30\" id=\"Page_30\"\u003e[30]\u003c/a\u003e\u003c/span\u003e(hyperbolic) system; if it be imaginary, we get either spherical\r\nGeometry or a new system, analogous to that of Helmholtz,\r\ncalled by Klein elliptic; if the Absolute be an imaginary\r\npoint-pair, we get parabolic Geometry, and if, in particular,\r\nthe point-pair be the circular points, we get ordinary Euclid.\r\nIn elliptic Geometry, two straight lines in the same plane meet\r\nin only one point, not two as in Helmholtz\u0027s system. The\r\ndistinction between the two kinds of Geometry is difficult,\r\nand will be discussed later.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e33.\u003c/b\u003e\u003ca name=\"N33\" id=\"N33\"\u003e\u003c/a\u003e\r\nSince these systems are all obtained from a Euclidean\r\nplane, by a mere alteration in the definition of distance, Cayley\r\nand Klein tend to regard the whole question as one, not of\r\nthe nature of space, but of the definition of distance. Since\r\nthis definition, on their view, is perfectly arbitrary, the philosophical\r\nproblem vanishes\u0026mdash;Euclidean \u003cem\u003espace\u003c/em\u003e is left in undisputed\r\npossession, and the only problem remaining is one\r\nof convention and mathematical convenience\u003ca name=\"FNanchor_42_42\" id=\"FNanchor_42_42\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_42_42\" class=\"fnanchor\"\u003e[42]\u003c/a\u003e. This view has\r\nbeen forcibly expressed by Poincaré: \"What ought one to\r\nthink,\" he says, \"of this question: Is the Euclidean Geometry\r\ntrue? The question is nonsense.\" Geometrical axioms, according\r\nto him, are mere conventions: they are \"definitions\r\nin disguise\u003ca name=\"FNanchor_43_43\" id=\"FNanchor_43_43\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_43_43\" class=\"fnanchor\"\u003e[43]\u003c/a\u003e.\" Thus Klein blames Beltrami for regarding his\r\nauxiliary plane as merely auxiliary, and remarks that, if he\r\nhad known Cayley\u0027s Memoir, he would have seen the relation\r\nbetween the plane and the pseudosphere to be far more intimate\r\nthan he supposed\u003ca name=\"FNanchor_44_44\" id=\"FNanchor_44_44\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_44_44\" class=\"fnanchor\"\u003e[44]\u003c/a\u003e. A view which removes the problem entirely\r\nfrom the arena of philosophy demands, plainly, a full discussion.\r\nTo this discussion we will now proceed.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e34.\u003c/b\u003e\u003ca name=\"N34\" id=\"N34\"\u003e\u003c/a\u003e\r\nThe view in question has arisen, it would seem, from\r\na natural confusion as to the nature of the coordinates employed.\r\nThose who hold the view have not adequately realised,\r\nI believe, that their coordinates are not \u003cem\u003espatial\u003c/em\u003e quantities, as\r\nin metrical Geometry, but mere conventional signs, by which\r\ndifferent points can be distinctly designated. There is no\r\nreason, therefore, until we already have metrical Geometry,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_31\" id=\"Page_31\"\u003e[31]\u003c/a\u003e\u003c/span\u003efor regarding one function of the coordinates as a better expression\r\nof distance than another, so long as the fundamental\r\naddition-equation\u003ca name=\"FNanchor_45_45\" id=\"FNanchor_45_45\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_45_45\" class=\"fnanchor\"\u003e[45]\u003c/a\u003e is preserved. Hence, if our coordinates are\r\nregarded as adequate for all Geometry, an indeterminateness\r\narises in the expression of distance, which can only be avoided\r\nby a convention. But projective coordinates\u0026mdash;so our argument\r\nwill contend\u0026mdash;though perfectly adequate for all projective\r\nproperties, and entirely free from any metrical presupposition,\r\nare inadequate to express metrical properties, just because they\r\nhave no metrical presupposition. Thus where metrical properties\r\nare in question, Beltrami remains justified as against\r\nKlein; the reduction of metrical to projective properties is\r\nonly apparent, though the independence of these last, as against\r\nmetrical Geometry, is perfectly real.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e35.\u003c/b\u003e\u003ca name=\"N35\" id=\"N35\"\u003e\u003c/a\u003e\r\nBut what are projective coordinates, and how are they\r\nintroduced? This question was not touched upon in Cayley\u0027s\r\nMemoir, and it seemed, therefore, as if a logical error were\r\ninvolved in using coordinates to define distance. For coordinates,\r\nin all previous systems, had been deduced from distance;\r\nto use any existing coordinate system in defining distance\r\nwas, accordingly, to incur a vicious circle. Cayley mentions\r\nthis difficulty in a note, where he only remarks, however,\r\nthat he had regarded his coordinates as numbers arbitrarily\r\nassigned, on some system not further investigated, to different\r\npoints. The difficulty has been treated at length by Sir R.\r\nBall (Theory of the Content, Trans. R. I. A. 1889), who urges\r\nthat if the values of our coordinates already involve the usual\r\nmeasure of distance, then to give a new definition, while retaining\r\nthe usual coordinates, is to incur a contradiction. He says\r\n(op. cit. p. 1): \"In the study of non-Euclidean Geometry I have\r\noften felt a difficulty which has, I know, been shared by others.\r\nIn that theory it seems as if we try to replace our ordinary\r\nnotion of distance between two points by the logarithm of\r\na certain anharmonic ratio\u003ca name=\"FNanchor_46_46\" id=\"FNanchor_46_46\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_46_46\" class=\"fnanchor\"\u003e[46]\u003c/a\u003e. But this ratio itself involves the\r\nnotion of distance measured in the ordinary way. How, then,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_32\" id=\"Page_32\"\u003e[32]\u003c/a\u003e\u003c/span\u003ecan we supersede our old notion of distance by the non-Euclidean\r\nnotion, inasmuch as the very definition of the latter\r\ninvolves the former?\"\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e36.\u003c/b\u003e\u003ca name=\"N36\" id=\"N36\"\u003e\u003c/a\u003e\r\nThis objection is valid, we must admit, so long as\r\nanharmonic ratio is defined in the ordinary metrical manner.\r\nIt would be valid, for example, against any attempt to found\r\na new definition of distance on Cremona\u0027s account of anharmonic\r\nratio\u003ca name=\"FNanchor_47_47\" id=\"FNanchor_47_47\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_47_47\" class=\"fnanchor\"\u003e[47]\u003c/a\u003e, in which it appears as a metrical property\r\nunaltered by projective transformation. If a logical error is\r\nto be avoided, in fact, all reference to spatial magnitude of\r\nany kind must be avoided; for all spatial magnitude, as will\r\nbe shown hereafter\u003ca name=\"FNanchor_48_48\" id=\"FNanchor_48_48\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_48_48\" class=\"fnanchor\"\u003e[48]\u003c/a\u003e, is logically dependent on the fundamental\r\nmagnitude of distance. Anharmonic ratio and coordinates\r\nmust alike be defined by purely descriptive properties, if the\r\nuse afterwards made of them is to be free from metrical presuppositions,\r\nand therefore from the objections of Sir R. Ball.\u003c/p\u003e\r\n\r\n\u003cp\u003eSuch a definition has been satisfactorily given by Klein\u003ca name=\"FNanchor_49_49\" id=\"FNanchor_49_49\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_49_49\" class=\"fnanchor\"\u003e[49]\u003c/a\u003e,\r\nwho appeals, for the purpose, to v. Staudt\u0027s quadrilateral construction\u003ca name=\"FNanchor_50_50\" id=\"FNanchor_50_50\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_50_50\" class=\"fnanchor\"\u003e[50]\u003c/a\u003e.\r\nBy this construction, which I have reproduced in\r\noutline in \u003ca href=\"#N112\"\u003eChapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Section A, § 112 ff.\u003c/a\u003e, we obtain a purely descriptive\r\ndefinition of harmonic and anharmonic ratio, and, given\r\na pair of points, we can obtain the harmonic conjugate to any\r\nthird point on the same straight line. On this construction, the\r\nintroduction of projective coordinates is based. Starting with\r\nany three points on a straight line, we assign to them arbitrarily\r\nthe numbers 0, 1, ∞. We then find the harmonic conjugate to\r\nthe first with respect to 1, ∞, and assign to it the number 2.\r\nThe object of assigning this number rather than any other, is\r\nto obtain the value \u0026ndash;1 for the anharmonic ratio of the four\r\nnumbers corresponding to the four points\u003ca name=\"FNanchor_51_51\" id=\"FNanchor_51_51\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_51_51\" class=\"fnanchor\"\u003e[51]\u003c/a\u003e. We then find the\r\nharmonic conjugate to the point 1, with respect to 2, ∞, and\r\nassign to it the number 3; and so on. Klein has shown that\r\nby this construction, we can obtain any number of points, and\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_33\" id=\"Page_33\"\u003e[33]\u003c/a\u003e\u003c/span\u003ecan construct a point corresponding to any given number,\r\nfractional or negative. Moreover, when two sets of four points\r\nhave the same anharmonic ratio, descriptively defined\u003ca name=\"FNanchor_52_52\" id=\"FNanchor_52_52\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_52_52\" class=\"fnanchor\"\u003e[52]\u003c/a\u003e, the\r\ncorresponding numbers also have the same anharmonic ratio.\r\nBy introducing such a numerical system on two straight lines,\r\nor on three, we obtain the coordinates of any point in a plane,\r\nor in space. By this construction, which is of fundamental\r\nimportance to projective Geometry, the logical error, upon\r\nwhich Sir R. Ball bases his criticism, is satisfactorily avoided.\r\nOur coordinates are introduced by a purely descriptive method,\r\nand involve no presupposition whatever as to the measurement\r\nof distance.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e37.\u003c/b\u003e\u003ca name=\"N37\" id=\"N37\"\u003e\u003c/a\u003e\r\nWith this coordinate system, then, to define distance\r\nas a certain function of the coordinates is not to be guilty of\r\na vicious circle. But it by no means follows that the definition\r\nof distance is arbitrary. All reference to distance has\r\nbeen hitherto excluded, to avoid metrical ideas; but when\r\ndistance is introduced, metrical ideas inevitably reappear, and\r\nwe have to remember that our coordinates give no information,\r\n\u003ci lang=\"la\" xml:lang=\"la\"\u003eprimâ facie\u003c/i\u003e, as to any of these metrical ideas. It is open to\r\nus, of course, if we choose, to continue to exclude distance in\r\nthe ordinary sense, as the quantity of a finite straight line,\r\nand to define the \u003cem\u003eword\u003c/em\u003e distance in any way we please. But\r\nthe conception, for which the word has hitherto stood, will\r\nthen require a new name, and the only result will be a confusion\r\nbetween the \u003cem\u003eapparent\u003c/em\u003e meaning of our propositions, to\r\nthose who retain the associations belonging to the old sense\r\nof the word, and the \u003cem\u003ereal\u003c/em\u003e meaning, resulting from the new\r\nsense in which the word is used.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis confusion, I believe, has actually occurred, in the case\r\nof those who regard the question between Euclid and Metageometry\r\nas one of the definition of distance. Distance is a\r\nquantitative relation, and as such presupposes identity of\r\nquality. But projective Geometry deals only with quality\u0026mdash;for\r\nwhich reason it is called descriptive\u0026mdash;and cannot distinguish\r\nbetween two figures which are qualitatively alike. Now the\r\nmeaning of qualitative likeness, in Geometry, is the possibility\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_34\" id=\"Page_34\"\u003e[34]\u003c/a\u003e\u003c/span\u003e\r\nof mutual transformation by a collineation\u003ca name=\"FNanchor_53_53\" id=\"FNanchor_53_53\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_53_53\" class=\"fnanchor\"\u003e[53]\u003c/a\u003e. Any two pairs of\r\npoints on the same straight line, therefore, are qualitatively\r\nalike; their only qualitative relation is the straight line, which\r\nboth pairs have in common; and it is exactly the qualitative\r\nidentity of the relations of the two pairs, which enables the\r\ndifference of their relations to be exhaustively dealt with by\r\nquantity, as a difference of distance. But where quantity is\r\nexcluded, any two pairs of points on the same straight line\r\nappear as alike, and even any two sets of three: for any three\r\npoints on a straight line can be projectively transformed into\r\nany other three. It is only with \u003cem\u003efour\u003c/em\u003e points in a line that we\r\nacquire a projective property distinguishing them from other\r\nsets of four, and this property is anharmonic ratio, descriptively\r\ndefined. The projective Geometer, therefore, sees no\r\nreason to give a name to the relation between two points, in so\r\nfar as this relation is anything over and above the unlimited\r\nstraight line on which they lie; and when he introduces the\r\nnotion of distance, he defines it, in the only way in which\r\nprojective principles allow him to define it, as a relation between\r\n\u003cem\u003efour\u003c/em\u003e points. As he nevertheless wishes the word to give him\r\nthe power of distinguishing between different \u003cem\u003epairs\u003c/em\u003e of points,\r\nhe agrees to take two out of the four points as fixed. In this\r\nway, the only variables in distance are the two remaining\r\npoints, and distance appears, therefore, as a function of \u003cem\u003etwo\u003c/em\u003e\r\nvariables, namely the coordinates of the two variable points.\r\nWhen we have further defined our function so that distance\r\nmay be additive, we have a function with many of the properties\r\nof distance in the ordinary sense. This function, therefore,\r\nthe projective Geometer regards as the only proper definition of\r\ndistance.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe can see, in fact, from the manner in which our projective\r\ncoordinates were introduced, that \u003cem\u003esome\u003c/em\u003e function of these\r\ncoordinates must express distance in the ordinary sense. For\r\nthey were introduced serially, so that, as we proceeded from the\r\nzero-point towards the infinity-point, our coordinates continually\r\ngrew. To every point, a definite coordinate corresponded: to\r\nthe distance between two variable points, therefore, as a\r\nfunction dependent on no other variables, must correspond\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_35\" id=\"Page_35\"\u003e[35]\u003c/a\u003e\u003c/span\u003e\r\nsome definite function of the coordinates, since these are\r\nthemselves functions of their points. The function discussed\r\nabove, therefore, must certainly include distance in the ordinary\r\nsense.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut the arbitrary and conventional nature of distance, as\r\nmaintained by Poincaré and Klein, arises from the fact that the\r\ntwo fixed points, required to determine our distance in the\r\nprojective sense, may be arbitrarily chosen, and although, when\r\nour choice is once made, any two points have a definite distance,\r\nyet, according as we make that choice, distance will become a\r\ndifferent function of the two variable points. The ambiguity\r\nthus introduced is unavoidable on projective principles; but\r\nare we to conclude, from this, that it is really unavoidable?\r\nMust we not rather conclude that projective Geometry cannot\r\nadequately deal with distance? If \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, be three different\r\npoints on a line, there must be \u003cem\u003esome\u003c/em\u003e difference between the\r\nrelation of \u003cem\u003eA\u003c/em\u003e to \u003cem\u003eB\u003c/em\u003e and of \u003cem\u003eA\u003c/em\u003e to \u003cem\u003eC\u003c/em\u003e, for otherwise, owing to the\r\nqualitative identity of all points, \u003cem\u003eB\u003c/em\u003e and \u003cem\u003eC\u003c/em\u003e could not be distinguished.\r\nBut such a difference involves a relation, between\r\n\u003cem\u003eA\u003c/em\u003e and \u003cem\u003eB\u003c/em\u003e, which is independent of other points on the line;\r\nfor unless we have such a relation, the other points cannot be\r\ndistinguished as different. Before we can distinguish the two\r\nfixed points, therefore, from which the projective definition\r\nstarts, we must already suppose some relation, between any\r\ntwo points on our line, in which they are independent of other\r\npoints; and this relation is distance in the ordinary sense\u003ca name=\"FNanchor_54_54\" id=\"FNanchor_54_54\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_54_54\" class=\"fnanchor\"\u003e[54]\u003c/a\u003e.\r\nWhen we have measured this quantitative relation by the\r\nordinary methods of metrical Geometry, we can proceed to\r\ndecide what base-points must be chosen, on our line, in order\r\nthat the projective function discussed above may have the\r\nsame value as ordinary distance. But the choice of these base-points,\r\nwhen we are discussing distance in the ordinary sense,\r\nis not arbitrary, and their introduction is only a technical\r\ndevice. Distance, in the ordinary sense, remains a relation\r\nbetween \u003cem\u003etwo\u003c/em\u003e points, not between \u003cem\u003efour\u003c/em\u003e; and it is the failure to\r\nperceive that the projective sense differs from, and cannot\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_36\" id=\"Page_36\"\u003e[36]\u003c/a\u003e\u003c/span\u003e\r\nsupersede, the ordinary sense, which has given rise to the views\r\nof Klein and Poincaré. The question is not one of convention,\r\nbut of the irreducible metrical properties of space. To sum\r\nup: Quantities, as used in projective Geometry, do not stand\r\nfor spatial magnitudes, but are conventional symbols for purely\r\nqualitative spatial relations. But distance, \u003cem\u003equâ\u003c/em\u003e quantity,\r\npresupposes identity of quality, as the condition of quantitative\r\ncomparison. Distance in the ordinary sense is, in short, that\r\nquantitative relation, between two points on a line, by which\r\ntheir difference from other points can be defined. The projective\r\ndefinition, however, being unable to distinguish a\r\ncollection of less than four points from any other on the same\r\nstraight line, makes distance depend on two other points\r\nbesides those whose relation it defines. No name remains,\r\ntherefore, for distance in the ordinary sense, and many projective\r\nGeometers, having abolished the name, believe the\r\nthing to be abolished also, and are inclined to deny that \u003cem\u003etwo\u003c/em\u003e\r\npoints have a unique relation at all. This confusion, in\r\nprojective Geometry, shows the importance of a name, and\r\nshould make us chary of allowing new meanings to obscure one\r\nof the fundamental properties of space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e38.\u003c/b\u003e\u003ca name=\"N38\" id=\"N38\"\u003e\u003c/a\u003e\r\nIt remains to discuss the manner in which non-Euclidean\r\nGeometries result from the projective definition of\r\ndistance, as also the true interpretation to be given to this view\r\nof Metageometry. It is to be observed that the projective\r\nmethods which follow Cayley deal throughout with a Euclidean\r\nplane, on which they introduce different measures of distance.\r\nHence arises, in any interpretation of these methods, an\r\napparent subordination of the non-Euclidean spaces, as though\r\nthese were less self-subsistent than Euclid\u0027s. This subordination\r\nis not intended in what follows; on the contrary, the\r\ncorrelation with Euclidean space is regarded as valuable, first,\r\nbecause Euclidean space has been longer studied and is more\r\nfamiliar, but secondly, because this correlation proves, when\r\ntruly interpreted, that the other spaces are self-subsistent.\r\nWe may confine ourselves chiefly, in discussing this interpretation,\r\nto distances measured along a single straight line.\r\nBut we must be careful to remember that the metrical definition\r\nof distance\u0026mdash;which, according to the view here advocated,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_37\" id=\"Page_37\"\u003e[37]\u003c/a\u003e\u003c/span\u003e\r\nis the only adequate definition\u0026mdash;is the same in Euclidean and\r\nin non-Euclidean spaces; to argue in its favour is not, therefore,\r\nto argue in favour of Euclid.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe projective scheme of coordinates consists of a series of\r\nnumbers, of which each represents a certain anharmonic ratio\r\nand denotes one and only one point, and which increase\r\nuniformly with the distance from a fixed origin, until they\r\nbecome infinite on reaching a certain point. Now Cayley\r\nshowed that, in Euclidean Geometry, distance may be expressed\r\nas the limit of the logarithm of the anharmonic\r\nratio of the two points and the (coincident) points at infinity\r\non their straight line; while, if we assumed that the points at\r\ninfinity were distinct, we obtained the formula for distance in\r\nhyperbolic or spherical Geometry, according as these points\r\nwere real or imaginary. Hence it follows that, with the\r\nprojective definition of distance, we shall obtain precisely the\r\nformulae of hyperbolic, parabolic or spherical Geometry, according\r\nas we choose the point, to which the value +∞ is assigned,\r\nat a finite, infinite or imaginary distance (in the ordinary sense)\r\nfrom the point to which we assign the value 0. Our straight\r\nline remains, all the while, an ordinary Euclidean straight line.\r\nBut we have seen that the projective definition of distance fits\r\nwith the true definition only when the two fixed points to\r\nwhich it refers are suitably chosen. Now the ordinary meaning\r\nof distance is required in non-Euclidean as in Euclidean\r\nGeometries\u0026mdash;indeed, it is only in metrical properties that these\r\nGeometries differ. Hence our \u003cem\u003eEuclidean\u003c/em\u003e straight line, though\r\nit may serve to illustrate other Geometries than Euclid\u0027s, can\r\nonly be dealt with correctly by Euclid. Where we give a\r\ndifferent definition of distance from Euclid\u0027s, we are still in the\r\ndomain of purely projective properties, and derive no information\r\nas to the metrical properties of our straight line. But the\r\nimportance, to Metageometry, of this new interpretation, lies in\r\nthe fact that, having independently established the metrical\r\nformulae of non-Euclidean spaces, we find, as in Beltrami\u0027s\r\nSaggio, that these spaces can be related, by a homographic\r\ncorrespondence, with the points of Euclidean space; and that\r\nthis can be effected in such a manner as to give, for the\r\ndistance between two points of our non-Euclidean space, the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_38\" id=\"Page_38\"\u003e[38]\u003c/a\u003e\u003c/span\u003e\r\nhyperbolic or spherical measure of distance for the corresponding\r\npoints of Euclidean space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e39.\u003c/b\u003e\u003ca name=\"N39\" id=\"N39\"\u003e\u003c/a\u003e\r\nOn the whole, then, a modification of Sir R. Ball\u0027s view,\r\nwhich is practically a generalized statement of Beltrami\u0027s method,\r\nseems the most tenable. He imagines what, with Grassmann, he\r\ncalls a Content, \u003cem\u003ei.e.\u003c/em\u003e a perfectly general three-dimensional manifold,\r\nand then correlates its elements, one by one, with points\r\nin Euclidean space. Thus every element of the Content acquires,\r\nas its coordinates, the ordinary Euclidean coordinates\r\nof the corresponding point in Euclidean space. By means of\r\nthis correlation, our calculations, though they refer to the\r\nContent, are carried on, as in Beltrami\u0027s Saggio, in ordinary\r\nEuclidean space. Thus the confusion disappears, but with it,\r\nthe supposed Euclidean interpretation also disappears. Sir\r\nR. Ball\u0027s Content, if it is to be a space at all, must be a space\r\nradically different from Euclid\u0027s\u003ca name=\"FNanchor_55_55\" id=\"FNanchor_55_55\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_55_55\" class=\"fnanchor\"\u003e[55]\u003c/a\u003e; to speak, as Klein does, of\r\nordinary planes with hyperbolic or elliptic measures of distance,\r\nis either to incur a contradiction, or to forego any metrical\r\nmeaning of distance. Instead of ordinary planes, we have surfaces\r\nlike Beltrami\u0027s, of constant measure of curvature; instead\r\nof Euclid\u0027s space, we have hyperbolic or spherical space. At\r\nthe same time, it remains true that we can, by Klein\u0027s method,\r\ngive a Euclidean meaning to every symbolic proposition in non-Euclidean\r\nGeometry. For by substituting, for distance, the\r\nlogarithm above alluded to, we obtain, from the non-Euclidean\r\nresult, a result which follows from the ordinary Euclidean\r\naxioms. This correspondence removes, once for all, the possibility\r\nof a lurking contradiction in Metageometry, since, to a\r\nproposition in the one, corresponds one and only one proposition\r\nin the other, and contradictory results in one system, therefore,\r\nwould correspond to contradictory results in the other. Hence\r\nMetageometry cannot lead to contradictions, unless Euclidean\r\nGeometry, at the same moment, leads to corresponding contradictions.\r\nThus the Euclidean plane with hyperbolic or elliptic\r\nmeasure of distance, though either contradictory or not metrical\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_39\" id=\"Page_39\"\u003e[39]\u003c/a\u003e\u003c/span\u003e\r\nas an independent notion, has, as a help in the interpretation of\r\nnon-Euclidean results, a very high degree of utility.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e40.\u003c/b\u003e\u003ca name=\"N40\" id=\"N40\"\u003e\u003c/a\u003e\r\nWe have still to discuss Klein\u0027s third kind of non-Euclidean\r\nGeometry, which he calls elliptic. The difference\r\nbetween this and spherical Geometry is difficult to grasp, but\r\nit may be illustrated by a simpler example. A plane, as every\r\none knows, can be wrapped, without stretching, on a cylinder,\r\nand straight lines in the plane become, by this operation,\r\ngeodesics on the cylinder. The Geometries of the plane and\r\nthe cylinder, therefore, have much in common. But since the\r\ngenerating circle of the cylinder, which is one of its geodesics,\r\nis finite, only a portion of the plane is used up in wrapping it\r\nonce round the cylinder. Hence, if we endeavour to establish\r\na point-to-point correspondence between the plane and the\r\ncylinder, we shall find an infinite series of points on the plane\r\nfor a single point on the cylinder. Thus it happens that\r\ngeodesics, though on the plane they have only one point in\r\ncommon, may on the cylinder have an infinite number of intersections.\r\nSomewhat similar to this is the relation between the\r\nspherical and elliptic Geometries. To any one point in elliptic\r\nspace, two points correspond in spherical space. Thus geodesics,\r\nwhich in spherical space may have two points in common, can\r\nnever, in elliptic space, have more than one intersection.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut Klein\u0027s method can only prove that elliptic Geometry\r\nholds of the ordinary Euclidean plane with elliptic measure\r\nof distance. Klein has made great endeavours to enforce the\r\ndistinction between the spherical and elliptic Geometries\u003ca name=\"FNanchor_56_56\" id=\"FNanchor_56_56\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_56_56\" class=\"fnanchor\"\u003e[56]\u003c/a\u003e, but\r\nit is not immediately evident that the latter, as distinct from\r\nthe former, is valid.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, Klein\u0027s elliptic Geometry, which arises as\r\none of the alternative metrical systems on a Euclidean plane or\r\nin a Euclidean space, does not by itself suffice, if the above\r\ndiscussion has been correct, to prove the possibility of an\r\nelliptic space, \u003cem\u003ei.e.\u003c/em\u003e of a space having a point-to-point correspondence\r\nwith the Euclidean space, and having as the ordinary\r\ndistance between two of its points the elliptic definition of the\r\ndistance between corresponding points of the Euclidean space.\r\nTo prove this possibility, we must adopt the direct method of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_40\" id=\"Page_40\"\u003e[40]\u003c/a\u003e\u003c/span\u003e\r\nNewcomb (Crelle\u0027s Journal, Vol. 83). Now in the first place\r\nNewcomb has not proved that his postulates are self-consistent;\r\nhe has only failed to prove that they are contradictory\u003ca name=\"FNanchor_57_57\" id=\"FNanchor_57_57\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_57_57\" class=\"fnanchor\"\u003e[57]\u003c/a\u003e. This\r\nwould leave elliptic space in the same position in which Lobatchewsky\r\nand Bolyai left hyperbolic space. But further there\r\nseems to be, at first sight, in \u003cem\u003etwo\u003c/em\u003e-dimensional elliptic space, a\r\npositive contradiction. To explain this, however, some account\r\nof the peculiarities of the elliptic plane will be necessary.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figright\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep041.jpg\" width=\"100\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003eThe elliptic plane, regarded as a figure in three-dimensional\r\nelliptic space, is what is called a double surface\u003ca name=\"FNanchor_58_58\" id=\"FNanchor_58_58\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_58_58\" class=\"fnanchor\"\u003e[58]\u003c/a\u003e, \u003cem\u003ei.e.\u003c/em\u003e as Newcomb\r\nsays (\u003ccite\u003eloc. cit.\u003c/cite\u003e p. 298): \"The two sides of a complete plane are\r\nnot distinct, as in a Euclidean surface…. If … a being should\r\ntravel to distance 2\u003cem\u003eD\u003c/em\u003e, he would, on his return, find himself on\r\nthe opposite surface to that on which he started, and would\r\nhave to repeat his journey in order to return to his original\r\nposition without leaving the surface.\" Now if we imagine a\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_41\" id=\"Page_41\"\u003e[41]\u003c/a\u003e\u003c/span\u003e\u003cem\u003etwo\u003c/em\u003e-dimensional elliptic space, the distinction between the sides\r\nof a plane becomes unmeaning, since it only acquires significance\r\nby reference to the third dimension. Nevertheless, some such\r\ndistinction would be forced upon us. Suppose, for example,\r\nthat we took a small circle provided with an arrow,\r\nas in the figure, and moved this circle once round\r\nthe universe. Then the sense of the arrow would\r\nbe reversed. We should thus be forced, either to\r\nregard the new position as distinct from the former,\r\nwhich transforms our plane into a spherical plane,\r\nor to attribute the reversal of the arrow to the action of a\r\nmotion which restores our circle to its original place. It is\r\nto be observed that nothing short of moving round the\r\nuniverse would suffice to reverse the sense of the arrow. This\r\nreversal \u003cem\u003eseems\u003c/em\u003e like an action of empty space, which would force\r\nus to regard the points which, from a three-dimensional point\r\nof view, are coincident though opposite, as really distinct, and\r\nso reduce the elliptic to the spherical plane. But motion, not\r\nspace, really causes the change, and the elliptic plane is therefore\r\nnot proved to be impossible. The question is not, however,\r\nof any great philosophic importance.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e41.\u003c/b\u003e\u003ca name=\"N41\" id=\"N41\"\u003e\u003c/a\u003e\r\nIn connection with the reduction of metrical to projective\r\nGeometry, we have one more topic for discussion. This\r\nis the geometrical use of imaginaries, by means of which, except\r\nin the case of hyperbolic space, the reduction is effected. I\r\nhave already contended, on other grounds, that this reduction,\r\nin spite of its immense technical importance, and in spite of\r\nthe complete logical freedom of projective Geometry from\r\nmetrical ideas, is \u003cem\u003epurely\u003c/em\u003e technical, and is not philosophically\r\nvalid. The same conclusion will appear, if we take up Cayley\u0027s\r\nchallenge at the British Association, in his Presidential Address\r\nof 1883.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn this address, Professor Cayley devoted most of his time\r\nto non-Euclidean systems. Non-Euclidean \u003cem\u003espaces\u003c/em\u003e, he declared,\r\nseemed to him mistaken \u003cem\u003eà priori\u003c/em\u003e\u003ca name=\"FNanchor_59_59\" id=\"FNanchor_59_59\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_59_59\" class=\"fnanchor\"\u003e[59]\u003c/a\u003e; but non-Euclidean \u003cem\u003eGeometries\u003c/em\u003e,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_42\" id=\"Page_42\"\u003e[42]\u003c/a\u003e\u003c/span\u003e\r\nhere as in his mathematical works, were accepted as flowing\r\nfrom a change in the definition of distance. This view has\r\nbeen already discussed, and need not, therefore, be further\r\ncriticised here. What I wish to speak about, is the question\r\nwith which Cayley himself opened his address, namely, the geometrical\r\nuse and meaning of imaginary quantities. From the\r\nmanner in which he spoke of this question, it becomes imperative\r\nto treat it somewhat at length. For he said (pp. 8\u0026ndash;9):\u003c/p\u003e\r\n\r\n\u003cp\u003e\"… The notion which is the really fundamental one (and\r\nI cannot too strongly emphasize the assertion) underlying and\r\npervading the whole notion of modern analysis and Geometry,\r\n[is] that of imaginary magnitude in analysis, and of imaginary\r\nspace (or space as the \u003ci lang=\"la\" xml:lang=\"la\"\u003elocus in quo\u003c/i\u003e of imaginary points and\r\nfigures) in Geometry: I use in each case the word imaginary\r\nas including real…. Say even the conclusion were that the\r\nnotion belongs to mere technical mathematics, or has reference\r\nto nonentities in regard to which no science is possible, still\r\nit seems to me that (as a subject of philosophical discussion)\r\nthe notion ought not to be thus ignored; it should at least\r\nbe shown that there is a right to ignore it.\"\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e42.\u003c/b\u003e\u003ca name=\"N42\" id=\"N42\"\u003e\u003c/a\u003e\r\nThis right it is now my purpose to demonstrate. But\r\nfor fear non-mathematicians should miss the point of Cayley\u0027s\r\nremark (which has sometimes been erroneously supposed to\r\nrefer to non-Euclidean spaces), I may as well explain, at the\r\noutset, that this question is radically distinct from, and only\r\nindirectly connected with, the validity or import of Metageometry.\r\nAn imaginary quantity is one which involves\r\n√\u003cspan class=\"over\"\u003e\u0026ndash;1\u003c/span\u003e : its most general form is \u003cem\u003ea\u003c/em\u003e + √\u003cspan class=\"over\"\u003e\u0026ndash;1\u003c/span\u003e\u0026nbsp;\u003cem\u003eb\u003c/em\u003e where \u003cem\u003ea\u003c/em\u003e and \u003cem\u003eb\u003c/em\u003e are\r\nreal; Cayley uses the word imaginary so as to include real, in\r\norder to cover the special case where \u003cem\u003eb\u003c/em\u003e = 0. It will be convenient,\r\nin what follows, to exclude this wider meaning, and\r\nassume that \u003cem\u003eb\u003c/em\u003e is not zero. An imaginary point is one whose\r\ncoordinates involve √\u003cspan class=\"over\"\u003e\u0026ndash;1\u003c/span\u003e, \u003cem\u003ei.e.\u003c/em\u003e whose coordinates are imaginary\r\nquantities. An imaginary curve is one whose points are imaginary\u0026mdash;or,\r\nin some special uses, one whose equation contains\r\nimaginary coefficients. The mathematical subtleties to which\r\nthis notion leads need not be here discussed; the reader who\r\nis interested in them will find an excellent elementary account\r\nof their geometrical uses in Klein\u0027s Nicht-Euklid, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 38\u0026ndash;46.\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_43\" id=\"Page_43\"\u003e[43]\u003c/a\u003e\u003c/span\u003e\r\nBut for our present purpose, we may confine ourselves to\r\nimaginary points. If these are found to have a merely technical\r\nimport, and to be destitute of any philosophical meaning, then\r\nthe same will hold of any collection of imaginary points, \u003cem\u003ei.e.\u003c/em\u003e\r\nof any imaginary curve or surface.\u003c/p\u003e\r\n\r\n\u003cp\u003eThat the notion of imaginary points is of supreme importance\r\nin Geometry, will be seen by any one who reflects\r\nthat the circular points are imaginary, and that the reduction\r\nof metrical to projective Geometry, which is one of Cayley\u0027s\r\ngreatest achievements, depends on these points. But to discuss\r\nadequately their philosophical import is difficult to me, since\r\nI am unacquainted with any satisfactory philosophy of imaginaries\r\nin pure Algebra. I will therefore adopt the most\r\nfavourable hypothesis, and assume that no objection can be\r\nsuccessfully urged against this use. Even on this hypothesis,\r\nI think, no case can be made out for imaginary points in\r\nGeometry.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, we must exclude, from the imaginary\r\npoints considered, those whose coordinates are only imaginary\r\nwith certain special systems of coordinates. For example, if\r\none of a point\u0027s coordinates be the tangent from it to a sphere,\r\nthis coordinate will be imaginary for any point inside the\r\nsphere, and yet the point is perfectly real. A point, then, is\r\nonly to be called imaginary, when, whatever real system of\r\ncoordinates we adopt, one or more of the quantities expressing\r\nthese coordinates remains imaginary. For this purpose, it is\r\nmathematically sufficient to suppose our coordinates Cartesian\u0026mdash;a\r\npoint whose Cartesian coordinates are imaginary, is a true\r\nimaginary point in the above sense.\u003c/p\u003e\r\n\r\n\u003cp\u003eTo discuss the meaning of such a point, it is necessary to\r\nconsider briefly the fundamental nature of the correspondence\r\nbetween a point and its coordinates. Assuming that elementary\r\nGeometry has proved\u0026mdash;what I think it does satisfactorily\r\nprove\u0026mdash;that spatial relations are susceptible of quantitative\r\nmeasurement, then a given point will have, with a suitable\r\nsystem of coordinates, in a space of \u003cem\u003en\u003c/em\u003e dimensions, \u003cem\u003en\u003c/em\u003e quantitative\r\nrelations to the fixed spatial figure forming the axes of coordinates,\r\nand these \u003cem\u003en\u003c/em\u003e quantitative relations will, under certain\r\nreservations, be unique\u0026mdash;\u003cem\u003ei.e.\u003c/em\u003e, no other point will have the same\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_44\" id=\"Page_44\"\u003e[44]\u003c/a\u003e\u003c/span\u003e\r\nquantities assigned to it. (With many possible coordinate\r\nsystems, this latter condition is not realized: but for that\r\nvery reason they are inconvenient, and employed only in special\r\nproblems.) Thus given a coordinate system, and given any set\r\nof quantities, these quantities, \u003cem\u003eif they determine a point at all\u003c/em\u003e,\r\ndetermine it uniquely. But, by a natural extension of the\r\nmethod, the above reservation is dropped, and it is assumed\r\nthat to \u003cem\u003eevery\u003c/em\u003e set of quantities some point must correspond.\r\nFor this assumption there seems to me no vestige of evidence.\r\nAs well might a postman assume that, because every house in a\r\nstreet is uniquely determined by its number, therefore there\r\nmust be a house for every imaginable number. We must\r\nknow, in fact, that a given set of quantities can be the coordinates\r\nof some point in space, before it is legitimate to give\r\nany spatial significance to these quantities: and this knowledge,\r\nobviously, cannot be derived from operations with coordinates\r\nalone, on pain of a vicious circle. We must, to return to the\r\nabove analogy, know the number of houses in Piccadilly, before\r\nwe know whether a given number has a corresponding house or\r\nnot; and arithmetic alone, however subtly employed, will never\r\ngive us this information.\u003c/p\u003e\r\n\r\n\u003cp\u003eThus the distinction which is important is, not the distinction\r\nbetween real and imaginary quantities, but between\r\nquantities to which points correspond and quantities to which\r\nno points correspond. We can conventionally agree to denote\r\nreal points by imaginary coordinates, as in the Gaussian method\r\nof denoting by the single quantity (\u003cem\u003ea\u003c/em\u003e + √\u003cspan class=\"over\"\u003e\u0026ndash;1\u003c/span\u003e\u0026nbsp;\u003cem\u003eb\u003c/em\u003e) the point whose\r\nordinary coordinates are \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e. But this does not touch Cayley\u0027s\r\nmeaning. Cayley means that it is of great utility in mathematics\r\nto regard, as points with a real existence in space, the\r\nassumed spatial correlates of quantities which, with the\r\ncoordinate system employed, have no correlates in every-day\r\nspace; and that this utility is supposed, by many mathematicians,\r\nto indicate the validity of so fruitful an assumption.\r\nTo fix our ideas, let us consider Cartesian axes in three-dimensional\r\nEuclidean space. Then it appears, by inspection,\r\nthat a point may be situated at any distance to right or left of\r\nany of the three coordinate planes; taking this distance as a\r\ncoordinate, therefore, it appears that real points correspond to\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_45\" id=\"Page_45\"\u003e[45]\u003c/a\u003e\u003c/span\u003e\r\nall quantities from -∞ to +∞. The same appears for the\r\nother two coordinates; and since elementary Geometry proves\r\ntheir variations mutually independent, we know that one and\r\nonly one real point corresponds to any three real quantities.\r\nBut we also know, from the exhaustive method pursued, that\r\nall space is covered by the range of these three variable\r\nquantities: a fresh set of quantities, therefore, such as is\r\nintroduced by the use of imaginaries, possesses no spatial\r\ncorrelate, and can be supposed to possess one only by a\r\nconvenient fiction.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e43.\u003c/b\u003e\u003ca name=\"N43\" id=\"N43\"\u003e\u003c/a\u003e\r\nThe fact that the fiction \u003cem\u003eis\u003c/em\u003e convenient, however, may\r\nbe thought to indicate that it is more than a fiction. But this\r\npresumption, I think, can be easily explained away. For all\r\nthe fruitful uses of imaginaries, in Geometry, are those which\r\nbegin and end with real quantities, and use imaginaries only\r\nfor the intermediate steps. Now in all such cases, we have a\r\nreal spatial interpretation at the beginning and end of our\r\nargument, where alone the spatial interpretation is important:\r\nin the intermediate links, we are dealing in a purely algebraical\r\nmanner with purely algebraical quantities, and may perform\r\nany operations which are algebraically permissible. If the\r\nquantities with which we end are capable of spatial interpretation,\r\nthen, and only then, our result may be regarded as\r\ngeometrical. To use geometrical language, in any other case,\r\nis only a convenient help to the imagination. To speak, for\r\nexample, of projective properties which refer to the circular\r\npoints, is a mere \u003ci lang=\"la\" xml:lang=\"la\"\u003ememoria technica\u003c/i\u003e for purely algebraical\r\nproperties; the circular points are not to be found in space,\r\nbut only in the auxiliary quantities by which geometrical\r\nequations are transformed. That no contradictions arise from\r\nthe geometrical interpretation of imaginaries, is not wonderful:\r\nfor they are interpreted solely by the rules of Algebra, which\r\nwe may admit as valid in their application to imaginaries. The\r\nperception of space being wholly absent, Algebra rules supreme,\r\nand no inconsistency can arise. Wherever, for a moment, we\r\nallow our ordinary spatial notions to intrude, the grossest\r\nabsurdities do arise\u0026mdash;every one can see that a circle, being a\r\nclosed curve, cannot get to infinity. The metaphysician, who\r\nshould invent anything so preposterous as the circular points,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_46\" id=\"Page_46\"\u003e[46]\u003c/a\u003e\u003c/span\u003e\r\nwould be hooted from the field. But the mathematician may\r\nsteal the horse with impunity.\u003c/p\u003e\r\n\r\n\u003cp\u003eFinally, then, only a knowledge of space, not a knowledge of\r\nAlgebra, can assure us that any given set of quantities will have\r\na spatial correlate, and in the absence of such a correlate,\r\noperations with these quantities have no geometrical import.\r\nThis is the case with imaginaries in Cayley\u0027s sense, and their\r\nuse in Geometry, great as are its technical advantages, and\r\nrigid as is its technical validity, is wholly destitute of philosophical\r\nimportance.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e44.\u003c/b\u003e\u003ca name=\"N44\" id=\"N44\"\u003e\u003c/a\u003e\r\nWe have now, I think, discussed most of the questions\r\nconcerning the scope and validity of the projective method. We\r\nhave seen that it is independent of all metrical presuppositions,\r\nand that its use of coordinates does not involve the assumption\r\nthat spatial magnitudes are measured or expressed by them.\r\nWe have seen that it is able to deal, by its own methods alone,\r\nwith the question of the qualitative likeness of geometrical\r\nfigures, which is logically prior to any comparison as to quantity,\r\nsince quantity presupposes qualitative likeness. We have seen\r\nalso that, so far as its legitimate use extends, it applies equally\r\nto all homogeneous spaces, and that its criterion of an independently\r\npossible space\u0026mdash;the determination of a straight line by\r\ntwo points\u003ca name=\"FNanchor_60_60\" id=\"FNanchor_60_60\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_60_60\" class=\"fnanchor\"\u003e[60]\u003c/a\u003e\u0026mdash;is not subject to the qualifications and limitations\r\nwhich belong, as we have seen in the case of the cylinder, to\r\nthe metrical criterion of constant curvature. But we have also\r\nseen that, when projective Geometry endeavours to grapple\r\nwith spatial magnitude, and bring distance and the measurement\r\nof angles beneath its sway, its success, though technically\r\nvalid and important, is philosophically an apparent success only.\r\nMetrical Geometry, therefore, if quantity is to be applied to\r\nspace at all, remains a separate, though logically subsequent\r\nbranch of Mathematics.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e45.\u003c/b\u003e\u003ca name=\"N45\" id=\"N45\"\u003e\u003c/a\u003e\r\nIt only remains to say a few words about \u003cem\u003eSophus Lie\u003c/em\u003e.\r\nAs a mathematician, as the inventor of a new and immensely\r\npowerful method of analysis, he cannot be too highly praised.\r\nGeometry is only one of the numerous subjects to which his\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_47\" id=\"Page_47\"\u003e[47]\u003c/a\u003e\u003c/span\u003e\r\ntheory of continuous groups applies, but its application to\r\nGeometry has made a revolution in method, and has rendered\r\npossible, in such problems as Helmholtz\u0027s, a treatment infinitely\r\nmore precise and exhaustive than any which was possible\r\nbefore.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe general definition of a group is as follows: If we have\r\nany number of independent variables \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e \u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e…\u003cem\u003ex\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u003cspan class=\"xs\"\u003en\u003c/span\u003e\u003c/em\u003e\u003c/sub\u003e, and any series\r\nof transformations of these into new variables\u0026mdash;the transformations\r\nbeing defined by equations of specified forms, with\r\nparameters varying from one transformation to another\u0026mdash;then\r\nthe series of transformations form a \u003cem\u003egroup\u003c/em\u003e, if the successive\r\napplication of any two is equivalent to a single member of the\r\noriginal series of transformations. The group is \u003cem\u003econtinuous\u003c/em\u003e,\r\nwhen we can pass, by infinitesimal gradations within the group,\r\nfrom any one of the transformations to any other.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow, in Geometry, the result of two successive motions\r\nor collineations of a figure can always be obtained by a single\r\nmotion or collineation, and any motion or collineation can be\r\nbuilt up of a series of infinitesimal motions or collineations.\r\nMoreover the analytical expression of either is a certain transformation\r\nof the coordinates of all the points of the figure\u003ca name=\"FNanchor_61_61\" id=\"FNanchor_61_61\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_61_61\" class=\"fnanchor\"\u003e[61]\u003c/a\u003e.\r\nHence the transformations determining a motion or a collineation\r\nare such as to form a continuous group. But the\r\nquestion of the projective equivalence of two figures, to which\r\nall projective Geometry is reducible, must always be dealt\r\nwith by a collineation; and the question of the equality of\r\ntwo figures, to which all metrical Geometry is reducible, must\r\nalways be decided by a motion such as to cause superposition;\r\nhence the whole subject of Geometry may be regarded as a\r\ntheory of the continuous groups which define all possible\r\ncollineations and motions.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow Sophus Lie has developed, at great length, the purely\r\nanalytical theory of groups; he has therefore, by this method\r\nof formulating the problem, a very powerful weapon ready for\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_48\" id=\"Page_48\"\u003e[48]\u003c/a\u003e\u003c/span\u003e\r\nthe attack. In two papers \"On the foundations of Geometry\u003ca name=\"FNanchor_62_62\" id=\"FNanchor_62_62\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_62_62\" class=\"fnanchor\"\u003e[62]\u003c/a\u003e,\"\r\nundertaken at Klein\u0027s urgent request, he takes premisses which\r\nroughly correspond to those of Helmholtz, omitting Monodromy,\r\nand applies the theory of groups to the deduction of\r\ntheir consequences\u003ca name=\"FNanchor_63_63\" id=\"FNanchor_63_63\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_63_63\" class=\"fnanchor\"\u003e[63]\u003c/a\u003e. Helmholtz\u0027s work, he says, can hardly be\r\nlooked upon as \u003cem\u003eproving\u003c/em\u003e its conclusions, and indeed the more\r\nsearching analysis of the group-theory reveals several possibilities\r\nunknown to Helmholtz. Nevertheless, as a pioneer,\r\ndevoid of Lie\u0027s machinery, Helmholtz deserves, I think, more\r\npraise than Lie is willing to give him\u003ca name=\"FNanchor_64_64\" id=\"FNanchor_64_64\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_64_64\" class=\"fnanchor\"\u003e[64]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eLie\u0027s method is perfectly exhaustive; omitting the premiss\r\nof Monodromy, the others show that a body has six degrees of\r\nfreedom, \u003cem\u003ei.e.\u003c/em\u003e that the group giving all possible motions of a\r\nbody will have six independent members; if we keep one point\r\nfixed, the number of independent members is reduced to three.\r\nHe then, from his general theory, enumerates all the groups\r\nwhich satisfy this condition. In order that such a group should\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_49\" id=\"Page_49\"\u003e[49]\u003c/a\u003e\u003c/span\u003egive possible motions, it is necessary, by Helmholtz\u0027s second\r\naxiom, that it should leave invariant some function of the\r\ncoordinates of any two points. This eliminates several of the\r\ngroups previously enumerated, each of which he discusses in\r\nturn. He is thus led to the following results:\u003c/p\u003e\r\n\r\n\u003cp\u003eI. \u003cem\u003eIn two dimensions\u003c/em\u003e, if free mobility is to hold \u003cem\u003euniversally\u003c/em\u003e,\r\nthere are no groups satisfying Helmholtz\u0027s first three\r\naxioms, except those which give the ordinary Euclidean and\r\nnon-Euclidean motions; but if it is to hold only \u003cem\u003ewithin a\r\ncertain region\u003c/em\u003e, there is also a possible group in which the\r\ncurve described by any point in a rotation is not closed, but\r\nan equiangular spiral. To exclude this possibility, Helmholtz\u0027s\r\naxiom of Monodromy is required.\u003c/p\u003e\r\n\r\n\u003cp\u003eII. \u003cem\u003eIn three dimensions\u003c/em\u003e, the results go still more against\r\nHelmholtz. Assuming free mobility only \u003cem\u003ewithin a certain region\u003c/em\u003e,\r\nwe have to distinguish two cases: \u003cem\u003eEither\u003c/em\u003e free mobility holds,\r\nwithin that region, absolutely without exception, \u003cem\u003ei.e.\u003c/em\u003e when one\r\npoint is held fast, \u003cem\u003eevery\u003c/em\u003e other point within the region can\r\nmove freely over a surface: in this case the axiom of Monodromy\r\nis unnecessary, and the first three axioms suffice to\r\ndefine our group as that of Euclidean and non-Euclidean motions.\r\n\u003cem\u003eOr\u003c/em\u003e free mobility, within the specified region, holds\r\nonly of every point \u003cem\u003eof general position\u003c/em\u003e, while the points of a\r\ncertain line, when one point is fixed, are only able to move\r\non that line, not on a surface: when this is the case, other\r\ngroups are possible, and can only be excluded by Helmholtz\u0027s\r\nfourth axiom.\u003c/p\u003e\r\n\r\n\u003cp\u003eHaving now stated the purely mathematical results of Lie\u0027s\r\ninvestigations, we may return to philosophical considerations,\r\nby which Helmholtz\u0027s work was mainly motived. It becomes\r\nobvious, not only that exceptions within a certain region, but\r\nalso that limitation to a certain region, of the axiom of Free\r\nMobility, are philosophically quite impossible and inconceivable.\r\nHow can a certain line, or a certain surface, form an impassable\r\nbarrier in space, or have any mobility different in kind from\r\nthat of all other lines or surfaces? The notion cannot, in\r\nphilosophy, be permitted for a moment, since it destroys that\r\nmost fundamental of all the axioms, the homogeneity of space.\r\nWe not only may, therefore, but must take Helmholtz\u0027s axiom\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_50\" id=\"Page_50\"\u003e[50]\u003c/a\u003e\u003c/span\u003e\r\nof Free Mobility in its very strictest sense; the axiom of\r\nMonodromy thus becomes mathematically, as well as philosophically,\r\nsuperfluous. This is, from a philosophical standpoint,\r\nthe most important of Lie\u0027s results.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e46.\u003c/b\u003e\u003ca name=\"N46\" id=\"N46\"\u003e\u003c/a\u003e\r\nI have now come to the end of my history of Metageometry.\r\nIt has not been my aim to give an exhaustive\r\naccount of even the important works on the subject\u0026mdash;in the\r\nthird period, especially, the names of Poincaré, Pasch, Cremona,\r\nVeronese, and others who might be mentioned, would have\r\ncried shame upon me, had I had any such object. But I have\r\ntried to set forth, as clearly as I could, the principles at work\r\nin the various periods, the motives and results of successive\r\ntheories. We have seen how the philosophical motive, at first\r\npredominant, has been gradually extruded by the purely mathematical\r\nand technical spirit of most recent Geometers. At\r\nfirst, to discredit the Transcendental Aesthetic seemed, to Metageometers,\r\nas important as to advance their science; but from\r\nthe works of Cayley, Klein or Lie, no reader could gather that\r\nKant had ever lived. We have also seen, however, that as\r\nthe interest \u003cem\u003ein\u003c/em\u003e philosophy waned, the interest \u003cem\u003efor\u003c/em\u003e philosophy\r\nincreased: as the mathematical results shook themselves free\r\nfrom philosophical controversies, they assumed gradually a\r\nstable form, from which further development, we may reasonably\r\nhope, will take the form of growth, rather than transformation.\r\nThe same gradual development out of philosophy\r\nmight, I believe, be traced in the infancy of most branches of\r\nmathematics; when philosophical motives cease to operate,\r\nthis is, in general, a sign that the stage of uncertainty as to\r\npremisses is past, so that the future belongs entirely to mathematical\r\ntechnique. When this stable stage has been attained,\r\nit is time for Philosophy to borrow of Science, accepting its\r\nfinal premisses as those imposed by a real necessity of fact\r\nor logic.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e47.\u003c/b\u003e\u003ca name=\"N47\" id=\"N47\"\u003e\u003c/a\u003e\r\nNow in discussing the systems of Metageometry, we\r\nhave found two kinds, radically distinct and subject to different\r\naxioms. The historically prior kind, which deals with metrical\r\nideas, discusses, to begin with, the conditions of Free Mobility,\r\nwhich is essential to all measurement of space. It finds the\r\nanalytical expression of these conditions in the existence of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_51\" id=\"Page_51\"\u003e[51]\u003c/a\u003e\u003c/span\u003e\r\na space-constant, or constant measure of curvature, which is\r\nequivalent to the homogeneity of space. This is its first\r\naxiom.\u003c/p\u003e\r\n\r\n\u003cp\u003eIts second axiom states that space has a finite integral\r\nnumber of dimensions, \u003cem\u003ei.e.\u003c/em\u003e in metrical terms, that the position\r\nof a point, relative to any other figure in space, is uniquely\r\ndetermined by a finite number of spatial magnitudes, called\r\ncoordinates.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe third axiom of metrical Geometry may be called, to\r\ndistinguish it from the corresponding projective axiom, the\r\naxiom of distance. There exists one relation, it says, between\r\nany two points, which can be preserved unaltered in a combined\r\nmotion of both points, and which, in any motion of a system\r\nas one rigid body, is always unaltered. This relation we call\r\ndistance.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe above statement of the three essential axioms of\r\nmetrical Geometry is taken from Helmholtz as amended by Lie.\r\nLie\u0027s own statement of the axioms, as quoted above, has been\r\ntoo much influenced by projective methods to give a historically\r\ncorrect rendering of the spirit of the second period; Helmholtz\u0027s\r\nstatement, on the other hand, requires, as Lie has shewn, very\r\nconsiderable modifications. The above compromise may, therefore,\r\nI hope be taken as accepting Lie\u0027s corrections while\r\nretaining Helmholtz\u0027s spirit.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e48.\u003c/b\u003e\u003ca name=\"N48\" id=\"N48\"\u003e\u003c/a\u003e\r\nBut metrical Geometry, though it is historically prior,\r\nis logically subsequent to projective Geometry. For projective\r\nGeometry deals directly with that qualitative likeness, which\r\nthe judgment of quantitative comparison requires as its basis.\r\nNow the above three axioms of metrical Geometry, as we shall\r\nsee in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Section B, do not presuppose measurement,\r\nbut are, on the contrary, the conditions presupposed by\r\nmeasurement. Without these axioms, which are common to\r\nall three spaces, measurement would be impossible; with them,\r\nso I shall contend, measurement is able, though only empirically,\r\nto decide approximately which of the three spaces is valid of\r\nour actual world. But if these three axioms themselves express,\r\nnot results, but conditions, of measurement, must they not be\r\nequivalent to the statement of that qualitative likeness on\r\nwhich quantitative comparison depends? And if so, must we\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_52\" id=\"Page_52\"\u003e[52]\u003c/a\u003e\u003c/span\u003e\r\nnot expect to find the same axioms, though perhaps under a\r\ndifferent form, in projective Geometry?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e49.\u003c/b\u003e\u003ca name=\"N49\" id=\"N49\"\u003e\u003c/a\u003e\r\nThis expectation will not be disappointed. The above\r\nthree axioms, as we shall see hereafter, are one and all\r\nphilosophically equivalent to the homogeneity of space, and\r\nthis in turn is equivalent to the axioms of projective Geometry.\r\nThe axioms of projective Geometry, in fact, may be roughly\r\nstated thus:\u003c/p\u003e\r\n\r\n\u003cp\u003eI. Space is continuous and infinitely divisible; the zero of\r\nextension, resulting from infinite division, is called a Point.\r\nAll points are qualitatively similar, and distinguished by the\r\nmere fact that they lie outside one another.\u003c/p\u003e\r\n\r\n\u003cp\u003eII. Any two points determine a unique figure, the straight\r\nline; two straight lines, like two points, are qualitatively\r\nsimilar, and distinguished by the mere fact that they are\r\nmutually external.\u003c/p\u003e\r\n\r\n\u003cp\u003eIII. Three points not in one straight line determine a\r\nunique figure, the plane, and four points not in one plane\r\ndetermine a figure of three dimensions. This process may, so\r\nfar as can be seen \u003cem\u003eà priori\u003c/em\u003e, be continued, without in any way\r\ninterfering with the possibility of projective Geometry, to five\r\nor to \u003cem\u003en\u003c/em\u003e points. But projective Geometry requires, as an axiom,\r\nthat the process should stop with some positive integral number\r\nof points, after which, any fresh point is contained in the\r\nfigure determined by those already given. If the process stops\r\nwith (\u003cem\u003en\u003c/em\u003e + 1) points, our space is said to have \u003cem\u003en\u003c/em\u003e dimensions.\u003c/p\u003e\r\n\r\n\u003cp\u003eThese three axioms, it will be seen, are the equivalents of\r\nthe three axioms of metrical Geometry\u003ca name=\"FNanchor_65_65\" id=\"FNanchor_65_65\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_65_65\" class=\"fnanchor\"\u003e[65]\u003c/a\u003e, expressed without\r\nreference to quantity. We shall find them to be deducible, as\r\nbefore, from the homogeneity of space, or, more generally still,\r\nfrom the possibility of experiencing externality. They will\r\ntherefore appear as \u003cem\u003eà priori\u003c/em\u003e, as essential to the existence of any\r\nGeometry and to experience of an external world as such.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e50.\u003c/b\u003e\u003ca name=\"N50\" id=\"N50\"\u003e\u003c/a\u003e\r\nThat some logical necessity is involved in these axioms\r\nmight, I think, be inferred as probable, from their historical\r\ndevelopment alone. For the systems of Metageometry have\r\nnot, in general, been set up as more likely to fit facts than the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_53\" id=\"Page_53\"\u003e[53]\u003c/a\u003e\u003c/span\u003e\r\nsystem of Euclid; with the exception of Zöllner, for example, I\r\nknow of no one who has regarded the fourth dimension as\r\nrequired to explain phenomena. As regards the space-constant\r\nagain, though a \u003cem\u003esmall\u003c/em\u003e space-constant is regarded as empirically\r\npossible, it is not usually regarded as probable; and the finite\r\nspace-constants, with which Metageometry is equally conversant,\r\nare not usually thought even possible, as explanations\r\nof empirical fact\u003ca name=\"FNanchor_66_66\" id=\"FNanchor_66_66\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_66_66\" class=\"fnanchor\"\u003e[66]\u003c/a\u003e. Thus the motive has been throughout not\r\none of fact, but one of logic. Does not this give a strong\r\npresumption, that those axioms which are retained, are retained\r\nbecause they are logically indispensable? If this be so, the\r\naxioms common to Euclid and Metageometry will be \u003cem\u003eà priori\u003c/em\u003e,\r\nwhile those peculiar to Euclid will be empirical. After a\r\ncriticism of some differing theories of Geometry, I shall proceed,\r\nin Chapters \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e, to the proof and consequences of this\r\nthesis, which will form the remainder of the present work.\u003c/p\u003e\r\n\r\n\r\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_5_5\" id=\"Footnote_5_5\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_5_5\"\u003e\u003cspan class=\"label\"\u003e[5]\u003c/span\u003e\u003c/a\u003e V. \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003eMémoires de l\u0027Académie royale des Sciences de l\u0027lnstitut de France\u003c/cite\u003e,\r\nT. \u003cspan class=\"fs70\"\u003eXII.\u003c/span\u003e 1833, for a full statement of his results, with references to former\r\nwritings.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_6_6\" id=\"Footnote_6_6\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_6_6\"\u003e\u003cspan class=\"label\"\u003e[6]\u003c/span\u003e\u003c/a\u003e This bolder method, it appears, had been suggested, nearly a century\r\nearlier, by an Italian, Saccheri. His work, which seems to have remained\r\ncompletely unknown until Beltrami rediscovered it in 1889, is called \u003ccite class=\"fsnormal\" lang=\"la\" xml:lang=\"la\"\u003e\"Euclides\r\nab omni naevo vindicatus, etc.\"\u003c/cite\u003e Mediolani, 1733. (See Veronese, Grundzüge\r\nder Geometrie, German translation, Leipzig, 1894, p. 636.) His results\r\nincluded spherical as well as hyperbolic space; but they alarmed him to such\r\nan extent that he devoted the last half of his book to disproving them.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_7_7\" id=\"Footnote_7_7\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_7_7\"\u003e\u003cspan class=\"label\"\u003e[7]\u003c/span\u003e\u003c/a\u003e Klein\u0027s first account of elliptic Geometry, as a result of Cayley\u0027s projective\r\ntheory of distance, appeared in two articles entitled \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Ueber die sogenannte\r\nNicht-Euklidische Geometrie, I, II,\"\u003c/cite\u003e Math. Annalen 4, 6 (1871\u0026ndash;2). It was\r\nafterwards independently discovered by Newcomb, in an article entitled \"Elementary\r\nTheorems relating to the geometry of a space of three dimensions, and\r\nof uniform positive curvature in the fourth dimension,\" Crelle\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eJournal für die\r\nreine und angewandte Mathematik\u003c/cite\u003e, Vol. 83 (1877). For an account of the\r\nmathematical controversies concerning elliptic Geometry, see Klein\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Vorlesungen\r\nüber Nicht-Euklidische Geometrie,\"\u003c/cite\u003e Göttingen 1893, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 284 ff. A\r\nbibliography of the relevant literature up to the year 1878 was given by Halsted\r\nin the American Journal of Mathematics, Vols. 1, 2.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_8_8\" id=\"Footnote_8_8\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_8_8\"\u003e\u003cspan class=\"label\"\u003e[8]\u003c/span\u003e\u003c/a\u003e Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of\r\na non-Euclidean system, though he admits him to have been the first to\r\nregard the axiom of parallels as indemonstrable. His grounds for the former\r\nassertion seem scarcely adequate: on the evidence against it, see Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e,\r\n\u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e pp. 171\u0026ndash;174.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_9_9\" id=\"Footnote_9_9\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_9_9\"\u003e\u003cspan class=\"label\"\u003e[9]\u003c/span\u003e\u003c/a\u003e V. \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eBriefwechsel mit Schumacher\u003c/cite\u003e, Bd. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 268.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_10_10\" id=\"Footnote_10_10\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_10_10\"\u003e\u003cspan class=\"label\"\u003e[10]\u003c/span\u003e\u003c/a\u003e f. Helmholtz, Wiss. Abh. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 611.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_11_11\" id=\"Footnote_11_11\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_11_11\"\u003e\u003cspan class=\"label\"\u003e[11]\u003c/span\u003e\u003c/a\u003e Crelle\u0027s Journal, 1837.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_12_12\" id=\"Footnote_12_12\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_12_12\"\u003e\u003cspan class=\"label\"\u003e[12]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eTheorie der Parallellinien\u003c/cite\u003e, Berlin, 1840. Republished, Berlin, 1887.\r\nTranslated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_13_13\" id=\"Footnote_13_13\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_13_13\"\u003e\u003cspan class=\"label\"\u003e[13]\u003c/span\u003e\u003c/a\u003e Frischauf, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAbsolute Geometrie, nach Johann Bolyai\u003c/cite\u003e, Leipzig, 1872. Halsted,\r\nThe Science Absolute of Space, translated from the Latin, 4th edition, Austin,\r\nTexas, U.S.A. 1896.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_14_14\" id=\"Footnote_14_14\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_14_14\"\u003e\u003cspan class=\"label\"\u003e[14]\u003c/span\u003e\u003c/a\u003e Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from\r\nthe point-pair than from distance. See Frischauf, Absolute Geometrie,\r\nAnhang.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_15_15\" id=\"Footnote_15_15\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_15_15\"\u003e\u003cspan class=\"label\"\u003e[15]\u003c/span\u003e\u003c/a\u003e Compare Stallo, \u003ccite class=\"fsnormal\"\u003eConcepts of Modern Physics\u003c/cite\u003e, p. 248.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_16_16\" id=\"Footnote_16_16\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_16_16\"\u003e\u003cspan class=\"label\"\u003e[16]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eGesammelte Werke\u003c/cite\u003e, pp. 255\u0026ndash;268.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_17_17\" id=\"Footnote_17_17\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_17_17\"\u003e\u003cspan class=\"label\"\u003e[17]\u003c/span\u003e\u003c/a\u003e On the history of this word, see Stallo, \u003ccite class=\"fsnormal\"\u003eConcepts of Modern Physics\u003c/cite\u003e,\r\np. 258. It was used by Kant, and adapted by Herbart to almost the same\r\nmeaning as it bears in Riemann. Herbart, however, also uses the word\r\n\u003ci lang=\"de\" xml:lang=\"de\"\u003eReihenform\u003c/i\u003e to express a similar idea. See \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003ePsychologie als Wissenschaft\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\n§ 100 and \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e § 139, where Riemann\u0027s analogy with colours is also suggested.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_18_18\" id=\"Footnote_18_18\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_18_18\"\u003e\u003cspan class=\"label\"\u003e[18]\u003c/span\u003e\u003c/a\u003e Compare Erdmann\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Grössenbegriff vom Raum.\"\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_19_19\" id=\"Footnote_19_19\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_19_19\"\u003e\u003cspan class=\"label\"\u003e[19]\u003c/span\u003e\u003c/a\u003e Compare Veronese, op. cit. p. 642: \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Riemann ist in seiner Definition\r\ndes Begriffs Grösse dunkel.\"\u003c/cite\u003e See also Veronese\u0027s whole following criticism.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_20_20\" id=\"Footnote_20_20\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_20_20\"\u003e\u003cspan class=\"label\"\u003e[20]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eVorträge und Reden\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 18.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_21_21\" id=\"Footnote_21_21\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_21_21\"\u003e\u003cspan class=\"label\"\u003e[21]\u003c/span\u003e\u003c/a\u003e Cf. Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 160.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_22_22\" id=\"Footnote_22_22\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_22_22\"\u003e\u003cspan class=\"label\"\u003e[22]\u003c/span\u003e\u003c/a\u003e Since we are considering the curvature at a point, we are only concerned\r\nwith the first infinitesimal elements of the geodesics that start from such a\r\npoint.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_23_23\" id=\"Footnote_23_23\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_23_23\"\u003e\u003cspan class=\"label\"\u003e[23]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"la\" xml:lang=\"la\"\u003eDisquisitiones generales circa superficies curvas\u003c/cite\u003e, Werke, Bd. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e SS. 219\u0026ndash;258,\r\n1827.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_24_24\" id=\"Footnote_24_24\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_24_24\"\u003e\u003cspan class=\"label\"\u003e[24]\u003c/span\u003e\u003c/a\u003e Nevertheless, the Geometries of different surfaces of equal curvature are\r\nliable to important differences. For example, the cylinder is a surface of zero\r\ncurvature, but since its lines of curvature in one direction are finite, its\r\nGeometry coincides with that of the plane only for lengths smaller than the\r\ncircumference of its generating circle (see Veronese, op. cit. p. 644). Two\r\ngeodesics on a cylinder may meet in many points. For surfaces of zero\r\ncurvature on which this is not possible, the identity with the plane may be\r\nallowed to stand. Otherwise, the identity extends only to the properties of\r\nfigures not exceeding a certain size.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_25_25\" id=\"Footnote_25_25\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_25_25\"\u003e\u003cspan class=\"label\"\u003e[25]\u003c/span\u003e\u003c/a\u003e For we may consider two different parts of the same surface as corresponding\r\nparts of different surfaces; the above proposition then shows that a\r\nfigure can be reproduced in one part when it has been drawn in another, if the\r\nmeasures of curvature correspond in the two parts.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_26_26\" id=\"Footnote_26_26\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_26_26\"\u003e\u003cspan class=\"label\"\u003e[26]\u003c/span\u003e\u003c/a\u003e Crelle, Vols, \u003cspan class=\"fs70\"\u003eXIX.\u003c/span\u003e, \u003cspan class=\"fs70\"\u003eXX.\u003c/span\u003e, 1839\u0026ndash;40.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_27_27\" id=\"Footnote_27_27\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_27_27\"\u003e\u003cspan class=\"label\"\u003e[27]\u003c/span\u003e\u003c/a\u003e In this formula, \u003cem\u003eu\u003c/em\u003e, \u003cem\u003ev\u003c/em\u003e may be the lengths of lines, or the angles between\r\nlines, drawn on the surface, and having thus no necessary reference to a third\r\ndimension.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_28_28\" id=\"Footnote_28_28\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_28_28\"\u003e\u003cspan class=\"label\"\u003e[28]\u003c/span\u003e\u003c/a\u003e In what follows, I have given rather Klein\u0027s exposition of Riemann, than\r\nRiemann\u0027s own account. The former is much clearer and fuller, and not\r\nsubstantially different in any way. V. Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e pp. 206 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_29_29\" id=\"Footnote_29_29\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_29_29\"\u003e\u003cspan class=\"label\"\u003e[29]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N69\"\u003e§§ 69\u0026ndash;73.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_30_30\" id=\"Footnote_30_30\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_30_30\"\u003e\u003cspan class=\"label\"\u003e[30]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eGrundlagen der Geometrie\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, Leipziger Berichte, 1890; v. end of\r\npresent chapter, \u003ca href=\"#N45\"\u003e§ 45.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_31_31\" id=\"Footnote_31_31\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_31_31\"\u003e\u003cspan class=\"label\"\u003e[31]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e pp. 258\u0026ndash;9.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_32_32\" id=\"Footnote_32_32\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_32_32\"\u003e\u003cspan class=\"label\"\u003e[32]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"it\" xml:lang=\"it\"\u003eGiornale di Matematiche\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eVI.\u003c/span\u003e, 1868. Translated into French by\r\nJ. Hoüel in the \u003ccite lang=\"fr\" xml:lang=\"fr\"\u003e\"Annales Scientifiques de l\u0027École Normale Supérieure,\"\u003c/cite\u003e\r\nVol. \u003cspan class=\"fs70\"\u003eVI.\u003c/span\u003e 1869.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_33_33\" id=\"Footnote_33_33\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_33_33\"\u003e\u003cspan class=\"label\"\u003e[33]\u003c/span\u003e\u003c/a\u003e Crelle\u0027s Journal, Vols. \u003cspan class=\"fs70\"\u003eXIX.\u003c/span\u003e \u003cspan class=\"fs70\"\u003eXX.\u003c/span\u003e, 1839\u0026ndash;40.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_34_34\" id=\"Footnote_34_34\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_34_34\"\u003e\u003cspan class=\"label\"\u003e[34]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 190.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_35_35\" id=\"Footnote_35_35\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_35_35\"\u003e\u003cspan class=\"label\"\u003e[35]\u003c/span\u003e\u003c/a\u003e This article is more trigonometrical and analytical than the German book,\r\nand therefore makes the above interpretation peculiarly evident.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_36_36\" id=\"Footnote_36_36\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_36_36\"\u003e\u003cspan class=\"label\"\u003e[36]\u003c/span\u003e\u003c/a\u003e Such surfaces are by no means particularly remote. One of them, for\r\nexample, is formed by the revolution of the common Tractrix\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003ex\u003c/em\u003e = \u003cem\u003ea\u003c/em\u003e sin \u003cem\u003eφ\u003c/em\u003e, \u003cspan class=\"pad2\"\u003e\u0026nbsp;\u003c/span\u003e\r\n\u003cem\u003ey\u003c/em\u003e = \u003cem\u003ea\u003c/em\u003e \u003csub\u003e\u003cspan class=\"xl\"\u003e(\u003c/span\u003e\u003c/sub\u003elog\r\ntan \u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003eφ\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e2\u003c/span\u003e\r\n \u003c/span\u003e\r\n+ cos \u003cem\u003eφ\u003c/em\u003e\u003csub\u003e\u003cspan class=\"xl\"\u003e)\u003c/span\u003e\u003c/sub\u003e.\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_37_37\" id=\"Footnote_37_37\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_37_37\"\u003e\u003cspan class=\"label\"\u003e[37]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"it\" xml:lang=\"it\"\u003e\"Teoria fondamentale degli spazii di curvatura costanta,\" Annali di\r\nMatematica\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Vol. 2, 1868\u0026ndash;9. Also translated by J. Hoüel, \u003ccite\u003eloc. cit.\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_38_38\" id=\"Footnote_38_38\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_38_38\"\u003e\u003cspan class=\"label\"\u003e[38]\u003c/span\u003e\u003c/a\u003e See Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 47 ff., and the references there given.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_39_39\" id=\"Footnote_39_39\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_39_39\"\u003e\u003cspan class=\"label\"\u003e[39]\u003c/span\u003e\u003c/a\u003e See quotation below, from his British Association Address.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_40_40\" id=\"Footnote_40_40\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_40_40\"\u003e\u003cspan class=\"label\"\u003e[40]\u003c/span\u003e\u003c/a\u003e Compare the opening sentence, due to Cayley, of Salmon\u0027s Higher Plane\r\nCurves.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_41_41\" id=\"Footnote_41_41\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_41_41\"\u003e\u003cspan class=\"label\"\u003e[41]\u003c/span\u003e\u003c/a\u003e V. \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Chaps. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_42_42\" id=\"Footnote_42_42\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_42_42\"\u003e\u003cspan class=\"label\"\u003e[42]\u003c/span\u003e\u003c/a\u003e See p. 9 of Cayley\u0027s address to the Brit. Ass. 1883. Also a quotation from\r\nKlein in Erdmann\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAxiome der Geometrie\u003c/cite\u003e, p. 124 note.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_43_43\" id=\"Footnote_43_43\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_43_43\"\u003e\u003cspan class=\"label\"\u003e[43]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eNature\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eXLV.\u003c/span\u003e p. 407.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_44_44\" id=\"Footnote_44_44\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_44_44\"\u003e\u003cspan class=\"label\"\u003e[44]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 200.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_45_45\" id=\"Footnote_45_45\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_45_45\"\u003e\u003cspan class=\"label\"\u003e[45]\u003c/span\u003e\u003c/a\u003e I.e. the equation \u003cem\u003eAB\u003c/em\u003e + \u003cem\u003eBC\u003c/em\u003e = \u003cem\u003eAC\u003c/em\u003e, for three points in one straight line.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_46_46\" id=\"Footnote_46_46\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_46_46\"\u003e\u003cspan class=\"label\"\u003e[46]\u003c/span\u003e\u003c/a\u003e The formula substituted by Klein for Cayley\u0027s inverse sine or cosine.\r\nThe two are equivalent, but Klein\u0027s is mathematically much the more\r\nconvenient.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_47_47\" id=\"Footnote_47_47\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_47_47\"\u003e\u003cspan class=\"label\"\u003e[47]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eElements of Projective Geometry\u003c/cite\u003e, Second Edition, Oxford, 1893,\r\nChap. \u003cspan class=\"fs70\"\u003eIX.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_48_48\" id=\"Footnote_48_48\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_48_48\"\u003e\u003cspan class=\"label\"\u003e[48]\u003c/span\u003e\u003c/a\u003e Chap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Section B.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_49_49\" id=\"Footnote_49_49\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_49_49\"\u003e\u003cspan class=\"label\"\u003e[49]\u003c/span\u003e\u003c/a\u003e See \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 338 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_50_50\" id=\"Footnote_50_50\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_50_50\"\u003e\u003cspan class=\"label\"\u003e[50]\u003c/span\u003e\u003c/a\u003e See his \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eGeometrie der Lage\u003c/cite\u003e, § 8, Harmonische Gebilde.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_51_51\" id=\"Footnote_51_51\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_51_51\"\u003e\u003cspan class=\"label\"\u003e[51]\u003c/span\u003e\u003c/a\u003e The anharmonic ratio of four numbers, \u003cem\u003ep\u003c/em\u003e, \u003cem\u003eq\u003c/em\u003e, \u003cem\u003er\u003c/em\u003e, \u003cem\u003es\u003c/em\u003e, is defined as\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n(\u003cem\u003ep\u003c/em\u003e – \u003cem\u003eq\u003c/em\u003e).(\u003cem\u003er\u003c/em\u003e – \u003cem\u003es\u003c/em\u003e) \u003csub\u003e\u003cspan class=\"xl\"\u003e/\u003c/span\u003e\u003c/sub\u003e (\u003cem\u003ep\u003c/em\u003e – \u003cem\u003er\u003c/em\u003e).(\u003cem\u003eq\u003c/em\u003e – \u003cem\u003es\u003c/em\u003e).\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_52_52\" id=\"Footnote_52_52\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_52_52\"\u003e\u003cspan class=\"label\"\u003e[52]\u003c/span\u003e\u003c/a\u003e \u003cem\u003eI.e.\u003c/em\u003e as transformable into each other by a collineation. See \u003ca href=\"#N110\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\r\nSec. A, § 110.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_53_53\" id=\"Footnote_53_53\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_53_53\"\u003e\u003cspan class=\"label\"\u003e[53]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#SA\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Sec. A.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_54_54\" id=\"Footnote_54_54\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_54_54\"\u003e\u003cspan class=\"label\"\u003e[54]\u003c/span\u003e\u003c/a\u003e It follows from this, that the reduction of metrical to projective properties,\r\neven when, as in hyperbolic Geometry, the Absolute is real, is only apparent,\r\nand has a merely technical validity.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_55_55\" id=\"Footnote_55_55\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_55_55\"\u003e\u003cspan class=\"label\"\u003e[55]\u003c/span\u003e\u003c/a\u003e Sir R. Ball does not regard his non-Euclidean content as a possible space\r\n(\u003ccite\u003ev. op. cit.\u003c/cite\u003e p. 151). In this important point I disagree with his interpretation,\r\nholding such a content to be a space as possible, \u003cem\u003eà priori\u003c/em\u003e, as Euclid\u0027s, and\r\nperhaps actually true within the margin due to errors of observation.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_56_56\" id=\"Footnote_56_56\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_56_56\"\u003e\u003cspan class=\"label\"\u003e[56]\u003c/span\u003e\u003c/a\u003e See \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 97 ff. and p. 292 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_57_57\" id=\"Footnote_57_57\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_57_57\"\u003e\u003cspan class=\"label\"\u003e[57]\u003c/span\u003e\u003c/a\u003e Newcomb says (\u003ccite\u003eloc. cit.\u003c/cite\u003e p. 293): \"The system here set forth is founded\r\non the following three postulates.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"1. I assume that space is triply extended, unbounded, without properties\r\ndependent either on position or direction, and possessing such planeness in its\r\nsmallest parts that both the postulates of the Euclidean Geometry, and our\r\ncommon conceptions of the relations of the parts of space are true for every\r\nindefinitely small region in space.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"2. I assume that this space is affected with such curvature that a right\r\nline shall always return into itself at the end of a finite and real distance 2\u003cem\u003eD\u003c/em\u003e\r\nwithout losing, in any part of its course, that symmetry with respect to space\r\non all sides of it which constitutes the fundamental property of our conception\r\nof it.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\n\"3. I assume that if two right lines emanate from the same point, making\r\nthe indefinitely small angle \u003cem\u003ea\u003c/em\u003e with each other, their distance apart at the\r\ndistance \u003cem\u003er\u003c/em\u003e from the point of intersection will be given by the equation\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003es\u003c/em\u003e =\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e2\u003cem\u003eaD\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003eπ\u003c/span\u003e\r\n\u003c/span\u003e\r\nsin\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003erπ\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e2\u003cem\u003eD\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e\r\n.\u003c/p\u003e\r\n\r\n\u003cp\u003e\r\nThe right line thus has this property in common with the Euclidean right line\r\nthat two such lines intersect only in a single point. It may be that the number\r\nof points in which two such lines can intersect admit of being determined from\r\nthe laws of curvature, but not being able so to determine it, I assume as a\r\npostulate the fundamental property of the Euclidean right line.\"\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is plain that in the absence of the determination spoken of, the possibility\r\nof elliptic space is not established. It may be possible, for example, to prove\r\nthat, in a space where there is a maximum to distance, there must be an infinite\r\nnumber of straight lines joining two points of maximum distance. In this\r\nevent, elliptic space would become impossible.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_58_58\" id=\"Footnote_58_58\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_58_58\"\u003e\u003cspan class=\"label\"\u003e[58]\u003c/span\u003e\u003c/a\u003e For an elucidation of this term, see Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNicht-Euklid\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 99 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_59_59\" id=\"Footnote_59_59\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_59_59\"\u003e\u003cspan class=\"label\"\u003e[59]\u003c/span\u003e\u003c/a\u003e Cf. p. 9 of Report: \"My own view is that Euclid\u0027s twelfth axiom, in\r\nPlayfair\u0027s form of it, does not need demonstration, but is part of our notion of\r\nspace, of the physical space of our experience, but which is the representation\r\nlying at the bottom of all external experience.\"\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_60_60\" id=\"Footnote_60_60\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_60_60\"\u003e\u003cspan class=\"label\"\u003e[60]\u003c/span\u003e\u003c/a\u003e The exception to this axiom, in spherical space, presupposes metrical\r\nGeometry, and does not destroy the validity of the axiom for projective\r\nGeometry. See \u003ca href=\"#N171\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Sec. B, § 171.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_61_61\" id=\"Footnote_61_61\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_61_61\"\u003e\u003cspan class=\"label\"\u003e[61]\u003c/span\u003e\u003c/a\u003e Mathematicians of Lie\u0027s school have a habit, at first somewhat confusing,\r\nof speaking of motions of space instead of motions of bodies, as though space\r\nas a whole could move. All that is meant is, of course, the equivalent\r\nmotion of the coordinate axes, \u003cem\u003ei.e.\u003c/em\u003e a change of axes in the usual elementary\r\nsense.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_62_62\" id=\"Footnote_62_62\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_62_62\"\u003e\u003cspan class=\"label\"\u003e[62]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Ueber die Grundlagen der Geometrie,\"\u003c/cite\u003e Leipziger Berichte, 1890. The\r\nproblem of these two papers is really metrical, since it is concerned, not with\r\ncollineations in general, but with motions. The problem, however, is dealt\r\nwith by the projective method, motions being regarded as collineations which\r\nleave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie\u0027s\r\nwork, until some account had been given of the projective method.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_63_63\" id=\"Footnote_63_63\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_63_63\"\u003e\u003cspan class=\"label\"\u003e[63]\u003c/span\u003e\u003c/a\u003e Lie\u0027s premisses, to be accurate, are the following:\r\n\u003c/p\u003e\r\n\u003cp\u003eLet\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e = \u003cem\u003ef\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e…)\u003cbr /\u003e\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e = \u003cem\u003eφ\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e…)\u003cbr /\u003e\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e = \u003cem\u003eψ\u003c/em\u003e (\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e…)\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\r\ngive an infinite family of real transformations of space, as to which we make the\r\nfollowing hypotheses:\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pad1\"\u003e\u0026nbsp;\u003c/span\u003e\r\nA. The functions \u003cem\u003ef\u003c/em\u003e, \u003cem\u003eφ\u003c/em\u003e, \u003cem\u003eψ\u003c/em\u003e, are \u003cem\u003eanalytical\u003c/em\u003e functions of\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\n\u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ea\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e….\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pad1\"\u003e\u0026nbsp;\u003c/span\u003e\r\nB. Two points \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003cem\u003ez\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003cem\u003ez\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\r\npossess an invariant, \u003cem\u003ei.e.\u003c/em\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"center\"\u003e\r\nΩ(\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e, \u003cem\u003ez\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e,\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e, \u003cem\u003ez\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e) =\r\nΩ(\u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e′, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e′, \u003cem\u003ez\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e′,\r\n\u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e′, \u003cem\u003ey\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e′, \u003cem\u003ez\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e′)\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\r\nwhere \u003cem\u003ex\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e′…, \u003cem\u003ex\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e′…, are the transformed coordinates of the two points.\r\n\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pad1\"\u003e\u0026nbsp;\u003c/span\u003e\r\nC. Free Mobility: \u003cem\u003ei.e.\u003c/em\u003e, any point can be moved into any other position;\r\nwhen one point is fixed, any other point of general position can take up ∞\u003csup\u003e2\u003c/sup\u003e\r\npositions; when two points are fixed, any other of general position can take up\r\n∞\u003csup\u003e1\u003c/sup\u003e positions; when three, no motion is possible\u0026mdash;these limitations being results\r\nof the equations given by the invariant Ω.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_64_64\" id=\"Footnote_64_64\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_64_64\"\u003e\u003cspan class=\"label\"\u003e[64]\u003c/span\u003e\u003c/a\u003e On this point, cf. Klein, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eHöhere Geometrie\u003c/cite\u003e, Göttingen, 1893, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 225\u0026ndash;244,\r\nespecially pp. 230\u0026ndash;1.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_65_65\" id=\"Footnote_65_65\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_65_65\"\u003e\u003cspan class=\"label\"\u003e[65]\u003c/span\u003e\u003c/a\u003e Axiom \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e of the metrical triad corresponds to Axiom \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e of the projective,\r\nand \u003ci lang=\"la\" xml:lang=\"la\"\u003evice versâ\u003c/i\u003e.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_66_66\" id=\"Footnote_66_66\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_66_66\"\u003e\u003cspan class=\"label\"\u003e[66]\u003c/span\u003e\u003c/a\u003e Cf. Helmholtz, Wiss. Abh. Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 640, note: \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Die Bearbeiter\r\nder Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie\r\nbehauptet.\"\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\r\n\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_54\" id=\"Page_54\"\u003e[54]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003e\u003ca name=\"CHAP_II\" id=\"CHAP_II\"\u003e\u003c/a\u003eCHAPTER II.\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"fs60\"\u003eCRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL\r\nTHEORIES OF GEOMETRY.\u003c/span\u003e\u003c/h2\u003e\r\n\r\n\r\n\u003cp\u003e\u003cb\u003e51.\u003c/b\u003e\u003ca name=\"N51\" id=\"N51\"\u003e\u003c/a\u003e\r\nWe have now traced the mathematical development\r\nof the theory of geometrical axioms, from the first revolt against\r\nEuclid to the present day. We may hope, therefore, to have\r\nat our command the technical knowledge required for the\r\nphilosophy of the subject. The importance of Geometry, in\r\nthe theories of knowledge which have arisen in the past, can\r\nscarcely be exaggerated. In Descartes, we find the whole\r\ntheory of method dominated by analytical Geometry, of whose\r\nfruitfulness he was justly proud. In Spinoza, the paramount\r\ninfluence of Geometry is too obvious to require comment.\r\nAmong mathematicians, Newton\u0027s belief in absolute space was\r\nlong supreme, and is still responsible for the current formulation\r\nof the laws of motion. Against this belief on the one\r\nhand, and against Leibnitz\u0027s theory of space on the other, and\r\nnot, as Caird has pointed out\u003ca name=\"FNanchor_67_67\" id=\"FNanchor_67_67\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_67_67\" class=\"fnanchor\"\u003e[67]\u003c/a\u003e, against Hume\u0027s empiricism,\r\nwas directed that keystone of the Critical Philosophy, the\r\nKantian doctrine of space. Thus Geometry has been, throughout,\r\nof supreme importance in the theory of knowledge.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut in a criticism of representative modern theories of\r\nGeometry, which is designed to be, not a history of the subject,\r\nbut an introduction to, and defence of, the views of the author,\r\nit will not be necessary to discuss any more ancient theory\r\nthan that of Kant. Kant\u0027s views on this subject, true or false,\r\nhave so dominated subsequent thought, that whether they were\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_55\" id=\"Page_55\"\u003e[55]\u003c/a\u003e\u003c/span\u003e\r\naccepted or rejected, they seemed equally potent in forming\r\nthe opinions, and the manner of exposition, of almost all later\r\nwriters.\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eKant.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e52.\u003c/b\u003e\u003ca name=\"N52\" id=\"N52\"\u003e\u003c/a\u003e\r\nIt is not my purpose, in this chapter, to add to the\r\nvoluminous literature of Kantian criticism, but only to discuss\r\nthe bearing of Metageometry on the argument of the Transcendental\r\nAesthetic, and the aspect under which this argument\r\nmust be viewed in a discussion of Geometry\u003ca name=\"FNanchor_68_68\" id=\"FNanchor_68_68\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_68_68\" class=\"fnanchor\"\u003e[68]\u003c/a\u003e. On this point\r\nseveral misunderstandings seem to me to have had wide prevalence,\r\nboth among friends and foes, and these misunderstandings\r\nI shall endeavour, if I can, to remove.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, what does Kant\u0027s doctrine mean for\r\nGeometry? Obviously not the aspect of the doctrine which\r\nhas been attacked by psychologists, the \"Kantian machine-shop\"\r\nas James calls it\u0026mdash;at any rate, if this can be clearly\r\nseparated from the logical aspect. The question whether space\r\nis given in sensation, or whether, as Kant maintained, it is\r\ngiven by an intuition to which no external matter corresponds,\r\nmay for the present be disregarded. If, indeed, we held the\r\nview which seems crudely to sum up the standpoint of the\r\nCritique, the view that all certain knowledge is self-knowledge,\r\nthen we should be committed, if we had decided that Geometry\r\nwas apodeictic, to the view that space is subjective. But even\r\nthen, the psychological question could only arise when the\r\nepistemological question had been solved, and could not, therefore,\r\nbe taken into account in our first investigation. The\r\nquestion before us is precisely the question whether, or how\r\nfar, Geometry is apodeictic, and for the moment we have only\r\nto investigate this question, without fear of psychological consequences.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e53.\u003c/b\u003e\u003ca name=\"N53\" id=\"N53\"\u003e\u003c/a\u003e\r\nNow on this question, as on almost all questions in the\r\nAesthetic or the Analytic, Kant\u0027s argument is twofold. On\r\nthe one hand, he says, Geometry is known to have apodeictic\r\ncertainty: therefore space must be \u003cem\u003eà priori\u003c/em\u003e and subjective.\r\nOn the other hand, it follows, from grounds independent of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_56\" id=\"Page_56\"\u003e[56]\u003c/a\u003e\u003c/span\u003e\r\nGeometry, that space is subjective and \u003cem\u003eà priori\u003c/em\u003e; therefore\r\nGeometry must have apodeictic certainty. These two arguments\r\nare not clearly distinguished in the Aesthetic, but a\r\nlittle analysis, I think, will disentangle them. Thus in the first\r\nedition, the first two arguments deduce, from non-geometrical\r\ngrounds, the apriority of space; the third deduces the apodeictic\r\ncertainty of Geometry, and maintains, conversely, that no other\r\nview can account for this certainty\u003ca name=\"FNanchor_69_69\" id=\"FNanchor_69_69\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_69_69\" class=\"fnanchor\"\u003e[69]\u003c/a\u003e; the last two arguments\r\nonly maintain that space is an intuition, not a concept. In\r\nthe second edition, the double argument is clearer, the apriority\r\nof space being proved independently of Geometry in the metaphysical\r\ndeduction, and deduced from the certainty of Geometry,\r\nas the only possible explanation of this, in the transcendental\r\ndeduction. In the Prolegomena, the latter argument alone is\r\nused, but in the Critique both are employed.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e54.\u003c/b\u003e\u003ca name=\"N54\" id=\"N54\"\u003e\u003c/a\u003e\r\nNow it must be admitted, I think, that Metageometry\r\nhas destroyed the legitimacy of the argument from Geometry\r\nto space; we can no longer affirm, on purely geometrical\r\ngrounds, the apodeictic certainty of Euclid. But unless Metageometry\r\nhas done more than this\u0026mdash;unless it has proved, what\r\nI believe it alone cannot prove, that Euclid has \u003cem\u003enot\u003c/em\u003e apodeictic\r\ncertainty\u0026mdash;then Kant\u0027s other line of argument retains what\r\nforce it may ever have had. The actual space we know, it may\r\nsay, is admittedly Euclidean, and is proved, without any reference\r\nto Geometry, to be \u003cem\u003eà priori\u003c/em\u003e; \u003cem\u003ehence\u003c/em\u003e Euclid has apodeictic\r\ncertainty, and non-Euclid stands condemned. To this it is no\r\nanswer to urge, with the Metageometers, that non-Euclidean\r\nsystems are \u003cem\u003elogically\u003c/em\u003e self-consistent; for Kant is careful to\r\nargue that geometrical reasoning, by virtue of our intuition\r\nof space, is synthetic, and cannot, though \u003cem\u003eà priori\u003c/em\u003e, be upheld\r\nby the principle of contradiction alone\u003ca name=\"FNanchor_70_70\" id=\"FNanchor_70_70\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_70_70\" class=\"fnanchor\"\u003e[70]\u003c/a\u003e. Unless non-Euclideans\r\ncan prove, what they have certainly failed to prove up to the\r\npresent, that we can frame an \u003cem\u003eintuition\u003c/em\u003e of non-Euclidean spaces,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_57\" id=\"Page_57\"\u003e[57]\u003c/a\u003e\u003c/span\u003eKant\u0027s position cannot be upset by Metageometry alone, but\r\nmust also be attacked, if it is to be successfully attacked, on\r\nits purely philosophical side.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e55.\u003c/b\u003e\u003ca name=\"N55\" id=\"N55\"\u003e\u003c/a\u003e\r\nFor such an attack, two roads lie open: either we may\r\ndisprove the first two arguments of the Aesthetic, or we may\r\ncriticize, from the standpoint of general logic, the Kantian doctrine\r\nof synthetic \u003cem\u003eà priori\u003c/em\u003e judgments and their connection with\r\nsubjectivity. Both these attacks, I believe, could be conducted\r\nwith some success; but if we are to disprove the apodeictic certainty\r\nof Geometry, one or other is essential, and both, I believe,\r\nwill be found only partially successful. It will be my aim to\r\nprove, in discussing these two lines of attack, (1) that the distinction\r\nof synthetic and analytic judgments is untenable, and\r\nfurther, that the principle of contradiction can only give fruitful\r\nresults on the assumption that experience in general, or, in a\r\nparticular science, some special branch of experience, is to be\r\nformally possible; (2) that the first two arguments of the Transcendental\r\nAesthetic suffice to prove, not Euclidean space,\r\nbut \u003cem\u003esome\u003c/em\u003e form of externality\u0026mdash;which may be sensational or\r\nintuitional, but not merely conceptual\u0026mdash;a necessary prerequisite\r\nof experience of an external world. In the third and fourth\r\nchapters, I shall contend, as a result of these conclusions, that\r\nthose axioms, which Euclid and Metageometry have in common,\r\ncoincide with those properties of any form of externality which\r\nare deducible, by the principle of contradiction, from the possibility\r\nof experience of an external world. These properties,\r\nthen, may be said, though not quite in the Kantian sense, to be\r\n\u003cem\u003eà priori\u003c/em\u003e properties of space, and as to these, I think, a modified\r\nKantian position may be maintained. But the question of the\r\nsubjective or objective nature of space may be left wholly out\r\nof account during the course of this discussion, which will gain\r\nby dealing exclusively with logical, as opposed to psychological\r\npoints of view.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e56.\u003c/b\u003e\u003ca name=\"N56\" id=\"N56\"\u003e\u003c/a\u003e\r\n(1) \u003cem\u003eKant\u0027s logical position.\u003c/em\u003e The doctrine of synthetic\r\nand analytic judgments\u0026mdash;at any rate if this is taken as the\r\ncorner-stone of Epistemology\u0026mdash;has been so completely rejected\r\nby most modern logicians\u003ca name=\"FNanchor_71_71\" id=\"FNanchor_71_71\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_71_71\" class=\"fnanchor\"\u003e[71]\u003c/a\u003e, that it would demand little attention\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_58\" id=\"Page_58\"\u003e[58]\u003c/a\u003e\u003c/span\u003e\r\nhere, but for the fact that an enthusiastic French Kantian,\r\nM. Renouvier, has recently appealed to it, with perfect confidence,\r\non the very question of Geometry\u003ca name=\"FNanchor_72_72\" id=\"FNanchor_72_72\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_72_72\" class=\"fnanchor\"\u003e[72]\u003c/a\u003e. And it must be\r\nowned, with M. Renouvier, that if such judgments existed, in\r\nthe Kantian sense, non-Euclidean Geometry, which makes no\r\nappeal to intuition, could have nothing to say against them.\r\nM. Renouvier\u0027s contention, therefore, forces us briefly to review\r\nthe arguments against Kant\u0027s doctrine, and briefly to discuss\r\nwhat logical canon is to replace it.\u003c/p\u003e\r\n\r\n\u003cp\u003eEvery judgment\u0026mdash;so modern logic contends\u0026mdash;is both synthetic\r\nand analytic; it combines parts into a whole, and analyses\r\na whole into parts\u003ca name=\"FNanchor_73_73\" id=\"FNanchor_73_73\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_73_73\" class=\"fnanchor\"\u003e[73]\u003c/a\u003e. If this be so, the distinction of analysis\r\nand synthesis, whatever may be its importance in pure Logic,\r\ncan have no value in Epistemology. But such a doctrine, it\r\nmust be observed, allows full scope to the principle of contradiction:\r\nthis criterion, since all judgments, in one aspect at\r\nleast, are analytic, is applicable to all judgments alike. On\r\nthe other hand, the whole which is analysed must be supposed\r\nalready given, before the parts can be mutually contradictory:\r\nfor only by connection in a given whole can two parts or\r\nadjectives be incompatible. Thus the principle of contradiction\r\nremains barren until we already have some judgments, and\r\neven some inference: for the parts may be regarded, to some\r\nextent, as an inference from the whole, or \u003ci lang=\"la\" xml:lang=\"la\"\u003evice versâ\u003c/i\u003e. When\r\nonce the arch of knowledge is constructed, the parts support\r\none another, and the principle of contradiction is the keystone:\r\nbut until the arch is built, the keystone remains suspended,\r\nunsupported and unsupporting, in the empty air. In other\r\nwords, knowledge once existent can be analysed, but knowledge\r\nwhich should have to win every inch of the way against a critical\r\nscepticism, could never begin, and could never attain that\r\ncircular condition in which alone it can stand.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut Kant\u0027s doctrine, if true, is designed to restrain a critical\r\nscepticism even where it might be effective. Certain fundamental\r\npropositions, he says, are not deducible from logic,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_59\" id=\"Page_59\"\u003e[59]\u003c/a\u003e\u003c/span\u003e\u003cem\u003ei.e.\u003c/em\u003e their contradictories are not self-contradictory; they combine\r\na subject and predicate which cannot, in any purely logical\r\nway, be shewn to have any connection, and yet these judgments\r\nhave apodeictic certainty. But concerning such judgments,\r\nKant is generally careful not to rely upon the mere subjective\r\nconviction that they are undeniable: he proves, with every\r\nprecaution, that without them experience would be impossible.\r\nExperience consists in the combination of terms which formal\r\nlogic leaves apart, and presupposes, therefore, certain judgments\r\nby which a framework is made for bringing such terms together.\r\nWithout these judgments\u0026mdash;so Kant contends\u0026mdash;all synthesis\r\nand all experience would be impossible. If, therefore, the\r\ndetail of the Kantian reasoning be sound, his results may be\r\nobtained by the principle of contradiction \u003cem\u003eplus\u003c/em\u003e the possibility\r\nof experience, as well as by his distinction of synthetic and\r\nanalytic judgments.\u003c/p\u003e\r\n\r\n\u003cp\u003eLogic, at the present day, arrogates to itself at once a wider\r\nand a narrower sphere than Kant allowed to it. Wider, because\r\nit believes itself capable of condemning any false principle or\r\npostulate; narrower, because it believes that its law of contradiction,\r\nwithout a given whole or a given hypothesis, is powerless,\r\nand that two terms, \u003cem\u003eper se\u003c/em\u003e, though they may be different,\r\ncannot be contradictories, but acquire this relation only by\r\ncombination in a whole about which something is known, or\r\nby connection with a postulate which, for some reason, must\r\nbe preserved. Thus no judgment, \u003cem\u003eper se\u003c/em\u003e, is either analytic or\r\nsynthetic, for the severance of a judgment from its context robs\r\nit of its vitality, and makes it not truly a judgment at all.\r\nBut in its proper context it is neither purely synthetic nor\r\npurely analytic; for while it is the further determination of a\r\ngiven whole, and thus in so far analytic, it also involves the\r\nemergence of \u003cem\u003enew\u003c/em\u003e relations within this whole, and is so far\r\nsynthetic.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e57.\u003c/b\u003e\u003ca name=\"N57\" id=\"N57\"\u003e\u003c/a\u003e\r\nWe may retain, however, a distinction roughly corresponding\r\nto the Kantian \u003cem\u003eà priori\u003c/em\u003e and \u003cem\u003eà posteriori\u003c/em\u003e, though\r\nless rigid, and more liable to change with the degree of organisation\r\nof knowledge. Kant usually endeavoured to prove,\r\nas observed above, that his synthetic \u003cem\u003eà priori\u003c/em\u003e propositions were\r\nnecessary prerequisites of experience; now although we cannot\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_60\" id=\"Page_60\"\u003e[60]\u003c/a\u003e\u003c/span\u003e\r\nretain the term synthetic, we can retain the term \u003cem\u003eà priori\u003c/em\u003e,\r\nfor those assumptions, or those postulates, from which alone\r\nthe possibility of experience follows. Whatever can be deduced\r\nfrom these postulates, without the aid of the matter of experience,\r\nwill also, of course, be \u003cem\u003eà priori\u003c/em\u003e. From the standpoint\r\nof general logic, the laws of thought and the categories, with\r\nthe indispensable conditions of their applicability, will be alone\r\n\u003cem\u003eà priori\u003c/em\u003e; but from the standpoint of any special science, we\r\nmay call \u003cem\u003eà priori\u003c/em\u003e whatever renders possible the experience\r\nwhich forms the subject-matter of our science. In Geometry,\r\nto particularize, we may call \u003cem\u003eà priori\u003c/em\u003e whatever renders possible\r\nexperience of externality as such.\u003c/p\u003e\r\n\r\n\u003cp\u003eIt is to be observed that this use of the term is at once\r\nmore rationalistic and less precise than that of Kant. Kant\r\nwould seem to have supposed himself immediately aware, by\r\ninspection, that some knowledge was apodeictic, and its subject-matter,\r\ntherefore, \u003cem\u003eà priori\u003c/em\u003e: but he did not always deduce its\r\napriority from any further principle. Here, however, it is to\r\nbe shown, before admitting apriority, that the falsehood of the\r\njudgment in question would not be effected by a mere change\r\nin the \u003cem\u003ematter\u003c/em\u003e of experience, but only by a change which should\r\nrender some branch of experience formally impossible, \u003cem\u003ei.e.\u003c/em\u003e inaccessible\r\nto our methods of cognition. The above use is also\r\nless precise, for it varies according to the specialization of the\r\nexperience we are assuming possible, and with every progress\r\nof knowledge some new connection is perceived, two previously\r\nisolated judgments are brought into logical relation, and the\r\n\u003cem\u003eà priori\u003c/em\u003e may thus, at any moment, enlarge its sphere, as more\r\nis found deducible from fundamental postulates.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e58.\u003c/b\u003e\u003ca name=\"N58\" id=\"N58\"\u003e\u003c/a\u003e\r\n(2) \u003cem\u003eKant\u0027s arguments for the apriority of space.\u003c/em\u003e\r\nHaving now discussed the logical canon to be used as regards\r\nthe \u003cem\u003eà priori\u003c/em\u003e, we may proceed to test Kant\u0027s arguments as\r\nregards space. The argument from Geometry, as remarked\r\nabove, is upset by Metageometry, at least so far as those\r\nproperties are concerned, which belong to Euclid but not to\r\nnon-Euclidean spaces; as regards the common properties of\r\nboth kinds of space, we cannot decide on their apriority till\r\nwe have discussed the consequences of denying them, which\r\nwill be done in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e As regards the two arguments\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_61\" id=\"Page_61\"\u003e[61]\u003c/a\u003e\u003c/span\u003e\r\nwhich prove that space is an intuition, not a concept, they\r\nwould call for much discussion in a special criticism of Kant,\r\nbut here they may be passed by with the obvious comment\r\nthat infinite homogeneous Euclidean space is a concept, not\r\nan intuition\u0026mdash;a concept invented to explain an intuition, it\r\nis true, but still a pure concept\u003ca name=\"FNanchor_74_74\" id=\"FNanchor_74_74\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_74_74\" class=\"fnanchor\"\u003e[74]\u003c/a\u003e. And it is this pure concept\r\nwhich, in all discussions of Geometry, is primarily to be dealt\r\nwith; the intuition need only be referred to where it throws\r\nlight on the functions or the nature of the concept. The\r\nsecond of Kant\u0027s arguments, that we can imagine empty space,\r\nthough not the absence of space, is false if it means a space\r\nwithout matter anywhere, and irrelevant if it merely means\r\na space between matters and regarded as empty\u003ca name=\"FNanchor_75_75\" id=\"FNanchor_75_75\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_75_75\" class=\"fnanchor\"\u003e[75]\u003c/a\u003e. The only\r\nargument of importance, then, is the first argument. But\r\nI must insist, at the outset, that our problem is purely logical,\r\nand that all psychological implications must be excluded to\r\nthe utmost possible extent. Moreover, as will be proved in\r\nChapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e, the proper function of space is to distinguish\r\nbetween different presented things, not between the Self and\r\nthe object of sensation or perception. The argument then\r\nbecomes the following: consciousness of a world of mutually\r\nexternal things demands, in presentations, a cognitive but non-inferential\r\nelement leading to the discrimination of the objects\r\npresented. This element must be non-inferential, for from\r\nwhatever number or combination of presentations, which did\r\nnot of themselves demand diversity in their objects, I could\r\nnever be led to infer the mutual externality of their objects.\r\nKant says: \"In order that sensations may be ascribed to something\r\nexternal to me … and similarly in order that I may be\r\nable to present them as outside and beside one another, …\r\nthe presentation of space must be already present.\" But\r\nthis goes rather too far: in the first place, the question\r\nshould be only as to the mutual externality of presented\r\nthings, not as to their externality to the Self\u003ca name=\"FNanchor_76_76\" id=\"FNanchor_76_76\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_76_76\" class=\"fnanchor\"\u003e[76]\u003c/a\u003e; and in the\r\nsecond place, things will appear mutually external if I have\r\nthe presentation of \u003cem\u003eany\u003c/em\u003e form of externality, whether Euclidean\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_62\" id=\"Page_62\"\u003e[62]\u003c/a\u003e\u003c/span\u003eor non-Euclidean. Whatever may be true of the \u003cem\u003epsychological\u003c/em\u003e\r\nscope of this argument\u0026mdash;whose validity is here irrelevant\u0026mdash;the\r\n\u003cem\u003elogical\u003c/em\u003e scope extends, not to Euclidean space, but only to\r\nany form of externality which could exist intuitively, and\r\npermit knowledge, in beings with our laws of thought, of a\r\nworld of diverse but interrelated things.\u003c/p\u003e\r\n\r\n\u003cp\u003eMoreover externality, to render the scope of the argument\r\nwholly logical, must not be left with a sensational or intuitional\r\nmeaning, though it must be supposed given in sensation or\r\nintuition. It must mean, in this argument, the fact of Otherness\u003ca name=\"FNanchor_77_77\" id=\"FNanchor_77_77\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_77_77\" class=\"fnanchor\"\u003e[77]\u003c/a\u003e,\r\nthe fact of being different from some other thing: it must\r\ninvolve the distinction between different things, and must be\r\nthat element, in a cognitive state, which leads us to discriminate\r\nconstituent parts in its object. So much, then, would\r\nappear to result from Kant\u0027s argument, that experience of\r\ndiverse but interrelated things demands, as a necessary prerequisite,\r\nsome sensational or intuitional element, in perception,\r\nby which we are led to attribute complexity to objects of\r\nperception\u003ca name=\"FNanchor_78_78\" id=\"FNanchor_78_78\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_78_78\" class=\"fnanchor\"\u003e[78]\u003c/a\u003e; that this element, in its isolation may be called\r\na form of externality; and that those properties of this form,\r\nif any such be found, which can be deduced from its mere\r\nfunction of rendering experience of interrelated diversity possible,\r\nare to be regarded as \u003cem\u003eà priori\u003c/em\u003e. What these properties\r\nare, and how the various lines of argument here suggested converge\r\nto a single result, we shall see in Chapters \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e59.\u003c/b\u003e\u003ca name=\"N59\" id=\"N59\"\u003e\u003c/a\u003e\r\nIn the philosophers who followed Kant, Metaphysics,\r\nfor the most part, so predominated over Epistemology, that\r\nlittle was added to the theory of Geometry. What was added,\r\ncame indirectly from the one philosopher who stood out against\r\nthe purely ontological speculations of his time, namely \u003cem\u003eHerbart\u003c/em\u003e.\r\nHerbart\u0027s actual views on Geometry, which are to be found\r\nchiefly in the first section of his \u003ccite lang=\"fr\" xml:lang=\"fr\"\u003eSynechologie\u003c/cite\u003e, are not of any\r\ngreat value, and have borne no great fruit in the development\r\nof the subject. But his psychological theory of space, his\r\nconstruction of extension out of series of points, his comparison\r\nof space with the tone and colour-series, his general preference\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_63\" id=\"Page_63\"\u003e[63]\u003c/a\u003e\u003c/span\u003efor the discrete above the continuous, and finally his belief\r\nin the great importance of classifying space with other forms\r\nof series (\u003ci lang=\"de\" xml:lang=\"de\"\u003eReihenformen\u003c/i\u003e\u003ca name=\"FNanchor_79_79\" id=\"FNanchor_79_79\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_79_79\" class=\"fnanchor\"\u003e[79]\u003c/a\u003e), gave rise to many of Riemann\u0027s epoch-making\r\nspeculations, and encouraged the attempt to explain\r\nthe nature of space by its analytical and quantitative aspect\r\nalone\u003ca name=\"FNanchor_80_80\" id=\"FNanchor_80_80\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_80_80\" class=\"fnanchor\"\u003e[80]\u003c/a\u003e. Through his influence on Riemann, he acquired, indirectly,\r\na great importance in geometrical philosophy. To\r\nRiemann\u0027s dissertation, which we have already discussed in\r\nits mathematical aspect, we must now return, considering, this\r\ntime, only its philosophical views.\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eRiemann.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e60.\u003c/b\u003e\u003ca name=\"N60\" id=\"N60\"\u003e\u003c/a\u003e\r\nThe aim of Riemann\u0027s dissertation, as we saw in\r\nChapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, was to define space as a species of manifold, \u003cem\u003ei.e.\u003c/em\u003e\r\nas a particular kind of collection of magnitudes. It was thus\r\nassumed, to begin with, that spatial figures could be regarded\r\nas magnitudes, and the axioms which emerged, accordingly,\r\ndetermined only the particular place of these among the many\r\nalgebraically possible varieties of magnitudes. The resulting\r\nformulation of the axioms\u0026mdash;while, from the mathematical\r\nstandpoint of metrical Geometry, it was almost wholly laudable\u0026mdash;must,\r\nfrom the standpoint of philosophy, be regarded,\r\nin my opinion, as a \u003ci lang=\"la\" xml:lang=\"la\"\u003epetitio principii\u003c/i\u003e. For when we have\r\narrived at regarding spatial figures as magnitudes, we have\r\nalready traversed the most difficult part of the ground. The\r\naxioms of metrical Geometry\u0026mdash;and it is metrical Geometry,\r\nexclusively, which is considered in Riemann\u0027s Essay\u0026mdash;will\r\nappear, in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, to be divisible into two classes. Of\r\nthese, the first class\u0026mdash;which contains the axioms common to\r\nEuclid and Metageometry, the only axioms seriously discussed\r\nby Riemann\u0026mdash;are not the results of measurement, nor of any\r\nconception of magnitude, but are conditions to be fulfilled\r\nbefore measurement becomes possible. The second class only\u0026mdash;those\r\nwhich express the difference between Euclidean and non-Euclidean\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_64\" id=\"Page_64\"\u003e[64]\u003c/a\u003e\u003c/span\u003espaces\u0026mdash;can be deduced as results of measurement\r\nor of conceptions of magnitude. As regards the first class, on\r\nthe contrary, we shall see that the relativity of position\u0026mdash;by\r\nwhich space is distinguished from all other known manifolds,\r\nexcept time\u0026mdash;leads logically to the necessity of three of the\r\nmost distinctive axioms of Geometry, and yet this relativity\r\ncannot be called a deduction from conceptions of magnitude.\r\nIn analytical Geometry, owing to the fact that coordinate\r\nsystems start from points, and hence build up lines and surfaces,\r\nit is easy to suppose that points can be given independently\r\nof lines and of each other, and thus the relativity of position\r\nis lost sight of. The error thus suggested by mathematics\r\nwas probably reinforced by Herbart\u0027s theory of space, which,\r\nby its serial character, as we have seen, appeared to him to\r\nfacilitate a construction out of successive points, and to which\r\nRiemann acknowledges his indebtedness both in his Dissertation\r\nand elsewhere. The same error reappears in Helmholtz,\r\nin whom it is probably due wholly to the methods of analytical\r\nGeometry. It is a striking fact that, throughout the writings\r\nof these two men, there is not, so far as I know, one allusion\r\nto the relativity of position, that property of space from which,\r\nas our next chapter will shew, the richest quarry of consequences\r\ncan be extracted. This is not a result of any conception\r\nof magnitude, but follows from the nature of our space-intuition;\r\nyet no one, surely, could call it empirical, since it\r\nis bound up in the very possibility of locating things \u003cem\u003ethere\u003c/em\u003e\r\nas opposed to \u003cem\u003ehere\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e61.\u003c/b\u003e\u003ca name=\"N61\" id=\"N61\"\u003e\u003c/a\u003e\r\nIndeed we can see, from a purely logical consideration\r\nof the judgment of quantity, that Riemann\u0027s manner of approaching\r\nthe problem can never, by legitimate methods, attain\r\nto a philosophically sound formulation of the axioms. For\r\nquantity is a result of comparison of two qualitatively similar\r\nobjects, and the judgment of quantity neglects altogether the\r\nqualitative aspect of the objects compared. Hence a knowledge\r\nof the essential properties of space can never be obtained from\r\njudgments of quantity, which neglect these properties, while\r\nthey yet presuppose them. As well might one hope to learn\r\nthe nature of man from a census. Moreover, the judgment of\r\nquantity is the result of comparison, and therefore presupposes\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_65\" id=\"Page_65\"\u003e[65]\u003c/a\u003e\u003c/span\u003e\r\nthe possibility of comparison. To know whether, or by what\r\nmeans, comparison is possible, we must know the qualities of\r\nthe things compared and of the medium in which comparison\r\nis effected; while to know that \u003cem\u003equantitative\u003c/em\u003e comparison is possible,\r\nwe must know that there is a qualitative identity between\r\nthe things compared, which again involves a previous\r\nqualitative knowledge. When spatial figures have once been\r\nreduced to quantity, their quality has already been neglected,\r\nas known and similar to the quality of other figures. To hope,\r\ntherefore, for the qualities of space, from a comparison of its\r\nexpression as pure quantity with other pure quantities, is an\r\nerror natural to an analytical geometer, but an error, none\r\nthe less, from which there is no return to the qualitative basis\r\nof spatial quantity.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e62.\u003c/b\u003e\u003ca name=\"N62\" id=\"N62\"\u003e\u003c/a\u003e\r\nWe must entirely dissent, therefore, from the disjunction\r\nwhich underlies Riemann\u0027s philosophy of space. Either\r\nthe axioms must be consequences of general conceptions of\r\nmagnitude, he thinks, or else they can only be proved by\r\nexperience (p. 255). Whatever \u003cem\u003ecan\u003c/em\u003e be derived from general\r\nconceptions of magnitude, we may retort, cannot be an \u003cem\u003eà priori\u003c/em\u003e\r\nadjective of space: for all the necessary adjectives of space are\r\npresupposed in any judgment of spatial quantity, and cannot,\r\ntherefore, be consequences of such a judgment. Riemann\u0027s disjunction,\r\naccordingly, since one of its alternatives is obviously\r\nimpossible, really begs the question. In formulating the axioms\r\nof metrical Geometry, our question should be: What axioms,\r\n\u003cem\u003ei.e.\u003c/em\u003e what adjectives of space, must be presupposed, in order that\r\nquantitative comparison of the parts of space may be possible\r\nat all? And only when we have determined these conditions,\r\nwhich are \u003cem\u003eà priori\u003c/em\u003e necessary to any quantitative science of\r\nspace, does the second question arise: what inferences can we\r\ndraw, as to space, from the observed results of this quantitative\r\nscience, \u003cem\u003ei.e.\u003c/em\u003e of this measurement of spatial figures? The conditions\r\nof measurement themselves, though not results of any\r\nconception of magnitude, will be \u003cem\u003eà priori\u003c/em\u003e, if it can be shown that,\r\nwithout them, experience of externality would be impossible.\u003c/p\u003e\r\n\r\n\u003cp\u003eAfter this initial protest against Riemann\u0027s general philosophical\r\nposition, let us proceed to examine, in detail, his use of\r\nthe notion of a manifold.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_66\" id=\"Page_66\"\u003e[66]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e63.\u003c/b\u003e\u003ca name=\"N63\" id=\"N63\"\u003e\u003c/a\u003e\r\nIn the first place there is, if I am not mistaken,\r\nconsiderable obscurity in the definition of a manifold, of\r\nwhich an almost verbal rendering was given in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\nWhat is meant, to begin with, by a general conception capable\r\nof various determinations? Does not this property belong to\r\nall conceptions? It affords, certainly, a basis for counting, but\r\nif continuous quantity is to arise, we must, surely, have some\r\nless discrete formulation. It might afford a basis, for example,\r\nfor the distinction of points in projective Geometry, but projective\r\nGeometry has nothing to do with quantity. Something\r\nmore fluid and flexible than a conception, one would think, is\r\nnecessary as the basis of continua. Then, again, what is meant\r\nby a quantum of a manifold? In space, the answer is obvious:\r\nwhat is meant is a piece of volume. But how about Riemann\u0027s\r\nother continuous manifold, colour? Does a quantum of colour\r\nmean a single line in the spectrum, or a band of finite thickness?\r\nIn either case, what are the magnitudes to be compared? And\r\nhow is superposition necessary, or even possible? A colour is\r\nfixed by its position in the spectrum: two lines in the same\r\nspectrum cannot be superposed, and two lines in different\r\nspectra need not be\u0026mdash;their positions in their respective spectra\r\nsuffice, or even, roughly, their immediate sense-quality. The\r\nfact is, Riemann had space in his mind from the start, and\r\nmany of the properties, which he enunciates as belonging to all\r\nmanifolds, belong, as a matter of fact, only to space. It is far\r\nfrom clear what the magnitudes are which the various determinations\r\nmake possible. Do these magnitudes measure the\r\nelements of the manifold, or the relations between elements?\r\nThis is surely a very fundamental point, but it is one which\r\nRiemann never touches on. In the former case, the superposition\r\nwhich he speaks of becomes unnecessary, since the\r\nmagnitude is inherent in the element considered. We do not\r\nrequire superposition to measure quantities corresponding to\r\ndifferent tones or colours; these can be discovered by analysis of\r\nsingle tones or colours. With space, on the other hand, if we\r\nseek for elements, we can find none except points, and no\r\nanalysis of a point will find magnitudes inherent in it\u0026mdash;such\r\nmagnitudes are a fiction of coordinate Geometry. The magnitudes\r\nwhich space deals with, as we shall see in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_67\" id=\"Page_67\"\u003e[67]\u003c/a\u003e\u003c/span\u003e\r\nare relations between points, and it is for this reason that superposition\r\nis essential to space-measurement. There is no inherent\r\nquality in a single point, as there is in a single colour, by which\r\nit can be quantitatively distinguished from another. Thus the\r\nconception of a manifold, as defined by Riemann, either does not\r\ninclude colours, or does not involve superposition as the only\r\nmeans of measurement. From this dilemma there is no escape.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e64.\u003c/b\u003e\u003ca name=\"N64\" id=\"N64\"\u003e\u003c/a\u003e\r\nBut if \"measurement \u003cem\u003econsists\u003c/em\u003e in a superposition of the\r\nmagnitudes compared\" (p. 256), does it not follow immediately\r\nthat measurement is logically possible \u003cem\u003eonly\u003c/em\u003e where such superposition\r\nleaves the magnitudes unchanged? And therefore that\r\nmeasurement, as above defined, involves, as an \u003cem\u003eà priori\u003c/em\u003e condition,\r\nthat magnitudes are unchanged by motion? This consequence\r\nis not drawn by Riemann; indeed he proceeds immediately\r\n(pp. 256\u0026ndash;7) to consider what he calls a general portion of the\r\ndoctrine of magnitude (\u003ci lang=\"de\" xml:lang=\"de\"\u003eGrössenlehre\u003c/i\u003e), independent of measurement.\r\nBut how is any doctrine of magnitude possible, in which\r\nthe magnitudes cannot be measured? The reason of the confusion\r\nis, that Riemann\u0027s definition of measurement is applicable\r\nto no single manifold except space, since it depends on the\r\nnoteworthy property that what we measure in Geometry is\r\nnot points, but relations between points, and the latter, though\r\nnot the former, may of course be unaltered by motion. Let us\r\ntry, in illustration, to apply Riemann\u0027s definition of measurement\r\nto colours. We must remember that motion, in dealing with\r\nthe colour manifold, means\u0026mdash;not motion in space but\u0026mdash;motion\r\nin the colour manifold itself. Now since every point of the\r\ncolour manifold is completely determined by three magnitudes,\r\nwhich are given in fact, and cannot be arbitrarily chosen, it is\r\nplain that measurement by superposition\u0026mdash;involving, as it does,\r\nmotion, and therefore change in these determining magnitudes\u0026mdash;is\r\ntotally out of the question. The superposition of one colour\r\non another, as a means of measurement, is sheer nonsense. And\r\nyet measurement is possible in the colour-manifold, by means of\r\nHelmholtz\u0027s law of mixture (\u003ci lang=\"de\" xml:lang=\"de\"\u003eMischungsgesetz\u003c/i\u003e); but the measurement\r\nis of every separate element, not of the relations between\r\nelements, and is thus radically different from space-measurement\u003ca name=\"FNanchor_81_81\" id=\"FNanchor_81_81\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_81_81\" class=\"fnanchor\"\u003e[81]\u003c/a\u003e.\r\nThe elements are not, like points in space, qualitatively\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_68\" id=\"Page_68\"\u003e[68]\u003c/a\u003e\u003c/span\u003e\r\nalike, and distinguished by the mere fact of their mutual\r\nexternality. What we have, in colours, is three fundamental\r\nqualitatively distinct elements, out of certain proportions of\r\nwhich we can build up all the other elements of the manifold\u0026mdash;each\r\nof the resulting elements having the same combination of\r\nqualitative diversity and similarity as the three original elements.\r\nBut in space, what could we make of such a procedure? Given\r\nthree points, how are we to combine them in certain proportions?\r\nThe phrase is meaningless. If some one makes the obvious\r\nretort, that we have to combine lines, not points, my rejoinder\r\nis equally obvious. To begin with, lines are not elements.\r\nMetaphysically, space has \u003cem\u003eno\u003c/em\u003e elements, being, as the sequel\r\nwill show, mere relations between non-spatial elements. Mathematically,\r\nthis fact exhibits itself in the self-contradictory notion\r\nof the point, or zero magnitude in space, as the limit in our\r\nvain search for spatial elements. But even if we allow the line\r\nto pass as the spatial element, what does the combination of\r\nthree lines in definite proportions give us? It gives us, simply,\r\nthe coordinates of a \u003cem\u003epoint\u003c/em\u003e. Here again we see a great difference\r\nbetween the colour and space-manifolds. In colours, the combination\r\nof magnitudes gives a new magnitude of the same\r\nkind; in space, it defines, not a magnitude at all, but a would-be\r\nelement of a different kind from the defining magnitudes.\r\nIn the tone-manifold, we should find still different conditions.\r\nHere, no one of the measuring magnitudes can vanish without\r\nthe tone vanishing too, and all three are so bound up together,\r\nin the single resulting sensation, that none can exist without\r\na finite quantity of the others. They are all qualitatively\r\ndifferent, both from each other, and from any possible tone,\r\nbeing constituents of it, as mass and velocity are constituents\r\nof momentum. All these different conditions require to be\r\nexamined, before a manifold can be completely defined; and\r\nuntil we have conducted such an examination in detail, we\r\ncannot pronounce as to the \u003cem\u003eà priori\u003c/em\u003e or empirical nature of the\r\nlaws of the manifold. As regards space, I have attempted such\r\nan examination in the third and fourth chapters of this Essay.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_69\" id=\"Page_69\"\u003e[69]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e65.\u003c/b\u003e\u003ca name=\"N65\" id=\"N65\"\u003e\u003c/a\u003e\r\nI do not wish to deny, however, the great value of the\r\nconception of space as a manifold. On the contrary, this conception\r\nseems to have become essential to any treatment of the\r\nquestion. I only wish to urge that the purely algebraical\r\ntreatment of any manifold, important as it may be in deducing\r\nfresh consequences from known premisses, tends rather to\r\nconceal than to make clear the basis of the premisses themselves,\r\nand is therefore misleading in a philosophical investigation.\r\nFor mathematics, where quantity reigns supreme,\r\nRiemann\u0027s conception has proved itself abundantly fruitful;\r\nfor philosophy, on the contrary, where quantity appears rather\r\nas a cloak to conceal the qualities it abstracts from, the\r\nconception seems to me more productive of error and confusion\r\nthan of sound doctrine.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe are thus brought back to the point from which we\r\nstarted, namely, the falsity of Riemann\u0027s initial disjunction,\r\nand the consequent fallacy in his proof of the empirical nature\r\nof the axioms. His philosophy is chiefly vitiated, to my mind,\r\nby this fallacy, and by the uncritical assumption that a metrical\r\ncoordinate system can be set up independently of any axioms\r\nas to space-measurement\u003ca name=\"FNanchor_82_82\" id=\"FNanchor_82_82\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_82_82\" class=\"fnanchor\"\u003e[82]\u003c/a\u003e. Riemann has failed to observe,\r\nwhat I have endeavoured to prove in the next chapter, that,\r\nunless space had a strictly constant measure of curvature,\r\nGeometry would become impossible; also that the absence\r\nof constant measure of curvature involves absolute position,\r\nwhich is an absurdity. Hence he is led to the conclusion\r\nthat all geometrical axioms are empirical, and may not hold\r\nin the infinitesimal, where observation is impossible. Thus he\r\nsays (p. 267): \"Now the empirical conceptions, on which\r\nspatial measurements are based, the conceptions of the rigid\r\nbody and the light-ray, appear to lose their validity in the\r\ninfinitesimal: it is therefore quite conceivable that the relations\r\nof spatial magnitudes in the infinitesimal do not correspond to\r\nthe presuppositions of Geometry, and this would, in fact, have\r\nto be assumed, as soon as it would enable us to explain the\r\nphenomena more simply.\" From this conclusion I must\r\nentirely dissent. In very large spaces, there might be a\r\ndeparture from Euclid; for they depend upon the axiom of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_70\" id=\"Page_70\"\u003e[70]\u003c/a\u003e\u003c/span\u003e\r\nparallels, which is not contained in the axiom of Free Mobility;\r\nbut in the infinitesimal, departures from Euclid could only be\r\ndue to the absence of Free Mobility, which, as I hope my third\r\nchapter will show, is once for all impossible.\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eHelmholtz.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e66.\u003c/b\u003e\u003ca name=\"N66\" id=\"N66\"\u003e\u003c/a\u003e\r\nHelmholtz, like Riemann, was important both in the\r\nmathematics and in the philosophy of Geometry. From the\r\nmathematical point of view, his work has been already considered\r\nin Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e; the consideration of his philosophy,\r\nwhich must occupy us here, will be a more serious task. Like\r\nRiemann, he endeavoured to prove that all the axioms are\r\nempirical, and like Riemann, he based his proof chiefly on\r\nMetageometry. He had an additional resource, however, in\r\nthe physiology of the senses, which first led him to reject the\r\nTranscendental Aesthetic, and enabled him to attack Kant\r\nfrom the psychological as well as the mathematical side\u003ca name=\"FNanchor_83_83\" id=\"FNanchor_83_83\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_83_83\" class=\"fnanchor\"\u003e[83]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe principal topics, for a criticism of Helmholtz, are three:\r\nFirst, his criterion of the \u003cem\u003eà priori\u003c/em\u003e; second, his discussion with\r\nLand as to the \"imaginability\" of non-Euclidean spaces; third\u0026mdash;and\r\nthis is by far the most important of the three\u0026mdash;his\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_71\" id=\"Page_71\"\u003e[71]\u003c/a\u003e\u003c/span\u003e\r\ntheory of the dependence of Geometry on Mechanics. Let us\r\ndiscuss these three points successively.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e67.\u003c/b\u003e\u003ca name=\"N67\" id=\"N67\"\u003e\u003c/a\u003e\r\nHelmholtz\u0027s criterion of apriority is difficult to discover,\r\nas he never, to my knowledge, gives a precise statement of it.\r\nFrom his discussion of physical and transcendental Geometry\u003ca name=\"FNanchor_84_84\" id=\"FNanchor_84_84\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_84_84\" class=\"fnanchor\"\u003e[84]\u003c/a\u003e,\r\nhowever, it would appear that he regards as empirical whatever\r\napplies to empirical matter. For he there maintains, that even\r\nif space were an \u003cem\u003eà priori\u003c/em\u003e form, yet any Geometry, which aimed\r\nat an application to Physics, would, since the actual places of\r\nbodies are not known \u003cem\u003eà priori\u003c/em\u003e, be necessarily empirical\u003ca name=\"FNanchor_85_85\" id=\"FNanchor_85_85\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_85_85\" class=\"fnanchor\"\u003e[85]\u003c/a\u003e. It\r\nseems the more probable that he regards this as a possible\r\ncriterion, as it is adopted, in several passages, by his disciple\r\nErdmann\u003ca name=\"FNanchor_86_86\" id=\"FNanchor_86_86\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_86_86\" class=\"fnanchor\"\u003e[86]\u003c/a\u003e, and so strange a test could hardly be accepted by\r\na philosopher, unless he had found it in his master. I have\r\ncalled this a strange test, because it seems to me completely\r\nto ignore the work of the Critical Philosophy. For if there\r\nis one thing which, one might have hoped, had been made\r\nsufficiently clear by Kant\u0027s Critique, it is this, that knowledge\r\nwhich is \u003cem\u003eà priori\u003c/em\u003e, being itself the condition of possible experience,\r\napplies\u0026mdash;and in Kant\u0027s view, applies only\u0026mdash;to empirical matter.\r\nHelmholtz and Erdmann, therefore, in setting up this test without\r\ndiscussion, simply ignore the existence of Kant and the\r\npossibility of a transcendental argument. Helmholtz assumes\r\nalways that empirical knowledge must be wholly empirical, that\r\nthere can be no \u003cem\u003eà priori\u003c/em\u003e conditions of the experience in question,\r\nthat experience will always be possible, and may give any kind\r\nof result. Thus in discussing \"physical\" Geometry, he assumes\r\nthat the possibility of empirical measurement involves no\r\n\u003cem\u003eà priori\u003c/em\u003e axioms, and that no \u003cem\u003eà priori\u003c/em\u003e element can be contained\r\nin the process. This assumption, as we shall see in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e,\r\nis quite unwarrantable: certain properties of \u003cem\u003espace\u003c/em\u003e, in fact, are\r\ninvolved in the possibility of measuring \u003cem\u003ematter\u003c/em\u003e. In spite of\r\nthe fact, therefore, that we apply measurement to empirical\r\nmatter, and that our results are therefore empirical, there\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_72\" id=\"Page_72\"\u003e[72]\u003c/a\u003e\u003c/span\u003emay well be an \u003cem\u003eà priori\u003c/em\u003e element in measurement, which is\r\npresupposed in its possibility. Such a criterion, therefore, must\r\npronounce everything empirical, but must itself be pronounced\r\nworthless.\u003c/p\u003e\r\n\r\n\u003cp\u003eAnother and a better criterion, it is true, is also to be found\r\nin Helmholtz, and has also been adopted by Erdmann. Whatever\r\nmight, by a different experience, have been rendered\r\ndifferent\u0026mdash;so this criterion contends\u0026mdash;must itself be dependent\r\non experience, and so empirical. This criterion seems perfectly\r\nsound, but Helmholtz\u0027s use of it is usually vitiated by his\r\nneglecting to prove the possibility of the different experience\r\nin question. He says, for example, that if our experience\r\nshowed us only bodies which changed their shapes in motion,\r\nwe should not arrive at the axiom of Congruence, which he\r\npronounces accordingly to be empirical. But I shall endeavour\r\nto prove, in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, that without the axiom of Congruence,\r\nexperience of spatial magnitude would be impossible. If my\r\nproof be correct, it follows that no experience can ever reveal\r\nspatial magnitudes which contradict this axiom\u0026mdash;a possibility\r\nwhich Helmholtz nowhere discusses, in setting up his hypothetical\r\nexperience. Thus this second criterion, though perfectly\r\nsound, requires always an accompanying transcendental\r\nargument, as to the conditions of possible experience. But\r\nthis accompaniment is seldom to be found in Helmholtz.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e68.\u003c/b\u003e\u003ca name=\"N68\" id=\"N68\"\u003e\u003c/a\u003e\r\nOne of the few cases, in which Helmholtz has attempted\r\nsuch an accompaniment, occurs in connection with\r\nour second point, the imaginability of non-Euclidean spaces.\r\nThe argument on this point was elicited by Helmholtz\u0027s Kantian\r\nopponents, who maintained that the merely logical possibility\r\nof these spaces was irrelevant, since the basis of Geometry was\r\nnot logic, but intuition. The axioms, they said, are synthetic\r\npropositions, and their contraries are, therefore, not self-contradictory;\r\nthey are nevertheless apodeictic propositions, since no\r\nother \u003cem\u003eintuition\u003c/em\u003e than the Euclidean is possible to us\u003ca name=\"FNanchor_87_87\" id=\"FNanchor_87_87\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_87_87\" class=\"fnanchor\"\u003e[87]\u003c/a\u003e. I have\r\nalready criticized this line of argument in the beginning of the\r\npresent chapter. Helmholtz\u0027s criticism, however, was different:\r\nadmitting the internal consistency of the argument, he denied\r\none of its premisses. We \u003cem\u003ecan\u003c/em\u003e imagine non-Euclidean spaces,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_73\" id=\"Page_73\"\u003e[73]\u003c/a\u003e\u003c/span\u003e\r\nhe said, though their unfamiliarity makes this difficult. From\r\nthis view it followed, of course, that Kant\u0027s argument, even if it\r\nwere formally valid, could not prove the apriority of Euclidean\r\nspace in particular, but only of that general space which included\r\nEuclid and non-Euclid alike\u003ca name=\"FNanchor_88_88\" id=\"FNanchor_88_88\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_88_88\" class=\"fnanchor\"\u003e[88]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eAlthough I agree with Helmholtz in thinking the distinction\r\nbetween Euclidean and non-Euclidean spaces empirical, I cannot\r\nthink his argument on the \"imaginability\" of the latter a very\r\nhappy one. The validity of any proof must turn, obviously, on\r\nthe definition of imaginability. The definition which Helmholtz\r\ngives in his answer to Land is as follows: Imaginability requires\r\n\u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"die vollständige Vorstellbarkeit derjenigen Sinneseindrücke,\r\nwelche das betreffende Object in uns nach den bekannten\r\nGesetzen unserer Sinnesorgane unter allen denkbaren Bedingungen\r\nder Beobachtung erregen, und wodurch es sich von\r\nanderen ähnlichen Objecten unterscheiden würde\"\u003c/span\u003e (Wiss. Abh.\r\n\u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 644). This definition is not very clear, owing to the ambiguity\r\nof the word \"\u003ci lang=\"de\" xml:lang=\"de\"\u003eVorstellbarkeit\u003c/i\u003e.\" The following definition\r\nseems less ambiguous: \u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"Wenn die Reihe der Sinneseindrücke\r\nvollständig und eindeutig angegeben werden kann, muss man\r\nm. E. die Sache für \u003cem\u003eanschaulich vorstellbar\u003c/em\u003e erklären\" (Vorträge\r\nund Reden,\u003c/span\u003e \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 234). This makes clear, what also appears from\r\nhis manner of proof, that he regards things as imaginable which\r\ncan be \u003cem\u003edescribed\u003c/em\u003e in conceptual terms. Such, as Land remarks\r\n(\u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 45), \"is not the sense required for argumentation\r\nin this case.\" That Land\u0027s criticism is just, is shown by\r\nHelmholtz\u0027s proof for non-Euclidean spaces, for it consists only\r\nin an analogy to the volume inside a sphere, which is mathematically\r\nobtained thus: We take the symbols representing\r\nmagnitudes in \"pseudo-spherical\" (hyperbolic) space, and give\r\nthem a new Euclidean meaning; thus all our symbolic propositions\r\nbecome capable of two interpretations, one for pseudo-spherical\r\nspace, and one for the volume inside a sphere. It is,\r\nhowever, sufficiently obvious that this procedure, though it\r\nenables us to \u003cem\u003edescribe\u003c/em\u003e our new space, does not enable us to\r\n\u003cem\u003eimagine\u003c/em\u003e it, in the sense of calling up images of the way things\r\nwould look in it. We really derive, from this analogy, no more\r\nknowledge than a man born blind may derive, as to light, from\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_74\" id=\"Page_74\"\u003e[74]\u003c/a\u003e\u003c/span\u003e\r\nan analogy with heat. The dictum \u003cspan lang=\"la\" xml:lang=\"la\"\u003e\"Nihil est in intellectu\r\nquod non fuerit ante in sensu,\"\u003c/span\u003e would unquestionably be true,\r\nif for \u003cem\u003eintellect\u003c/em\u003e we were to substitute \u003cem\u003eimagination\u003c/em\u003e; it is vain,\r\ntherefore, \u003cem\u003eif\u003c/em\u003e our actual space be Euclidean, to hope for a power\r\nof \u003cem\u003eimagining\u003c/em\u003e a non-Euclidean space. What Helmholtz might,\r\nI believe with perfect truth, have urged against Land, is that\r\nthe image we actually have of space is not sufficiently accurate\r\nto exclude, in the actual space we know, all possibility of a\r\nslight departure from the Euclidean type. But in maintaining\r\nthat we cannot imagine, though we can conceive and describe,\r\na space different from that we actually have, Land is, in my\r\nopinion, unquestionably in the right. For a pure Kantian,\r\nwho maintains, with Land, that none of the axioms can be\r\nproved, this question is of great importance. But if, as I have\r\nmaintained, some of the axioms are susceptible of a transcendental\r\nproof, while the others can be verified empirically, the\r\nquestion is freed from psychological implications, and the\r\nimaginability or non-imaginability of metageometrical spaces\r\nbecomes unimportant.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e69.\u003c/b\u003e\u003ca name=\"N69\" id=\"N69\"\u003e\u003c/a\u003e\r\nWe come now to the third and most important question,\r\nthe relation of Geometry to Mechanics. There are three\r\nsenses in which Helmholtz\u0027s appeal to rigid bodies may be\r\ntaken: the first, I think, is the sense in which he originally\r\nintended it; the second seems to be the sense which he adopted\r\nin his defence against Land; while the third is admitted by\r\nLand, and will be admitted in the following argument. These\r\nthree senses are as follows:\u003c/p\u003e\r\n\r\n\u003cp\u003e(1) It may be asserted that the actual meaning of the\r\naxiom of Free Mobility lies in the assertion of empirical rigid\r\nbodies, and that the two propositions are equivalent to one\r\nanother. This is certainly false.\u003c/p\u003e\r\n\r\n\u003cp\u003e(2) The axiom of Free Mobility, it may be said, is logically\r\ndistinguishable from the assertion of rigid bodies, and may\r\neven be not empirical; but it is barren, even for pure Geometry,\r\nwithout the aid of measures, which must themselves be empirical\r\nrigid bodies. This sense is more plausible than the\r\nfirst, but I believe we can show that, in this sense also, the\r\nproposition is false.\u003c/p\u003e\r\n\r\n\u003cp\u003e(3) For pure Geometry and the abstract study of space,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_75\" id=\"Page_75\"\u003e[75]\u003c/a\u003e\u003c/span\u003e\r\nit may be said, Free Mobility, as applied to an abstract geometrical\r\nmatter, gives a sufficient possibility of quantitative\r\ncomparison; but the moment we extend our results to mixed\r\nmathematics, and apply them to empirically given matter, we\r\nrequire also, as measures, empirically given rigid bodies, or\r\nbodies, at least, whose departures from rigidity are empirically\r\nknown. In this sense, I admit, the proposition is correct\u003ca name=\"FNanchor_89_89\" id=\"FNanchor_89_89\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_89_89\" class=\"fnanchor\"\u003e[89]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn discussing these three meanings, I shall not confine\r\nmyself strictly to the text of Helmholtz or Land: if I endeavoured\r\nto do so, I should be met by the difficulty that\r\nneither of them defines the \u003cem\u003eà priori\u003c/em\u003e, and that each is too much\r\ninclined, in my opinion, to test it by psychological criteria.\r\nI shall, therefore, take the three meanings in turn, without\r\nlaying stress on their historical adequacy to the views of Land\r\nor Helmholtz.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e70.\u003c/b\u003e\u003ca name=\"N70\" id=\"N70\"\u003e\u003c/a\u003e\r\n(1) Congruence may be taken to mean\u0026mdash;as Helmholtz\r\nwould certainly seem to desire\u0026mdash;that we find actual\r\nbodies, in our mechanical experience, to preserve their shapes\r\nwith approximate constancy, and that we infer, from this\r\nexperience, the homogeneity of space. This view, in my\r\nopinion, radically misconceives the nature of measurement,\r\nand of the axioms involved in it. For what is meant by the\r\nnon-rigidity of a body? We mean, simply, that it has changed\r\nits shape. But this involves the possibility of comparison with\r\nits former shape, in other words, of measurement. In order,\r\ntherefore, that there may be any question of rigidity or non-rigidity,\r\nthe measurement of spatial magnitudes must be\r\nalready possible. It follows that measurement cannot, without\r\na vicious circle, be itself derived from experience of rigid bodies.\r\nGeometrical measurement, in fact, is the comparison of spatial\r\nmagnitudes, and such comparison involves, as will be proved\r\nat length in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, the homogeneity of space. This is,\r\ntherefore, the logical prerequisite of all experience of rigid\r\nbodies, and cannot be the result of such experience. Without\r\nthe homogeneity of space, the very notion of rigidity or non-rigidity\r\ncould not exist, since these mean, respectively, the\r\nconstancy or inconstancy of spatial magnitude in pieces of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_76\" id=\"Page_76\"\u003e[76]\u003c/a\u003e\u003c/span\u003e\r\nmatter, and both alike, therefore, presuppose the possibility\r\nof spatial measurement. From the homogeneity of space, we\r\nlearn that a body, when it moves, will not change its shape\r\nwithout some physical cause; that it actually does not change\r\nits shape, is never asserted, and is indeed known to be false.\r\nAs soon as measurement is possible, actual changes of shape\r\ncan be estimated, and their empirical causes can be sought.\r\nBut if space were not homogeneous, measurement would be\r\nimpossible, constant shape would be a meaningless phrase, and\r\nrigidity could never be experienced. Congruence asserts, in\r\nshort, that a body can, so far as mere space is concerned, move\r\nwithout change of shape; rigidity asserts that it actually does\r\nso move\u0026mdash;a very different proposition, involving obviously, as\r\nits logical prius, the former geometrical proposition.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis argument may be summed up by the following disjunction:\r\nIf bodies change their shapes in motion\u0026mdash;and to some\r\nextent, since no body is perfectly rigid, they must all do so\u0026mdash;then\r\none of two cases must occur. \u003cem\u003eEither\u003c/em\u003e the changes of\r\nshape, as bodies move from place to place, follow no geometrical\r\nlaw, are not, for instance, functions of the amount\r\nor direction of motion; in which case the law of causation\r\nrequires that they should not be effects of the change of place,\r\nbut of some simultaneous non-geometrical change, such as\r\ntemperature. \u003cem\u003eOr\u003c/em\u003e the changes are regular, and the shape \u003cem\u003eS\u003c/em\u003e\r\nbecomes, in a new position \u003cem\u003ep\u003c/em\u003e, \u003cem\u003eSf\u003c/em\u003e(\u003cem\u003ep\u003c/em\u003e). In this case, the law\r\nof concomitant variations leads us to attribute the change of\r\nshape to the mere motion, and shape thus becomes a function\r\nof absolute position. But this is absurd, for position \u003cem\u003emeans\u003c/em\u003e\r\nmerely a relation or set of relations; it is impossible, therefore,\r\nthat mere position should be able to effect changes in a body.\r\nPosition is one term in a relation, not a thing \u003cem\u003eper se\u003c/em\u003e; it\r\ncannot, therefore, act on a thing, nor exist by itself, apart from\r\nthe other terms of the relation. Thus Helmholtz\u0027s view, that\r\nCongruence depends on the existence of rigid bodies, must, since\r\nit involves absolute position, be condemned as a logical fallacy.\r\nCongruence, in fact, as I shall prove more fully in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e,\r\nis an \u003cem\u003eà priori\u003c/em\u003e deduction from the relativity of position.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e71.\u003c/b\u003e\u003ca name=\"N71\" id=\"N71\"\u003e\u003c/a\u003e\r\n(2) The above argument seems to me to answer\r\nsatisfactorily Helmholtz\u0027s contention in the precise form which\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_77\" id=\"Page_77\"\u003e[77]\u003c/a\u003e\u003c/span\u003e\r\nhe first gave it. The axiom of Congruence, we must agree,\r\nis logically distinguishable from the existence of rigid bodies.\r\nNevertheless some reference to matter is logically involved in\r\nGeometry\u003ca name=\"FNanchor_90_90\" id=\"FNanchor_90_90\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_90_90\" class=\"fnanchor\"\u003e[90]\u003c/a\u003e, but whether this reference makes Geometry empirical,\r\nor does not, rather, show an \u003cem\u003eà priori\u003c/em\u003e element in\r\ndynamics, is a further question.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe reference to matter is necessitated by the homogeneity\r\nof empty space. For so long as we leave matter out of account,\r\none position is perfectly indistinguishable from another, and\r\na science of the relations of positions is impossible. Indeed,\r\nbefore spatial relations can arise at all, the homogeneity of\r\nempty space must be destroyed, and this destruction must be\r\neffected by matter. The blank page is useless to the geometer\r\nuntil he defaces its homogeneity by lines in ink or pencil.\r\nNo spatial figures, in short, are conceivable, without a reference\r\nto a not purely spatial matter. Again, if Congruence is ever\r\nto be used, there must be motion: but a purely geometrical\r\npoint, being defined solely by its spatial attributes, cannot be\r\nsupposed to move without a contradiction in terms. What\r\nmoves, therefore, must be matter. Hence, in order that motion\r\nmay afford a test of equality, we must have some \u003cem\u003ematter\u003c/em\u003e which\r\nis known to be unaffected throughout the motion, that is, we\r\nmust have some rigid bodies. And the difficulty is, that these\r\nbodies must not only undergo no change due solely to the\r\nnature of space, but must, further, be unchanged by their\r\nchanging relation to other bodies. And here we have a requisite\r\nwhich can no longer be fulfilled \u003cem\u003eà priori\u003c/em\u003e: which, indeed,\r\nwe know to be, in strictness, untrue. For the forces acting on\r\na body depend upon its spatial relations to other bodies, and\r\nchanging forces are liable to produce changing configuration.\r\nHence, it would seem, actual measurement must be purely\r\nempirical, and must depend on the degree of rigidity to be\r\nobtained, during the process of measurement, in the bodies\r\nwith which we are conversant.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis conclusion, I believe, is valid of all actual measurement.\r\nBut the possibility of such empirical and approximate rigidity,\r\nI must insist, depends on the \u003cem\u003eà priori\u003c/em\u003e law that \u003cem\u003emere\u003c/em\u003e motion,\r\napart from the action of other matter, cannot effect a change\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_78\" id=\"Page_78\"\u003e[78]\u003c/a\u003e\u003c/span\u003e\r\nof shape. For without this law, the effect of other matter\r\nwould not be discoverable; the laws of motion would be absurd,\r\nand Physics would be impossible. Consider the second law, for\r\nexample: How could we measure the change of motion, if motion\r\nitself produced a change in our measures? Or consider the law\r\nof gravitation: How could we establish the inverse square, unless\r\nwe were able, independently of Dynamics, to measure distances?\r\nThe whole science of Dynamics, in short, is fundamentally dependent\r\non Geometry, and but for the independent possibility\r\nof measuring spatial magnitudes, none of the magnitudes of\r\nDynamics could be measured. Time, force, and mass are alike\r\nmeasured by spatial correlates: these correlates are given, for\r\ntime, by the first law, for force and mass, by the second and\r\nthird. It is true, then, that an empirical element appears\r\nunavoidably in all actual measurement, inasmuch as we can\r\nonly know empirically that a given piece of matter preserves\r\nits shape throughout the necessary change of dynamical relations\r\nto other matter involved in motion; but it is further true that,\r\nfor Geometry\u0026mdash;which regards matter simply as supplying the\r\nnecessary breach in the homogeneity of space, and the necessary\r\nterm for spatial relations, not as the bearer of forces which change\r\nthe configuration of other material systems\u0026mdash;for Geometry, which\r\ndeals with this abstract and merely kinematical matter, rigidity\r\nis \u003cem\u003eà priori\u003c/em\u003e, in so far as the only changes with which it is cognizant\u0026mdash;changes\r\nof mere position, namely\u0026mdash;are incapable of\r\naffecting the shapes of the imaginary and abstract bodies with\r\nwhich it deals. To use a scholastic distinction, we may say that\r\nmatter is the \u003ci lang=\"la\" xml:lang=\"la\"\u003ecausa essendi\u003c/i\u003e of space, but Geometry is the \u003ci lang=\"la\" xml:lang=\"la\"\u003ecausa\r\ncognoscendi\u003c/i\u003e of Physics. Without a Geometry independent of\r\nPhysics, Physics itself, which necessarily assumes the results\r\nof Geometry, could never arise; but when Geometry is used in\r\nPhysics, it loses some of its \u003cem\u003eà priori\u003c/em\u003e certainty, and acquires the\r\nempirical and approximate character which belongs to all\r\naccounts of actual phenomena.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e72.\u003c/b\u003e\u003ca name=\"N72\" id=\"N72\"\u003e\u003c/a\u003e\r\n(3) This argument leads us to Land\u0027s distinction of\r\nphysical and geometrical rigidity. The distinction may be\r\nexpressed\u0026mdash;and I think it is better expressed\u0026mdash;by distinguishing\r\nbetween the conceptions of matter proper to Dynamics and\r\nto Geometry respectively. In Dynamics, we are concerned with\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_79\" id=\"Page_79\"\u003e[79]\u003c/a\u003e\u003c/span\u003e\r\nmatter as subject to and as causing motion, as affected by and\r\nas exerting \u003cem\u003eforce\u003c/em\u003e. We are therefore concerned with the changes\r\nof spatial configuration to which material systems are liable:\r\nthe description and explanation of these changes is the proper\r\nsubject-matter of all Dynamics. But in order that such a\r\nscience may exist, it is obviously necessary that spatial configuration\r\nshould be already measurable. If this were not\r\nthe case, motion, acceleration and force would remain perfectly\r\nindeterminate. Geometry, therefore, must already exist before\r\nDynamics becomes possible: to make Geometry dependent for\r\nits possibility on the laws of motion or any of their consequences,\r\nis a gross \u003cspan title=\"hysteron proteron\"\u003eὕστερον πρότερον\u003c/span\u003e. Nevertheless, as we have seen,\r\nsome sort of matter is essential to Geometry. But this geometrical\r\nmatter is a more abstract and wholly different matter from\r\nthat of Dynamics. In order to study space by itself, we reduce\r\nthe properties of matter to a bare minimum: we avoid entirely\r\nthe category of causation, so essential to Dynamics, and retain\r\nnothing, in our matter, but its spatial adjectives\u003ca name=\"FNanchor_91_91\" id=\"FNanchor_91_91\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_91_91\" class=\"fnanchor\"\u003e[91]\u003c/a\u003e. The kind of\r\nrigidity affirmed of this abstract matter\u0026mdash;a kind which suffices\r\nfor the theory of our science, though not for its application to\r\nthe objects of daily life\u0026mdash;is purely geometrical, and asserts no\r\nmore than this: That since our matter is devoid, \u003ci lang=\"la\" xml:lang=\"la\"\u003eex hypothesi\u003c/i\u003e,\r\nof causal properties, there remains nothing, in mere empty space,\r\nwhich is capable of changing the configuration of any geometrical\r\nsystem. A change of absolute position, it asserts, is nothing;\r\ntherefore the only real change involved in motion is a change of\r\nrelation to other matter; but such other matter, for the purposes\r\nof our science, is regarded as destitute of causal powers; hence\r\nno change can occur, in the configuration of our system, by the\r\nmere effect of motion through empty space. The necessity of\r\nsuch a principle may be shown by a simple \u003ci lang=\"la\" xml:lang=\"la\"\u003ereductio ad absurdum\u003c/i\u003e,\r\nas follows. A motion of translation of the universe as a whole,\r\nwith constant direction and velocity, is dynamically negligeable;\r\nindeed it is, philosophically, no motion at all, for it involves no\r\nchange in the condition or mutual relations of the things in the\r\nuniverse. But if our geometrical rigidity were denied, the\r\nchange in the parameter of space might cause all bodies to\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_80\" id=\"Page_80\"\u003e[80]\u003c/a\u003e\u003c/span\u003e\r\nchange their shapes owing to the mere change of absolute\r\nposition, which is obviously absurd.\u003c/p\u003e\r\n\r\n\u003cp\u003eTo make quite plain the function of rigid bodies in Geometry,\r\nlet us suppose a liquid geometer in a liquid world. We cannot\r\nsuppose the liquid perfectly homogeneous and undifferentiated,\r\nin the first place because such a liquid would be indistinguishable\r\nfrom empty space, in the second place because our geometer\u0027s\r\nbody\u0026mdash;unless he be a disembodied spirit\u0026mdash;will itself constitute\r\na differentiation for him. We may therefore assume\u003c/p\u003e\r\n\r\n\u003cdiv class=\"poetry-container\"\u003e\u003cdiv class=\"poetry\"\u003e\r\n\u003cp class=\"verse8\"\u003e\"dim beams,\u003c/p\u003e\r\n\u003cp class=\"verse4\"\u003eWhich amid the streams\u003c/p\u003e\r\n\u003cp class=\"verse\"\u003eWeave a network of coloured light,\"\u003c/p\u003e\r\n\u003c/div\u003e\u003c/div\u003e\r\n\r\n\u003cp class=\"noindent\"\u003eand we may suppose this network to form the occasion for our\r\ngeometer\u0027s reflections. Then he will be able to imagine a\r\nnetwork in which the lines are straight, or circular, or parabolic,\r\nor any other shape, and he will be able to infer that such a\r\nnetwork, if it can be woven in one part of the fluid, can be\r\nwoven in another. This will form sufficient basis for his\r\ndeductions. The superposition he is concerned with\u0026mdash;since\r\nnot actual equality, but only the formal conditions of equality,\r\nare the subject-matter of Geometry\u0026mdash;is purely ideal, and is\r\nunaffected by the impossibility of congealing any actual network.\r\nBut in order to apply his Geometry to the exigencies of\r\nlife, he would need some standard of comparison between actual\r\nnetworks, and here, it is true, he would need either a rigid body,\r\nor a knowledge of the conditions under which similar networks\r\narose. Moreover these conditions, being necessarily empirical,\r\ncould hardly be known apart from previous measurement. Hence\r\nfor applied, though not for pure Geometry, one rigid body at\r\nleast seems essential.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e73.\u003c/b\u003e\u003ca name=\"N73\" id=\"N73\"\u003e\u003c/a\u003e\r\nThe utility, for Dynamics, of our abstract geometrical\r\nmatter, is sufficiently evident. For having, by its means, a\r\npower of determining the configurations of material systems in\r\nwhatever part of space, and knowing that changes of configuration\r\nare not due to mere change of place, we are able to attribute\r\nthese changes to the action of other matter, and thus to establish\r\nthe notion of force, which would be impossible if change of shape\r\nmight be due to empty space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_81\" id=\"Page_81\"\u003e[81]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eThus, to conclude: Geometry requires, if it is to be \u003cem\u003epractically\u003c/em\u003e\r\npossible, some body or bodies which are either rigid (in\r\nthe dynamical sense), or known to undergo some definite\r\nchanges of shape according to some definite law. (These\r\nchanges, we may suppose, are known by the laws of Physics,\r\nwhich have been experimentally established, and which throughout\r\nassume the truth of Geometry.) One or more such bodies\r\nare necessary to applied Geometry\u0026mdash;but only in the sense in\r\nwhich rulers and compasses are necessary. They are necessary\r\nas, in making the Ordnance Survey, an elaborate apparatus was\r\nnecessary for measuring the base line on Salisbury Plain. But\r\nfor the \u003cem\u003etheory\u003c/em\u003e of Geometry, geometrical rigidity suffices, and\r\ngeometrical rigidity means only that a shape, which is possible\r\nin one part of space, is possible in any other. The empirical\r\nelement in practice, arising from the purely empirical nature of\r\nphysical rigidity, is comparable to the empirical inaccuracies\r\narising from the failure to find straight lines or circles in the\r\nworld\u0026mdash;which no one but Mill has regarded as rendering\r\nGeometry itself empirical or inaccurate. But to make Geometry\r\nawait the perfection of Physics, is to make Physics,\r\nwhich depends throughout on Geometry, forever impossible.\r\nAs well might we leave the formation of numbers until we had\r\ncounted the houses in Piccadilly.\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eErdmann.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e74.\u003c/b\u003e\u003ca name=\"N74\" id=\"N74\"\u003e\u003c/a\u003e\r\nIn connection with Riemann and Helmholtz, it is\r\nnatural to consider Erdmann\u0027s philosophical work on their\r\ntheories\u003ca name=\"FNanchor_92_92\" id=\"FNanchor_92_92\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_92_92\" class=\"fnanchor\"\u003e[92]\u003c/a\u003e. This is certainly the most important book on the\r\nsubject which has appeared from the philosophical side, and in\r\nspite of the fact that, like the whole theory of Riemann and\r\nHelmholtz, it is inapplicable to projective Geometry, it still\r\ndeserves a very full discussion.\u003c/p\u003e\r\n\r\n\u003cp\u003eErdmann agrees throughout with the conclusions of Riemann\r\nand Helmholtz, except on a few points of minor importance;\r\nand his views, as this agreement would lead one to expect, are\r\nultra-empirical. Indeed his logic seems\u0026mdash;though I say this with\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_82\" id=\"Page_82\"\u003e[82]\u003c/a\u003e\u003c/span\u003e\r\nhesitation\u0026mdash;to be incompatible with any system but that of\r\nMill: there is apparently no distinction, to him, between the\r\ngeneral and the universal, and consequently no concept not\r\nembodied in a series of instances. Such a theory of logic, to\r\nmy mind, vitiates most of his work, as it vitiated Riemann\u0027s\r\nphilosophy\u003ca name=\"FNanchor_93_93\" id=\"FNanchor_93_93\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_93_93\" class=\"fnanchor\"\u003e[93]\u003c/a\u003e. This general criticism will find abundant illustration\r\nin the course of our account of Erdmann\u0027s views.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e75.\u003c/b\u003e\u003ca name=\"N75\" id=\"N75\"\u003e\u003c/a\u003e\r\nAfter a general introduction, and a short history of\r\nthe development of Metageometry, Erdmann proceeds, in his\r\nsecond chapter, to discuss what are the axioms of Euclidean\r\nGeometry. The arithmetical axioms, as they are called, he\r\nleaves aside, as applying to magnitude in general; what we\r\nwant here, he says, is a definition of space, for which the geometrical\r\naxioms are alone relevant. But a definition of space,\r\nhe says\u0026mdash;following Riemann\u0026mdash;demands a genus of which space\r\nshall be a species, and this, since our space is psychologically\r\nunique, can only be furnished by analytical mathematics (p. 36).\r\nNow the space-forms dealt with by Geometry are magnitudes,\r\nand conceptions of magnitude are everywhere applied in Geometry.\r\nBut before Riemann, only particular determinations of\r\nspace could be exhibited as magnitudes, and thus the desired\r\ndefinition was impossible to obtain. Now, however, we can\r\nsubsume space as a whole under a general conception of magnitude,\r\nand thus obtain, besides the space-intuition and the\r\nspace-conception, a third form, namely, the conception of space\r\nas a magnitude (\u003ccite lang=\"de\" xml:lang=\"de\"\u003eGrössenbegriff vom Raum\u003c/cite\u003e, pp. 38\u0026ndash;39). The\r\ndefinition of this will give us the complete, but not redundant,\r\nsystem of axioms, which could not be obtained by transforming\r\nthe general intuition of space into the space-conception, for\r\nwant of a plurality of instances (p. 40).\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e76.\u003c/b\u003e\u003ca name=\"N76\" id=\"N76\"\u003e\u003c/a\u003e\r\nBefore considering the subsequent method of definition,\r\nlet us reflect on the theories involved in the above\r\naccount of the conception of space as a magnitude. In the\r\nfirst place, it is assumed that conceptions cannot be formed\r\nunless we have a series of separate objects from which to abstract\r\na common property\u0026mdash;in other words, that the universal is\r\nalways the general. In the second place, it is assumed that all\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_83\" id=\"Page_83\"\u003e[83]\u003c/a\u003e\u003c/span\u003e\r\ndefinition is classification under a genus. In the third place, the\r\nconception of magnitude, if I am not mistaken, is fundamentally\r\nmisunderstood when it is supposed applicable to space as a\r\nwhole. But in the fourth place, even if such a conception\r\nexisted, it could give none of the essential properties of space.\r\nLet us consider these four points successively.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e77.\u003c/b\u003e\u003ca name=\"N77\" id=\"N77\"\u003e\u003c/a\u003e\r\nAs regards the first point, it is to be observed that\r\npeople certainly had some conception of space before Riemann\r\ninvented the notion of a manifold, and that this conception\r\nwas certainly something other than the common qualities of\r\nall the points, lines or figures in space. In the second place,\r\nErdmann\u0027s view would make it impossible to conceive God,\r\nunless one were a polytheist, or the universe\u0026mdash;unless, like\r\nLeibnitz, one imagined a series of possible worlds, set over\r\nagainst God, and none of them, therefore, a true Universe\u0026mdash;or,\r\nto take an instance more likely to appeal to an empiricist,\r\nthe necessarily unique centre of mass of the material universe.\r\nAny universal, in short, which is a bond or unity between things,\r\nand not merely a common property among independent objects,\r\nbecomes impossible on Erdmann\u0027s view. We cannot, therefore,\r\nunless we adopt Mill\u0027s philosophy intact, regard the conception\r\nof space as demanding a series of instances from which to\r\nabstract. But even if we did so regard it, Riemann\u0027s manifolds\r\nwould leave us without resources. For Euclidean space still\r\nappears as unique, at the end of his series of determinations.\r\nWe have instances of manifolds, but not instances of Euclidean\r\nspace. Thus if Erdmann\u0027s theory of conceptions were correct,\r\nhe would still be left searching in vain for the conception of\r\nEuclidean space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e78.\u003c/b\u003e\u003ca name=\"N78\" id=\"N78\"\u003e\u003c/a\u003e\r\nThe second point, the view that all definition is classification,\r\nis closely allied to the first, and the two together\r\nplunge us into the depths of scholastic formal logic. The same\r\ninstances of things which could not, on Erdmann\u0027s view, be\r\nconceived, may now be adduced as things which cannot be\r\ndefined. Whatever was said above applies here also, and the\r\npoint need not, therefore, be further discussed\u003ca name=\"FNanchor_94_94\" id=\"FNanchor_94_94\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_94_94\" class=\"fnanchor\"\u003e[94]\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_84\" id=\"Page_84\"\u003e[84]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp\u003e\u003cb\u003e79.\u003c/b\u003e\u003ca name=\"N79\" id=\"N79\"\u003e\u003c/a\u003e\r\nAs regards the third point, the impossibility of applying\r\nconceptions of magnitude to space as a whole, a longer\r\nargument will be necessary, for we are concerned, here, with\r\nthe whole question of the logical nature of judgments of magnitude.\r\nAs we had before too much comparison for our needs,\r\nso we have now too little. I will endeavour to explain this\r\npoint, which is of great importance, and underlies, I think,\r\nmost of the philosophical fallacies of Riemann\u0027s school.\u003c/p\u003e\r\n\r\n\u003cp\u003eA judgment of magnitude is always a judgment of comparison,\r\nand what is more, the comparison is never concerned\r\nwith quality, but only with quantity. Quality, in the judgment\r\nof magnitude, is supposed identical, in the object whose\r\nmagnitude is stated, and in the unit with which it is compared.\r\nBut quality, except in pure number, and in pure quantity as\r\ndealt with by the Calculus, is always present, and is partly\r\nabsorbed into quantity, partly untouched by the judgment\r\nof magnitude. As Bosanquet says (Logic, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 124);\r\n\"Quantitative comparison is not \u003ci lang=\"la\" xml:lang=\"la\"\u003eprima facie\u003c/i\u003e coordinate with\r\nqualitative, but rather stands in its place as the \u003cem\u003eeffect of\r\ncomparison on quality\u003c/em\u003e, which so far as comparable \u003cem\u003ebecomes\r\nquantity\u003c/em\u003e, and so far as incomparable furnishes the distinction\r\nof parts essential to the quantitative whole\" (italics in the\r\noriginal). Thus, if we are to regard space as a magnitude, we\r\nmust be able to adduce all those series of instances of which\r\nErdmann speaks, and which, for the conception of space, seemed\r\nirrelevant. But it remains to be proved that the comparison,\r\nwhich we \u003cem\u003ecan\u003c/em\u003e institute between various spaces, is capable of\r\nexpression in a quantitative form. Rather it would seem that\r\nthe difference of quality is such as to preclude quantitative\r\ncomparison between different spaces, and therefore also to\r\npreclude all judgments of magnitude about space as a whole.\r\nHere an exception might seem to be demanded by non-Euclidean\r\nspaces, whose space-constants give a definite magnitude,\r\ninherent in space as a whole, and therefore, one might\r\nthink, characterizing space as a magnitude. But this is a\r\nmistake. For the space-constant, in such spaces, is the ultimate\r\nunit, the fixed term in all quantitative comparison; it is itself,\r\ntherefore, destitute of quantity, since there is no independently\r\ngiven magnitude with which to compare it. A non-Euclidean\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_85\" id=\"Page_85\"\u003e[85]\u003c/a\u003e\u003c/span\u003e\r\nworld, in which the space-constant and all lines and figures\r\nwere suddenly multiplied in a constant ratio, would be wholly\r\nunchanged; the lines, as measured against the space-constant,\r\nwould have the same magnitude as before, and the space-constant\r\nitself, having no outside standard of comparison, would\r\nbe destitute of quantity, and therefore not subject to change\r\nof quantity. Such an enlargement of a non-Euclidean world,\r\nin other words, is unmeaning; and this proves how inapplicable\r\nis the notion of quantity to space as a whole.\u003c/p\u003e\r\n\r\n\u003cp\u003eIt might be objected that this only proves the absence\r\nof quantitative difference between different spaces of positive\r\nspace-constant, or between those of negative space-constant:\r\nthe quantitative difference persists, it might be said, between\r\nthose of positive curvature in general and those of negative\r\ncurvature in general, or between both together and Euclidean\r\nspace. This I entirely deny. There is no qualitatively similar\r\nunit, in the three kinds of space, by which quantitative\r\ncomparison could be effected. The straight lines of one space\r\ncannot be put into the other: the two straight lines, in one\r\nspace, whose product is the reciprocal of the measure of curvature,\r\nhave no corresponding curves in the other space, and\r\nthe measures of curvature cannot, therefore, be quantitatively\r\ncompared with each other. That the one may be regarded as\r\npositive, the other negative, I admit, but their values are\r\nindeterminate, and the units in the two cases are qualitatively\r\ndifferent. A debt of £300 may be represented as the asset\r\nof -£300, and the height of the Eiffel Tower is +300 metres;\r\nbut it does not follow that the two are quantitatively comparable.\r\nSo with space-constants: the space-constant is itself the unit\r\nfor magnitudes in its own space, and differs qualitatively from\r\nthe space-constant of another kind of space.\u003c/p\u003e\r\n\r\n\u003cp\u003eAgain, to proceed to a more philosophical argument, two\r\ndifferent spaces cannot co-exist in the same world: we may\r\nbe unable to decide between the alternatives of the disjunction,\r\nbut they remain, none the less, absolutely incompatible alternatives.\r\nHence we cannot get that coexistence of two\r\nspaces which is essential to comparison. The fact seems to be\r\nthat Erdmann, in his admiration for Riemann and Helmholtz,\r\nhas fallen in with their mathematical bias, and assumed, as\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_86\" id=\"Page_86\"\u003e[86]\u003c/a\u003e\u003c/span\u003e\r\nmathematicians naturally tend to assume, that quantity is\r\neverywhere and always applicable and adequate, and can deal\r\nwith more than the mere comparison of things whose qualities\r\nare already known as similar\u003ca name=\"FNanchor_95_95\" id=\"FNanchor_95_95\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_95_95\" class=\"fnanchor\"\u003e[95]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e80.\u003c/b\u003e\u003ca name=\"N80\" id=\"N80\"\u003e\u003c/a\u003e\r\nThis suggests the fourth and last of the above points,\r\nthat the \u003cem\u003equalities\u003c/em\u003e of space, even if space could be successfully\r\nregarded as a magnitude, would have to be entirely omitted\r\nin such a manner of regarding it, and that, therefore, none of\r\nits important or essential properties would emerge from such\r\ntreatment. For to regard space as a magnitude involves, as\r\nwe saw, a comparison with something qualitatively similar,\r\nand an abstraction from the similar qualities. To some extent\r\nand by the help of certain doubtful arguments, such a comparison\r\nis instituted by Riemann and Erdmann; but when they\r\nhave instituted it, they forget all about the common qualities\r\non which its possibility depends. But these are precisely the\r\nfundamental properties of space, and those from which, as I\r\nshall endeavour to prove in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, the axioms common\r\nto Euclid and Metageometry follow \u003cem\u003eà priori\u003c/em\u003e. Such are the\r\ndangers of the quantitative bias.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e81.\u003c/b\u003e\u003ca name=\"N81\" id=\"N81\"\u003e\u003c/a\u003e\r\nAfter this protest against the initial assumptions in\r\nErdmann\u0027s deduction of space, let us return to consider the\r\nmanner, in which this deduction is carried out. Here there\r\nwill be less ground for criticism, as the deduction, given its\r\npresuppositions, is, I think, as good as such a deduction can be.\r\nTo define space as a magnitude, he says, let us start with two\r\nof its most obvious properties, continuity and the three dimensions.\r\nTones and colours afford other instances of a manifold\r\nwith these two properties, but differ from space in that their\r\ndimensions are not homogeneous and interchangeable. To\r\ndesignate this difference, Erdmann introduces a useful pair of\r\nterms: in the general case, he calls a manifold \u003cem\u003en\u003c/em\u003e-determined\r\n(n-\u003ci lang=\"de\" xml:lang=\"de\"\u003ebestimmt\u003c/i\u003e); in the case where, as in space, the dimensions are\r\nhomogeneous, he calls the manifold \u003cem\u003en\u003c/em\u003e-extended (n-\u003ci lang=\"de\" xml:lang=\"de\"\u003eausgedehnt\u003c/i\u003e).\r\nManifolds of the latter sort he calls extents (\u003ci lang=\"de\" xml:lang=\"de\"\u003eAusgedehntheiten\u003c/i\u003e).\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_87\" id=\"Page_87\"\u003e[87]\u003c/a\u003e\u003c/span\u003eThat the difference between the two kinds is one of quality,\r\nnot of quantity, he seems not to perceive; he also overlooks\r\nthe fact that, in the second kind, from its very definition,\r\nthe axiom of Congruence must hold, on account of the qualitative\r\nsimilarity of different parts. In spite of this fact, he\r\ndefines space as an extent, and then regards Congruence as\r\nempirical, and as possibly false in the infinitesimal. This is\r\nthe more strange, as he actually proves (p. 50) that measurement\r\nis impossible, in an extent, unless the parts are independent\r\nof their place, and can be carried about unaltered as\r\nmeasures. In spite of this, he proceeds immediately to discuss\r\nwhether the measure of curvature is constant or variable,\r\nwithout investigating how, in the latter case, Geometry could\r\nexist. We cannot know, he says, from geometrical superposition,\r\nthat geometrical bodies are independent of place,\r\nfor if their dimensions altered in motion according to any\r\nfixed law, two bodies which could be superposed in one place\r\ncould be superposed in any other. That such a hypothesis\r\ninvolves absolute position, and denies the qualitative similarity\r\nof the parts of space, which he declares (p. 171) to be\r\nthe principle of his theory of Geometry, is nowhere perceived.\r\nBut what is more, his notion that magnitude is something\r\nabsolute, independent of comparison, has prevented him from\r\nseeing that such a hypothesis is unmeaning. He says himself\r\nthat, even on this hypothesis, a geometrical body can be defined\r\nas one whose points retain constant distances from each other,\r\nfor, since we have no absolute measure, measurement could not\r\nreveal to us the change of absolute magnitude (p. 60). But is\r\nnot this a \u003ci lang=\"la\" xml:lang=\"la\"\u003ereductio ad absurdum\u003c/i\u003e? For magnitude is nothing\r\napart from comparison, and the comparison here can only be\r\neffected by superposition; if, then, as on the above hypothesis,\r\nsuperposition always gives the same result, by whatever motion\r\nit is effected, there is no sense in speaking of magnitudes as\r\nno longer equal when separated: absolute magnitude is an\r\nabsurdity, and the magnitude resulting from comparison does\r\nnot differ from that which would result if the dimensions of\r\nbodies were unchanged in motion. Therefore, since magnitude\r\nis only intelligible as the result of comparison, the dimensions\r\nof bodies \u003cem\u003eare\u003c/em\u003e unchanged in motion, and the suggested hypothesis\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_88\" id=\"Page_88\"\u003e[88]\u003c/a\u003e\u003c/span\u003e\r\nis unmeaning. On this subject I shall have more to say in\r\nChapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003ca name=\"FNanchor_96_96\" id=\"FNanchor_96_96\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_96_96\" class=\"fnanchor\"\u003e[96]\u003c/a\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e82.\u003c/b\u003e\u003ca name=\"N82\" id=\"N82\"\u003e\u003c/a\u003e\r\nThis hypothesis, however, is not introduced for its\r\nown sake, but only to usher in the Helmholtzian \u003ci lang=\"la\" xml:lang=\"la\"\u003edeus ex\r\nmachina\u003c/i\u003e, Mechanics. For Mechanics proves\u0026mdash;\u003cins class=\"corr\" title=\"Transcriber\u0027s Note\u0026mdash;Original text: \u0027so Erdmannn confidently\u0027\"\u003eso Erdmann confidently\u003c/ins\u003e\r\ncontinues\u0026mdash;that rigidity must hold, not merely as to\r\nratios, in the above restricted geometrical sense, but as to\r\nabsolute magnitudes (p. 62). Hence we get at last true Congruence,\r\nempirical as Mechanics is empirical, and impossible to\r\nprove apart from Mechanics. I have already criticized Helmholtz\u0027s\r\nview of the dependence of Geometry on Mechanics, and\r\nneed not here speak of it at length. It is a pity that Erdmann\r\nhas in no way specified the procedure by which Mechanics\r\ndecides the geometrical alternatives\u0026mdash;indeed he seems to rely\r\non the \u003ci lang=\"la\" xml:lang=\"la\"\u003eipse dixit\u003c/i\u003e of Helmholtz. How, if Geometry would be\r\ntotally unable to discover a change in dimensions of the kind\r\nsuggested, the Laws of Motion, which throughout depend on\r\nGeometry, should be able to discover it if it existed, I am\r\nwholly at a loss to understand. Uniform motion in a straight\r\nline, for example, presupposes geometrical measurement; if\r\nthis measurement is mistaken, what Mechanics imagines to be\r\nuniform motion is not really such, but Mechanics can never\r\ndiscover the discrepancy. If the Laws of Motion had been\r\nregarded as \u003cem\u003eà priori\u003c/em\u003e, Geometry might possibly have been reinforced\r\nby them; but so long as they are empirical, they presuppose\r\ngeometrical measurement, and cannot therefore condition\r\nor affect it.\u003c/p\u003e\r\n\r\n\u003cp\u003eErdmann\u0027s conclusion, in the second chapter, is that Congruence\r\nis probable, but cannot be verified in the infinitesimal;\r\nthat its truth involves the actual existence of rigid bodies\r\n(though, by the way, we know these to be, strictly speaking,\r\nnon-existent), that rigid bodies are freely moveable, and do\r\nnot alter their size in rotation (Helmholtz\u0027s Monodromy); that\r\nthe axiom of three dimensions is certain, since small errors are\r\nimpossible; and that the remaining axioms of Euclid\u0026mdash;those\r\nof the straight line and of parallels\u0026mdash;are approximately, if not\r\naccurately, true of our actual space (pp. 78, 83). He does not\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_89\" id=\"Page_89\"\u003e[89]\u003c/a\u003e\u003c/span\u003e\r\ndiscuss how Congruence, on the above view, is compatible with\r\nthe atomic theory, or even with the observed deformations of\r\napproximately rigid bodies; nor how, if space, as he assumes,\r\nis homogeneous, rigid bodies can fail to be freely moveable\r\nthrough space. The axioms are all lumped together as empirical,\r\nand it appears, in the following chapters, that Erdmann\r\nregards their empirical nature as sufficiently proved by their\r\napplicability to empirical material (cf. pp. 159, 165)\u0026mdash;a strange\r\ncriterion, which would prove the same conclusion, with equal\r\nfacility, of Arithmetic and of the laws of thought.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e83.\u003c/b\u003e\u003ca name=\"N83\" id=\"N83\"\u003e\u003c/a\u003e\r\nThe third chapter, on the philosophical consequences\r\nof Metageometry, need not be discussed at length, since it\r\ndeals rather with space than with Geometry. At the same\r\ntime, it will be worth while to treat briefly of Erdmann\u0027s\r\ncriterion of apriority. On this subject it is very difficult to\r\ndiscover his meaning, since it seems to vary with the topic he\r\nis discussing. Thus at one time (p. 147) he rejects most\r\nemphatically the Kantian connection of the \u003cem\u003eà priori\u003c/em\u003e and the\r\nsubjective\u003ca name=\"FNanchor_97_97\" id=\"FNanchor_97_97\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_97_97\" class=\"fnanchor\"\u003e[97]\u003c/a\u003e, and yet at another time (p. 96) he regards every\r\npresentation of external things as partly \u003cem\u003eà priori\u003c/em\u003e, partly\r\nempirical, merely because such a presentation is due to an\r\ninteraction between ourselves and things, and is therefore\r\npartly due to subjective activity, partly due to outside objects.\r\nHence, he says, the distinction is not between different presentations,\r\nbut between different aspects of one and the same\r\npresentation. This seems to return wholly to the Kantian\r\npsychological criterion of subjectivity, with the added disadvantage\r\nthat it makes the distinction, like that of analytic\r\nand synthetic, epistemologically worthless. And yet he never\r\nhesitates to pronounce every piece of knowledge in turn empirical.\r\nThe fact seems to be, that where he wants a more\r\nlogical criterion, he adopts a modification of Helmholtz\u0027s criterion\r\nfor sensations. If space be an \u003cem\u003eà priori\u003c/em\u003e form, he says,\r\nno experience could possibly change it (p. 108); but this Metageometry\r\nhas proved not to be the case, since we can intuit the\r\nperceptions which non-Euclidean space would give us (p. 115).\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_90\" id=\"Page_90\"\u003e[90]\u003c/a\u003e\u003c/span\u003eI have criticised this argument in discussing Helmholtz; at\r\npresent we are concerned with Erdmann\u0027s criterion of apriority.\r\nThe subjectivity-criterion\u0026mdash;though he certainly uses it in discussing\r\nthe apriority of space, and solemnly decides, by its\r\nmeans, that space is both \u003cem\u003eà priori\u003c/em\u003e and empirical since a change\r\neither in us or in the outer world could change it (p. 97)\u0026mdash;would\r\nseem, like several of his other tests, to be a lapse on\r\nhis part: the criterion which he means to use is Helmholtz\u0027s.\r\nThis criterion, I think, with a slight change of wording, might\r\nbe accepted; it seems to me a necessary, but not a sufficient\r\ncondition. The \u003cem\u003eà priori\u003c/em\u003e, we may say, is not only that which\r\nno experience can change, but that without which experience\r\nwould become impossible. It is the omission to discuss the conditions\r\nwhich render geometrical (and mechanical) experience\r\npossible, to my mind, which vitiates the empirical conclusions\r\nof Helmholtz and Erdmann. Why certain conditions should\r\nbe necessary for experience\u0026mdash;whether on account of the constitution\r\nof the mind, or for some other reason\u0026mdash;is a further\r\nquestion, which introduces the relation of the \u003cem\u003eà priori\u003c/em\u003e to the\r\nsubjective. But in discussing the question as to what knowledge\r\nis \u003cem\u003eà priori\u003c/em\u003e, as opposed to the question concerning the\r\nfurther consequences of apriority, it is well to keep to the\r\npurely logical criterion, and so preserve our independence of\r\npsychological controversies. The fact, if it be a fact, that the\r\nworld might be such as to defy our attempts to know it, will\r\nnot, with the above criterion, invalidate the conclusion that\r\ncertain elements in knowledge are \u003cem\u003eà priori\u003c/em\u003e; for whether fulfilled\r\nor not, they remain necessary conditions for the existence\r\nof any knowledge at all.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e84.\u003c/b\u003e\u003ca name=\"N84\" id=\"N84\"\u003e\u003c/a\u003e\r\nWith this caution as to the meaning of apriority, we\r\nshall find, I think, that the conclusions of Erdmann\u0027s final\r\nchapter, on the principles of a theory of Geometry, are largely\r\ninvalidated by the diversity and inadequacy of his tests of the\r\n\u003cem\u003eà priori\u003c/em\u003e. He begins by asserting, in conformity with the\r\nquantitative bias noticed above, that the question as to the\r\nnature of geometrical axioms is completely analogous to the\r\ncorresponding question of the foundations of pure mathematics\r\n(p. 138). This is, I think, a radical error: for the function of\r\nthe axioms seems to be, to establish that qualitative basis on\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_91\" id=\"Page_91\"\u003e[91]\u003c/a\u003e\u003c/span\u003e\r\nwhich, as we saw, all qualitative comparison must rest. But\r\nin pure mathematics, this qualitative basis is irrelevant, for\r\nwe deal there with pure quantity, \u003cem\u003ei.e.\u003c/em\u003e with the merely quantitative\r\nresult of quantitative comparison, wherever it is possible,\r\nindependently of the qualities underlying the comparison.\r\nGeometry, as Grassmann insists\u003ca name=\"FNanchor_98_98\" id=\"FNanchor_98_98\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_98_98\" class=\"fnanchor\"\u003e[98]\u003c/a\u003e, ought not to be classed with\r\npure mathematics, for it deals with a matter which is given\r\nto the intellect, not created by it. The axioms give the\r\nmeans by which this matter is made amenable to quantity,\r\nand cannot, therefore, be themselves deduced from purely\r\nquantitative considerations.\u003c/p\u003e\r\n\r\n\u003cp\u003eLeaving this point aside, however, let us return to Erdmann.\r\nHe distinguishes, within space, a form and a matter: the form\r\nis to contain the properties common to all extents, the matter\r\nthe properties which distinguish space from other extents. This\r\ndistinction, he says, is purely logical, and does not correspond\r\nwith Kant\u0027s: matter and form, for Erdmann, are alike empirical.\r\nThe axioms and definitions of Geometry, he says, deal exclusively\r\nwith the matter of space. It seems a pity, having made this\r\ndistinction, to put it to so little use: after a few pages, it is\r\ndropped, and no epistemological consequences are drawn from\r\nit. The reason is, I think, that Erdmann has not perceived how\r\nmuch can be deduced from his definition of an extent, as a\r\nmanifold in which the dimensions are homogeneous and interchangeable.\r\nFor this property suffices to prove the complete\r\nhomogeneity of an extent, and hence\u0026mdash;from the absence of qualitative\r\ndifferences among elements\u0026mdash;the relativity of position and\r\nthe axiom of Congruence. This deduction will be made at length\r\nin the sequel\u003ca name=\"FNanchor_99_99\" id=\"FNanchor_99_99\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_99_99\" class=\"fnanchor\"\u003e[99]\u003c/a\u003e; at present, I have only to observe that every\r\nextent, on this view, possesses all the properties (except the\r\nthree dimensions) common to Euclidean and non-Euclidean\r\nspaces. The axioms which express these properties, therefore,\r\napply to the form of space, and follow from homogeneity alone,\r\nwhich Erdmann allows (p. 171) as the principle of any theory\r\nof space. The above distinction of form and matter, therefore,\r\ncorresponds, when its full consequences are deduced, to\r\nthe distinction between the axioms which follow from the\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_92\" id=\"Page_92\"\u003e[92]\u003c/a\u003e\u003c/span\u003ehomogeneity of space and those which do not. Since, then,\r\nhomogeneity is equivalent to the relativity of position, and\r\nthe relativity of position is of the very essence of a form of\r\nexternality, it would seem that his distinction of form and\r\nmatter can also be made coextensive with the distinction\r\nof the \u003cem\u003eà priori\u003c/em\u003e and empirical in Geometry. On this subject,\r\nI shall have more to say in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the remainder of the chapter, Erdmann insists that the\r\nstraight line, etc., though not abstracted from experience, which\r\nnowhere presents straight lines, must yet, as applicable to\r\nadmittedly empirical sciences, be empirical (p. 159)\u0026mdash;a criterion\r\nwhich he appears to employ only when all other grounds for an\r\nempirical opinion fail, and one which, obviously, can never refuse\r\nto do its work, since all elements of knowledge are susceptible\r\nof employment on some empirical material. He also defines the\r\nstraight line (p. 155) as a line of constant curvature zero, as\r\nthough curvature could be measured independently of the\r\nstraight line. Even the arithmetical axioms are declared\r\nempirical (p. 165), since in a world where things were all\r\nhopelessly different from one another, these axioms could not be\r\napplied. After this reminder of Mill, we are not surprised, a few\r\npages later (p. 172), at a vague appeal to \"English logicians\" as\r\nhaving proved Geometry to be an inductive science. Nevertheless,\r\nErdmann declares, almost on the last page of his book\r\n(p. 173), that Geometry is distinguished from all other sciences\r\nby the homogeneity of its material: a principle of which no\r\nsingle application occurs throughout his book, and which, as we\r\nshall see in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, flatly contradicts the philosophical\r\ntheories advocated throughout his preceding pages.\u003c/p\u003e\r\n\r\n\u003cp\u003eOn the whole, then, it cannot be said that Erdmann has\r\ndone much to strengthen the philosophical position of Riemann\r\nand Helmholtz. I have criticized him at length, because his\r\nbook has the appearance of great thoroughness, and because it\r\nis undoubtedly the best defence extant of the position which it\r\ntakes up. We shall now have the opposite task to perform, in\r\ndefending Metageometry, on its mathematical side, from the\r\nattacks of Lotze and others, and in vindicating for it that\r\nmeasure of philosophical importance\u0026mdash;far inferior, indeed, to\r\nthe hopes of Erdmann\u0026mdash;which it seems really to possess.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_93\" id=\"Page_93\"\u003e[93]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eLotze.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e85.\u003c/b\u003e\u003ca name=\"N85\" id=\"N85\"\u003e\u003c/a\u003e\r\nLotze\u0027s argument as regards Geometry\u003ca name=\"FNanchor_100_100\" id=\"FNanchor_100_100\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_100_100\" class=\"fnanchor\"\u003e[100]\u003c/a\u003e\u0026mdash;which follows\r\na metaphysical argument as to the ontological nature of space,\r\nand assumes the results of this argument\u0026mdash;consists of two\r\nparts: the first discusses the various meanings logically assignable\r\n(pp. 233\u0026ndash;247) to the proposition that other spaces than\r\nEuclid\u0027s are possible, and the second criticizes, in detail, the\r\nprocedure of Metageometry. The first of these questions is\r\nvery important, and demands considerable care as to the logical\r\nimport of a judgment of possibility. Although Lotze\u0027s discussion\r\nis excellent in many respects, I cannot persuade myself\r\nthat he has hit on the only true sense in which non-Euclidean\r\nspaces are possible. I shall endeavour to make good this statement\r\nin the following pages.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e86.\u003c/b\u003e\u003ca name=\"N86\" id=\"N86\"\u003e\u003c/a\u003e\r\nLotze opens with a somewhat startling statement,\r\nwhich, though philosophically worthy to be true, does not\r\nappear to be historically borne out. Euclidean Geometry has\r\nbeen chiefly shaken, he says, by the Kantian notion of the\r\nexclusive subjectivity of space\u0026mdash;if space is only our private\r\nform of intuition, to which there exists no analogue in the\r\nobjective world, then other beings may have other spaces,\r\nwithout supposing any difference in the world which they\r\narrange in these spaces (p. 233). This certainly seems a\r\nlegitimate deduction from the subjectivity of space, which, so\r\nfar from establishing the universal validity of Euclid, establishes\r\nhis validity only after an empirical investigation of the nature of\r\nspace as intuited by Tom, Dick or Harry. But as a matter of\r\nfact, those who have done most to further non-Euclidean Geometry\u0026mdash;with\r\nthe exception of Riemann, who was a disciple of\r\nHerbart\u0026mdash;have usually inherited from Newton a naïve realism\r\nas regards absolute space. I might instance the passage quoted\r\nfrom Bolyai in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, or Clifford, who seems to have thought\r\nthat we actually see the images of things on the retina\u003ca name=\"FNanchor_101_101\" id=\"FNanchor_101_101\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_101_101\" class=\"fnanchor\"\u003e[101]\u003c/a\u003e, or again\r\nHelmholtz\u0027s belief in the dependence of Geometry on the behaviour\r\nof rigid bodies. This belief led to the view that\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_94\" id=\"Page_94\"\u003e[94]\u003c/a\u003e\u003c/span\u003eGeometry, like Physics, is an experimental science, in which\r\nobjective truth can be attained, it is true, but only by empirical\r\nmethods. However, Lotze\u0027s ground for uncertainty about Euclid\r\nis a philosophically tenable ground, and it will be instructive to\r\nobserve the various possibilities which arise from it.\u003c/p\u003e\r\n\r\n\u003cp\u003eIf space is only a subjective form\u0026mdash;so Lotze opens his\r\nargument\u0026mdash;other beings may have a different form. If this\r\ncorresponds to a different world, the difference, he says, is\r\nuninteresting: for our world alone is relevant to any metaphysical\r\ndiscussion. But if this different space corresponds\r\nto the same world which we know under the Euclidean form,\r\nthen, in his opinion, we get a question of genuine philosophic\r\ninterest. And here he distinguishes two cases: \u003cem\u003eeither\u003c/em\u003e the\r\nrelations between things, which are presented to these hypothetical\r\nbeings under the form of some different space, are\r\nrelations which do not appear to us, or at any rate do not\r\nappear spatial; \u003cem\u003eor\u003c/em\u003e they are the same relations which appear\r\nto us as figures in Euclidean space (p. 235). The first possibility\r\nwould be illustrated, he says, by beings to whom the tone\r\nor colour-manifolds appeared extended; but we cannot, in his\r\nopinion, imagine a manifold, such as is required for this case, to\r\nhave its dimensions homogeneous and comparable \u003ci lang=\"la\" xml:lang=\"la\"\u003einter se\u003c/i\u003e, and\r\ntherefore the contents of the various presentations constituting\r\nsuch a manifold could not be combined into a single content\r\ncontaining them all. But the possibility of such a combination\r\nis of the essence of anything worth calling a space: therefore the\r\nfirst of the above possibilities is unmotived and uninteresting.\r\nLotze\u0027s conclusion on this point, I think, is undeniable, but I\r\ndoubt whether his argument is very cogent. However, as this\r\npossibility has no connection with that contemplated by non-Euclideans,\r\nit is not worth while to discuss it further.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe second possibility also, Lotze thinks, is not that of\r\nMetageometry, but in truth it comes nearer to it than any\r\nof the other possibilities discussed. If a non-Euclidean were\r\nat the same time a believer in the subjectivity of space, he\r\nwould have to be an adherent of this view. Let us see more\r\nprecisely what the view is. In Book \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, Lotze has\r\naccepted the argument of the Transcendental Aesthetic, but\r\nrejected that of the mathematical antinomies: he has decided\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_95\" id=\"Page_95\"\u003e[95]\u003c/a\u003e\u003c/span\u003e\r\nthat space is, as Kant believed, subjective, but possesses nevertheless,\r\nwhat Kant denied it, an objective counterpart. The\r\nrelation of presented space to its objective counterpart, as\r\nconceived by Lotze, is rather hard to understand. It seems\r\nscarcely to resemble the relation of sensation to its object\u0026mdash;\u003cem\u003ee.g.\u003c/em\u003e\r\nof light to ether-vibrations\u0026mdash;for if it did, space would not\r\nbe in any peculiar sense subjective. It seems rather to resemble\r\nthe relation of a perceived bodily motion to the state of mind\r\nof the person willing the motion. However this may be, the\r\nobjective counterpart of space is supposed to consist of certain\r\nimmediate interactions of monads, who experience the interactions\r\nas modifications of their internal states. Such interactions,\r\nit is plain, do not form the subject-matter of Geometry,\r\nwhich deals only with our resulting perceptions of spatial figures.\r\nNow if Lotze\u0027s construction of space be correct, there seems\r\ncertainly no reason why these resulting perceptions should\r\nnot, for one and the same interaction between monads, be\r\nvery different in beings differently constituted from ourselves.\r\nBut if they were different, says Lotze, they would have to be\r\nutterly different\u0026mdash;as different, for example, as the interval\r\nbetween two notes is from a straight line. The possibility\r\nis, therefore, in his opinion, one about which we can know\r\nnothing, and one which must remain always a mere empty\r\nidea. This seems to me to go too far: for whatever the\r\nobjective counterpart may be, any argument which gives us\r\ninformation about it must, when reversed, give us information\r\nabout any possible form of intuition in which this counterpart\r\nis presented. The argument which Lotze has used in his former\r\nchapter, for example, deducing, from the relativity of position,\r\nthe merely relational nature of the objective counterpart, allows\r\nus, conversely, to infer, from this relational nature, the complete\r\nrelativity of position in any possible space-intuition\u0026mdash;unless,\r\nindeed, it bore a wholly deceitful relation to those interactions\r\nof monads which form its objective counterpart. But the\r\ncomplete relativity of position, as I shall endeavour to establish\r\nin Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, suffices to prove that our Geometry must be\r\nEuclidean, elliptic, spherical or pseudo-spherical. We have,\r\ntherefore, it would seem, very considerable knowledge, on Lotze\u0027s\r\ntheory of space, of the manner in which what appears to us as\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_96\" id=\"Page_96\"\u003e[96]\u003c/a\u003e\u003c/span\u003e\r\nspace \u003cem\u003emust\u003c/em\u003e appear to any beings with our laws of thought. We\r\ncannot know, it is true, what \u003cem\u003epsychological\u003c/em\u003e theory of space-perception\r\nwould apply to such beings: they might have a\r\nsense different from any of ours, and they might have no\r\nsense in any way resembling ours, but yet their Geometry\r\nwould have points of resemblance to ours, as that of the blind\r\ncoincides with that of the seeing. If space has any objective\r\ncounterpart whatever, in short, and if any inference is possible,\r\nas Lotze holds it to be, from space to its counterpart, then a\r\nconverse argument is also possible, though it may give some\r\nonly of the qualities of Euclidean space, since some only of\r\nthese qualities may be found to have a necessary analogue in\r\nthe counterpart.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e87.\u003c/b\u003e\u003ca name=\"N87\" id=\"N87\"\u003e\u003c/a\u003e\r\nAdmitting, then, in Lotze\u0027s sense, the subjectivity of\r\nspace, the above possibility does not seem so empty as he\r\nimagines. He discusses it briefly, however, in order to pass\r\non to what he regards as the real meaning of Metageometry.\r\nIn this he is guilty of a mathematical mistake, which causes\r\nmuch irrelevant reasoning. For he believes that Metageometry\r\nconstructs its spaces out of straight lines and angles in all\r\nrespects similar to Euclid\u0027s, whence he derives an easy victory\r\nin proving that these elements can lead only to the one space.\r\nIn this he has been misled by the phraseology of non-Euclideans,\r\nas well as by Euclid\u0027s separation of definitions and axioms.\r\nFor the fact is, of course, that straight lines are only fully\r\ndefined when we add to the formal definition the axioms of\r\nthe straight line and of parallels. Within Euclidean space,\r\nEuclid\u0027s definition suffices to distinguish the straight line from\r\nall other curves; the two axioms referred to are then absorbed\r\ninto the definition of space. But apart from the restriction\r\nto Euclidean space, the definition has to be supplemented by\r\nthe two axioms, in order to define completely the Euclidean\r\nstraight line. Thus Lotze has misconceived the bearing of\r\nnon-Euclidean constructions, and has simply missed the point\r\nin arguing as he does. The possibility contemplated by a\r\nnon-Euclidean, if it fell under any of Lotze\u0027s cases, would fall\r\nunder the second case discussed above.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e88.\u003c/b\u003e\u003ca name=\"N88\" id=\"N88\"\u003e\u003c/a\u003e\r\nBut the bearing of Metageometry is really, I think,\r\ndifferent from anything imagined by Lotze; and as few writers\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_97\" id=\"Page_97\"\u003e[97]\u003c/a\u003e\u003c/span\u003e\r\nseem clear on this point, I will enter somewhat fully into what\r\nI conceive to be its purpose.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, there are some writers\u0026mdash;notably Clifford\u0026mdash;who,\r\nbeing naïve realists as regards space, hold that our\r\nevidence is wholly insufficient, as yet, to decide as to its nature\r\nin the infinite or in the infinitesimal (cf. Essays, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 320):\r\nthese writers are not concerned with any possibility of beings\r\ndifferent from ourselves, but simply with the everyday space we\r\nknow, which they investigate in the spirit of a chemist discussing\r\nwhether hydrogen is a metal, or an astronomer discussing\r\nthe nebular hypothesis.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut these are a minority: most, more cautious, admit that\r\nour space, so far as observation extends, is Euclidean, and if\r\nnot accurately Euclidean, must be only slightly spherical or\r\npseudo-spherical. Here again, it is the space of daily life which\r\nis under discussion, and here further the discussion is, I think,\r\nindependent of any philosophical assumption as to the nature\r\nof our space-intuition. For even if this be purely subjective,\r\nthe translation of an intuition into a conception can only be\r\naccomplished approximately, within the errors of observation\r\nincident to self-analysis; and until the intuition of space has\r\nbecome a conception, we get no scientific Geometry. The\r\napodeictic certainty of the axiom of parallels shrinks to an\r\nunmotived subjective conviction, and vanishes altogether in\r\nthose who entertain non-Euclidean doubts. To reinforce the\r\nEuclidean faith, reason must now be brought to the aid of\r\nintuition; but reason, unfortunately, abandons us, and we are\r\nleft to the mercy of approximate observations of stellar triangles\u0026mdash;a\r\nmeagre support, indeed, for the cherished religion of our\r\nchildhood.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e89.\u003c/b\u003e\u003ca name=\"N89\" id=\"N89\"\u003e\u003c/a\u003e\r\nBut the possibility of an inaccuracy so slight, that\r\nour finest instruments and our most distant parallaxes show\r\nno trace of it, would trouble men\u0027s minds no more than the\r\nanalogous chance of inaccuracy in the law of gravitation, were\r\nit not for the philosophical import of even the slenderest possibility\r\nin this sphere. And it is the philosophical bearing of\r\nMetageometry alone, I think, which constitutes its real importance.\r\nEven if, as we will suppose for the moment, observation\r\nhad established, beyond the possibility of doubt, that\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_98\" id=\"Page_98\"\u003e[98]\u003c/a\u003e\u003c/span\u003e\r\nour space might be safely regarded as Euclidean, still Metageometry\r\nwould have shown a philosophical possibility, and\r\non this ground alone it could claim, I think, very nearly all\r\nthe attention which it at present deserves.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut what is this possibility? A thing is possible, according\r\nto Bradley (Logic, p. 187), when it would follow from a certain\r\nnumber of conditions, some of which are known to be realized.\r\nNow the conditions to which a form of externality must conform,\r\nin order to be affirmed, are: first, of course, that it should\r\nbe experienced, or legitimately inferred from something experienced;\r\nbut secondly, that it should conform to certain\r\nlogical conditions, detailed in Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, which may be summed\r\nup in the relativity of position. Now what Metageometry\r\nhas done, in any case, is to suggest the proof that the second\r\nof these conditions is fulfilled by non-Euclidean spaces. Euclid\r\nis affirmed, therefore, on the ground of immediate experience\r\nalone, and his truth, as unmediated by logical necessity, is\r\nmerely assertorical, or, if we prefer it, empirical. This is the\r\nmost important sense, it seems to me, in which non-Euclidean\r\nspaces are possible. They are, in short, a step in a philosophical\r\nargument, rather than in the investigation of fact:\r\nthey throw light on the nature of the grounds for Euclid, rather\r\nthan on the actual conformation of space\u003ca name=\"FNanchor_102_102\" id=\"FNanchor_102_102\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_102_102\" class=\"fnanchor\"\u003e[102]\u003c/a\u003e. This import of\r\nMetageometry is denied by Lotze, on the ground that non-Euclidean\r\nlogic is faulty, a ground which he endeavours, by\r\nmuch detail and through many pages, to make good\u0026mdash;with\r\nwhat success, we will now proceed to examine.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e90.\u003c/b\u003e\u003ca name=\"N90\" id=\"N90\"\u003e\u003c/a\u003e\r\nLotze\u0027s attack on Metageometry\u0026mdash;although it remains,\r\nso far as I know, the best hostile criticism extant, and although\r\nits arguments have become part of the regular stock-in-trade\r\nof Euclidean philosophers\u0026mdash;contains, if I am not mistaken,\r\nseveral misunderstandings due to insufficient mathematical\r\nknowledge of the subject. As these misunderstandings have\r\nbeen widely spread among philosophers, and cannot be easily\r\nremoved except by a critic who has gone into non-Euclidean\r\nGeometry with some care, it seems desirable to discuss Lotze\u0027s\r\nstrictures point by point.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_99\" id=\"Page_99\"\u003e[99]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep099.jpg\" width=\"350\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e91.\u003c/b\u003e\u003ca name=\"N91\" id=\"N91\"\u003e\u003c/a\u003e\r\nThe mathematical criticism begins (§ 131) with a\r\nsomewhat question-begging definition of parallel straight lines.\r\nTwo straight lines \u003cem\u003eaα\u003c/em\u003e, \u003cem\u003ebβ\u003c/em\u003e, according to this definition, are\r\nparallel when\u0026mdash;\u003cem\u003ea\u003c/em\u003e and \u003cem\u003eb\u003c/em\u003e being arbitrary points on the two\r\nlines\u0026mdash;if \u003cem\u003eaα\u003c/em\u003e = \u003cem\u003ebβ\u003c/em\u003e, then \u003cem\u003eab\u003c/em\u003e = \u003cem\u003eαβ\u003c/em\u003e, where \u003cem\u003eα\u003c/em\u003e, \u003cem\u003eβ\u003c/em\u003e are two other points\r\non the two straight lines respectively. This definition\u0026mdash;which\r\ncontains Euclid\u0027s axiom and definition combined in a very\r\nconvenient and enticing form\u0026mdash;is of course thoroughly suitable\r\nto Euclidean Geometry, and leads immediately to all the\r\nEuclidean propositions about parallels. But it is perhaps\r\nmore honest to follow Euclid\u0027s course; when an axiom is thus\r\nburied in a definition, it is apt to seem, since definitions are\r\nsupposed to be arbitrary, as though the difficulty had been\r\novercome, while in reality, the possibility of parallels, as above\r\ndefined, involves the very point in question, namely, the disputed\r\naxiom of parallels. For what this axiom asserts is\r\nsimply the existence of lines conforming to Lotze\u0027s definition.\r\nThe deduction of the principal propositions on parallels, with\r\nwhich Lotze follows up his definition, is of course a very simple\r\nproceeding\u0026mdash;a proceeding, however, in which the first step begs\r\nthe question.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e92.\u003c/b\u003e\u003ca name=\"N92\" id=\"N92\"\u003e\u003c/a\u003e\r\nThe next argument for the apriority of Euclidean\r\nGeometry has, oddly enough, an exactly opposite bearing,\r\nalthough it is a great favourite with opponents of Metageometry.\r\nMeasurements of stellar triangles, and all similar\r\nattempts at an empirical determination of the space-constant\r\nare, according to Lotze, beside the mark; for any observed\r\ndeparture from two right angles, or any finite annual parallax\r\nfor distant stars, would be attributed to some new kind of\r\nrefraction, or, as in the case of aberration, to some other physical\r\ncause, and never to the geometrical nature of space. This is a\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_100\" id=\"Page_100\"\u003e[100]\u003c/a\u003e\u003c/span\u003e\r\nstrong argument for the empirical validity of Euclid, but as an\r\nargument for the apodeictic certainty of the orthodox system, it\r\nhas an opposite tendency. For observations of the kind contemplated\r\nwould have to be due to departures from Euclidean\r\nstraightness, hitherto unknown, on the part of stellar light-rays.\r\nSuch departure could, in certain cases, be accounted for by a\r\nfinite space-constant, but it could also, probably, be accounted\r\nfor by a change in Optics, for example, by attributing refractive\r\nproperties to the ether. Such properties could only exist if ether\r\nwere of varying density, if (say) it were denser in the neighbourhood\r\nof any of the heavenly bodies. But such an assumption\r\nwould, I believe, destroy the utility of ether for Physics;\r\na slight alteration in our Geometry, so slight as not appreciably\r\nto affect distances within the Solar System, would probably\r\nbe in the end, therefore, should such errors ever be discovered,\r\na simpler explanation than any that Physics could offer. But\r\nthis is not the point of my contention. The point is that, if\r\nthe physical explanation, as Lotze holds, be possible in the\r\nabove case, the converse must also hold: it must be possible\r\nto explain the present phenomena by supposing ether refractive\r\nand space non-Euclidean. From this conclusion there is no\r\nescape. If every conceivable behaviour of light-rays can be\r\nexplained, within Euclid, by physical causes, it must also be\r\npossible, by a suitable choice of hypothetical physical causes,\r\nto explain the actual phenomena as belonging to a non-Euclidean\r\nspace. Such a hypothesis would be rightly rejected\r\nby Science, for the present, on account of its unnecessary\r\ncomplexity. Nevertheless it would remain, for philosophy, a\r\npossibility to be reckoned with, and the choice could only be\r\ndecided upon empirical grounds of simplicity. It may well\r\nbe doubted whether, in the world we know, the phenomena\r\ncould be attributed to a distinctly non-Euclidean space, but\r\nthis conclusion follows inevitably from the contention that no\r\nphenomena could force us to assume such a space. Lotze\u0027s\r\nargument, therefore, if pushed home, disproves his own view,\r\nand puts Euclidean space, as an empirical explanation of phenomena,\r\non a level with luminiferous ether\u003ca name=\"FNanchor_103_103\" id=\"FNanchor_103_103\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_103_103\" class=\"fnanchor\"\u003e[103]\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_101\" id=\"Page_101\"\u003e[101]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp\u003e\u003cb\u003e93.\u003c/b\u003e\u003ca name=\"N93\" id=\"N93\"\u003e\u003c/a\u003e\r\nLotze now proceeds (§ 132) to a detailed criticism of\r\nHelmholtz, whom he regards as a typical exponent of Metageometry.\r\nIt is possible that, at the time when he wrote,\r\nHelmholtz really did occupy this position; but it is unfortunate\r\nthat, in the minds of philosophers, he should still continue\r\nto do so, after the very material advances brought about by\r\nthe projective treatment of the subject. It is also unfortunate\r\nthat his somewhat careless attempts to popularise mathematical\r\nresults have so often been disposed of, without due attention\r\nto his more technical and solid contributions. Thus his romances\r\nabout Flatland and Sphereland\u0026mdash;at best only fairy-tale\r\nanalogies of doubtful value\u0026mdash;have been attacked as if they\r\nformed an essential feature of Metageometry.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut to proceed to particulars: Lotze readily allows that\r\nthe Flatlanders would set up Plane Geometry, as we know it,\r\nbut refuses to admit that the Spherelanders could, without\r\ninferring the third dimension, set up a two-dimensional spherical\r\nGeometry which should be free from contradictions. I will\r\nendeavour to give a free rendering of Lotze\u0027s argument on\r\nthis point.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figright\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep101.jpg\" width=\"150\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003eSuppose, he says, a north and south pole, \u003cem\u003eN\u003c/em\u003e and \u003cem\u003eS\u003c/em\u003e, arbitrarily\r\nfixed, and an equator \u003cem\u003eEW\u003c/em\u003e. Suppose\r\na being, \u003cem\u003eB\u003c/em\u003e, capable of impressions\r\nonly from things on the surface of the\r\nsphere, to move in a meridian \u003cem\u003eNBS\u003c/em\u003e. Let\r\n\u003cem\u003eB\u003c/em\u003e start from some point \u003cem\u003ea\u003c/em\u003e, and finally,\r\nafter describing a great circle, return to\r\nthe same point \u003cem\u003ea\u003c/em\u003e. If \u003cem\u003ea\u003c/em\u003e is known only by\r\nthe quality of the impression it makes on\r\n\u003cem\u003eB\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e may imagine he has not reached the same point \u003cem\u003ea\u003c/em\u003e, but\r\nanother similar point \u003cem\u003ea′\u003c/em\u003e, bearing a relation to \u003cem\u003ea\u003c/em\u003e similar to that\r\nof the octave in singing: he might even not arrange his impressions\r\nspatially at all. In order that this may occur, we\r\nrequire the further assumption, that every difference in the\r\nabove-mentioned feelings (as he describes the meridian) may\r\nbe presented as a spatial distance between two places. Even\r\nnow, \u003cem\u003eB\u003c/em\u003e may think he is describing a Euclidean straight line,\r\ncontaining similar points at certain intervals. Allowing, however,\r\nthat he realizes the identity of \u003cem\u003ea\u003c/em\u003e with his initial position,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_102\" id=\"Page_102\"\u003e[102]\u003c/a\u003e\u003c/span\u003e\r\nhe will now seem, by motion in a straight line, to have returned\r\nto the point from which he started, for his motion cannot,\r\nwithout the third dimension, seem to him other than rectilinear.\u003c/p\u003e\r\n\r\n\u003cp\u003eUp to this point, there seems little ground for objection,\r\nexcept, perhaps, to the idea of a straight line with periodical\r\nsimilar points\u0026mdash;if \u003cem\u003eB\u003c/em\u003e were as philosophical as, in these discussions,\r\nwe usually suppose him to be, he would probably\r\nobject to this interpretation of his experiences, on the ground\r\nthat it regards empty space as something independent of the\r\nobjects in it. It is worth pointing out, also, that \u003cem\u003eB\u003c/em\u003e would\r\nnot need to describe the whole circle, in order suddenly to\r\nfind himself home again with his old friends. Accurate measurements\r\nof small triangles would suffice to determine his\r\nspace-constant, and show him the length of a great circle (or\r\nstraight line, as he would call it). We must admit, also, that\r\nso hypothetical a being as \u003cem\u003eB\u003c/em\u003e might form no space-intuition at\r\nall, but as he is introduced solely for the purposes of the\r\nanalogy, it is convenient to allow him all possible qualifications\r\nfor his post. But these points do not touch the kernel of the\r\nargument, which lies in the statement that such a straight\r\nline, returning into itself after a finite time, would appear to\r\n\u003cem\u003eB\u003c/em\u003e as an \"unendurable contradiction,\" and thus force him, for\r\nlogical though not for sensational purposes, into the assumption\r\nof a third dimension. This assertion seems to me quite unwarranted:\r\nthe whole of Metageometry is a solid array in\r\ndisproof of it. Helmholtz\u0027s argument is, it must be remembered,\r\nonly an analogy, and the contradiction would exist \u003cem\u003eonly\u003c/em\u003e\r\nfor a Euclidean. A complete \u003cem\u003ethree\u003c/em\u003e-dimensional Geometry has,\r\nwe have seen in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, been developed on the assumption\r\nthat straight lines are of finite length. A \u003cem\u003econstant\u003c/em\u003e value for\r\nthe measure of curvature, as our discussion of Riemann showed,\r\ninvolves neither reference to the fourth dimension, nor any\r\nkind of internal contradiction. This fact disproves Lotze\u0027s\r\ncontention, which arises solely from inability to divest his\r\nimagination of Euclidean ideas.\u003c/p\u003e\r\n\r\n\u003cp\u003eLotze next attacks Helmholtz for the assertion that \u003cem\u003eB\u003c/em\u003e would\r\nknow nothing of parallel lines\u0026mdash;parallel \u003cem\u003estraight\u003c/em\u003e lines, as the\r\ncontext shows, he meant to say\u003ca name=\"FNanchor_104_104\" id=\"FNanchor_104_104\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_104_104\" class=\"fnanchor\"\u003e[104]\u003c/a\u003e. Lotze, however, takes him\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_103\" id=\"Page_103\"\u003e[103]\u003c/a\u003e\u003c/span\u003e\r\nas meaning, apparently, mere curves of constant distance from\r\na given straight line, which are part of the regular stock-in-trade\r\nof Metageometry. Parallels of latitude, in the geographical\r\nsense, would not\u0026mdash;with the exception of the equator\u0026mdash;appear to\r\n\u003cem\u003eB\u003c/em\u003e as straight lines, but as circles. \u003cem\u003eGreat\u003c/em\u003e circles he \u003cem\u003ewould\u003c/em\u003e call\r\nstraight, and this fact seems to have misled Lotze into thinking\r\n\u003cem\u003eall\u003c/em\u003e circles were to be treated as straight lines. Parallels of\r\nlatitude, therefore, though \u003cem\u003eB\u003c/em\u003e might call them parallels, would\r\nnot invalidate Helmholtz\u0027s contention, which applies only to\r\nstraight lines.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe argument that such small circles would be parallel,\r\nwhich we have just disposed of, is only the preface to another\r\nproof that \u003cem\u003eB\u003c/em\u003e would need a third dimension. Let us call two\r\nof these parallels of latitude \u003cem\u003el\u003csub\u003en\u003c/sub\u003e\u003c/em\u003e and \u003cem\u003el\u003csub\u003es\u003c/sub\u003e\u003c/em\u003e, and let them be equidistant\r\nfrom the equator, one in the northern, one in the southern\r\nhemisphere. Consecutive tangent planes, along these parallels,\r\nconverge, in the one case northwards, in the other southwards.\r\nEither \u003cem\u003eB\u003c/em\u003e could become aware of their difference, says Lotze,\r\nor he could not. In the former case, which he regards as the\r\nmore probable, he easily proves that \u003cem\u003eB\u003c/em\u003e would infer a third dimension.\r\nBut this alternative is, I think, wholly inadmissible.\r\nTangent planes, like Euclidean planes in general, would have\r\nno meaning to \u003cem\u003eB\u003c/em\u003e; unless, indeed, he were a metageometrician,\r\nwhich, with all his metaphysical and mathematical subtlety, the\r\nargument supposes him not to be\u0026mdash;and to such a supposition\r\nLotze, surely, is the last person who has a right to object.\r\nLotze\u0027s attempted proof that this is the right alternative rests,\r\nif I understand him aright, on a sheer error in ordinary spherical\r\nGeometry. \u003cem\u003eB\u003c/em\u003e would observe, he says, that the meridians made\r\nsmaller angles with his path towards the nearer than towards\r\nthe further pole\u0026mdash;as a matter of fact, they would be simply\r\nperpendicular to his path in both directions. What Lotze\r\nmeans is, perhaps, that all the meridians would meet sooner\r\nin one direction than in the other, and this, of course, is true.\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_104\" id=\"Page_104\"\u003e[104]\u003c/a\u003e\u003c/span\u003e\r\nBut the poles, in which the meridians meet, would appear to\r\n\u003cem\u003eB\u003c/em\u003e as the centres of the respective parallels, while the parallels\r\nthemselves would appear to be circles. Now I am at a loss to\r\nsee what difficulty would arise, to \u003cem\u003eB\u003c/em\u003e, in supposing two different\r\ncircles to have different centres\u003ca name=\"FNanchor_105_105\" id=\"FNanchor_105_105\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_105_105\" class=\"fnanchor\"\u003e[105]\u003c/a\u003e. We must, therefore, take\r\nthe first alternative, that \u003cem\u003eB\u003c/em\u003e would have no sort of knowledge\r\nas to the direction in which the tangent planes converged.\r\nHere Lotze attempts, if I have not misunderstood him, to prove\r\na \u003ci lang=\"la\" xml:lang=\"la\"\u003ereductio ad absurdum\u003c/i\u003e: \u003cem\u003eB\u003c/em\u003e would think, he says, that he was\r\ndescribing two paths wholly the same in direction, and then\r\nhe \u003cem\u003emight\u003c/em\u003e regard both paths as circles in a plane. It may be\r\nobserved that direction, when applied to a circle as a whole,\r\nis meaningless; indeed direction, in all Metageometry, can only\r\nmean, even when applied to straight lines, direction towards\r\na point. To speak of two lines, which do not meet, as having\r\nthe same direction, is a surreptitious introduction of the axiom\r\nof parallels. Apart from this, I cannot conceive any objection,\r\non \u003cem\u003eB\u003c/em\u003e\u0027s part, to such a view\u0026mdash;one should say \u003cem\u003emust\u003c/em\u003e, not \u003cem\u003emight\u003c/em\u003e.\r\nThe whole argumentation, therefore, unless its obscurity has\r\nled me astray, must be pronounced fruitless and inconclusive.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e94.\u003c/b\u003e\u003ca name=\"N94\" id=\"N94\"\u003e\u003c/a\u003e\r\nAfter this preliminary discussion of Sphereland, Lotze\r\nproceeds to the question of a fourth dimension, and thence to\r\nspherical and pseudo-spherical space. As before, he appears to\r\nknow only the more careless and popular utterances of Helmholtz\r\nand Riemann, and to have taken no trouble to understand\r\neven the foundations of mathematical Metageometry. By this\r\nneglect, much of what he says is rendered wholly worthless.\r\nTo begin with, he regards, as the purpose of Helmholtz\u0027s fairy\r\ntale, the suggestion of a possible fourth dimension, whereas\r\nthe real purpose was quite the opposite\u0026mdash;to make intelligible\r\na purely three-dimensional non-Euclidean space. Helmholtz\r\nintroduced Flatland only because its relation to Sphereland\r\nis analogous to the relation of ours to spherical space\u003ca name=\"FNanchor_106_106\" id=\"FNanchor_106_106\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_106_106\" class=\"fnanchor\"\u003e[106]\u003c/a\u003e. But\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_105\" id=\"Page_105\"\u003e[105]\u003c/a\u003e\u003c/span\u003eLotze says: The Flatlanders would find no difficulty in a third\r\ndimension, since it would in no way contradict their own\r\nGeometry, while the people in Sphereland, from the contradictions\r\nin their two-dimensional system, would already have\r\nbeen led to it. The latter contention I have already tried to\r\nanswer; the former has an odd sound, in view of the attempt,\r\na few pages later, to prove \u003cem\u003eà priori\u003c/em\u003e that all forms of intuition,\r\nin any way analogous to space, \u003cem\u003emust\u003c/em\u003e have three dimensions.\r\nOne cannot help suspecting that the Flatlanders, with two\r\ninstead of three dimensions, would make a similar attempt.\r\nBut to return to Lotze\u0027s argument: Neither analogy can be\r\nused, he says, to prove that we ought perhaps to set up a\r\nfourth dimension, since, for us, no contradictions or otherwise\r\ninexplicable phenomena exist. The only people, so far as I\r\nknow, who have used this analogy, are Dr Abbot and a few\r\nSpiritualists\u0026mdash;the former in joke, the latter to explain certain\r\nphenomena more simply explained, perhaps, by Maskelyne and\r\nCooke. But although Lotze\u0027s conclusion in this matter is\r\nsound, and one with which Helmholtz might have agreed, his\r\narguments, to my mind, are irrelevant and unconvincing.\r\nThere is this difference, he says, between us and the Spherelanders:\r\nthe latter were logically forced to a new dimension,\r\nand found it possible; we are not forced to it, and find it, in\r\nour space, impossible. I have contended that, on the contrary,\r\nnothing would force the Spherelanders to assume a third dimension,\r\nwhile they would find it impossible exactly as we find a\r\nfourth impossible\u0026mdash;not logically, that is to say, but only as\r\na presentable construction in given space.\u003c/p\u003e\r\n\r\n\u003cp\u003eAfter a somewhat elephantine piece of humour, about\r\nsocialistic whales in a four-dimensional sea of Fourrier\u0027s \u003ci lang=\"fr\" xml:lang=\"fr\"\u003eeau\r\nsucrée\u003c/i\u003e, Lotze proceeds to a proof, by logic, that every form of\r\nintuition, which embraces the whole system of ordered relations\r\nof a coexisting manifold, \u003cem\u003emust\u003c/em\u003e have three dimensions. One\r\nmight object, on \u003cem\u003eà priori\u003c/em\u003e grounds, to any such attempt:\r\nwhat belongs to pure intuition could hardly, one would have\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_106\" id=\"Page_106\"\u003e[106]\u003c/a\u003e\u003c/span\u003e\r\nthought, be determined by \u003cem\u003eà priori\u003c/em\u003e reasoning\u003ca name=\"FNanchor_107_107\" id=\"FNanchor_107_107\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_107_107\" class=\"fnanchor\"\u003e[107]\u003c/a\u003e. I will not,\r\nhowever, develop this argument here, but endeavour to point\r\nout, as far as its obscurity will allow, the particular fallacy of\r\nthe proof in question.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep106.jpg\" width=\"400\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003eLotze\u0027s argument is as follows. In this discussion, though\r\nour terminology is necessarily taken from space, we are really\r\nconcerned with a much more general conception. We assume,\r\nin order to preserve the homogeneity of dimensions, that the\r\ndifference (distance) between any two elements (points) of our\r\nmanifold\u0026mdash;to borrow Riemann\u0027s word\u0026mdash;is of the same kind as,\r\nand commensurable with, the difference between any other\r\ntwo elements. Let us take a series of elements at successive\r\ndistances \u003cem\u003ex\u003c/em\u003e such that the distance between any two is the sum\r\nof the distances between intermediate elements. Such a series\r\ncorresponds to a straight line, which is taken as the \u003cem\u003ex\u003c/em\u003e-axis.\r\nThen a series \u003cem\u003eOY\u003c/em\u003e is called perpendicular to the \u003cem\u003ex\u003c/em\u003e-axis \u003cem\u003eOX\u003c/em\u003e,\r\nwhen the distances of any element \u003cem\u003ey\u003c/em\u003e, on \u003cem\u003eOY\u003c/em\u003e, from +\u003cem\u003emx\u003c/em\u003e and\r\n-\u003cem\u003emx\u003c/em\u003e are equal. By our hypothesis, these distances are comparable\r\nwith, and qualitatively similar to, \u003cem\u003ex\u003c/em\u003e and \u003cem\u003ey\u003c/em\u003e. So long\r\nas \u003cem\u003eOY\u003c/em\u003e is defined only by relation to \u003cem\u003eOX\u003c/em\u003e, it is conceptually\r\nunique. But now let us suppose the same relation as that\r\nbetween \u003cem\u003eOX\u003c/em\u003e and \u003cem\u003eOY\u003c/em\u003e, to be possible between \u003cem\u003eOY\u003c/em\u003e and a new\r\nseries \u003cem\u003eOZ\u003c/em\u003e; we then get a third series \u003cem\u003eOZ\u003c/em\u003e perpendicular to \u003cem\u003eOY\u003c/em\u003e,\r\nand again conceptually unique, so long as it is defined by\r\nrelation to \u003cem\u003eOY\u003c/em\u003e alone. We might proceed, in the same way,\r\nto a fourth line \u003cem\u003eOU\u003c/em\u003e perpendicular to \u003cem\u003eOZ\u003c/em\u003e. But it is necessary,\r\nfor our purposes, that \u003cem\u003eOZ\u003c/em\u003e should be perpendicular to \u003cem\u003eOX\u003c/em\u003e as\r\nwell as \u003cem\u003eOY\u003c/em\u003e. Without this condition, \u003cem\u003eOZ\u003c/em\u003e might extend into\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_107\" id=\"Page_107\"\u003e[107]\u003c/a\u003e\u003c/span\u003e\r\nanother world, and have no corresponding relation to \u003cem\u003eOX\u003c/em\u003e\u0026mdash;this\r\nis a possibility only excluded by our unavoidable spatial images.\r\nAt this point comes the crux of the argument. \u003cem\u003eThat OZ\u003c/em\u003e, says\r\nLotze, which, besides being perpendicular to \u003cem\u003eOY\u003c/em\u003e, is also perpendicular\r\nto \u003cem\u003eOX\u003c/em\u003e, must be among the series of \u003cem\u003eOY\u003c/em\u003e\u0027s, for these\r\nwere defined only by perpendicularity to \u003cem\u003eOX\u003c/em\u003e. \u003cem\u003eHence\u003c/em\u003e, he concludes,\r\nthere can only be even a third dimension if \u003cem\u003eOZ\u003c/em\u003e coincides\r\nwith one, and\u0026mdash;as soon as \u003cem\u003eOX\u003c/em\u003e is considered fixed\u0026mdash;with \u003cem\u003eonly\u003c/em\u003e\r\none, of the many members of the \u003cem\u003eOY\u003c/em\u003e series.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn this argument it is difficult\u0026mdash;to me at any rate\u0026mdash;to see\r\nany force at all. The only way I can account for it is, to\r\nsuppose that Lotze has neglected the possibility of any but\r\nsingle infinities. On this interpretation, the argument might\r\nbe stated thus: There is an infinite series of continuously\r\nvarying \u003cem\u003eOY\u003c/em\u003e\u0027s; to the common property of these, we add another\r\nproperty, which will divide their total number by infinity. The\r\nremaining \u003cem\u003eOZ\u003c/em\u003e, therefore, must be uniquely determined. The\r\nsame form of argument, however, would prove that two surfaces\r\ncan only cut one another in a single point, and numberless\r\nother absurdities. The fact is, that infinities may be of different\r\norders. For example, the number of points in a line may be\r\ntaken as a single infinity, and so may the number of lines\r\nin a plane through any point; hence, by multiplication, the\r\nnumber of points in a plane is a double infinity, ∞\u003csup\u003e2\u003c/sup\u003e, and if we\r\ndivide this number by a single infinity, we get still an infinite\r\nnumber left. Thus Lotze\u0027s argument assumes what he has\r\nto prove, that the number of lines perpendicular to a given\r\nline, through any point, is a single infinity, which is equivalent\r\nto the axiom of three dimensions. The whole passage is so\r\nobscure, that its meaning may have escaped me. It is obvious\r\n\u003cem\u003eà priori\u003c/em\u003e, however, as I pointed out in the beginning, that any\r\nproof of the axiom must be fallacious somewhere, and the above\r\ninterpretation of the argument is the only one I have been\r\nable to find.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e95.\u003c/b\u003e\u003ca name=\"N95\" id=\"N95\"\u003e\u003c/a\u003e\r\nThe rest of the Chapter is devoted to an attack on\r\nspherical and pseudo-spherical space, on the ground that they\r\ninterfere with the homogeneity of the three dimensions, and\r\nwith the similarity of all parts of space. This is simply false.\r\nSuch spaces, like the surface of a sphere, \u003cem\u003eare\u003c/em\u003e exactly alike\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_108\" id=\"Page_108\"\u003e[108]\u003c/a\u003e\u003c/span\u003e\r\nthroughout. Lotze shows, here and elsewhere, that he has\r\nnot taken the pains to find out what Metageometry really is.\r\nI hold myself, and have tried to prove in this Essay, that\r\nCongruence is an \u003cem\u003eà priori\u003c/em\u003e axiom, without which Geometry\r\nwould be impossible; but the wish to uphold this axiom is,\r\nas Lotze ought to have known, the precise motive which led\r\nMetageometry to limit itself to spaces of constant measure\r\nof curvature. We see here the importance of distinguishing\r\nbetween Helmholtz the philosopher and Helmholtz the mathematician.\r\nThough the philosopher wished to dispense with\r\nCongruence, the mathematician, as we saw in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e,\r\nretained and strongly emphasized it. A little later Lotze\r\nshows, again, how he has been misled by the unfortunate\r\nanalogy of Sphereland. A spherical \u003cem\u003esurface\u003c/em\u003e, he says, he can\r\nunderstand; but how are we to pass from this to a spherical\r\nspace? Either this surface is the whole of our space, as in\r\nSphereland, or it generates space by a gradually growing radius.\r\nSuch concentric spheres, as Lotze triumphantly points out, of\r\ncourse generate Euclidean space. His disjunction, however, is\r\nutterly and entirely false, and could never have been suggested\r\nby any one with even a superficial knowledge of Metageometry.\r\nThis point is less laboured than the former, which, in all its\r\nnakedness, is thus re-stated in the last sentence of the Chapter:\r\n\"I cannot persuade myself that one could, without the elements\r\nof homogeneous space, even form or define the presentation of\r\nheterogeneous spaces, or of such as had variable measures of\r\ncurvature.\" As though such spaces were ever set up by non-Euclidean\r\nmathematics!\u003c/p\u003e\r\n\r\n\u003cp\u003eIn conclusion, Lotze expresses a hope that Philosophy, on\r\nthis point, will not allow itself to be imposed upon by Mathematics.\r\nI must, instead, rejoice that Mathematics has not been\r\nimposed upon by Philosophy, but has developed freely an\r\nimportant and self-consistent system, which deserves, for its\r\nsubtle analysis into logical and factual elements, the gratitude\r\nof all who seek for a philosophy of space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e96.\u003c/b\u003e\u003ca name=\"N96\" id=\"N96\"\u003e\u003c/a\u003e\r\nThe objections to non-Euclidean Geometry which have\r\njust been discussed fall under four heads:\u003c/p\u003e\r\n\r\n\u003cp\u003eI. Non-Euclidean spaces are not homogeneous; Metageometry\r\ntherefore unduly reifies space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_109\" id=\"Page_109\"\u003e[109]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eII. They involve a reference to a fourth dimension.\u003c/p\u003e\r\n\r\n\u003cp\u003eIII. They cannot be set up without an implicit reference\r\nto Euclidean space, or to the Euclidean straight line, on which\r\nthey are therefore dependent.\u003c/p\u003e\r\n\r\n\u003cp\u003eIV. They are self-contradictory in one or more ways.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe reader who has followed me in regarding these four objections\r\nas fallacious, will have no difficulty in disposing of any other\r\ncritic of Metageometry, as these are the only mathematical\r\narguments, so far as I know, ever urged against non-Euclideans\u003ca name=\"FNanchor_108_108\" id=\"FNanchor_108_108\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_108_108\" class=\"fnanchor\"\u003e[108]\u003c/a\u003e.\r\nThe logical validity of Metageometry, and the mathematical\r\npossibility of three-dimensional non-Euclidean spaces, will therefore\r\nbe regarded, throughout the remainder of the work, as\r\nsufficiently established.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e97.\u003c/b\u003e\u003ca name=\"N97\" id=\"N97\"\u003e\u003c/a\u003e\r\nTwo other objections may, indeed, be urged against\r\nMetageometry, but these are rather of a philosophical than of\r\na strictly mathematical import. The first of these, which has\r\nbeen made the base of operations by Delbœuf, applies equally\r\nto all non-Euclidean spaces. The second, which has not, so far\r\nas I know, been much employed, but yet seems to me deserving\r\nof notice, bears directly against spaces of positive curvature\r\nalone; but if it could discredit these, it might throw doubt on\r\nthe method by which all alike are obtained. The two objections\r\nare:\u003c/p\u003e\r\n\r\n\u003cp\u003eI. Space must be such as to allow of similarity, \u003cem\u003ei.e.\u003c/em\u003e of the\r\nincrease or diminution, in a constant ratio, of all the lines in a\r\nfigure, without change of angles; whereas in non-Euclid, lines,\r\nlike angles, have absolute magnitude.\u003c/p\u003e\r\n\r\n\u003cp\u003eII. Space must be infinite, whereas spherical and elliptic\r\nspaces are finite.\u003c/p\u003e\r\n\r\n\u003cp\u003eI will discuss the first objection in connection with Delbœuf\u0027s\r\narticles referred to above. The second, which has not, to my\r\nknowledge, been widely used in criticism, will be better deferred\r\nto Chapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_110\" id=\"Page_110\"\u003e[110]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003eDelbœuf.\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e98.\u003c/b\u003e\u003ca name=\"N98\" id=\"N98\"\u003e\u003c/a\u003e\r\nM. Delbœuf\u0027s four articles in the Revue Philosophique\r\ncontain much matter that has already been dealt with in the\r\ncriticism of Lotze, and much that is irrelevant for our present\r\npurpose. The only point, which I wish to discuss here, is the\r\nquestion of absolute magnitude, as it is called\u0026mdash;the question,\r\nthat is, whether the possibility of similar but unequal geometrical\r\nfigures can be known \u003cem\u003eà priori\u003c/em\u003e\u003ca name=\"FNanchor_109_109\" id=\"FNanchor_109_109\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_109_109\" class=\"fnanchor\"\u003e[109]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn discussing this question, it is important, to begin with, to\r\ndistinguish clearly the sense in which absolute magnitude \u003cem\u003eis\u003c/em\u003e\r\nrequired in non-Euclidean Geometry, from another sense, in\r\nwhich it would be absurd to regard any magnitude as absolute.\r\nJudgments of magnitude can only result from comparison, and\r\nif Metageometry required magnitudes which could be determined\r\nwithout comparison, it would certainly deserve condemnation.\r\nBut this is not required. All we require is, that it\r\nshall be impossible, while the rest of space is unaffected, to alter\r\nthe magnitude of any figure, as compared with other figures,\r\nwhile leaving the relative internal magnitudes of its parts\r\nunchanged. This construction, which is possible in Euclid, is\r\nimpossible in Metageometry. We have to discuss whether such\r\nan impossibility renders non-Euclidean spaces logically faulty.\u003c/p\u003e\r\n\r\n\u003cp\u003eM. Delbœuf\u0027s position on this axiom\u0026mdash;which he calls the\r\npostulate of homogeneity\u003ca name=\"FNanchor_110_110\" id=\"FNanchor_110_110\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_110_110\" class=\"fnanchor\"\u003e[110]\u003c/a\u003e\u0026mdash;is, that all Geometry must presuppose\r\nit, and that Metageometry, consequently, though logically\r\nsound, is logically subsequent to Euclid, and can only make its\r\nconstructions within a Euclidean \"homogeneous\" space (Rev.\r\nPhil. Vol. \u003cspan class=\"fs70\"\u003eXXXVII.\u003c/span\u003e, pp. 380\u0026ndash;1). He would appear to think,\r\nnevertheless, that homogeneity (in his sense) is learnt from\r\nexperience, though on this point he is not very explicit. (See\r\nVol. \u003cspan class=\"fs70\"\u003eXXXVIII.\u003c/span\u003e, p. 129.) No \u003cem\u003eà priori\u003c/em\u003e proof, at any rate, is offered\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_111\" id=\"Page_111\"\u003e[111]\u003c/a\u003e\u003c/span\u003ein his articles. As a result of experience, every one would\r\nadmit, similarity is known to be possible within the limits of\r\nobservation; but the fact that this possibility extends to\r\nOrdnance maps, which deal with a spherical surface, should\r\nmake us chary of inferring, from such a datum, the certainty\r\nof Euclid for large spaces. Moreover if homogeneity be empirical,\r\nMetageometry, which dispenses with it, is not necessarily\r\nin \u003cem\u003elogical\u003c/em\u003e dependence upon Euclid, since homogeneity and\r\nisogeneity are \u003cem\u003elogically\u003c/em\u003e separable. I shall assume, therefore,\r\nas the only contention which can be interesting to our argument,\r\nthat homogeneity is regarded as \u003cem\u003eà priori\u003c/em\u003e, and as logically\r\nessential to Geometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e99.\u003c/b\u003e\u003ca name=\"N99\" id=\"N99\"\u003e\u003c/a\u003e\r\nNow we saw, in discussing Erdmann\u0027s views of the\r\njudgment of quantity, that in non-Euclidean space, as in\r\nEuclidean, a change of all spatial magnitudes, in the same\r\nratio, would be no change at all; the ratios of all magnitudes\r\nto the space-constant would be unchanged, and the space-constant,\r\nas the ultimate standard of comparison, cannot, in\r\nany intelligible sense, be said to have any particular magnitude.\r\nThe absolute magnitudes of Metageometry, therefore, are absolute\r\nonly as against any other \u003cem\u003eparticular\u003c/em\u003e magnitude, not as against\r\nother magnitudes in general. If this were not the case, the\r\ncomparative nature of the judgment of magnitude would\r\nbe contradicted, and metrical Metageometry would become\r\nabsurd. But as it is, the difference from Euclid consists only\r\nin this: that in Metageometry we have, while in Euclid we\r\nhave not, a standard of comparison involved in the nature\r\nof our space as a whole, which we call the space-constant.\r\nWe have to discuss whether the assertion of such a standard\r\ninvolves an undue reification of space.\u003c/p\u003e\r\n\r\n\u003cp\u003eI do not believe that this is the case. For an undue reification\r\nof space would only arise, if we were no longer able to\r\nregard position as wholly relative, and as geometrically definable\r\nonly by departure from other positions. But the relativity of\r\nposition, as we have abundantly seen, is preserved by all spaces\r\nof constant curvature\u0026mdash;in all of these, positions can only be\r\ndefined, geometrically, by relations to fresh positions\u003ca name=\"FNanchor_111_111\" id=\"FNanchor_111_111\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_111_111\" class=\"fnanchor\"\u003e[111]\u003c/a\u003e. This\r\nseries of definitions may lead to an infinite regress, but it may\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_112\" id=\"Page_112\"\u003e[112]\u003c/a\u003e\u003c/span\u003e\r\nalso, as in spherical space, form a vicious circle, and return\r\nagain to the position from which it started. No reification of\r\nspace, no independent existence of mere relations, seems involved\r\nin such a procedure. The whole of Metageometry, in short, is a\r\nproof that the relativity of position is compatible with absolute\r\nmagnitude, in the only sense required by non-Euclidean spaces.\r\nWe must conclude, therefore, that there is nothing incompatible,\r\nin a denial of homogeneity (in Delbœuf\u0027s sense), either with the\r\nrelational nature of space, or with the comparative nature of\r\nmagnitude. This last \u003cem\u003eà priori\u003c/em\u003e objection to Metageometry, therefore,\r\ncannot be maintained, and the issue must be decided on\r\nempirical grounds alone.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e100.\u003c/b\u003e\u003ca name=\"N100\" id=\"N100\"\u003e\u003c/a\u003e\r\nThe foundations of Geometry have been the subject of\r\nmuch recent speculation in France, and this seems to demand\r\nsome notice. But in spite of the splendid work which the\r\nFrench have done on the allied question of number and\r\ncontinuous quantity, I cannot persuade myself that they have\r\nsucceeded in greatly advancing the subject of geometrical\r\nphilosophy. The chief writers have been, from the mathematical\r\nside, \u003cem\u003eCalinon\u003c/em\u003e and \u003cem\u003ePoincaré\u003c/em\u003e, from the philosophical,\r\n\u003cem\u003eRenouvier\u003c/em\u003e and \u003cem\u003eDelbœuf\u003c/em\u003e; as a mediator between mathematics\r\nand philosophy, \u003cem\u003eLechalas\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003eCalinon\u003c/em\u003e, in an interesting article on the geometrical indeterminateness\r\nof the universe, maintains that any Geometry\r\nmay be applied to the actual world by a suitable hypothesis as\r\nto the course of light-rays. For the earth only is known to us\r\notherwise than by Optics, and the earth is an infinitesimal part\r\nof the universe. This line of argument has been already discussed\r\nin connection with Lotze, but Calinon adds a new suggestion,\r\nthat the space-constant may perhaps vary with the time. This\r\nwould involve a causal connection between space and other\r\nthings, which seems hardly conceivable, and which, if regarded as\r\npossible, must surely destroy Geometry, since Geometry depends\r\nthroughout on the irrelevance of Causation\u003ca name=\"FNanchor_112_112\" id=\"FNanchor_112_112\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_112_112\" class=\"fnanchor\"\u003e[112]\u003c/a\u003e. Moreover, in\r\nall operations of measurement, some time is spent; unless\r\nwe knew that space was unchanging throughout the operation,\r\nit is hard to see how our results could be trustworthy,\r\nand how, consequently, a change in the parameter could be\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_113\" id=\"Page_113\"\u003e[113]\u003c/a\u003e\u003c/span\u003e\r\ndiscovered. The same difficulties would arise, in fact, as those\r\nwhich result from supposing space not homogeneous.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003ePoincaré\u003c/em\u003e maintains that the question, whether Euclid or\r\nMetageometry should be accepted, is one of convenience and\r\nconvention, not of truth; axioms are definitions in disguise, and\r\nthe choice between definitions is arbitrary. This view has been\r\ndiscussed in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, in connection with Cayley\u0027s theory of\r\ndistance, on which it depends.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003eLechalas\u003c/em\u003e is a philosophical disciple of Calinon. He is a\r\nrationalist of the pre-Kantian type, but a believer in the\r\nvalidity of Metageometry. He holds that Geometry can dispense\r\nwith all purely spatial postulates, and work with axioms\r\nof magnitude alone\u003ca name=\"FNanchor_113_113\" id=\"FNanchor_113_113\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_113_113\" class=\"fnanchor\"\u003e[113]\u003c/a\u003e, which, in his opinion, are purely analytic.\r\nThe principle of contradiction, to him, is the sole and only test\r\nof truth; we make long chains of reasoning from our premisses\r\nto see if contradictions will emerge. It might be objected that\r\nthis view, though it saves general Geometry from being logically\r\nempirical, leaves it only empirically logical; this must, in fact,\r\nbe the fate of every piece of \u003cem\u003eà priori\u003c/em\u003e knowledge, if M. Lechalas\u0027s\r\nwere the only test of truth. However, he concludes that general\r\nGeometry is apodeictic, while the space of our actual world, like\r\nall other phenomena, is contingent.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003eDelbœuf\u003c/em\u003e criticizes non-Euclidean space from an ultra-realist\r\nstandpoint: he holds that \u003cem\u003ereal\u003c/em\u003e space is neither homogeneous\r\nnor isogeneous, but that \u003cem\u003econceived\u003c/em\u003e space, as abstracted from real\r\nspace, has both these properties. He offers no justification for\r\nhis real space, which seems to be maintained in the spirit of\r\nnaïve realism, nor does he show how he has acquired his intimate\r\nknowledge of its constitution\u003ca name=\"FNanchor_114_114\" id=\"FNanchor_114_114\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_114_114\" class=\"fnanchor\"\u003e[114]\u003c/a\u003e. His arguments against Metageometry,\r\nin so far as they are not repetitions of Lotze, have\r\nbeen discussed above.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cem\u003eRenouvier\u003c/em\u003e, finally, is a pure Kantian, of the most orthodox\r\ntype. His views as to the importance, for Geometry, of the\r\ndistinction between synthetic and analytic judgments, have\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_114\" id=\"Page_114\"\u003e[114]\u003c/a\u003e\u003c/span\u003ebeen discussed, in connection with Kant, at the beginning of\r\nthe present Chapter\u003ca name=\"FNanchor_115_115\" id=\"FNanchor_115_115\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_115_115\" class=\"fnanchor\"\u003e[115]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e101.\u003c/b\u003e\u003ca name=\"N101\" id=\"N101\"\u003e\u003c/a\u003e\r\nBefore beginning the constructive argument of the\r\nnext Chapter, let us endeavour briefly to sum up the theories\r\nwhich have been polemically advocated throughout the criticisms\r\nwe have just concluded. We agreed to accept, with Kant, necessity\r\nfor any possible experience as the test of the \u003cem\u003eà priori\u003c/em\u003e, but\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_115\" id=\"Page_115\"\u003e[115]\u003c/a\u003e\u003c/span\u003e\r\nwe refused, for the present, to discuss the connection of the\r\n\u003cem\u003eà priori\u003c/em\u003e with the subjective, regarding the purely logical test\r\nas sufficient for our immediate purpose. We also refused to\r\nattach importance to the distinction of analytic and synthetic,\r\nsince it seemed to apply, not to different judgments, but only to\r\ndifferent aspects of any judgment.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe then discussed Riemann\u0027s attempt to identify the\r\nempirical element in Geometry with the element not deducible\r\nfrom ideas of magnitude, and we decided that this\r\nidentification was due to a confusion as to the nature of magnitude.\r\nFor judgments of magnitude, we said, require always\r\nsome qualitative basis, which is not quantitatively expressible.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn criticizing Helmholtz, we decided that Mechanics logically\r\npresupposes Geometry, though space presupposes matter; but\r\nthat the matter which space presupposes, and to which Geometry\r\nindirectly refers, is a more abstract matter than that of Mechanics,\r\na matter destitute of force and of causal attributes, and possessed\r\nonly of the purely spatial attributes required for the possibility\r\nof spatial figures. But we conceded that Geometry, when applied\r\nto mixed mathematics or to daily life, demands more than this,\r\ndemands, in fact, some means of discovering, in the more concrete\r\nmatter of Mechanics, either a rigid body, or a body whose departure\r\nfrom rigidity follows some empirically discoverable law.\r\n\u003cem\u003eActual\u003c/em\u003e measurement, therefore, we agreed to regard as empirical.\u003c/p\u003e\r\n\r\n\u003cp\u003eOur conclusions, as regards the empiricism of Riemann and\r\nHelmholtz, were reinforced by a criticism of Erdmann. We then\r\nhad an opposite task to perform, in defending Metageometry\r\nagainst Lotze. Here we saw that there are two senses in which\r\nMetageometry is possible. The first concerns our actual space,\r\nand asserts that it may have a very small space-constant; the\r\nsecond concerns philosophical theories of space, and asserts a\r\npurely logical possibility, which leaves the decision to experience.\r\nWe saw also that Lotze\u0027s mathematical strictures arose\r\nfrom insufficient knowledge of the subject, and could all be\r\nrefuted by a better acquaintance with Metageometry.\u003c/p\u003e\r\n\r\n\u003cp\u003eFinally, we discussed the question of absolute magnitude,\r\nand found in it no logical obstacle to non-Euclidean spaces.\r\nOur conclusion, then, in so far as we are as yet entitled to a\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_116\" id=\"Page_116\"\u003e[116]\u003c/a\u003e\u003c/span\u003e\r\nconclusion, is that all spaces with a space-constant are \u003cem\u003eà priori\u003c/em\u003e\r\njustifiable, and that the decision between them must be the\r\nwork of experience. Spaces without a space-constant, on\r\nthe other hand, spaces, that is, which are not homogeneous\r\nthroughout, we found logically unsound and impossible to know,\r\nand therefore to be condemned \u003cem\u003eà priori\u003c/em\u003e. The constructive\r\nproof of this thesis will form the argument of the following\r\nchapter.\u003c/p\u003e\r\n\r\n\r\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_67_67\" id=\"Footnote_67_67\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_67_67\"\u003e\u003cspan class=\"label\"\u003e[67]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eThe Critical Philosophy of Kant\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 287.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_68_68\" id=\"Footnote_68_68\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_68_68\"\u003e\u003cspan class=\"label\"\u003e[68]\u003c/span\u003e\u003c/a\u003e For a discussion of Kant from a less purely mathematical standpoint, see\r\nChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_69_69\" id=\"Footnote_69_69\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_69_69\"\u003e\u003cspan class=\"label\"\u003e[69]\u003c/span\u003e\u003c/a\u003e Cf. Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 202, 265. Also p. 336 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_70_70\" id=\"Footnote_70_70\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_70_70\"\u003e\u003cspan class=\"label\"\u003e[70]\u003c/span\u003e\u003c/a\u003e E.g. second edition, p. 39: \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"So werden auch alle geometrischen Grundsätze,\r\nz. B. dass in einem Triangel zwei Seiten zusammen grösser sind als die\r\ndritte, niemals aus allgemeinen Begriffen von Linie und Triangel, sondern\r\naus der Anschauung, und zwar \u003cem\u003eà priori\u003c/em\u003e mit apodiktischer Gewissheit\r\nabgeleitet.\"\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_71_71\" id=\"Footnote_71_71\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_71_71\"\u003e\u003cspan class=\"label\"\u003e[71]\u003c/span\u003e\u003c/a\u003e Cf. Bradley\u0027s \u003ccite class=\"fsnormal\"\u003eLogic\u003c/cite\u003e, Bk. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Pt. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eVI.\u003c/span\u003e; Bosanquet\u0027s Logic, Bk. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\nChap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e pp. 97\u0026ndash;103.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_72_72\" id=\"Footnote_72_72\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_72_72\"\u003e\u003cspan class=\"label\"\u003e[72]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003ePhilosophie de la Règle et du Compas, Année Philosophique\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 1\u0026ndash;66.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_73_73\" id=\"Footnote_73_73\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_73_73\"\u003e\u003cspan class=\"label\"\u003e[73]\u003c/span\u003e\u003c/a\u003e I have stated this doctrine dogmatically, as a proof would require a whole\r\ntreatise on Logic. I accept the proofs offered by Bradley and Bosanquet, to\r\nwhich the reader is referred.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_74_74\" id=\"Footnote_74_74\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_74_74\"\u003e\u003cspan class=\"label\"\u003e[74]\u003c/span\u003e\u003c/a\u003e For a further discussion of this point, see \u003ca href=\"#CHAP_III\"\u003eChaps. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/a\u003e and \u003cspan class=\"fs70\"\u003e\u003ca href=\"#CHAP_IV\"\u003eIV.\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_75_75\" id=\"Footnote_75_75\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_75_75\"\u003e\u003cspan class=\"label\"\u003e[75]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#CHAP_IV\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003c/a\u003e for a discussion of this argument.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_76_76\" id=\"Footnote_76_76\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_76_76\"\u003e\u003cspan class=\"label\"\u003e[76]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N185\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e § 185.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_77_77\" id=\"Footnote_77_77\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_77_77\"\u003e\u003cspan class=\"label\"\u003e[77]\u003c/span\u003e\u003c/a\u003e An Otherness of substance, rather than of attribute, is here intended; an\r\nOtherness which may perhaps be called real as opposed to logical diversity.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_78_78\" id=\"Footnote_78_78\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_78_78\"\u003e\u003cspan class=\"label\"\u003e[78]\u003c/span\u003e\u003c/a\u003e This proposition will be argued at length in Chap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_79_79\" id=\"Footnote_79_79\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_79_79\"\u003e\u003cspan class=\"label\"\u003e[79]\u003c/span\u003e\u003c/a\u003e See \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003ePsychologie als Wissenschaft\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Section \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eVII.\u003c/span\u003e; \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Section \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\nChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e and Section \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Compare also \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003eSynechologie\u003c/cite\u003e, Section \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\nChaps. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_80_80\" id=\"Footnote_80_80\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_80_80\"\u003e\u003cspan class=\"label\"\u003e[80]\u003c/span\u003e\u003c/a\u003e On the influence of Herbart on Riemann, compare Erdmann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eDie Axiome\r\nder Geometrie\u003c/cite\u003e, p. 30.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_81_81\" id=\"Footnote_81_81\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_81_81\"\u003e\u003cspan class=\"label\"\u003e[81]\u003c/span\u003e\u003c/a\u003e I do not mean that measurement of colours is effected without reference to\r\ntheir relations, since all measurement is essentially comparison. But in\r\ncolours, it is the elements which are compared, while in space, it is the relations\r\nbetween elements.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_82_82\" id=\"Footnote_82_82\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_82_82\"\u003e\u003cspan class=\"label\"\u003e[82]\u003c/span\u003e\u003c/a\u003e For a discussion of this point, see \u003ca href=\"#N176\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Sec. B, § 176.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_83_83\" id=\"Footnote_83_83\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_83_83\"\u003e\u003cspan class=\"label\"\u003e[83]\u003c/span\u003e\u003c/a\u003e The works of Helmholtz on geometrical philosophy comprise, in addition\r\nto the articles quoted in Chap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, the following articles: \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Ursprung und Sinn\r\nder geometrischen Axiome, gegen Land,\"\u003c/cite\u003e Wiss. Abh. Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 640, 1878.\r\n(Also \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e: an answer to Land in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e) \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Ursprung und\r\nBedeutung der geometrischen Axiome,\"\u003c/cite\u003e 1870, Vorträge und Reden, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 1.\r\n(Also \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e) Two Appendices to \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Die Thatsachen in der Wahrnehmung,\"\u003c/cite\u003e\r\nentitled: II. \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Der Raum kann transcendental sein, ohne dass es die\r\nAxiome sind\"\u003c/cite\u003e; and III. \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Die Anwendbarkeit der Axiome auf die physische\r\nWelt,\"\u003c/cite\u003e 1878, Vorträge und Reden, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 256 ff.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nThe two Appendices last mentioned are popularizings and expansions of the\r\narticle in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e The most widely read, though also, to my mind, the\r\nleast valuable, of all Helmholtz\u0027s writings on Geometry, is the article in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e,\r\nVol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e This contains the famous and much misunderstood analogies of\r\nFlatland and Sphereland, which will be discussed, and as far as possible\r\ndefended, in answering Lotze\u0027s attack on Metageometry\u0026mdash;an attack based,\r\napparently, almost entirely on this one popular article. The present discussion,\r\ntherefore, may be confined almost entirely to \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, and the philosophical\r\nportions of the two papers quoted in Chap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, \u003cem\u003ei.e.\u003c/em\u003e to the articles in\r\nWiss. Abh. Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 610\u0026ndash;660. His other works are popular, and important\r\nonly because of the large public to which they appeal.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_84_84\" id=\"Footnote_84_84\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_84_84\"\u003e\u003cspan class=\"label\"\u003e[84]\u003c/span\u003e\u003c/a\u003e In the answer to Land, \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e and Wiss. Abh. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 640.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_85_85\" id=\"Footnote_85_85\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_85_85\"\u003e\u003cspan class=\"label\"\u003e[85]\u003c/span\u003e\u003c/a\u003e See also \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eDie Thatsachen in der Wahrnehmung, Zusatz \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, Der Raum kann\r\ntranscendental sein, ohne dass es die Axiome sind. Vorträge und Reden,\u003c/cite\u003e\r\nVol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_86_86\" id=\"Footnote_86_86\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_86_86\"\u003e\u003cspan class=\"label\"\u003e[86]\u003c/span\u003e\u003c/a\u003e See below, criticism of Erdmann, \u003ca href=\"#N84\"\u003e§ 84.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_87_87\" id=\"Footnote_87_87\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_87_87\"\u003e\u003cspan class=\"label\"\u003e[87]\u003c/span\u003e\u003c/a\u003e See Prof. Land, in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_88_88\" id=\"Footnote_88_88\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_88_88\"\u003e\u003cspan class=\"label\"\u003e[88]\u003c/span\u003e\u003c/a\u003e See concluding paragraph of Helmholtz\u0027s article in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_89_89\" id=\"Footnote_89_89\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_89_89\"\u003e\u003cspan class=\"label\"\u003e[89]\u003c/span\u003e\u003c/a\u003e Cf. Veronese, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eGrundzüge der Geometrie\u003c/cite\u003e (German translation), p. ix.\r\nAlso pp. xxxiv, 304, and Note \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 692\u0026ndash;4.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_90_90\" id=\"Footnote_90_90\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_90_90\"\u003e\u003cspan class=\"label\"\u003e[90]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N197\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e § 197 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_91_91\" id=\"Footnote_91_91\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_91_91\"\u003e\u003cspan class=\"label\"\u003e[91]\u003c/span\u003e\u003c/a\u003e Cf. the opinion of Bolyai, quoted by Erdmann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAxiome\u003c/cite\u003e, p. 26; cf. also ib.\r\np. 60.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_92_92\" id=\"Footnote_92_92\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_92_92\"\u003e\u003cspan class=\"label\"\u003e[92]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eDie Axiome der Geometrie: Eine philosophische Untersuchung der\r\nRiemann-Helmholtz\u0027schen Raumtheorie\u003c/cite\u003e, Leipzig, 1877.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_93_93\" id=\"Footnote_93_93\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_93_93\"\u003e\u003cspan class=\"label\"\u003e[93]\u003c/span\u003e\u003c/a\u003e On the influence of Mill, cf. Stallo, \u003ccite class=\"fsnormal\"\u003eConcepts of Modern Physics\u003c/cite\u003e, p. 216.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_94_94\" id=\"Footnote_94_94\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_94_94\"\u003e\u003cspan class=\"label\"\u003e[94]\u003c/span\u003e\u003c/a\u003e This view seems to be derived, through Riemann, from Herbart. See\r\n\u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003ePsych. als Wiss.\u003c/cite\u003e ed. Hart. Vol. \u003cspan class=\"fs70\"\u003eV.\u003c/span\u003e p. 262.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_95_95\" id=\"Footnote_95_95\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_95_95\"\u003e\u003cspan class=\"label\"\u003e[95]\u003c/span\u003e\u003c/a\u003e The same irreducibility of space to mere magnitude is proved by Kant\u0027s\r\nhands and spherical triangles, in which a difference persists in spite of complete\r\nquantitative equality.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_96_96\" id=\"Footnote_96_96\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_96_96\"\u003e\u003cspan class=\"label\"\u003e[96]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N146\"\u003e§§ 146\u0026ndash;7.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_97_97\" id=\"Footnote_97_97\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_97_97\"\u003e\u003cspan class=\"label\"\u003e[97]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Jeder Versuch, Kant\u0027s Lehre von der Apriorität als des subjectiven, von\r\naller Erfahrung absolut unabhängigen Erkenntnissfactors, trotzdem zu halten,\r\nist deshalb von voruherein aussichtslos.\"\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_98_98\" id=\"Footnote_98_98\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_98_98\"\u003e\u003cspan class=\"label\"\u003e[98]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAusdehnungslehre von 1844\u003c/cite\u003e, 2nd edition, pp. xxii. xxiii.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_99_99\" id=\"Footnote_99_99\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_99_99\"\u003e\u003cspan class=\"label\"\u003e[99]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N129\"\u003e§ 129 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_100_100\" id=\"Footnote_100_100\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_100_100\"\u003e\u003cspan class=\"label\"\u003e[100]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eMetaphysik\u003c/cite\u003e, Book \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e My references are to the original.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_101_101\" id=\"Footnote_101_101\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_101_101\"\u003e\u003cspan class=\"label\"\u003e[101]\u003c/span\u003e\u003c/a\u003e See \u003ccite class=\"fsnormal\"\u003eLectures and Essays\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 261.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_102_102\" id=\"Footnote_102_102\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_102_102\"\u003e\u003cspan class=\"label\"\u003e[102]\u003c/span\u003e\u003c/a\u003e On the meaning of geometrical possibility, cf. Veronese, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eGrundzüge der\r\nGeometrie\u003c/cite\u003e (German translation), pp. xi.-xiii.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_103_103\" id=\"Footnote_103_103\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_103_103\"\u003e\u003cspan class=\"label\"\u003e[103]\u003c/span\u003e\u003c/a\u003e Compare Calinon, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003e\"Sur l\u0027Indétermination géométrique de l\u0027Univers,\"\u003c/cite\u003e Revue\r\nPhilosophique, 1893, Vol. \u003cspan class=\"fs70\"\u003eXXXVI.\u003c/span\u003e pp. 595\u0026ndash;607.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_104_104\" id=\"Footnote_104_104\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_104_104\"\u003e\u003cspan class=\"label\"\u003e[104]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eVorträge und Reden\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 9:\r\n\u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Parallele Linien würden die\r\nBewohner der Kugel gar nicht kennen. Sie würden behaupten, dass jede\r\nbeliebige zwei \u003cem\u003egeradeste\u003c/em\u003e Linien, gehörig verlangert, sich schliesslich nicht nur\r\nin einem, sondern in zwei Punkten schneiden müssten.\"\u003c/cite\u003e (The italics are\r\nmine.) The omission of \u003cem\u003estraight\u003c/em\u003e in such phrases is a frequent laxity of\r\nmathematicians.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_105_105\" id=\"Footnote_105_105\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_105_105\"\u003e\u003cspan class=\"label\"\u003e[105]\u003c/span\u003e\u003c/a\u003e It has been suggested to me that Lotze regards the meridians as projected\r\non to a plane, as in a map. If this be so, there is an obviously illegitimate\r\nintroduction of the third dimension.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_106_106\" id=\"Footnote_106_106\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_106_106\"\u003e\u003cspan class=\"label\"\u003e[106]\u003c/span\u003e\u003c/a\u003e This is proved by Helmholtz\u0027s remark at the end of a detailed attempt to\r\nmake spherical and pseudo-spherical spaces imaginable (l.c. p. 28): \u003cspan lang=\"de\" xml:lang=\"de\"\u003e\"Anders\r\nist es mit den drei Dimensionen des Raumes. Da alle unsere Mittel sinnlicher\r\nAnschauung sich nur auf einen Raum von drei Dimensionen erstrecken, und\r\ndie vierte Dimension nicht bloss eine Abänderung von Vorhandenem, sondern\r\netwas vollkommen Neues wäre, so befinden wir uns schon wegen unserer\r\nkörperlichen Organisation in der absoluten Unmöglichkeit, uns eine Anschauungsweise\r\neiner vierten Dimension vorzustellen.\"\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_107_107\" id=\"Footnote_107_107\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_107_107\"\u003e\u003cspan class=\"label\"\u003e[107]\u003c/span\u003e\u003c/a\u003e Cf. Grassmann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAusdehnungslehre von 1844\u003c/cite\u003e, 2nd Edition, p. xxiii.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_108_108\" id=\"Footnote_108_108\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_108_108\"\u003e\u003cspan class=\"label\"\u003e[108]\u003c/span\u003e\u003c/a\u003e See especially Stallo, \u003ccite class=\"fsnormal\"\u003eConcepts of Modern Physics, International Science\r\nSeries\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eXLII.\u003c/span\u003e Chaps. \u003cspan class=\"fs70\"\u003eXIII.\u003c/span\u003e and \u003cspan class=\"fs70\"\u003eXIV.\u003c/span\u003e; Renouvier, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003e\"Philosophie de la règle et\r\ndu compas,\" Année Philosophique, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e; Delbœuf, \"L\u0027ancienne et les nouvelles\r\ngéométries,\" Revue Philosophique,\u003c/cite\u003e Vols. \u003cspan class=\"fs70\"\u003eXXXVI.-XXXIX.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_109_109\" id=\"Footnote_109_109\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_109_109\"\u003e\u003cspan class=\"label\"\u003e[109]\u003c/span\u003e\u003c/a\u003e M. Delbœuf deserves credit for having based Euclid, already in 1860, in\r\nhis \u003ccite lang=\"fr\" xml:lang=\"fr\"\u003e\"Prolégomènes Philosophiques de la Géométrie,\"\u003c/cite\u003e on this axiom\u0026mdash;certainly\r\na better basis, at first sight, than the axiom of parallels.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_110_110\" id=\"Footnote_110_110\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_110_110\"\u003e\u003cspan class=\"label\"\u003e[110]\u003c/span\u003e\u003c/a\u003e This meaning of homogeneity must not be confounded with the sense in\r\nwhich I have used the word. In Delbœuf\u0027s sense, it means that figures may be\r\nsimilar though of different sizes; in my sense it means that figures may be\r\nsimilar though in different places. This property of space is called by Delbœuf\r\nisogeneity.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_111_111\" id=\"Footnote_111_111\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_111_111\"\u003e\u003cspan class=\"label\"\u003e[111]\u003c/span\u003e\u003c/a\u003e For a full proof of this proposition, see \u003ca href=\"#CHAP_III\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_112_112\" id=\"Footnote_112_112\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_112_112\"\u003e\u003cspan class=\"label\"\u003e[112]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N133\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, especially § 133.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_113_113\" id=\"Footnote_113_113\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_113_113\"\u003e\u003cspan class=\"label\"\u003e[113]\u003c/span\u003e\u003c/a\u003e For a criticism of this view, see the above discussions on Riemann and\r\nErdmann.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_114_114\" id=\"Footnote_114_114\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_114_114\"\u003e\u003cspan class=\"label\"\u003e[114]\u003c/span\u003e\u003c/a\u003e Cf. Couturat, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003e\"De l\u0027Infini Mathématique,\"\u003c/cite\u003e Paris, Félix Alcan, 1896,\r\np. 544.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_115_115\" id=\"Footnote_115_115\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_115_115\"\u003e\u003cspan class=\"label\"\u003e[115]\u003c/span\u003e\u003c/a\u003e The following is a list of the most important recent French philosophical\r\nwritings on Geometry, so far as I am acquainted with them.\u003c/p\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnotex\" lang=\"fr\" xml:lang=\"fr\"\u003e\r\n\u003cp class=\"negin2\"\u003eAndrade: \"Les bases expérimentales de la géométrie euclidienne\"; Rev. Phil.\r\n1890, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, and 1891, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nBonnel: \"Les hypothèses dans la géométrie\"; Gauthier-Villars, 1897.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nL\u0027Abbé de Broglie: \"La géométrie non-euclidienne,\" two articles; Annales de\r\nPhil. Chrét. 1890.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nCalinon: \"Les espaces géométriques\"; Rev. Phil. 1889, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, and 1891, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\r\n\"Sur l\u0027indétermination géométrique de l\u0027univers\"; ib. 1893, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nCouturat: \"L\u0027Année Philosophique de F. Pillon,\" Rev. de Mét. et de Morale,\r\nJan. 1893.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"Note sur la géométrie non-euclidienne et la relativité de l\u0027espace\";\r\nib., May, 1893.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"Études sur l\u0027espace et le temps,\" ib. Sep. 1896.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nDelbœuf: \"L\u0027ancienne et les nouvelles géométries,\" four articles; Rev. Phil.\r\n1893\u0026ndash;5.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nLechalas: \"La géométrie générale\"; Crit. Phil. 1889.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"La géométrie générale et les jugements synthétiques à priori\" and\r\n\"Les bases expérimentales de la géométrie\"; Rev. Phil. 1890, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"M. Delbœuf et Le problème des mondes semblables\"; ib. 1894, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"Note sur la géométrie non-euclidienne et le principe de similitude\";\r\nRev. de Mét. et de Morale, March, 1893.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"La courbure et la distance en géométrie générale\"; ib., March,\r\n1896.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"La géométrie générale et l\u0027intuition\"; Annales de Phil. Chrét., 1890.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"Etude sur l\u0027espace et le temps\"; Paris, Alcan, 1896.\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"negin2\"\u003e\r\nLiard: \"Des définitions géométrie et des définitions empiriques,\" 2nd ed.;\r\nParis, Alcan, 1888.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nMansion: \"Premiers principes de la métagéométrie\"; two articles in Rev.\r\nNéo-Scholastique, 1896. Separately published, Gauthier-Villars, 1896.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nMilhaud: \"La géométrie non-euclidienne et la théorie de la connaissance\";\r\nRev. Phil. 1888, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"negin2\"\u003e\r\nPoincaré: \"Non-Euclidian Geometry\"; Nature, Vol. \u003cspan class=\"fs70\"\u003eXLV.\u003c/span\u003e, 1891\u0026ndash;2.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"L\u0027espace et la géométrie\"; Rev. de Mét. et de Morale, Nov. 1895.\u003cbr /\u003e\r\n\r\n\u0026nbsp; \u0026nbsp; \"Résponse à quelques critiques,\" ib. Jan. 1897.\r\n\u003c/p\u003e\r\n\r\n\u003cp class=\"negin2\"\u003e\r\nRenouvier: \"Philosophie de la règle et du compas\"; Crit. Phil., 1889, and\r\nL\u0027Année Phil., \u003cspan class=\"fs70\"\u003eII\u003c/span\u003e\u003csup\u003eme\u003c/sup\u003e année, 1891.\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nSorel: \"Sur la géométrie non-euclidienne\"; Rev. Phil., 1891, \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003cp class=\"negin2\"\u003e\r\nTannery: \"Théorie de la connaissance mathématique\"; Rev. Phil., 1894, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003c/div\u003e\r\n\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_117\" id=\"Page_117\"\u003e[117]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003e\u003ca name=\"CHAP_III\" id=\"CHAP_III\"\u003e\u003c/a\u003eCHAPTER III.\u003c/h2\u003e\r\n\r\n\r\n\u003ch3\u003e\u003ca name=\"SA\" id=\"SA\"\u003e\u003c/a\u003eSection A.\u003cbr /\u003e\u003cbr /\u003e\r\n\r\n\u003cspan class=\"fwnormal fs80\"\u003eTHE AXIOMS OF PROJECTIVE GEOMETRY.\u003c/span\u003e\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e102.\u003c/b\u003e\u003ca name=\"N102\" id=\"N102\"\u003e\u003c/a\u003e\r\nProjective Geometry proper, as we saw in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e,\r\ndoes not employ the conception of magnitude, and does not,\r\ntherefore, require those axioms which, in the systems of the\r\nsecond or metrical period, were required solely to render possible\r\nthe application of magnitude to space. But we saw, also, that\r\nCayley\u0027s reduction of metrical to projective properties was\r\npurely technical and philosophically irrelevant. Now it is in\r\nmetrical properties alone\u0026mdash;apart from the exception to the\r\naxiom of the straight line, which itself, however, presupposes\r\nmetrical properties\u003ca name=\"FNanchor_116_116\" id=\"FNanchor_116_116\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_116_116\" class=\"fnanchor\"\u003e[116]\u003c/a\u003e\u0026mdash;that non-Euclidean and Euclidean spaces\r\ndiffer. The properties dealt with by projective Geometry,\r\ntherefore, in so far as these are obtained without the use of\r\nimaginaries, are properties common to all spaces. Finally, the\r\ndifferences which appear between the Geometries of different\r\nspaces of the same curvature\u0026mdash;\u003cem\u003ee.g.\u003c/em\u003e between the Geometries of\r\nthe plane and the cylinder\u0026mdash;are differences in projective properties\u003ca name=\"FNanchor_117_117\" id=\"FNanchor_117_117\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_117_117\" class=\"fnanchor\"\u003e[117]\u003c/a\u003e.\r\nThus the necessity which arises, in metrical Geometry,\r\nfor further qualifications besides those of constant curvature,\r\ndisappears when our general space is defined by purely projective\r\nproperties.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e103.\u003c/b\u003e\u003ca name=\"N103\" id=\"N103\"\u003e\u003c/a\u003e\r\nWe have good ground for expecting, therefore, that\r\nthe axioms of projective Geometry will be the simplest and\r\nmost complete expression of the indispensable requisites of\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_118\" id=\"Page_118\"\u003e[118]\u003c/a\u003e\u003c/span\u003eany geometrical reasoning: and this expectation, I hope, will\r\nnot be disappointed. Projective Geometry, in so far as it\r\ndeals only with the properties common to all spaces, will be\r\nfound, if I am not mistaken, to be wholly \u003cem\u003eà priori\u003c/em\u003e, to take\r\nnothing from experience, and to have, like Arithmetic, a creature\r\nof the pure intellect for its object. If this be so, it is that\r\nbranch of pure mathematics which Grassmann, in his \u003ccite lang=\"de\" xml:lang=\"de\"\u003eAusdehnungslehre\u003c/cite\u003e\r\nof 1844, felt to be possible, and endeavoured, in a\r\nbrilliant failure, to construct without any appeal to the space of\r\nintuition.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e104.\u003c/b\u003e\u003ca name=\"N104\" id=\"N104\"\u003e\u003c/a\u003e\r\nBut unfortunately, the task of discovering the axioms\r\nof projective Geometry is far from easy. They have, as yet,\r\nfound no Riemann or Helmholtz to formulate them philosophically.\r\nMany geometers have constructed systems, which\r\nthey intended to be, and which, with sufficient care in interpretation,\r\nreally are, free from metrical presuppositions. But\r\nthese presuppositions are so rooted in all the very elements\r\nof Geometry, that the task of eliminating them demands a\r\nreconstruction of the whole geometrical edifice. Thus Euclid,\r\nfor example, deals, from the start, with spatial equality\u0026mdash;he\r\nemploys the circle, which is necessarily defined by means of\r\nequality, and he bases all his later propositions on the congruence\r\nof triangles as discussed in Book \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\u003ca name=\"FNanchor_118_118\" id=\"FNanchor_118_118\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_118_118\" class=\"fnanchor\"\u003e[118]\u003c/a\u003e Before we can\r\nuse any elementary proposition of Euclid, therefore, even if\r\nthis expresses a projective property, we have to prove that the\r\nproperty in question can be deduced by projective methods.\r\nThis has not, in general, been done by projective geometers,\r\nwho have too often assumed, for example, that the quadrilateral\r\nconstruction\u0026mdash;by which, as we saw in Chap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, they introduce\r\nprojective coordinates\u0026mdash;or anharmonic ratio, which is \u003ci lang=\"la\" xml:lang=\"la\"\u003eprimâ\r\nfacie\u003c/i\u003e metrical, could be satisfactorily established on their principles.\r\nBoth these assumptions, however, can be justified, and\r\nwe may admit, therefore, that the claims of projective Geometry\r\nto logical independence of measurement or congruence are\r\nvalid. Let us see, then, how it proceeds.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e105.\u003c/b\u003e\u003ca name=\"N105\" id=\"N105\"\u003e\u003c/a\u003e\r\nIn the first place, it is important to realize that\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_119\" id=\"Page_119\"\u003e[119]\u003c/a\u003e\u003c/span\u003e\r\nwhen coordinates are used, in projective Geometry, they are\r\nnot coordinates in the ordinary metrical sense, \u003cem\u003ei.e.\u003c/em\u003e the numerical\r\nmeasures of certain spatial magnitudes. On the contrary, they\r\nare a set of numbers, arbitrarily but systematically assigned\r\nto different points, like the numbers of houses in a street, and\r\nserving only, from a philosophical standpoint, as convenient\r\ndesignations for points which the investigation wishes to distinguish.\r\nBut for the brevity of the alphabet, in fact, they\r\nmight, as in Euclid, be replaced by letters. How they are\r\nintroduced, and what they mean, has been discussed in Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e\r\nHere we have only to repeat a caution, whose neglect has led\r\nto much misunderstanding.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e106.\u003c/b\u003e\u003ca name=\"N106\" id=\"N106\"\u003e\u003c/a\u003e\r\nThe distinction between various points, then, is not\r\na result, but a condition, of the projective coordinate system.\r\nThe coordinate system is a wholly extraneous, and merely convenient,\r\nset of marks, which in no way touches the essence\r\nof projective Geometry. What we must begin with, in this\r\ndomain, is the possibility of distinguishing various points from\r\none another. This may be designated, with Veronese, as the\r\nfirst axiom of Geometry\u003ca name=\"FNanchor_119_119\" id=\"FNanchor_119_119\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_119_119\" class=\"fnanchor\"\u003e[119]\u003c/a\u003e. How we are to define a point, and\r\nhow we distinguish it from other points, is for the moment\r\nirrelevant; for here we only wish to discover the nature of\r\nprojective Geometry, and the kind of properties which it uses\r\nand demonstrates. How, and with what justification, it uses\r\nand demonstrates them, we will discuss later.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e107.\u003c/b\u003e\u003ca name=\"N107\" id=\"N107\"\u003e\u003c/a\u003e\r\nNow it is obvious that a mere collection of points,\r\ndistinguished one from another, cannot found a Geometry:\r\nwe must have some idea of the manner in which the points\r\nare interrelated, in order to have an adequate subject-matter\r\nfor discussion. But since all ideas of quantity are excluded,\r\nthe relations of points cannot be relations of distance in the\r\nordinary sense, nor even, in the sense of ordinary Geometry,\r\nanharmonic ratios, for anharmonic ratios are usually defined\r\nas the ratios of four distances, or of four sines, and are thus\r\nquantitative. But since all quantitative comparison presupposes\r\nan identity of quality, we may expect to find, in projective Geometry,\r\nthe qualitative substrata of the metrical superstructure.\u003c/p\u003e\r\n\r\n\u003cp\u003eAnd this, we shall see, is actually the case. We have not\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_120\" id=\"Page_120\"\u003e[120]\u003c/a\u003e\u003c/span\u003e\r\ndistance, but we \u003cem\u003ehave\u003c/em\u003e the straight line; we have not quantitative\r\nanharmonic ratio, but we \u003cem\u003ehave\u003c/em\u003e the property, in any four points\r\non a line, of being the intersections with the rays of a given\r\npencil. And from this basis, we can build up a qualitative\r\nscience of abstract externality, which is projective Geometry.\r\nHow this happens, I shall now proceed to show.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e108.\u003c/b\u003e\u003ca name=\"N108\" id=\"N108\"\u003e\u003c/a\u003e\r\nAll geometrical reasoning is, in the last resort, circular:\r\nif we start by assuming points, they can only be defined\r\nby the lines or planes which relate them; and if we start\r\nby assuming lines or planes, they can only be defined by the\r\npoints through which they pass. This is an inevitable circle,\r\nwhose ground of necessity will appear as we proceed. It is,\r\ntherefore, somewhat arbitrary to start either with points or\r\nwith lines, as the eminently projective principle of duality\r\nmathematically illustrates; nevertheless we will elect, with\r\nmost geometers, to start with points\u003ca name=\"FNanchor_120_120\" id=\"FNanchor_120_120\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_120_120\" class=\"fnanchor\"\u003e[120]\u003c/a\u003e. We suppose, therefore,\r\nas our datum, a set of discrete points, for the moment without\r\nregard to their interconnections. But since connections are\r\nessential to any reasoning about them as a system, we introduce,\r\nto begin with, the axiom of the straight line. Any two\r\nof our points, we say, lie on a line which those two points completely\r\ndefine. This line, being determined by the two points,\r\nmay be regarded as a relation of the two points, or an adjective\r\nof the system formed by both together. This is the only purely\r\nqualitative adjective\u0026mdash;as will be proved later\u0026mdash;of a system of\r\ntwo points. Now projective Geometry can only take account\r\nof qualitative adjectives, and can distinguish between different\r\npoints only by their relations to other points, since all points,\r\n\u003cem\u003eper se\u003c/em\u003e, are qualitatively similar. Hence it comes that, for\r\nprojective Geometry, when two points only are given, they are\r\nqualitatively indistinguishable from any two other points on\r\nthe same straight line, since any two such other points have\r\nthe same qualitative relation. Reciprocally, since one straight\r\nline is a figure determined by any two of its points, and all\r\npoints are qualitatively similar, it follows that all straight lines\r\nare qualitatively similar. We may regard a point, therefore,\r\nas determined by two straight lines which meet in it, and the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_121\" id=\"Page_121\"\u003e[121]\u003c/a\u003e\u003c/span\u003e\r\npoint, on this view, becomes the only qualitative relation\r\nbetween the two straight lines. Hence, if the point only be\r\nregarded as given, the two straight lines are qualitatively\r\nindistinguishable from any other pair through the point.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e109.\u003c/b\u003e\u003ca name=\"N109\" id=\"N109\"\u003e\u003c/a\u003e\r\nThe extension of these two reciprocal principles is\r\nthe essence of all projective transformations, and indeed of all\r\nprojective Geometry. The fundamental operations, by which\r\nfigures are projectively transformed, are called projection and\r\nsection. The various forms of projection and section are defined\r\nin Cremona\u0027s \"Projective Geometry,\" Chapter \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, from which\r\nI quote the following account.\u003c/p\u003e\r\n\r\n\u003cp\u003e\"\u003cem\u003eTo project from a fixed point S\u003c/em\u003e (the \u003cem\u003ecentre of projection\u003c/em\u003e)\r\na figure (\u003cem\u003eABCD\u003c/em\u003e … \u003cem\u003eabcd\u003c/em\u003e …) composed of points and straight lines,\r\nis to construct the straight lines or \u003cem\u003eprojecting rays SA\u003c/em\u003e, \u003cem\u003eSB\u003c/em\u003e, \u003cem\u003eSC\u003c/em\u003e,\r\n\u003cem\u003eSD\u003c/em\u003e, … and the planes (\u003cem\u003eprojecting planes\u003c/em\u003e) \u003cem\u003eSa\u003c/em\u003e, \u003cem\u003eSb\u003c/em\u003e, \u003cem\u003eSc\u003c/em\u003e, \u003cem\u003eSd\u003c/em\u003e, … We\r\nthus obtain a new figure composed of straight lines and planes\r\nwhich all pass through the centre \u003cem\u003eS\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\"\u003cem\u003eTo cut by a fixed plane σ (transversal plane\u003c/em\u003e) a figure (\u003cem\u003eαβγδ\u003c/em\u003e …\r\n\u003cem\u003eabcd\u003c/em\u003e …) made up of planes and straight lines, is to construct\r\nthe straight lines or \u003cem\u003etraces σα, σβ, σγ\u003c/em\u003e … and the points or \u003cem\u003etraces\r\nσa, σb, σc\u003c/em\u003e….\u003ca name=\"FNanchor_121_121\" id=\"FNanchor_121_121\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_121_121\" class=\"fnanchor\"\u003e[121]\u003c/a\u003e By this means we obtain a new figure composed\r\nof straight lines and points lying in the plane \u003cem\u003eσ\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\"\u003cem\u003eTo project from a fixed straight line s\u003c/em\u003e (the \u003cem\u003eaxis\u003c/em\u003e) a figure\r\n\u003cem\u003eABCD\u003c/em\u003e composed of points, is to construct the planes \u003cem\u003esA\u003c/em\u003e, \u003cem\u003esB\u003c/em\u003e,\r\n\u003cem\u003esC\u003c/em\u003e…. The figure thus obtained is composed of planes which\r\nall pass through the axis \u003cem\u003es\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\"\u003cem\u003eTo cut by a fixed straight line s\u003c/em\u003e (a \u003cem\u003etransversal\u003c/em\u003e) a figure\r\n\u003cem\u003eαβγδ\u003c/em\u003e … composed of planes, is to construct the points \u003cem\u003esα\u003c/em\u003e, \u003cem\u003esβ\u003c/em\u003e,\r\n\u003cem\u003esγ\u003c/em\u003e…. In this way a new figure is obtained, composed of points\r\nall lying on the fixed transversal \u003cem\u003es\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\"If a figure is composed of straight lines \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e … which all\r\npass through a fixed point or \u003cem\u003ecentre S\u003c/em\u003e, it can be \u003cem\u003eprojected\u003c/em\u003e from\r\na straight line or \u003cem\u003eaxis s\u003c/em\u003e passing through \u003cem\u003eS\u003c/em\u003e; the result is a figure\r\ncomposed of planes \u003cem\u003esa\u003c/em\u003e, \u003cem\u003esb\u003c/em\u003e, \u003cem\u003esc\u003c/em\u003e….\u003c/p\u003e\r\n\r\n\u003cp\u003e\"If a figure is composed of straight lines \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e … all lying\r\nin a fixed plane, it may be cut by a straight line (transversal)\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_122\" id=\"Page_122\"\u003e[122]\u003c/a\u003e\u003c/span\u003e\r\n\u003cem\u003es\u003c/em\u003e lying in the same plane; the figure which results is formed\r\nby the points \u003cem\u003esa\u003c/em\u003e, \u003cem\u003esb\u003c/em\u003e, \u003cem\u003esc\u003c/em\u003e….\"\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e110.\u003c/b\u003e\u003ca name=\"N110\" id=\"N110\"\u003e\u003c/a\u003e\r\nThe successive application, to any figure, of two\r\nreciprocal operations of projection and section, is regarded as\r\nproducing a figure protectively indistinguishable from the first,\r\nprovided only that the dimensions of the original figure were\r\nthe same as those of the resulting figure, that, for example,\r\nif the second operation be section by a plane, the original\r\nfigure shall have been a plane figure. The figures obtained\r\nfrom a given figure, by projection or section alone, are related\r\nto that figure by the principle of duality, of which we shall\r\nhave to speak later on.\u003c/p\u003e\r\n\r\n\u003cp\u003eI shall endeavour to show, in what follows, first, in what\r\nsense figures obtained from each other by projective transformation\r\nare qualitatively alike; secondly, what axioms, or\r\nadjectives of space, are involved in the principle of projective\r\ntransformation; and thirdly, that these adjectives must belong\r\nto any form of externality with more than one dimension, and\r\nare, therefore, \u003cem\u003eà priori\u003c/em\u003e properties of any possible space.\u003c/p\u003e\r\n\r\n\u003cp\u003eFor the sake of simplicity, I shall in general confine myself\r\nto two dimensions. In so doing, I shall introduce no important\r\ndifference of principle, and shall greatly simplify the mathematics\r\ninvolved.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e111.\u003c/b\u003e\u003ca name=\"N111\" id=\"N111\"\u003e\u003c/a\u003e\r\nThe two mathematically fundamental things in projective\r\nGeometry are anharmonic ratio, and the quadrilateral\r\nconstruction. Everything else follows mathematically from\r\nthese two. Now what is meant, in projective Geometry, by\r\nanharmonic ratio?\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep123.jpg\" width=\"400\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003eIf we start from anharmonic ratio as ordinarily defined,\r\nwe are met by the difficulty of its quantitative nature\u003ca name=\"FNanchor_122_122\" id=\"FNanchor_122_122\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_122_122\" class=\"fnanchor\"\u003e[122]\u003c/a\u003e. But\r\namong the properties deduced from this definition, many, if\r\nnot most, are purely qualitative. The most fundamental of\r\nthese is that, if through any four points in a straight line\r\nwe draw four straight lines which meet in a point, and if we\r\nthen draw a new straight line meeting these four, the four new\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_123\" id=\"Page_123\"\u003e[123]\u003c/a\u003e\u003c/span\u003e\r\npoints of intersection have the same anharmonic ratio as the\r\nfour points we started with. Thus, in the figure, \u003cem\u003eabcd\u003c/em\u003e, \u003cem\u003ea′b′c′d′\u003c/em\u003e,\r\n\u003cem\u003ea″b″c″d″\u003c/em\u003e, all have the same anharmonic ratio. The reciprocal\r\nrelation holds for the anharmonic ratio of four straight lines.\r\nHere we have, plainly, the required basis for a qualitative\r\ndefinition. The definition must be as follows:\u003c/p\u003e\r\n\r\n\u003cp\u003eTwo sets of four points each are defined as having the same\r\nanharmonic ratio, when (1) each set of four lies in one straight\r\nline, and (2) corresponding points of different sets lie two by two\r\non four straight lines through a single point, or when both sets\r\nhave this relation to any third set\u003ca name=\"FNanchor_123_123\" id=\"FNanchor_123_123\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_123_123\" class=\"fnanchor\"\u003e[123]\u003c/a\u003e. And reciprocally: Two sets\r\nof four straight lines are defined as having the same anharmonic\r\nratio when (1) each set of four passes through a single point,\r\nand (2) corresponding lines of different sets pass, two by two,\r\nthrough four points in one straight line, or when both sets have\r\nthis relation to any third set.\u003c/p\u003e\r\n\r\n\u003cp\u003eTwo sets of points or of lines, which have the same anharmonic\r\nratio, are treated by projective Geometry as equivalent:\r\nthis qualitative equivalence replaces the quantitative equality\r\nof metrical Geometry, and is obviously included, by its definition,\r\nin the above account of projective transformations in\r\ngeneral.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e112.\u003c/b\u003e\u003ca name=\"N112\" id=\"N112\"\u003e\u003c/a\u003e\r\nWe have next to consider the quadrilateral\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_124\" id=\"Page_124\"\u003e[124]\u003c/a\u003e\u003c/span\u003e\r\nconstruction\u003ca name=\"FNanchor_124_124\" id=\"FNanchor_124_124\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_124_124\" class=\"fnanchor\"\u003e[124]\u003c/a\u003e. This has a double purpose: first, to define the\r\nimportant special case known as a harmonic range; and secondly,\r\nto afford an unambiguous and exhaustive method of assigning\r\ndifferent numbers to different points. This last method has,\r\nagain, a double purpose: first, the purpose of giving a convenient\r\nsymbolism for describing and distinguishing different\r\npoints, and of thus affording a means for the introduction of\r\nanalysis; and secondly, of so assigning these numbers that, if\r\nthey had the ordinary metrical significance, as distances from\r\nsome point on the numbered straight line, they would yield\r\n\u0026ndash;1 as the anharmonic ratio of a harmonic range, and that,\r\nif four points have the same anharmonic ratio as four others,\r\nso have the corresponding numbers. This last purpose is due\r\nto purely technical motives: it avoids the confusion with our\r\npreconceptions which would result from any other value for\r\na harmonic range; it allows us, when metrical interpretations\r\nof projective results are desired, to make these interpretations\r\nwithout tedious numerical transformations, and it enables us\r\nto perform projective transformations by algebraical methods.\r\nAt the same time, from the strictly projective point of view,\r\nas observed above, the numbers introduced have a purely\r\nconventional meaning; and until we pass to metrical Geometry,\r\nno reason can be shown for assigning the value \u0026ndash;1 to a harmonic\r\nrange. With this preliminary, let us see in what the\r\nquadrilateral construction consists.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep124.jpg\" width=\"400\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e113.\u003c/b\u003e\u003ca name=\"N113\" id=\"N113\"\u003e\u003c/a\u003e\r\nA harmonic range, in elementary Geometry, is one\r\nwhose anharmonic ratio is \u0026ndash;1, or one in which the three\r\nsegments formed by the four points are in harmonic progression,\r\nor again, one in which the ratio of the two internal\r\nsegments is equal to the ratio of the two external segments.\r\nIf \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e be the four points, it is easily seen that these\r\ndefinitions are equivalent to one another: they give respectively:\u003c/p\u003e\r\n\r\n\u003cp class=\"noindent pad2\"\u003e\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003eab\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003ebc\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e\r\n\u003csub\u003e\u003csub\u003e\u003cspan class=\"xxl\"\u003e/\u003c/span\u003e\u003c/sub\u003e\u003c/sub\u003e\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003ead\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003edc\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e = \u0026ndash; 1 , \u0026nbsp;\u0026nbsp;\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e1\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003eab\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e \u0026ndash;\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e1\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003eac\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e =\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e1\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003eac\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e \u0026ndash;\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e1\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003ead\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e , \u0026nbsp;\u0026nbsp; and \u0026nbsp;\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003eab\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003ebc\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e =\r\n\u003cspan class=\"blksum\"\u003e\r\n \u003cspan class=\"blka borbtm\"\u003e\u003cem\u003ead\u003c/em\u003e\u003c/span\u003e\r\n \u003cspan class=\"blka\"\u003e\u003cem\u003ecd\u003c/em\u003e\u003c/span\u003e\r\n\u003c/span\u003e\r\n.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_125\" id=\"Page_125\"\u003e[125]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eBut as they are all quantitative, they cannot be used for our\r\npresent purpose. Nor are any definitions which involve bisection\r\nof lines or angles available. We must have a definition\r\nwhich proceeds entirely by the help of straight lines and\r\npoints, without measurement of distances or angles. Now from\r\nthe above definitions of a harmonic range, we see that \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e\r\nhave the same anharmonic ratio as \u003cem\u003ec\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ea\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e. This gives us\r\nthe property we require for our definition. For it shows that,\r\nin a harmonic range, we can find a projective transformation\r\nwhich will interchange \u003cem\u003ea\u003c/em\u003e and \u003cem\u003ec\u003c/em\u003e. This is a necessary and sufficient\r\ncondition for a harmonic range, and the quadrilateral\r\nconstruction is the general method for giving effect to it.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep125.jpg\" width=\"400\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003eGiven any three points \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e in one straight line, the\r\nquadrilateral construction finds the point \u003cem\u003eC\u003c/em\u003e harmonic to \u003cem\u003eA\u003c/em\u003e\r\nwith respect to \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e by the following method: Take any point\r\n\u003cem\u003eO\u003c/em\u003e outside the straight line \u003cem\u003eABD\u003c/em\u003e, and join it to \u003cem\u003eB\u003c/em\u003e and \u003cem\u003eD\u003c/em\u003e.\r\nThrough \u003cem\u003eA\u003c/em\u003e draw any straight line cutting \u003cem\u003eOD\u003c/em\u003e, \u003cem\u003eOB\u003c/em\u003e in \u003cem\u003eP\u003c/em\u003e and \u003cem\u003eQ\u003c/em\u003e.\r\nJoin \u003cem\u003eDQ\u003c/em\u003e, \u003cem\u003eBP\u003c/em\u003e, and let them intersect in \u003cem\u003eR\u003c/em\u003e. Join \u003cem\u003eOR\u003c/em\u003e, and let\r\n\u003cem\u003eOR\u003c/em\u003e meet \u003cem\u003eABD\u003c/em\u003e in \u003cem\u003eC\u003c/em\u003e. Then \u003cem\u003eC\u003c/em\u003e is the point required.\u003c/p\u003e\r\n\r\n\u003cp\u003eTo prove this, let \u003cem\u003eDRQ\u003c/em\u003e meet \u003cem\u003eOA\u003c/em\u003e in \u003cem\u003eT\u003c/em\u003e, and draw \u003cem\u003eAR\u003c/em\u003e,\r\nmeeting \u003cem\u003eOD\u003c/em\u003e in \u003cem\u003eS\u003c/em\u003e. Then a projective transformation of \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e\r\nfrom \u003cem\u003eR\u003c/em\u003e on to \u003cem\u003eOD\u003c/em\u003e gives the points \u003cem\u003eS\u003c/em\u003e, \u003cem\u003eP\u003c/em\u003e, \u003cem\u003eO\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e, which, projected\r\nfrom \u003cem\u003eA\u003c/em\u003e on to \u003cem\u003eDQ\u003c/em\u003e, give \u003cem\u003eR\u003c/em\u003e, \u003cem\u003eQ\u003c/em\u003e, \u003cem\u003eT\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e. But these again, projected\r\nfrom \u003cem\u003eO\u003c/em\u003e on to \u003cem\u003eABD\u003c/em\u003e, give \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e. Hence \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e can\r\nbe projectively transformed into \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e, and therefore\r\nform a harmonic range. From this point, the proof that the\r\nconstruction is unique and general follows simply\u003ca name=\"FNanchor_125_125\" id=\"FNanchor_125_125\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_125_125\" class=\"fnanchor\"\u003e[125]\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_126\" id=\"Page_126\"\u003e[126]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp\u003eThe introduction of numbers, by this construction, offers\r\nno difficulties of principle\u0026mdash;except, indeed, those which always\r\nattend the application of number to continua\u0026mdash;and may be\r\nstudied satisfactorily in Klein\u0027s Nicht-Euklid (I. p. 337 ff.). The\r\nprinciple of it is, to assign the numbers 0, 1, ∞ to \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e and\r\ntherefore the number 2 to \u003cem\u003eC\u003c/em\u003e, in order that the differences \u003cem\u003eAB\u003c/em\u003e,\r\n\u003cem\u003eAC\u003c/em\u003e, \u003cem\u003eAD\u003c/em\u003e may be in harmonic progression. By taking \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e as\r\na new triad corresponding to \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e, we find a point harmonic\r\nto \u003cem\u003eB\u003c/em\u003e with respect to \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e and assign to it the number 3, and so\r\non. In this way, we can obtain any number of points, and\r\nwe are sure of having no number and no point twice over,\r\nso that our coordinates have the essential property of a unique\r\ncorrespondence with the points they denote, and \u003ci lang=\"la\" xml:lang=\"la\"\u003evice versa\u003c/i\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e114.\u003c/b\u003e\u003ca name=\"N114\" id=\"N114\"\u003e\u003c/a\u003e\r\nThe point of importance in the above construction,\r\nhowever, and the reason why I have reproduced it in detail,\r\nis that it proceeds entirely by means of the general principles\r\nof transformation enunciated above. From this stage onwards,\r\neverything is effected by means of the two fundamental ideas\r\nwe have just discussed, and everything, therefore, depends on\r\nour general principle of projective equivalence. This principle,\r\nas regards two dimensions, may be stated more simply than\r\nin the passage quoted from Cremona. It starts, in two\r\ndimensions, from the following definitions:\u003c/p\u003e\r\n\r\n\u003cp\u003eTo project the points \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, \u003cem\u003eD\u003c/em\u003e … from a centre \u003cem\u003eO\u003c/em\u003e, is to\r\nconstruct the straight lines \u003cem\u003eOA\u003c/em\u003e, \u003cem\u003eOB\u003c/em\u003e, \u003cem\u003eOC\u003c/em\u003e, \u003cem\u003eOD\u003c/em\u003e….\u003c/p\u003e\r\n\r\n\u003cp\u003eTo cut a number of straight lines \u003cem\u003ea\u003c/em\u003e, \u003cem\u003eb\u003c/em\u003e, \u003cem\u003ec\u003c/em\u003e, \u003cem\u003ed\u003c/em\u003e … by a transversal\r\n\u003cem\u003es\u003c/em\u003e, is to construct the points \u003cem\u003esa\u003c/em\u003e, \u003cem\u003esb\u003c/em\u003e, \u003cem\u003esc\u003c/em\u003e, \u003cem\u003esd\u003c/em\u003e….\u003ca name=\"FNanchor_126_126\" id=\"FNanchor_126_126\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_126_126\" class=\"fnanchor\"\u003e[126]\u003c/a\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eThe successive application of these two operations, provided\r\nthe original figure consisted of points on one straight line or\r\nof straight lines through one point, gives a figure projectively\r\nindistinguishable from the former figure; and hence, by extension,\r\nif any points in one straight line in the original figure\r\nlie in one straight line in the derived figure, and reciprocally\r\nfor straight lines through points, the two operations have\r\ngiven projectively similar figures. This general principle may\r\nbe regarded as consisting of two parts, according to the order\r\nof the operations: if we begin with projection and end with\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_127\" id=\"Page_127\"\u003e[127]\u003c/a\u003e\u003c/span\u003e\r\nsection, we transform a figure of points into another figure of\r\npoints; by the converse order, we transform a figure of lines\r\ninto another figure of lines.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e115.\u003c/b\u003e\u003ca name=\"N115\" id=\"N115\"\u003e\u003c/a\u003e\r\nBefore we can be clear as to the meaning of our\r\nprinciple, we must have some notion as to our definition of\r\npoints and straight lines. But this definition, in projective\r\nGeometry, cannot be given without some discussion of the\r\nprinciple of duality, the mathematical form of the philosophical\r\ncircle involved in geometrical definitions.\u003c/p\u003e\r\n\r\n\u003cp\u003eConfining ourselves for the moment to two dimensions,\r\nthe principle asserts, roughly speaking, that any theorem,\r\ndealing with lines through a point and points on a line, remains\r\ntrue if these two terms, wherever they occur, are interchanged.\r\nThus: two points lie on one straight line which they completely\r\ndetermine; and two straight lines meet in one point, which\r\nthey completely determine. The four points of intersection of\r\na transversal with four lines through a point have an anharmonic\r\nratio independent of the particular transversal; and\r\nthe four lines joining four points on one straight line to a\r\nfifth point have an anharmonic ratio independent of that fifth\r\npoint. So also our general principle of projective transformation\r\nhas two sides: one in which points move along fixed lines,\r\nand one in which lines turn about fixed points.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis duality suggests that any definition of points must\r\nbe effected by means of the straight line, and any definition\r\nof the straight line must be effected by means of points. When\r\nwe take the third dimension into account, it is true, the duality\r\nis no longer so simple; we have now to take account also of\r\nthe plane, but this only introduces a circle of three terms,\r\nwhich is scarcely preferable to a circle of two terms. We now\r\nsay: Three points, or a line and a point, determine a plane:\r\nbut conversely, three planes, or a line and plane, determine\r\na point. We may regard the straight line as a relation between\r\ntwo of its points, but we may also regard the point as a relation\r\nbetween two straight lines through it. We may regard the\r\nplane as a relation between three points, or between a point\r\nand a line, but we may also regard the point as a relation\r\nbetween three planes, or between a line and a plane, which\r\nmeet in it.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_128\" id=\"Page_128\"\u003e[128]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e116.\u003c/b\u003e\u003ca name=\"N116\" id=\"N116\"\u003e\u003c/a\u003e\r\nHow are we to get outside this circle? The fact\r\nis that, in pure Geometry, we cannot get outside it. For space,\r\nas we shall see more fully hereafter, is nothing but relations;\r\nif, therefore, we take any spatial figure, and seek for the terms\r\nbetween which it is a relation, we are compelled, in Geometry,\r\nto seek these terms within space, since we have nowhere else\r\nto seek them, but we are doomed, since anything purely spatial\r\nis a mere relation, to find our terms melting away as we grasp\r\nthem.\u003c/p\u003e\r\n\r\n\u003cp\u003eThus the relativity of space, while it is the essence of the\r\nprinciple of duality, at the same time renders impossible the\r\nexpression of that principle, or of any other principle of pure\r\nGeometry, in a manner which shall be free from contradictions.\r\nNevertheless, if we are to advance at all with our analysis of\r\ngeometrical reasoning and with our definitions of lines and\r\npoints, we must, for a while, ignore this contradiction; we\r\nmust argue as though it did not exist, so as to free our science\r\nfrom any contradictions which are not inevitable.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e117.\u003c/b\u003e\u003ca name=\"N117\" id=\"N117\"\u003e\u003c/a\u003e\r\nIn accordance with this procedure, then, let us define\r\nour points as the terms of spatial relations, regarding whatever\r\nis not a point as a relation between points. What, on this\r\nview, must our points be taken to be? Obviously, if extension\r\nis mere relativity, they must be taken to contain no extension;\r\nbut if they are to supply the terms for spatial relations, \u003cem\u003ee.g.\u003c/em\u003e\r\nfor straight lines, these relations must exhibit them as the\r\nterms of the figures they relate. In other words, since what\r\ncan really be taken, without contradiction, as the term of\r\na spatial relation, is unextended, we must take, as the term\r\nto be used in Geometry, where we cannot go outside space,\r\nthe least spatial thing which Geometry can deal with, the\r\nthing which, though \u003cem\u003ein\u003c/em\u003e space, \u003cem\u003econtains\u003c/em\u003e no space; and this\r\nthing we define as the point\u003ca name=\"FNanchor_127_127\" id=\"FNanchor_127_127\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_127_127\" class=\"fnanchor\"\u003e[127]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eNeglecting, then, the fundamental contradiction in this\r\ndefinition, the rest of our definitions follow without difficulty.\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_129\" id=\"Page_129\"\u003e[129]\u003c/a\u003e\u003c/span\u003e\r\nThe straight line is the relation between two points, and the\r\nplane is the relation between three. These definitions will be\r\nargued and defended at length in section B of this Chapter\u003ca name=\"FNanchor_128_128\" id=\"FNanchor_128_128\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_128_128\" class=\"fnanchor\"\u003e[128]\u003c/a\u003e,\r\nwhere we can discuss at the same time the alternative\r\nmetrical definitions; for our present purpose, it is sufficient\r\nto observe that projective Geometry, from the first, regards\r\nthe straight line as determined by two points, and the plane\r\nas determined by three, from which it follows, if we take points\r\nas possible terms for spatial relations, that the straight line\r\nand the plane may be regarded as relations between two and\r\nthree points respectively. If we agree on these definitions,\r\nwe can proceed to discuss the fundamental principle of projective\r\nGeometry, and to analyse the axioms implicated in\r\nits truth.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e118.\u003c/b\u003e\u003ca name=\"N118\" id=\"N118\"\u003e\u003c/a\u003e\r\nProjective Geometry, we have seen, does not deal\r\nwith quantity, and therefore recognizes no difference where\r\nthe difference is purely quantitative. Now quantitative comparison\r\ndepends on a recognized identity of quality; the recognition\r\nof qualitative identity, therefore, is logically prior to\r\nquantity, and presupposed by every judgment of quantity.\r\nHence all figures, whose differences can be exhaustively described\r\nby quantity, \u003cem\u003ei.e.\u003c/em\u003e by pure measurement, must have an\r\nidentity of quality, and this must be recognizable without\r\nappeal to quantity. It follows that, by defining the word\r\nquality in geometrical matters, we shall discover what sets\r\nof figures are projectively indiscernible. If our definition is\r\ncorrect, it ought to yield the general projective principle with\r\nwhich we set out.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e119.\u003c/b\u003e\u003ca name=\"N119\" id=\"N119\"\u003e\u003c/a\u003e\r\nWe agreed to regard points as the terms of spatial\r\nrelations, and we agreed that different points could be distinguished.\r\nBut we postponed the discussion of the conditions\r\nunder which this distinction could be effected. This discussion\r\nwill yield us the definition of quality and the proof of our\r\ngeneral projective principle.\u003c/p\u003e\r\n\r\n\u003cp\u003ePoints, to begin with, have been defined as nothing but\r\nthe terms for spatial relations. They have, therefore, no intrinsic\r\nproperties; but are distinguished solely by means of\r\ntheir relations. Now the relation between two points, we said,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_130\" id=\"Page_130\"\u003e[130]\u003c/a\u003e\u003c/span\u003e\r\nis the straight line on which they lie. This gives that identity\r\nof quality for all pairs of points on the same straight line,\r\nwhich is required both by our projective principle and by\r\nmetrical Geometry. (For only where there is identity of\r\nquality can quantity be properly applied.) If only two points\r\nare given, they cannot, without the use of quantity, be distinguished\r\nfrom any two other points on the same straight\r\nline; for the qualitative relation between any two such points\r\nis the same as for the original pair, and only by a difference\r\nof relation can points be distinguished from one another.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut conversely, one straight line is nothing but the relation\r\nbetween two of its points, and all points are qualitatively alike.\r\nHence there can be nothing to distinguish one straight line\r\nfrom another except the points through which it passes, and\r\nthese are distinguished from other points only by the fact that\r\nit passes through them. Thus we get the reciprocal transformation:\r\nif we are given only one point, any pair of straight\r\nlines through that point is qualitatively indistinguishable from\r\nany other. This again is, on the one hand, the basis of the\r\nsecond part of our general projective principle, and on the\r\nother hand the condition of applying quantity, in the measurement\r\nof angles, to the departure of two intersecting straight\r\nlines.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e120.\u003c/b\u003e\u003ca name=\"N120\" id=\"N120\"\u003e\u003c/a\u003e\r\nWe can now see the reason for what may have\r\nhitherto seemed a somewhat arbitrary fact, namely, the necessity\r\nof \u003cem\u003efour\u003c/em\u003e collinear points for anharmonic ratio. Recurring\r\nto the quadrilateral construction and the consequent introduction\r\nof number, we see that anharmonic ratio is an intrinsic\r\nprojective relation of four collinear points or concurrent straight\r\nlines, such that given three terms and the relation, the fourth\r\nterm can be uniquely determined by projective methods. Now\r\nconsider first a pair of points. Since all straight lines are\r\nprojectively equivalent, the relation between one pair of points\r\nis precisely equivalent to that between another pair. Given\r\none point only, therefore, no projective relation, to any second\r\npoint, can be assigned, which shall in any way limit our choice\r\nof the second point. Given two points, however, there is such\r\na relation\u0026mdash;the third point may be given collinear with the\r\nfirst two. This limits its position to one straight line, but\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_131\" id=\"Page_131\"\u003e[131]\u003c/a\u003e\u003c/span\u003e\r\nsince two points determine nothing but one straight line, the\r\nthird point cannot be further limited. Thus we see why no\r\nintrinsic projective relation can be found between three points,\r\nwhich shall enable us, from two, uniquely to determine the third.\r\nWith three given collinear points, however, we have more\r\ngiven than a mere straight line, and the quadrilateral construction\r\nenables us uniquely to determine any number of\r\nfresh collinear points. This shows why anharmonic ratio\r\nmust be a relation between four points, rather than between\r\nthree.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e121.\u003c/b\u003e\u003ca name=\"N121\" id=\"N121\"\u003e\u003c/a\u003e\r\nWe can now prove, I think, that two figures, which\r\nare projectively related, are qualitatively similar. Let us begin\r\nwith a collection of points on a straight line. So long as these\r\nare considered without reference to other points or figures, they\r\nare all qualitatively similar. They can be distinguished by\r\nimmediate intuition, but when we endeavour, without quantity,\r\nto distinguish them conceptually, we find the task impossible,\r\nsince the only qualitative relation of any two of them, the\r\nstraight line, is the same for any other two. But now let us\r\nchoose, at hap-hazard, some point outside the straight line.\r\nThe points of our line now acquire new adjectives, namely their\r\nrelations to the new point, \u003cem\u003ei.e.\u003c/em\u003e the straight lines joining them\r\nto this new point. But these straight lines, reciprocally, alone\r\ndefine our external point, and all straight lines are qualitatively\r\nsimilar. If we take some other external point, therefore,\r\nand join it to the same points of our original straight line, we\r\nobtain a figure in which, so long as quantity is excluded, there\r\nis no conceptual difference from the former figure. Immediate\r\nintuition can distinguish the two figures, but qualitative discrimination\r\ncannot do so. Thus we obtain a projective transformation\r\nof four lines into four other lines, as giving a figure\r\nqualitatively indistinguishable from the original figure. A\r\nsimilar argument applies to the other projective transformations.\r\nThus the only reason, within projective Geometry, for\r\nnot regarding projective figures as actually identical, is the\r\nintuitive perception of difference of position. This is fundamental,\r\nand must be accepted as a \u003cem\u003edatum\u003c/em\u003e. It is presupposed\r\nin the distinction of various points, and forms the very life of\r\nGeometry. It is, in fact, the essence of the notion of a form of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_132\" id=\"Page_132\"\u003e[132]\u003c/a\u003e\u003c/span\u003e\r\nexternality, which notion forms the subject-matter of projective\r\nGeometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e122.\u003c/b\u003e\u003ca name=\"N122\" id=\"N122\"\u003e\u003c/a\u003e\r\nWe may now sum up the results of our analysis of\r\nprojective Geometry, and state the axioms on which its reasoning\r\nis based. We shall then have to prove that these axioms\r\nare necessary to any form of externality, with which we shall\r\npass, from mere analysis, to a transcendental argument.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe axioms which have been assumed in the above analysis,\r\nand which, it would seem, suffice to found projective Geometry,\r\nmay be roughly stated as follows:\u003c/p\u003e\r\n\r\n\u003cp\u003eI. We can distinguish different parts of space, but all parts\r\nare qualitatively similar, and are distinguished only by the\r\nimmediate fact that they lie outside one another.\u003c/p\u003e\r\n\r\n\u003cp\u003eII. Space is continuous and infinitely divisible; the result\r\nof infinite division, the zero of extension, is called a \u003cem\u003epoint\u003c/em\u003e\u003ca name=\"FNanchor_129_129\" id=\"FNanchor_129_129\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_129_129\" class=\"fnanchor\"\u003e[129]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eIII. Any two points determine a unique figure, called a\r\nstraight line, any three in general determine a unique figure,\r\nthe plane. Any four determine a corresponding figure of three\r\ndimensions, and for aught that appears to the contrary, the\r\nsame may be true of any number of points. But this process\r\ncomes to an end, sooner or later, with some number of points\r\nwhich determine the whole of space. For if this were not the\r\ncase, no number of relations of a point to a collection of given\r\npoints could ever determine its relation to fresh points, and\r\nGeometry would become impossible\u003ca name=\"FNanchor_130_130\" id=\"FNanchor_130_130\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_130_130\" class=\"fnanchor\"\u003e[130]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis statement of the axioms is not intended to have any\r\nexclusive precision: other statements equally valid could easily\r\nbe made. For all these axioms, as we shall see hereafter, are\r\nphilosophically interdependent, and may, therefore, be enunciated\r\nin many ways. The above statement, however, includes,\r\nif I am not mistaken, everything essential to projective\r\nGeometry, and everything required to prove the principle of\r\nprojective transformation. Before discussing the apriority of\r\nthese axioms, let us once more briefly recapitulate the ends\r\nwhich they are intended to attain.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e123.\u003c/b\u003e\u003ca name=\"N123\" id=\"N123\"\u003e\u003c/a\u003e\r\nFrom the exclusively mathematical standpoint, as we\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_133\" id=\"Page_133\"\u003e[133]\u003c/a\u003e\u003c/span\u003ehave seen, projective Geometry discusses only what figures can\r\nbe obtained from each other by projective transformations, \u003cem\u003ei.e.\u003c/em\u003e\r\nby the operations of projection and section. These operations,\r\nin all their forms, presuppose the point, straight line, and\r\nplane\u003ca name=\"FNanchor_131_131\" id=\"FNanchor_131_131\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_131_131\" class=\"fnanchor\"\u003e[131]\u003c/a\u003e, whose necessity for projective Geometry, from the purely\r\nmathematical point of view, is thus self-evident from the start.\r\nBut philosophically, projective Geometry has, as we saw, a\r\nwider aim. This wider aim, which gives, to the investigation\r\nof projectively equivalent figures, its chief importance, consists\r\nin the determination of qualitative spatial similarity, in the\r\ndetermination, that is, of all the figures which, when any one\r\nfigure is given, can be distinguished from the given figure, so\r\nlong as quantity is excluded, only by the mere fact that they\r\nare external to it.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e124.\u003c/b\u003e\u003ca name=\"N124\" id=\"N124\"\u003e\u003c/a\u003e\r\nNow when we consider what is involved in such\r\nabsolute qualitative equivalence, we find at once, as its most\r\nobvious prerequisite, the perfect homogeneity of space. For it\r\nis assumed that a figure can be completely defined by its\r\ninternal relations, and that the external relations, which constitute\r\nits position, though they suffice to distinguish it from\r\nother figures, in no way affect its internal properties, which are\r\nregarded as qualitatively identical with those of figures with\r\nquite different external relations. If this were not the case,\r\nanything analogous to projective transformation would be impossible.\r\nFor such transformation always alters the position,\r\n\u003cem\u003ei.e.\u003c/em\u003e the external relations, of a figure, and could not, therefore,\r\nif figures were dependent on their relations to other figures or\r\nto empty space, be studied without reference to other figures,\r\nor to the absolute position of the original figure. We require\r\nfor our principle, in short, what may be called the mutual\r\npassivity and reciprocal independence of two parts or figures of\r\nspace.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis passivity and this independence involve the homogeneity\r\nof space, or its equivalent, the relativity of position.\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_134\" id=\"Page_134\"\u003e[134]\u003c/a\u003e\u003c/span\u003eFor if the internal properties of a figure are the same, whatever\r\nits external relations may be, it follows that all parts of\r\nspace are qualitatively similar, since a change of external\r\nrelation is a change in the part of space occupied. It follows,\r\nalso, that all position is relative and extrinsic, \u003cem\u003ei.e.\u003c/em\u003e, that the\r\nposition of a point, or the part of space occupied by a figure,\r\nis not, and has no effect upon, any intrinsic property of the\r\npoint or figure, but is exclusively a relation to other points\r\nor figures in space, and remains without effect except where\r\nsuch relations are considered.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e125.\u003c/b\u003e\u003ca name=\"N125\" id=\"N125\"\u003e\u003c/a\u003e\r\nThe homogeneity of space and the relativity of\r\nposition, therefore, are presupposed in the qualitative spatial\r\ncomparison with which projective Geometry deals. The latter,\r\nas we saw, is also the basis of the principle of duality. But\r\nthese properties, as I shall now endeavour to prove, belong of\r\nnecessity to any form of externality, and are thus \u003cem\u003eà priori\u003c/em\u003e\r\nproperties of all possible spaces. To prove this, however, we\r\nmust first define the notion of a form of externality in general.\u003c/p\u003e\r\n\r\n\u003cp\u003eLet us observe, to begin with, that the distinction between\r\nEuclidean and non-Euclidean Geometries, so important in metrical\r\ninvestigations, disappears in projective Geometry proper.\r\nThis suggests that projective Geometry, though originally\r\ninvented as the science of Euclidean space, and subsequently of\r\nnon-Euclidean spaces also, deals really with a wider conception,\r\na conception which includes both, and neglects the attributes\r\nin which they differ. This conception I shall speak of as a\r\nform of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e126.\u003c/b\u003e\u003ca name=\"N126\" id=\"N126\"\u003e\u003c/a\u003e\r\nIn Grassmann\u0027s profound philosophical introduction\r\nto his \u003ccite lang=\"de\" xml:lang=\"de\"\u003eAusdehnungslehre\u003c/cite\u003e of 1844, he suggested that Geometry,\r\nthough improperly regarded as pure, was really a branch of\r\napplied mathematics, since it dealt with a subject-matter not\r\ncreated, like number, by the intellect, but given to it, and therefore\r\nnot wholly subject to its laws alone. But it must be possible\u0026mdash;so\r\nhe contended\u0026mdash;to construct a branch of pure mathematics,\r\na science, that is, in which our object should be wholly a creature\r\nof the intellect, which should yet deal, as Geometry does, with\r\nextension\u0026mdash;extension as conceived, however, not as empirically\r\nperceived in sensation or intuition.\u003c/p\u003e\r\n\r\n\u003cp\u003eFrom this point of view, the controversy between Kantians\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_135\" id=\"Page_135\"\u003e[135]\u003c/a\u003e\u003c/span\u003e\r\nand anti-Kantians becomes wholly irrelevant, since the distinction\r\nbetween pure and mixed mathematics does not lie in the\r\ndistinction between the subjective and the objective, but\r\nbetween the purely intellectual on the one hand, and everything\r\nelse on the other. Now Kant had contended, with great emphasis,\r\nthat space was not an intellectual construction, but a\r\nsubjective intuition. Geometry, therefore, with Grassmann\u0027s\r\ndistinction, belongs to mixed mathematics as much on Kant\u0027s\r\nview as on that of his opponents. And Grassmann\u0027s distinction,\r\nI contend, is the more important for Epistemology, and the one\r\nto be adopted in distinguishing the \u003cem\u003eà priori\u003c/em\u003e from the empirical.\r\nFor what is merely intuitional can change, without upsetting\r\nthe laws of thought, without making knowledge formally impossible:\r\nbut what is purely intellectual cannot change, unless\r\nthe laws of thought should change, and all our knowledge\r\nsimultaneously collapse. I shall therefore follow Grassmann\u0027s\r\ndistinction in constructing an \u003cem\u003eà priori\u003c/em\u003e and purely conceptual\r\nform of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e127.\u003c/b\u003e\u003ca name=\"N127\" id=\"N127\"\u003e\u003c/a\u003e\r\nThe pure doctrine of extension, as constructed by\r\nGrassmann, need not be discussed\u0026mdash;it included much empirical\r\nmaterial, and was philosophically a failure. But his principles,\r\nI think, will enable us to prove that projective Geometry,\r\nabstractly interpreted, is the science which he foresaw, and\r\ndeals with a matter which can be constructed by the pure\r\nintellect alone. If this be so, however, it must be observed\r\nthat projective Geometry, for the moment, is rendered purely\r\nhypothetical\u003ca name=\"FNanchor_132_132\" id=\"FNanchor_132_132\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_132_132\" class=\"fnanchor\"\u003e[132]\u003c/a\u003e. All necessary truth, as Bradley has shown, is\r\nhypothetical\u003ca name=\"FNanchor_133_133\" id=\"FNanchor_133_133\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_133_133\" class=\"fnanchor\"\u003e[133]\u003c/a\u003e, and asserts, \u003ci lang=\"la\" xml:lang=\"la\"\u003eprimâ facie\u003c/i\u003e, only the ground on\r\nwhich rests the necessary connection of premisses and conclusion.\r\nIf we construct a mere conception of externality, and\r\nthus abandon our actually given space, the result of our construction,\r\nuntil we return to something actually given, remains\r\nwithout existential import\u0026mdash;if there \u003cem\u003ebe\u003c/em\u003e experienced externality,\r\nit asserts, then there must be a form of externality with such\r\nand such properties. That there must be experienced externality,\r\nKant\u0027s first argument about space proves, I think, to\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_136\" id=\"Page_136\"\u003e[136]\u003c/a\u003e\u003c/span\u003ethose who admit experience of a world of diverse but interrelated\r\nthings. But this is a question which belongs to the\r\nnext Chapter.\u003c/p\u003e\r\n\r\n\u003cp\u003eWhat we have to do here is, not to discuss whether there is\r\na form of externality, but whether, if there be such a form, it\r\nmust possess the properties embodied in the axioms of projective\r\nGeometry. Now first of all, what do we mean by such\r\na form?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e128.\u003c/b\u003e\u003ca name=\"N128\" id=\"N128\"\u003e\u003c/a\u003e\r\nIn any world in which perception presents us with\r\nvarious things, with discriminated and differentiated contents,\r\nthere must be, in perception, at least one \"principle of differentiation\u003ca name=\"FNanchor_134_134\" id=\"FNanchor_134_134\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_134_134\" class=\"fnanchor\"\u003e[134]\u003c/a\u003e,\"\r\nan element, that is, by which the things presented are\r\ndistinguished as various. This element, taken in isolation, and\r\nabstracted from the content which it differentiates, we may call\r\na form of externality. That it must, when taken in isolation,\r\nappear as a form, and not as a mere diversity of material\r\ncontent, is, I think, fairly obvious. For a diversity of material\r\ncontent cannot be studied apart from that material content;\r\nwhat we wish to study here, on the contrary, is the bare\r\npossibility of such diversity, which forms the residuum, as I\r\nshall try to prove hereafter\u003ca name=\"FNanchor_135_135\" id=\"FNanchor_135_135\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_135_135\" class=\"fnanchor\"\u003e[135]\u003c/a\u003e, when we abstract from any sense-perception\r\nall that is distinctive of its particular matter. This\r\npossibility, then, this principle of bare diversity, is our form of\r\nexternality. How far it is necessary to assume such a form, as\r\ndistinct from interrelated things, I shall consider later on\u003ca name=\"FNanchor_136_136\" id=\"FNanchor_136_136\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_136_136\" class=\"fnanchor\"\u003e[136]\u003c/a\u003e.\r\nFor the present, since space, as dealt with by Geometry, is\r\ncertainly a form of this kind, we have only to ask: What\r\nproperties must such a form, when studied in abstraction,\r\nnecessarily possess?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e129.\u003c/b\u003e\u003ca name=\"N129\" id=\"N129\"\u003e\u003c/a\u003e\r\nIn the first place, externality is an essentially relative\r\nconception\u0026mdash;nothing can be external to itself. To be external\r\nto something is to be another with some relation to that thing.\r\nHence, when we abstract a form of externality from all material\r\ncontent, and study it in isolation, position will appear, of\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_137\" id=\"Page_137\"\u003e[137]\u003c/a\u003e\u003c/span\u003enecessity, as purely relative\u0026mdash;a position can have no intrinsic\r\nquality, for our form consists of pure externality, and externality\r\ncontains no shadow or trace of an intrinsic quality. Thus we\r\nobtain our fundamental postulate, the relativity of position, or,\r\nas we may put it, the complete absence, on the part of our\r\nform, of any vestige of thinghood.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe same argument may also be stated as follows: If we\r\nabstract the conception of externality, and endeavour to deal\r\nwith it \u003cem\u003eper se\u003c/em\u003e, it is evident that we must obtain an object alike\r\ndestitute of elements and of totality. For we have abstracted\r\nfrom the diverse matter which filled our form, while any\r\nelement, or any whole, would retain some of the qualities of a\r\nmatter. Either an element or a whole, in fact, would have to\r\nbe a thing not external to itself, and would thus contain something\r\nnot pure externality. Hence arise infinite divisibility,\r\nwith the self-contradictory notion of the point, in the search\r\nfor elements, and unbounded extension, with the contradiction\r\nof an infinite regress or a vicious circle, in the search for a\r\ncompleted whole. Thus again, our form contains neither\r\nelements nor totality, but only endless relations\u0026mdash;the terms of\r\nthese relations being excluded by our abstraction from the\r\nmatter which fills our form.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e130.\u003c/b\u003e\u003ca name=\"N130\" id=\"N130\"\u003e\u003c/a\u003e\r\nIn like manner we can deduce the homogeneity of\r\nour form. The diversity of content, which was possible only\r\nwithin the form of externality, has been abstracted from,\r\nleaving nothing but the bare possibility of diversity, the bare\r\nprinciple of differentiation, itself uniform and undifferentiated.\r\nFor if diversity presupposes such a form, the form cannot,\r\nunless it were contained in a fresh form, be itself diverse or\r\ndifferentiated.\u003c/p\u003e\r\n\r\n\u003cp\u003eOr we may deduce the same property from the relativity of\r\nposition. For any quality in one position, by which it was\r\nmarked out from another, would be necessarily more or less\r\nintrinsic, and would contradict the pure relativity. Hence all\r\npositions are qualitatively alike, \u003cem\u003ei.e.\u003c/em\u003e the form is homogeneous\r\nthroughout.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e131.\u003c/b\u003e\u003ca name=\"N131\" id=\"N131\"\u003e\u003c/a\u003e\r\nFrom what has been said of homogeneity and relativity,\r\nfollows one of the strangest properties of a form of\r\nexternality. This property is, that the relation of externality\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_138\" id=\"Page_138\"\u003e[138]\u003c/a\u003e\u003c/span\u003e\r\nbetween any two things is infinitely divisible, and may be\r\nregarded, consequently, as made up of an infinite number of the\r\nwould-be elements of our form, or again as the sum of two\r\nrelations of externality\u003ca name=\"FNanchor_137_137\" id=\"FNanchor_137_137\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_137_137\" class=\"fnanchor\"\u003e[137]\u003c/a\u003e. To speak of dividing or adding relations\r\nmay well sound absurd\u0026mdash;indeed it reveals the impropriety\r\nof the word relation in this connexion. It is difficult,\r\nhowever, to find an expression which shall be less improper.\r\nThe fact seems to be, that externality is not so much a relation\r\nas bare relativity, or the bare possibility of a relation. On this\r\nsubject, I shall enlarge in Chapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003ca name=\"FNanchor_138_138\" id=\"FNanchor_138_138\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_138_138\" class=\"fnanchor\"\u003e[138]\u003c/a\u003e At this point it is\r\nonly important to realize, what the subsequent argument will\r\nassume, that the relation\u0026mdash;if we may so call it\u0026mdash;of externality\r\nbetween two or more things must, since our form is homogeneous,\r\nbe capable of continuous alteration, and must, since\r\nour infinitely divisible form is constituted by such relations, be\r\ncapable of infinite division. But the result of infinite division\r\nis defined as the element of our form. (Our form has no\r\nelements, but we have to imagine elements in order to reason\r\nabout it, as will be shown more fully in Chapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e) Hence\r\nit follows, that every relation of externality may be regarded,\r\nfor scientific purposes, as an infinite congeries of elements,\r\nthough philosophically, the relations alone are valid, and the\r\nelements are a self-contradictory result of hypostatizing the\r\nform of externality. This way of regarding relations of externality\r\nis important in understanding the meaning of such\r\nideas as three or four collinear points.\u003c/p\u003e\r\n\r\n\u003cp\u003eAs this point is difficult and important, I will repeat, in\r\nsomewhat greater detail, the explanation of the manner in\r\nwhich straight lines and planes come to be regarded as congeries\r\nof points. From the strictly projective standpoint, though all\r\nother figures \u003cem\u003eare\u003c/em\u003e merely a collection of any required number of\r\npoints, lines or planes, given by some projective construction,\r\nstraight lines and planes themselves are given integrally, and\r\nare not to be considered as divisible or composed of parts. To\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_139\" id=\"Page_139\"\u003e[139]\u003c/a\u003e\u003c/span\u003esay that a point lies on a straight line means, for projective\r\nGeometry proper, that the straight line is a relation between\r\nthis and some other point. Here the points concerned, if our\r\nstatement is to be freed from contradictions, must be regarded,\r\nif I may use such an expression, as \u003cem\u003ereal\u003c/em\u003e points\u0026mdash;\u003cem\u003ei.e.\u003c/em\u003e as unextended\r\nmaterial centres\u003ca name=\"FNanchor_139_139\" id=\"FNanchor_139_139\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_139_139\" class=\"fnanchor\"\u003e[139]\u003c/a\u003e. Straight lines and planes are then\r\nrelations between these material atoms. They are relations,\r\nhowever, which may undergo a metrical alteration while remaining\r\nprojectively unchanged. When the projective relation\r\nbetween the two points \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e is the same as that between the\r\ntwo points \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e, while the metrical relation (distance) is\r\ndifferent, the three points \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e are said to be collinear.\r\nNow the metrical manner of regarding spatial figures demands\r\nthat they should be hypostatized, and no longer regarded as\r\nmere relations. For when we regard a quantity as extensive,\r\n\u003cem\u003ei.e.\u003c/em\u003e as divisible into parts, we necessarily regard it as more than\r\na mere relation or adjective, since no mere relation or adjective\r\ncan be divided. For quantitative treatment, therefore, spatial\r\nrelations must be hypostatized\u003ca name=\"FNanchor_140_140\" id=\"FNanchor_140_140\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_140_140\" class=\"fnanchor\"\u003e[140]\u003c/a\u003e. When this is done, we obtain,\r\nas we saw above, a homogeneous and infinitely divisible form of\r\nexternality. We find now that distance, for example, may be\r\ncontinuously altered without changing the straight line on\r\nwhich it is measured. We thus obtain, on the straight line in\r\nquestion, a continuous series of points, which, since it is\r\ncontinuous, we regard as constituting our straight line. It is\r\nthus solely from the hypostatizing of relations, which metrical\r\nGeometry requires, that the view of straight lines and planes\r\nas \u003cem\u003ecomposed\u003c/em\u003e of points arises, and it is from this hypostatizing\r\nthat the difficulties of metrical Geometry spring.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e132.\u003c/b\u003e\u003ca name=\"N132\" id=\"N132\"\u003e\u003c/a\u003e\r\nThe next step, in defining a form of externality, is\r\nobtained from the idea of \u003cem\u003edimensions\u003c/em\u003e. Positions, we have seen,\r\nare defined solely by their relations to other positions. But in\r\norder that such definition may be possible, a finite number of\r\nrelations must suffice, since infinite numbers are philosophically\r\ninadmissible. A position must be definable, therefore, if knowledge\r\nof our form is to be possible at all, by some finite integral\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_140\" id=\"Page_140\"\u003e[140]\u003c/a\u003e\u003c/span\u003enumber of relations to other positions. Every relation thus\r\nnecessary for definition we call a dimension. Hence we obtain\r\nthe proposition: \u003cem\u003eAny form of externality must have a finite\r\nintegral number of dimensions\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e133.\u003c/b\u003e\u003ca name=\"N133\" id=\"N133\"\u003e\u003c/a\u003e\r\nThe above argument, it may be urged, has overlooked\r\na possibility. It has used a transcendental argument, so an\r\nopponent may contend, without sufficiently proving that knowledge\r\nabout externality must be possible without reference\r\nto the matters external to each other. The definition of a\r\nposition may be impossible, so long as we neglect the matter\r\nwhich fills the form, but may become possible when this matter\r\nis taken into account. Such an objection can, I think, be\r\nsuccessfully met, by a reference to the passivity and homogeneity\r\nof our form. For any dependence of the definition of\r\na position on the particular matter filling that position, would\r\ninvolve some kind of interaction between the matter and its\r\nposition, some effect of the diverse content on the homogeneous\r\nform. But since the form is totally destitute of thinghood,\r\nperfectly impassive, and perfectly void of differences between its\r\nparts, any such effect is inconceivable. An effect on a position\r\nwould have to alter it in some way, but how could it be altered?\r\nIt has no qualities except those which make it the position it\r\nis, as opposed to other positions; it cannot change, therefore,\r\nwithout becoming a different position. But such a change\r\ncontradicts the law of identity. Hence it is not the position\r\nwhich has changed, but the content which has moved in the\r\nform. Thus it must be possible, if knowledge of our form can\r\nbe obtained at all, to obtain this knowledge in logical independence\r\nof the particular matter which fills it. The above\r\nargument, therefore, granted the possibility of knowledge in\r\nthe department in question, shows the necessity of a finite\r\nintegral number of dimensions.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e134.\u003c/b\u003e\u003ca name=\"N134\" id=\"N134\"\u003e\u003c/a\u003e\r\nLet us repeat our original argument in the light of\r\nthis elucidation. A position is completely defined when, and\r\nonly when, enough relations are known to enable us to determine\r\nits relation to any fresh known position. Only by relations\r\nwithin the form of externality, as we have just seen, and never\r\nby relations which involve a reference to the particular matter\r\nfilling the form, can such a definition be effected. But the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_141\" id=\"Page_141\"\u003e[141]\u003c/a\u003e\u003c/span\u003e\r\npossibility of such a definition follows from the Law of Excluded\r\nMiddle, when this law is interpreted to mean, as Bosanquet\r\nmakes it mean, that \"Reality … is a system of reciprocally\r\ndeterminate parts\u003ca name=\"FNanchor_141_141\" id=\"FNanchor_141_141\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_141_141\" class=\"fnanchor\"\u003e[141]\u003c/a\u003e.\" For this implies that, given the relations\r\nof a part \u003cem\u003eA\u003c/em\u003e to other parts \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e …, a sufficient wealth of such\r\nrelations throws light on the relations of \u003cem\u003eB\u003c/em\u003e to \u003cem\u003eC\u003c/em\u003e, etc. If this\r\nwere not the case, the parts \u003cem\u003eA\u003c/em\u003e, \u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e … could not be said to\r\nform such a system; for in such a system, to define \u003cem\u003eA\u003c/em\u003e is to\r\ndefine, at the same time, all the other members, and to give\r\nan adjective to \u003cem\u003eA\u003c/em\u003e, is to give an adjective to \u003cem\u003eB\u003c/em\u003e and \u003cem\u003eC\u003c/em\u003e. But the\r\nrelations between positions are, when we restore the matter\r\nfrom which the positions were abstracted, relations between\r\nthe things occupying those positions, and these relations, we\r\nhave seen, can be studied without reference to the particular\r\nnature, in other respects, of the related things. It follows that,\r\nwhen we apply the general principle of systematic unity to\r\nthese relations in particular, we find these relations to be\r\ndependent on each other, since they are not dependent, for\r\ntheir definition, on anything else. This gives the axiom of\r\ndimensions, in the above general form, as the result, on our\r\nabstract geometrical level, of the relativity of position and the\r\nlaw of excluded middle.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e135.\u003c/b\u003e\u003ca name=\"N135\" id=\"N135\"\u003e\u003c/a\u003e\r\nBefore proceeding further, it is necessary to discuss\r\nthe important special case where a form of externality has only\r\none dimension. Of the two such forms, given in experience,\r\none, namely time, presents an instance of this special case.\r\nBut it may be shown, I think, that the function, in constituting\r\nthe possibility of experience, which we demand of such forms,\r\ncould not be accomplished by a one-dimensional form alone.\r\nFor in a one-dimensional form, the various contents may be\r\narranged in a series, and cannot, without interpenetration,\r\nchange the order of contents in the series. But interpenetration\r\nis impossible, since a form of externality is the mere\r\nexpression of diversity among things, from which it follows\r\nthat things cannot occupy the same position in a form, unless\r\nthere is another form by which to differentiate them. For\r\nwithout externality, there is no diversity\u003ca name=\"FNanchor_142_142\" id=\"FNanchor_142_142\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_142_142\" class=\"fnanchor\"\u003e[142]\u003c/a\u003e. Thus two bodies\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_142\" id=\"Page_142\"\u003e[142]\u003c/a\u003e\u003c/span\u003emay occupy the same space, but only at different times: two\r\nthings may exist simultaneously, but only at different places.\r\nA form of one dimension, therefore, could not, by itself, allow\r\nthat change of the relations of externality, by which alone\r\na varied world of interrelated things can be brought into\r\nconsciousness. In a one-dimensional space, for example, only a\r\nsingle object, which must appear as a point, or two objects at\r\nmost, one in front and one behind, could ever be perceived.\r\nThus two or more dimensions seem an essential condition of\r\nanything worth calling an experience of interrelated things.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e136.\u003c/b\u003e\u003ca name=\"N136\" id=\"N136\"\u003e\u003c/a\u003e\r\nIt may be objected, to this argument, that its\r\nvalidity depends upon the assumption that the change of a\r\nrelation of externality must be continuous. Both to make and\r\nto meet this objection, in a manner which shall not imply time,\r\nseems almost impossible. For we cannot speak of change,\r\nwhether continuous or discrete, without imagining time. Let\r\nus, therefore, allow time to be known, and discuss whether the\r\ntemporal change, in any other form of externality, is necessarily\r\ncontinuous\u003ca name=\"FNanchor_143_143\" id=\"FNanchor_143_143\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_143_143\" class=\"fnanchor\"\u003e[143]\u003c/a\u003e. We must reply, I think, that continuity is\r\nnecessary. The change of relation, in our non-temporal form,\r\nmay be safely described as motion, and the law of Causality\u0026mdash;since\r\nwe have already assumed time\u0026mdash;may be applied to this\r\nmotion. It then follows that discrete motion would involve a\r\nfinite effect from an infinitesimal cause, for a cause acting only\r\nfor a moment of time would be infinitesimal. It involves, also,\r\na validity in the point of time, whereas what is valid in any\r\nform of externality is not, as we have already seen, the\r\ninfinitesimal and self-contradictory element resulting from\r\ninfinite division, but the finite relation which mathematics\r\nanalyzes into vanishing elements. Hence change must be\r\ncontinuous, and the possibility of serial arrangement holds\r\ngood.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn a one-dimensional form other than time, the same\r\nargument must hold. For something analogous to Causality\r\nwould be necessary to experience, and the relativity of the\r\nform would still necessarily hold. Hence, since only these two\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_143\" id=\"Page_143\"\u003e[143]\u003c/a\u003e\u003c/span\u003e\r\nproperties of time have been assumed, the above contention\r\nwould remain valid of any second form whose relations were\r\ncorrelated with those of the first, as the analogue of Causality\r\nwould require them to be.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e137.\u003c/b\u003e\u003ca name=\"N137\" id=\"N137\"\u003e\u003c/a\u003e\r\nThe next step in the argument, which assumes two\r\nor more dimensions, is concerned with the general analogues of\r\nstraight lines and planes, \u003cem\u003ei.e.\u003c/em\u003e with figures\u0026mdash;which may be\r\nregarded either as relations between positions or as series of\r\npositions\u0026mdash;uniquely determined by two or by three positions.\r\nIf this step can be successfully taken, our deduction of the\r\nabove projective axioms will be complete, and descriptive\r\nGeometry will be established as the abstract \u003cem\u003eà priori\u003c/em\u003e doctrine\r\nof forms of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003eTo prove this contention, consider of what nature the\r\nrelations can be by which positions are defined. We have seen\r\nalready that our form is purely relational and infinitely\r\ndivisible, and that positions (points) are the self-contradictory\r\noutcome of the search for something other than relations.\r\nWhat we really mean, therefore, by the relations defining a\r\nposition, is, when we undo our previous abstraction, the\r\nrelations of externality by which some thing is related to other\r\nthings. But how, when we remain in the abstract form, must\r\nsuch relations appear?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e138.\u003c/b\u003e\u003ca name=\"N138\" id=\"N138\"\u003e\u003c/a\u003e\r\nWe have to prove that two positions must have a\r\nrelation independent of any reference to other positions. To\r\nprove this, let us recur to what was said, in connection with\r\ndimensions, as to the passivity and homogeneity of our form.\r\nSince positions are defined only by relations, there must be\r\nrelations, within the form, between positions. But if there are\r\nsuch relations, there must be a relation which is intrinsic to\r\ntwo positions. For to suppose the contrary, is to attribute an\r\ninteraction or causal connection, of some kind, between those\r\ntwo positions and other positions\u0026mdash;a supposition which the\r\nperfect homogeneity of our form renders absurd, since all\r\npositions are qualitatively similar, and cannot be changed\r\nwithout losing their identity. We may put this argument\r\nthus: since positions are only defined by their relations, such\r\ndefinition could never begin, unless it began with a relation\r\nbetween only two positions. For suppose three positions \u003cem\u003eA\u003c/em\u003e,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_144\" id=\"Page_144\"\u003e[144]\u003c/a\u003e\u003c/span\u003e\r\n\u003cem\u003eB\u003c/em\u003e, \u003cem\u003eC\u003c/em\u003e were necessary, and gave rise to the relation \u003cem\u003eabc\u003c/em\u003e between\r\nthe three. Then there would remain no means of defining the\r\ndifferent pairs \u003cem\u003eBC\u003c/em\u003e, \u003cem\u003eCA\u003c/em\u003e, \u003cem\u003eAB\u003c/em\u003e, since the only relation defining\r\nthem would be one common to all three pairs. Nothing would\r\nbe gained, in this case, by reference to fresh points, for it\r\nfollows, from the homogeneity and passivity of the form, that\r\nthese fresh points could not affect the internal relations of our\r\ntriad, which relations, if they can give definiteness at all, must\r\ngive it without the aid of external reference. Two positions\r\nmust, therefore, if definition is to be possible, have some\r\nrelation which they by themselves suffice to define. Precisely\r\nthe same argument applies to three positions, or to four; the\r\nargument loses its scope only when we have exhausted the\r\ndimensions of the form considered. Thus, in three dimensions,\r\nfive positions have no fresh relation, not deducible from those\r\nalready known, for by the definition of dimensions, all the\r\nrelations involved can be deduced from those of the fourth point\r\nto the first three, together with those of the fifth to the first three.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe may give the argument a more concrete, and perhaps a\r\nmore convincing shape, by considering the matter arranged in\r\nour form. If two things are mutually external, they must\r\nsince they belong to the same world, have some relation of\r\nexternality; there is, therefore, a relation of externality between\r\ntwo things. But since our form is homogeneous, the same\r\nrelation of externality may subsist in other parts of the form,\r\n\u003cem\u003ei.e.\u003c/em\u003e while the two things considered alter their relations of externality\r\nto other things. The relation of externality between\r\ntwo things is, therefore, independent of other things. Hence,\r\nwhen we return to the abstract language of the form, two\r\npositions have a relation determined by those two positions\r\nalone, and independent of other positions.\u003c/p\u003e\r\n\r\n\u003cp\u003ePrecisely the same argument applies to the relations of\r\nthree positions, and in each case the relation must appear in\r\nthe form as not a mere inference from the positions it relates.\r\nFor relations, as we have seen, actually constitute a form of\r\nexternality, and are not mere inferences from terms, which are\r\nnowhere to be found in the form\u003ca name=\"FNanchor_144_144\" id=\"FNanchor_144_144\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_144_144\" class=\"fnanchor\"\u003e[144]\u003c/a\u003e.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_145\" id=\"Page_145\"\u003e[145]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp\u003eTo sum up: Since position is relative, two positions must\r\nhave \u003cem\u003esome\u003c/em\u003e relation to each other; and since our form of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_146\" id=\"Page_146\"\u003e[146]\u003c/a\u003e\u003c/span\u003e\r\nexternality is homogeneous, this relation can be kept unchanged\r\nwhile the two positions change their relations to other\r\npositions. Hence their relation is intrinsic, and independent of\r\nother positions. Since the form is a mere complex of relations,\r\nthe relation in question must, if the form is sensuous or\r\nintuitive, be itself sensuous or intuitive, and not a mere\r\ninference. In this case, a unique relation must be a unique\r\nfigure\u0026mdash;in spatial terms, the straight line joining the two\r\npoints.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e139.\u003c/b\u003e\u003ca name=\"N139\" id=\"N139\"\u003e\u003c/a\u003e\r\nWith this, our deduction of projective Geometry\r\nfrom the \u003cem\u003eà priori\u003c/em\u003e conceptual properties of a form of externality\r\nis completed. That such a form, when regarded as an independent\r\nthing, is self-contradictory, has been abundantly\r\nevident throughout the discussion. But the science of the\r\nform has been founded on the opposite way of regarding it: we\r\nhave held it throughout to be a mere complex of relations, and\r\nhave deduced its properties exclusively from this view of it.\r\nThe many difficulties, in applying such an \u003cem\u003eà priori\u003c/em\u003e deduction\r\nto intuitive space, and in explaining, as logical necessities,\r\nproperties which appear as sensuous or intuitional data, must\r\nbe postponed to Chapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e For the present, I wish to point\r\nout that projective Geometry is wholly \u003cem\u003eà priori\u003c/em\u003e; that it deals\r\nwith an object whose properties are logically deduced from its\r\ndefinition, not empirically discovered from data; that its\r\ndefinition, again, is founded on the possibility of experiencing\r\ndiversity in relation, or multiplicity in unity; and that our\r\nwhole science, therefore, is logically implied in, and deducible\r\nfrom, the possibility of such experience.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e140.\u003c/b\u003e\u003ca name=\"N140\" id=\"N140\"\u003e\u003c/a\u003e\r\nIn metrical Geometry, on the contrary, we shall find\r\na very different result. Although the geometrical conditions\r\nwhich render spatial measurement possible, will be found\r\nidentical, except for slight differences in the form of statement,\r\nwith the \u003cem\u003eà priori\u003c/em\u003e axioms discussed above, yet the actual\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_147\" id=\"Page_147\"\u003e[147]\u003c/a\u003e\u003c/span\u003e\r\nmeasurement\u0026mdash;which deals with actually given space, not the\r\nmere intellectual construction we have been just discussing\u0026mdash;gives\r\nresults which can only be known empirically and\r\napproximately, and can be deduced by no necessity of thought.\r\nThe Euclidean and non-Euclidean spaces give the various\r\nresults which are \u003cem\u003eà priori\u003c/em\u003e possible; the axioms peculiar to\r\nEuclid\u0026mdash;which are properly not axioms, but empirical results\r\nof measurement\u0026mdash;determine, within the errors of observation,\r\nwhich of these \u003cem\u003eà priori\u003c/em\u003e possibilities is realized in our actual\r\nspace. Thus measurement deals throughout with an empirically\r\ngiven matter, not with a creature of the intellect, and\r\nits \u003cem\u003eà priori\u003c/em\u003e elements are only the conditions presupposed in the\r\npossibility of measurement. What these conditions are, we\r\nshall see in the second section of this chapter.\u003c/p\u003e\r\n\r\n\r\n \u003cdiv class=\"chapter\"\u003e\u003c/div\u003e\r\n\u003ch3\u003e\u003ca name=\"SB\" id=\"SB\"\u003e\u003c/a\u003eSection B.\u003cbr /\u003e\u003cbr /\u003e\r\n\r\n\u003cspan class=\"fwnormal fs80\"\u003eTHE AXIOMS OF METRICAL GEOMETRY.\u003c/span\u003e\u003c/h3\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e141.\u003c/b\u003e\u003ca name=\"N141\" id=\"N141\"\u003e\u003c/a\u003e\r\nWe have now reviewed the axioms of projective\r\nGeometry, and have seen that they are \u003cem\u003eà priori\u003c/em\u003e deductions\r\nfrom the fact that we can experience externality, \u003cem\u003ei.e.\u003c/em\u003e a coexistent\r\nmultiplicity of different but interrelated things. But\r\nprojective Geometry, in spite of its claims, is not the whole\r\nscience of space, as is sufficiently proved by the fact that it\r\ncannot discriminate between Euclidean and non-Euclidean\r\nspaces\u003ca name=\"FNanchor_145_145\" id=\"FNanchor_145_145\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_145_145\" class=\"fnanchor\"\u003e[145]\u003c/a\u003e. For this purpose, spatial measurement is required:\r\nmetrical Geometry, with its quantitative tests, can alone effect\r\nthe discrimination. For all application of Geometry to physics,\r\nalso, measurement is required; the law of gravitation, for\r\nexample, requires the determination of actual distances. For\r\nmany purposes, in short, projective Geometry is wholly insufficient:\r\nthus it is unable to distinguish between different\r\nkinds of conics, though their distinction is of fundamental importance\r\nin many departments of knowledge.\u003c/p\u003e\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_148\" id=\"Page_148\"\u003e[148]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp\u003eMetrical Geometry is, then, a necessary part of the science\r\nof space, and a part not included in descriptive Geometry.\r\nIts \u003cem\u003eà priori\u003c/em\u003e element, nevertheless, so far as this is spatial\r\nand not arithmetical, is the same as the postulate of projective\r\nGeometry, namely, the homogeneity of space, or its\r\nequivalent, the relativity of position. We can see, in fact, that\r\nthe \u003cem\u003eà priori\u003c/em\u003e element in both is likely to be the same. For\r\nthe \u003cem\u003eà priori\u003c/em\u003e in metrical Geometry will be whatever is presupposed\r\nin the possibility of spatial measurement, \u003cem\u003ei.e.\u003c/em\u003e of\r\nquantitative spatial comparison. But such comparison presupposes\r\nsimply a known identity of quality, the determination\r\nof which is precisely the problem of projective Geometry.\r\nHence the conditions for the possibility of measurement, in\r\nso far as they are not arithmetical, will be precisely the same\r\nas those for projective Geometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e142.\u003c/b\u003e\u003ca name=\"N142\" id=\"N142\"\u003e\u003c/a\u003e\r\nMetrical Geometry, therefore, though distinct from\r\nprojective Geometry, is not independent of it, but presupposes\r\nit, and arises from its combination with the extraneous idea\r\nof \u003cem\u003equantity\u003c/em\u003e. Nevertheless the mathematical form of the axioms,\r\nin metrical Geometry, is slightly different from their form in\r\nprojective Geometry. The homogeneity of space is replaced\r\nby its equivalent, the axiom of Free Mobility. The axiom of\r\nthe straight line is replaced by the axiom of distance: Two\r\npoints determine a unique quantity, distance, which is unaltered\r\nin any motion of the two points as a single figure. This axiom,\r\nindeed, will be found to involve the axiom of the straight line\u0026mdash;such\r\na quantity could not exist unless the two points determined\r\na unique curve\u0026mdash;but its mathematical form is changed.\r\nAnother important change is the collapse of the principle of\r\nduality: quantity can be applied to the straight line, because\r\nit is divisible into similar parts, but cannot be applied to the\r\nindivisible point. We thus obtain a reason, which was wanting\r\nin descriptive Geometry, for preferring points, as spatial elements,\r\nto straight lines or planes\u003ca name=\"FNanchor_146_146\" id=\"FNanchor_146_146\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_146_146\" class=\"fnanchor\"\u003e[146]\u003c/a\u003e. Finally, an entirely new\r\nidea is introduced with quantity, namely, the idea of \u003cem\u003eMotion\u003c/em\u003e.\r\nNot that we study motion, or that any of our results have\r\nreference to motion, but that they cannot, though in projective\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_149\" id=\"Page_149\"\u003e[149]\u003c/a\u003e\u003c/span\u003e\r\nGeometry they could, be obtained without at least an ideal\r\nmotion of our figures through space.\u003c/p\u003e\r\n\r\n\u003cp\u003eLet us now examine in detail the prerequisites of spatial\r\nmeasurement. We shall find three axioms, without which such\r\nmeasurement would be impossible, but with which it is adequate\r\nto decide, empirically and approximately, the Euclidean\r\nor non-Euclidean nature of our actual space. We shall find,\r\nfurther, that these three axioms can be deduced from the conception\r\nof a form of externality, and owe nothing to the\r\nevidence of intuition. They are, therefore, like their equivalents\r\nthe axioms of projective Geometry, \u003cem\u003eà priori\u003c/em\u003e, and deducible from\r\nthe conditions of spatial experience. This experience, accordingly,\r\ncan never disprove them, since its very existence\r\npresupposes them.\u003c/p\u003e\r\n\r\n\r\n\u003ch4\u003eI. \u003cem\u003eThe Axiom of Free Mobility.\u003c/em\u003e\u003c/h4\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e143.\u003c/b\u003e\u003ca name=\"N143\" id=\"N143\"\u003e\u003c/a\u003e\r\nMetrical Geometry, to begin with, may be defined as\r\nthe science which deals with the comparison and relations of\r\nspatial magnitudes. The conception of magnitude, therefore, is\r\nnecessary from the start. Some of Euclid\u0027s axioms, accordingly,\r\nhave been classed as arithmetical, and have been supposed to\r\nhave nothing particular to do with space. Such are the axioms\r\nthat equals added to or subtracted from equals give equals, and\r\nthat things which are equal to the same thing are equal to one\r\nanother. These axioms, it is said, are purely arithmetical, and\r\ndo not, like the others, ascribe an adjective to space. As regards\r\ntheir use in arithmetic, this is of course true. But if an arithmetical\r\naxiom is to be applied to spatial magnitudes, it must have\r\nsome spatial import\u003ca name=\"FNanchor_147_147\" id=\"FNanchor_147_147\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_147_147\" class=\"fnanchor\"\u003e[147]\u003c/a\u003e, and thus even this class is not, in Geometry,\r\n\u003cem\u003emerely\u003c/em\u003e arithmetical. Fortunately, the geometrical element is\r\nthe same in all the axioms of this class\u0026mdash;we can see at once, in\r\nfact, that it can amount to no more than a definition of spatial\r\nmagnitude\u003ca name=\"FNanchor_148_148\" id=\"FNanchor_148_148\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_148_148\" class=\"fnanchor\"\u003e[148]\u003c/a\u003e. Again, since the space with which Geometry\r\ndeals is infinitely divisible, a definition of spatial magnitude\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_150\" id=\"Page_150\"\u003e[150]\u003c/a\u003e\u003c/span\u003ereduces itself to a definition of spatial equality, for, as soon as\r\nwe have this last, we can compare two spatial magnitudes by\r\ndividing each into a number of equal units, and counting the\r\nnumber of such units in each\u003ca name=\"FNanchor_149_149\" id=\"FNanchor_149_149\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_149_149\" class=\"fnanchor\"\u003e[149]\u003c/a\u003e. The ratio of the number of\r\nunits is, of course, the ratio of the two magnitudes.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e144.\u003c/b\u003e\u003ca name=\"N144\" id=\"N144\"\u003e\u003c/a\u003e\r\nWe require, then, at the very outset, some criterion\r\nof spatial equality: without such a criterion metrical Geometry\r\nwould become wholly impossible. It might appear, at first\r\nsight, as though this need not be an axiom, but might be a\r\nmere definition. In part this is true, but not wholly. The part\r\nwhich is merely a definition is given in Euclid\u0027s eighth axiom:\r\n\"Magnitudes which exactly coincide are equal.\" But this gives\r\na sufficient criterion only when the magnitudes to be compared\r\nalready occupy the same position. When, as will normally be\r\nthe case, the two spatial magnitudes are external to one another\u0026mdash;as,\r\nindeed, must be the case, if they are distinct, and not\r\nwhole and part\u0026mdash;the two magnitudes can only be made to\r\ncoincide by a motion of one or both of them. In order, therefore,\r\nthat our definition of spatial magnitude may give unambiguous\r\nresults, coincidence when superposed, if it can ever\r\noccur, must occur always, whatever path be pursued in bringing\r\nit about. Hence, if mere motion could alter shapes, our criterion\r\nof equality would break down. It follows that the\r\napplication of the conception of magnitude to figures in space\r\ninvolves the following axiom\u003ca name=\"FNanchor_150_150\" id=\"FNanchor_150_150\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_150_150\" class=\"fnanchor\"\u003e[150]\u003c/a\u003e: \u003cem\u003eSpatial magnitudes can be moved\r\nfrom place to place without distortion\u003c/em\u003e; or, as it may be put,\r\n\u003cem\u003eShapes do not in any way depend upon absolute position in\r\nspace\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe above axiom is the axiom of Free Mobility\u003ca name=\"FNanchor_151_151\" id=\"FNanchor_151_151\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_151_151\" class=\"fnanchor\"\u003e[151]\u003c/a\u003e. I propose\r\nto prove (1) that the denial of this axiom would involve logical\r\nand philosophical absurdities, so that it must be classed as\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_151\" id=\"Page_151\"\u003e[151]\u003c/a\u003e\u003c/span\u003ewholly \u003cem\u003eà priori\u003c/em\u003e; (2) that metrical Geometry, if it refused this\r\naxiom, would be unable, without a logical absurdity, to establish\r\nthe notion of spatial magnitude at all. The conclusion will be,\r\nthat the axiom cannot be proved or disproved by experience,\r\nbut is an \u003cem\u003eà priori\u003c/em\u003e condition of metrical Geometry. As I shall\r\nthus be maintaining a position which has been much controverted,\r\nespecially by Helmholtz and Erdmann, I shall have to\r\nenter into the arguments at some length.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e145.\u003c/b\u003e\u003ca name=\"N145\" id=\"N145\"\u003e\u003c/a\u003e\r\nA. \u003cem\u003ePhilosophical Argument.\u003c/em\u003e The denial of the axiom\r\ninvolves absolute position, and an action of mere space, \u003cem\u003eper se\u003c/em\u003e,\r\non things. For the axiom does not assert that real bodies, as a\r\nmatter of empirical fact, never change their shape in any way\r\nduring their passage from place to place: on the contrary, we\r\nknow that such changes do occur, sometimes in a very noticeable\r\ndegree, and always to some extent. But such changes are\r\nattributed, not to the change of place as such, but to physical\r\ncauses: changes of temperature, pressure, etc. What our\r\naxiom has to deal with is not actual material bodies, but\r\ngeometrical figures\u003ca name=\"FNanchor_152_152\" id=\"FNanchor_152_152\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_152_152\" class=\"fnanchor\"\u003e[152]\u003c/a\u003e, and it asserts that a figure which is possible\r\nin any one position in space is possible in every other. Its\r\nmeaning will become clearer by reference to a case where it\r\ndoes not hold, say the space formed by the surface of an egg.\r\nHere, a triangle drawn near the equator cannot be moved\r\nwithout distortion to the point, as it would no longer fit the\r\ngreater curvature of the new position: a triangle drawn near\r\nthe point cannot be fitted on to the flatter end, and so on.\r\nThus the method of superposition, such as Euclid employs in\r\nBook \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Prop. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e, becomes impossible; figures cannot be freely\r\nmoved about, indeed, given any figure, we can determine a\r\ncertain series of possible positions for it on the egg, outside\r\nwhich it becomes impossible. What I assert is, then, that\r\nthere is a philosophic absurdity in supposing space in general\r\nto be of this nature. On the egg we have marked points, such\r\nas the two ends; the space formed by its surface is not homogeneous,\r\nand if things are moved about in it, it must of itself\r\nexercise a distorting effect upon them, quite independently of\r\nphysical causes; if it did not exercise such an effect, the things\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_152\" id=\"Page_152\"\u003e[152]\u003c/a\u003e\u003c/span\u003e\r\ncould not be moved. Thus such a space would not be homogeneous,\r\nbut would have marked points, by reference to which\r\nbodies would have absolute position, quite independently of\r\nany other bodies. Space would no longer be passive, but\r\nwould exercise a definite effect upon things, and we should\r\nhave to accommodate ourselves to the notion of marked points\r\nin empty space; these points being marked, not by the bodies\r\nwhich occupied them, but by their effects on any bodies which\r\nmight from time to time occupy them. This want of homogeneity\r\nand passivity is, however, absurd; space must, since it\r\nis a form of externality, allow only of relative, not of absolute,\r\nposition, and must be completely homogeneous throughout.\r\nTo suppose it otherwise, is to give it a thinghood which no\r\nform of externality can possibly possess. We must, then, on\r\npurely philosophical grounds, admit that a geometrical figure\r\nwhich is possible anywhere is possible everywhere, which is the\r\naxiom of Free Mobility.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e146.\u003c/b\u003e\u003ca name=\"N146\" id=\"N146\"\u003e\u003c/a\u003e\r\nB. \u003cem\u003eGeometrical Argument.\u003c/em\u003e Let us see next what sort\r\nof Geometry we could construct without this axiom. The ultimate\r\nstandard of comparison of spatial magnitudes must, as we\r\nsaw in introducing the axiom, be equality when superposed; but\r\nneed we, from this equality, infer equality when separated? It\r\nhas been urged by Erdmann that, for the more immediate purposes\r\nof Geometry, this would be unnecessary\u003ca name=\"FNanchor_153_153\" id=\"FNanchor_153_153\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_153_153\" class=\"fnanchor\"\u003e[153]\u003c/a\u003e. We might\r\nconstruct a new Geometry, he thinks, in which sizes varied with\r\nmotion on any definite law. Such a view, as I shall show below,\r\ninvolves a logical error as to the nature of magnitude. But\r\nbefore pointing this out, let us discuss the geometrical consequences\r\nof assuming its truth. Suppose the length of an infinitesimal\r\narc in some standard position were \u003cem\u003eds\u003c/em\u003e; then in any\r\nother position \u003cem\u003ep\u003c/em\u003e its length would be \u003cem\u003eds.f(p)\u003c/em\u003e, where the form of\r\nthe function \u003cem\u003ef(p)\u003c/em\u003e must be supposed known. But how are we to\r\ndetermine the position \u003cem\u003ep\u003c/em\u003e? For this purpose, we require \u003cem\u003ep\u003c/em\u003e\u0027s\r\ncoordinates, \u003cem\u003ei.e.\u003c/em\u003e, some measurement of distance from the origin.\r\nBut the distance from the origin could only be measured if we\r\nassumed our law \u003cem\u003ef(p)\u003c/em\u003e to measure it by. For suppose the\r\norigin to be \u003cem\u003eO\u003c/em\u003e, and \u003cem\u003eOp\u003c/em\u003e to be a straight line whose length is\r\nrequired. If we have a measuring rod with which we travel\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_153\" id=\"Page_153\"\u003e[153]\u003c/a\u003e\u003c/span\u003e\r\nalong the line and measure successive infinitesimal arcs, the\r\nmeasuring rod will change its size as we move, so that an arc\r\nwhich appears by the measure to be \u003cem\u003eds\u003c/em\u003e will really be \u003cem\u003ef(s).ds\u003c/em\u003e,\r\nwhere \u003cem\u003es\u003c/em\u003e is the previously traversed distance. If, on the\r\nother hand, we move our line \u003cem\u003eOp\u003c/em\u003e slowly through the origin, and\r\nmeasure each piece as it passes through, our measure, it is true,\r\nwill not alter, but now we have no means of discovering the law\r\nby which any element has changed its length in coming to the\r\norigin. Hence, until we assume our function \u003cem\u003ef(p)\u003c/em\u003e, we have\r\nno means of determining \u003cem\u003ep\u003c/em\u003e, for we have just seen that distances\r\nfrom the origin can only be estimated by means of the law\r\n\u003cem\u003ef(p)\u003c/em\u003e. It follows that experience can neither prove nor disprove\r\nthe constancy of shapes throughout motion, since, if shapes\r\nwere not constant, we should have to \u003cem\u003eassume\u003c/em\u003e a law of their\r\nvariation before measurement became possible, and therefore\r\nmeasurement could not itself reveal that variation to us\u003ca name=\"FNanchor_154_154\" id=\"FNanchor_154_154\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_154_154\" class=\"fnanchor\"\u003e[154]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eNevertheless, such an arbitrarily assumed law \u003cem\u003edoes\u003c/em\u003e, at first\r\nsight, give a mathematically possible Geometry. The fundamental\r\nproposition, that two magnitudes which can be superposed\r\nin any one position can be superposed in any other, still\r\nholds. For two infinitesimal arcs, whose lengths in the standard\r\nposition are \u003cem\u003eds\u003csub\u003e1\u003c/sub\u003e\u003c/em\u003e and \u003cem\u003eds\u003csub\u003e2\u003c/sub\u003e\u003c/em\u003e, would, in any other position \u003cem\u003ep\u003c/em\u003e, have\r\nlengths \u003cem\u003ef(p).ds\u003csub\u003e1\u003c/sub\u003e\u003c/em\u003e and \u003cem\u003ef(p).ds\u003csub\u003e2\u003c/sub\u003e\u003c/em\u003e, so that their ratio would be\r\nunaltered. From this constancy of ratio, as we know through\r\nRiemann and Helmholtz, the above proposition follows. Hence\r\nall that Geometry requires, it would seem, as a basis for\r\nmeasurement, is an axiom that the alteration of shapes during\r\nmotion follows a definite known law, such as that assumed\r\nabove.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e147.\u003c/b\u003e\u003ca name=\"N147\" id=\"N147\"\u003e\u003c/a\u003e\r\nThere is, however, in such a view, as I remarked above,\r\na logical error as to the nature of magnitude. This error has\r\nbeen already pointed out in dealing with Erdmann\u003ca name=\"FNanchor_155_155\" id=\"FNanchor_155_155\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_155_155\" class=\"fnanchor\"\u003e[155]\u003c/a\u003e, and need\r\nonly be briefly repeated here. A judgment of magnitude is\r\nessentially a judgment of comparison: in unmeasured quantity,\r\ncomparison as to the mere more or less, but in measured magnitude,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_154\" id=\"Page_154\"\u003e[154]\u003c/a\u003e\u003c/span\u003ecomparison as to the precise how many times. To\r\nspeak of differences of magnitude, therefore, in a case where\r\ncomparison cannot reveal them, is logically absurd. Now in\r\nthe case contemplated above, two magnitudes, which appear\r\nequal in one position, appear equal also when compared in\r\nanother position. There is no sense, therefore, in supposing\r\nthe two magnitudes unequal when separated, nor in supposing,\r\nconsequently, that they have changed their magnitudes in\r\nmotion. This senselessness of our hypothesis is the logical\r\nground of the mathematical indeterminateness as to the law of\r\nvariation. Since, then, there is no means of comparing two\r\nspatial figures, as regards magnitude, except superposition, the\r\nonly logically possible axiom, if spatial magnitude is to be self-consistent,\r\nis the axiom of Free Mobility in the form first given\r\nabove.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e148.\u003c/b\u003e\u003ca name=\"N148\" id=\"N148\"\u003e\u003c/a\u003e\r\nAlthough this axiom is \u003cem\u003eà priori\u003c/em\u003e, its application to the\r\nmeasurement of actual bodies, as we found in discussing Helmholtz\u0027s\r\nviews, always involves an empirical element\u003ca name=\"FNanchor_156_156\" id=\"FNanchor_156_156\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_156_156\" class=\"fnanchor\"\u003e[156]\u003c/a\u003e. Our\r\naxiom, then, only supplies the \u003cem\u003eà priori\u003c/em\u003e condition for carrying\r\nout an operation which, in the concrete, is empirical\u0026mdash;just as\r\narithmetic supplies the \u003cem\u003eà priori\u003c/em\u003e condition for a census. As\r\nthis topic has been discussed at length in Chapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, I shall\r\nsay no more about it here.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e149.\u003c/b\u003e\u003ca name=\"N149\" id=\"N149\"\u003e\u003c/a\u003e\r\nThere remain, however, a few objections and difficulties\r\nto be discussed. First, how do we obtain equality in\r\nsolids, and in Kant\u0027s cases of right and left hands, or of right\r\nand left-handed screws, where actual superposition is impossible?\r\nSecondly, how can we take congruence as the only\r\npossible basis of spatial measurement, when we have before us\r\nthe case of time, where no such thing as congruence is conceivable?\r\nThirdly, it might be urged that we can immediately\r\nestimate spatial equality by the eye, with more or less accuracy,\r\nand thus have a measure independent of congruence. Fourthly,\r\nhow is metrical Geometry possible on non-congruent surfaces,\r\nif congruence be the basis of spatial measurement? I will\r\ndiscuss these objections successively.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e150.\u003c/b\u003e\u003ca name=\"N150\" id=\"N150\"\u003e\u003c/a\u003e\r\n(1) How do we measure the equality of solids?\r\nThese could only be brought into actual congruence if we had\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_155\" id=\"Page_155\"\u003e[155]\u003c/a\u003e\u003c/span\u003e\r\na fourth dimension to operate in\u003ca name=\"FNanchor_157_157\" id=\"FNanchor_157_157\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_157_157\" class=\"fnanchor\"\u003e[157]\u003c/a\u003e, and from what I have said\r\nbefore of the absolute necessity of this test, it might seem as\r\nthough we should be left here in utter ignorance. Euclid is\r\nsilent on the subject, and in all works on Geometry it is assumed\r\nas self-evident that two cubes of equal side are equal. This assumption\r\nsuggests that we are not so badly off as we should have\r\nbeen without congruence, as a test of equality in one or two\r\ndimensions; for now we can at least be sure that two cubes have\r\nall their sides and all their faces equal. Two such cubes differ,\r\nthen, in no sensible spatial quality save position, for volume, in\r\nthis case at any rate, is not a sensible quality. They are,\r\ntherefore, as far as such qualities are concerned, indiscernible.\r\nIf their places were interchanged, we might know the change\r\nby their colour, or by some other non-geometrical property;\r\nbut so far as any property of which Geometry can take cognisance\r\nis concerned, everything would seem as before. To\r\nsuppose a difference of volume, then, would be to ascribe an\r\neffect to mere position, which we saw to be inadmissible while\r\ndiscussing Free Mobility. Except as regards position, they are\r\ngeometrically indiscernible, and we may call to our aid the\r\nIdentity of Indiscernibles to establish their agreement in the\r\none remaining geometrical property of volume. This may\r\nseem rather a strange principle to use in Mathematics, and for\r\nGeometry their equality is, perhaps, best regarded as a definition;\r\nbut if we demand a philosophical ground for this definition,\r\nit is, I believe, only to be found in the Identity of Indiscernibles.\r\nWe can, without error, make our \u003cem\u003edefinition\u003c/em\u003e of three-dimensional\r\nequality rest on two-dimensional congruence. For\r\nsince direct comparison as to volume is impossible, we are at\r\nliberty to \u003cem\u003edefine\u003c/em\u003e two volumes as equal, when all their various\r\nlines, surfaces, angles and solid angles are congruent, since\r\nthere remains, in such a case, no \u003cem\u003emeasurable\u003c/em\u003e difference between\r\nthe figures composing the two volumes. Of course, as soon as\r\nwe have established this one case of equality of volumes, the\r\nrest of the theory follows; as appears from the ordinary method\r\nof integrating volumes, by dividing them into small cubes.\u003c/p\u003e\r\n\r\n\u003cp\u003eThus congruence \u003cem\u003ehelps\u003c/em\u003e to establish three-dimensional equality,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_156\" id=\"Page_156\"\u003e[156]\u003c/a\u003e\u003c/span\u003e\r\nthough it cannot directly \u003cem\u003eprove\u003c/em\u003e such equality; and the same\r\nphilosophical principle, of the homogeneity of space, by which\r\ncongruence was proved, comes to our rescue here. But how\r\nabout right-handed and left-handed screws? Here we can no\r\nlonger apply the Identity of Indiscernibles, for the two are very\r\nwell discernible. But as with solids, so here, Free Mobility can\r\nhelp us much. It can enable us, by ordinary measurement, to\r\nshow that the internal relations of both screws are the same,\r\nand that the difference lies only in their relation to other\r\nthings in space. Knowing these internal relations, we can\r\ncalculate, by the Geometry which Free Mobility has rendered\r\npossible, all the geometrical properties of these screws\u0026mdash;radius,\r\npitch, etc.\u0026mdash;and can show them to be severally equal in both.\r\nBut this is all we require. Mediate comparison is possible,\r\nthough immediate comparison is not. Both can, for instance,\r\nbe compared with the cylinder on which both would fit, and\r\nthus their equality can be proved. A precisely similar proof\r\nholds, of course, for the other cases, right and left hands,\r\nspherical triangles, etc. On the whole, these cases confirm my\r\nargument; for they show, as Kant intended them to show\u003ca name=\"FNanchor_158_158\" id=\"FNanchor_158_158\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_158_158\" class=\"fnanchor\"\u003e[158]\u003c/a\u003e, the\r\nessential relativity of space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e151.\u003c/b\u003e\u003ca name=\"N151\" id=\"N151\"\u003e\u003c/a\u003e\r\n(2) As regards time, no congruence is here conceivable,\r\nfor to effect congruence requires always\u0026mdash;as we saw in\r\nthe case of solids\u0026mdash;one more dimension than belongs to the\r\nmagnitudes compared. No day can be brought into temporal\r\ncoincidence with any other day, to show that the two exactly\r\ncover each other; we are therefore reduced to the arbitrary\r\nassumption that some motion or set of motions, given us in experience,\r\nis uniform. Fortunately, we have a large set of motions\r\nwhich all roughly agree; the swing of the pendulum, the\r\nrotation and revolution of the earth and the planets, etc. These\r\ndo not exactly agree, but they lead us to the laws of motion, by\r\nwhich we are able, on our arbitrary hypothesis, to estimate\r\ntheir small departures from uniformity; just as the assumption\r\nof Free Mobility enabled us to measure the departures of actual\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_157\" id=\"Page_157\"\u003e[157]\u003c/a\u003e\u003c/span\u003e\r\nbodies from rigidity. But here, as there, another possibility is\r\nmathematically open to us, and can only be excluded by its\r\nphilosophic absurdity; we might have assumed that the above\r\nset of approximately agreeing motions all had velocities which\r\nvaried approximately as some arbitrarily assumed function of\r\nthe time, \u003cem\u003ef(t)\u003c/em\u003e say, measured from some arbitrary origin. Such\r\nan assumption would still keep them as nearly synchronous as\r\nbefore, and would give an equally possible, though more complex,\r\nsystem of Mechanics; instead of the first law of motion,\r\nwe should have the following: A particle perseveres in its\r\nstate of rest, or of rectilinear motion with velocity varying as\r\n\u003cem\u003ef(t)\u003c/em\u003e, except in so far as it is compelled to alter that state by\r\nthe action of external forces. Such a hypothesis \u003cem\u003eis\u003c/em\u003e mathematically\r\npossible, but, like the similar one for space, it is\r\nexcluded logically by the comparative nature of the judgment\r\nof quantity, and philosophically by the fact that it involves\r\nabsolute time, as a determining agent in change, whereas time\r\ncan never, philosophically, be anything but a passive form,\r\nabstracted from change. I have introduced this parallel from\r\ntime, not as directly bearing on the argument, but as a simpler\r\ncase which may serve to illustrate my reasoning in the more\r\ncomplex case of space. For since time, in mathematics, is one-dimensional,\r\nthe mathematical difficulties are simpler than in\r\nGeometry; and although nothing accurately corresponds to\r\ncongruence, there is a very similar mixture of mathematical\r\nand philosophical necessity, giving, finally, a thoroughly definite\r\naxiom as the basis of time-measurement, corresponding to\r\ncongruence as the basis of space-measurement\u003ca name=\"FNanchor_159_159\" id=\"FNanchor_159_159\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_159_159\" class=\"fnanchor\"\u003e[159]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e152.\u003c/b\u003e\u003ca name=\"N152\" id=\"N152\"\u003e\u003c/a\u003e\r\n(3) The case of time-measurement suggests the third\r\nof the above objections to the absolute necessity of the axiom of\r\nFree Mobility. Psycho-physics has shown that we have an\r\napproximate power, by means of what may be called the sense\r\nof duration, of immediately estimating equal short times. This\r\nestablishes a rough measure independent of any assumed\r\nuniform motion, and in space also, it may be said, we have a\r\nsimilar power of immediate comparison. We can see, by immediate\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_158\" id=\"Page_158\"\u003e[158]\u003c/a\u003e\u003c/span\u003e\r\ninspection, that the sub-divisions on a foot rule are\r\nnot grossly inaccurate; and so, it may be said, we both have a\r\nmeasure independent of congruence, and also could discover, by\r\nexperience, any gross departure from Free Mobility. Against\r\nthis view, however, there is at the outset a very fundamental\r\npsychological objection. It has been urged that all our comparison\r\nof spatial magnitudes proceeds by ideal superposition.\r\nThus James says (Psychology, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 152): \"Even where we\r\nonly feel one sub-division to be vaguely larger or less, the mind\r\nmust pass rapidly between it and the other sub-division, and\r\nreceive the immediate sensible shock of the more,\" and \"so far as\r\nthe sub-divisions of a sense-space are to be \u003cem\u003emeasured\u003c/em\u003e exactly\r\nagainst each other, objective forms occupying \u003cins class=\"corr\" title=\"Transcriber\u0027s Note\u0026mdash;Original text: \u0027one subdivison must\u0027\"\u003eone sub-division\r\nmust\u003c/ins\u003e be directly or indirectly superposed upon the other\u003ca name=\"FNanchor_160_160\" id=\"FNanchor_160_160\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_160_160\" class=\"fnanchor\"\u003e[160]\u003c/a\u003e.\"\u003c/p\u003e\r\n\r\n\u003cp\u003eEven if we waive this fundamental objection, however, others\r\nremain. To begin with, such judgments of equality are only\r\nvery rough approximations, and cannot be applied to lines of\r\nmore than a certain length, if only for the reason that such\r\nlines cannot well be seen together. Thus this method can only\r\ngive us any security in our own immediate neighbourhood,\r\nand could in no wise warrant such operations as would be\r\nrequired for the construction of maps \u0026amp;c., much less the measurement\r\nof astronomical distances. They might just enable\r\nus to say that some lines were longer than others, but they\r\nwould leave Geometry in a position no better than that of the\r\nHedonical Calculus, in which we depend on a purely subjective\r\nmeasure. So inaccurate, in fact, is such a method acknowledged\r\nto be, that the foot-rule is as much a need of daily life\r\nas of science. Besides, no one would trust such immediate\r\njudgments, but for the fact that the stricter test of congruence\r\nto some extent confirms them; if we could not apply this test,\r\nwe should have no ground for trusting them even as much as\r\nwe do. Thus we should have, here, no real escape from our\r\nabsolute dependence upon the axiom of Free Mobility.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e153.\u003c/b\u003e\u003ca name=\"N153\" id=\"N153\"\u003e\u003c/a\u003e\r\n(4) One last elucidatory remark is necessary before\r\nour proof of this axiom can be considered complete. We spoke\r\nabove of the Geometry on an egg, where Free Mobility does not\r\nhold. What, I may be asked, is there about a thoroughly non-congruent\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_159\" id=\"Page_159\"\u003e[159]\u003c/a\u003e\u003c/span\u003e\r\nGeometry, more impossible than this Geometry on\r\nthe egg? The answer is obvious. The Geometry of non-congruent\r\nsurfaces is \u003cem\u003eonly\u003c/em\u003e possible by the use of infinitesimals,\r\nand in the infinitesimal all surfaces become plane. The fundamental\r\nformula, that for the length of an infinitesimal arc, is\r\nonly obtained on the assumption that such an arc may be treated\r\nas a straight line, and that Euclidean Plane Geometry may be\r\napplied in the immediate neighbourhood of any point. If we\r\nhad not our Euclidean measure, which could be moved without\r\ndistortion, we should have no method of comparing small arcs\r\nin different places, and the Geometry of non-congruent surfaces\r\nwould break down. Thus the axiom of Free Mobility, as\r\nregards three-dimensional space, is necessarily implied and\r\npresupposed in the Geometry of non-congruent surfaces; the\r\npossibility of the latter, therefore, is a dependent and derivative\r\npossibility, and can form no argument against the \u003cem\u003eà priori\u003c/em\u003e\r\nnecessity of congruence as the test of equality.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e154.\u003c/b\u003e\u003ca name=\"N154\" id=\"N154\"\u003e\u003c/a\u003e\r\nIt is to be observed that the axiom of Free Mobility,\r\nas I have enunciated it, includes also the axiom to which\r\nHelmholtz gives the name of Monodromy. This asserts that\r\na body does not alter its dimensions in consequence of a\r\ncomplete revolution through four right angles, but occupies\r\nat the end the same position as at the beginning. The supposed\r\nmathematical necessity of making a separate axiom of\r\nthis property of space has been disproved by Sophus Lie (v.\r\n\u003ca href=\"#N45\"\u003eChap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e § 45\u003c/a\u003e); philosophically, it is plainly a particular case\r\nof Free Mobility\u003ca name=\"FNanchor_161_161\" id=\"FNanchor_161_161\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_161_161\" class=\"fnanchor\"\u003e[161]\u003c/a\u003e, and indeed a particularly obvious case, for\r\na translation really does make some change in a body, namely,\r\na change in position, but a rotation through four right angles\r\nmay be supposed to have been performed any number of times\r\nwithout appearing in the result, and the absurdity of ascribing\r\nto space the power of making bodies grow in the process is\r\npalpable; everything that was said above on congruence in\r\ngeneral applies with even greater evidence to this special\r\ncase.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e155.\u003c/b\u003e\u003ca name=\"N155\" id=\"N155\"\u003e\u003c/a\u003e\r\nThe axiom of Free Mobility involves, if it is to be\r\ntrue, the homogeneity of space, or the complete relativity of\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_160\" id=\"Page_160\"\u003e[160]\u003c/a\u003e\u003c/span\u003e\r\nposition. For if any shape, which is possible in one part of\r\nspace, be always possible in another, it follows that all parts\r\nof space are qualitatively similar, and cannot, therefore, be\r\ndistinguished by any intrinsic property. Hence positions in\r\nspace, if our axiom be true, must be wholly defined by external\r\nrelations, \u003cem\u003ei.e.\u003c/em\u003e \u003cem\u003ePosition is not an intrinsic, but a purely relative,\r\nproperty of things in space\u003c/em\u003e. If there could be such a thing\r\nas absolute position, in short, metrical Geometry would be\r\nimpossible. This relativity of position is the fundamental postulate\r\nof all Geometry, to which each of the necessary metrical\r\naxioms leads, and from which, conversely, each of these axioms\r\ncan be deduced.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e156.\u003c/b\u003e\u003ca name=\"N156\" id=\"N156\"\u003e\u003c/a\u003e\r\nThis converse deduction, as regards Free Mobility, is\r\nnot very difficult, and follows from the argument of Section A\u003ca name=\"FNanchor_162_162\" id=\"FNanchor_162_162\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_162_162\" class=\"fnanchor\"\u003e[162]\u003c/a\u003e,\r\nwhich I will briefly recapitulate. In the first place, externality\r\nis an essentially relative conception\u0026mdash;nothing can be external\r\nto itself. To be external to something is to be an other with\r\nsome relation to that thing. Hence, when we abstract a form\r\nof externality from all material content, and study it in isolation,\r\nposition will appear of necessity as purely relative\u0026mdash;it\r\ncan have no intrinsic quality, for our form consists of pure\r\nexternality, and externality contains no shadow or trace of\r\nan intrinsic quality. Hence we derive our fundamental postulate,\r\nthe relativity of position. From this follows the homogeneity\r\nof our form, for any quality in one position, which\r\nmarked out that position from another, would be necessarily\r\nmore or less intrinsic, and would contradict the pure relativity.\r\nFinally Free Mobility follows from homogeneity, for our form\r\nwould not be homogeneous unless it allowed, in every part,\r\nshapes or systems of relations, which it allowed in any other\r\npart. Free Mobility, therefore, is a necessary property of every\r\npossible form of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e157.\u003c/b\u003e\u003ca name=\"N157\" id=\"N157\"\u003e\u003c/a\u003e\r\nIn summing up the argument we have just concluded,\r\nwe may exhibit it, in consequence of the two preceding\r\nparagraphs, in the form of a completed circle. Starting from\r\nthe conditions of spatial measurement, we found that the comparison,\r\nrequired for measurement, could only be effected by\r\nsuperposition. But we found, further, that the result of such\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_161\" id=\"Page_161\"\u003e[161]\u003c/a\u003e\u003c/span\u003e\r\ncomparison will only be unambiguous, if spatial magnitudes and\r\nshapes are unaltered by motion in space, if, in other words,\r\nshapes do not depend upon absolute position in space. But\r\nthis axiom can only be true if space is homogeneous and\r\nposition merely relative. Conversely, if position is assumed\r\nto be merely relative, a change of magnitude in motion\u0026mdash;involving\r\nas it does, the assertion of absolute position\u0026mdash;is\r\nimpossible, and our test of spatial equality is therefore adequate.\r\nBut position in any form of externality must be purely\r\nrelative, since externality cannot be an intrinsic property of\r\nanything. Our axiom, therefore, is \u003cem\u003eà priori\u003c/em\u003e in a double sense.\r\nIt is presupposed in all spatial measurement, and it is a\r\nnecessary property of any form of externality. A similar double\r\napriority, we shall see, appears in our other necessary axioms.\u003c/p\u003e\r\n\r\n\r\n\u003ch4\u003eII. \u003cem\u003eThe Axiom of Dimensions\u003ca name=\"FNanchor_163_163\" id=\"FNanchor_163_163\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_163_163\" class=\"fnanchor\"\u003e[163]\u003c/a\u003e.\u003c/em\u003e\u003c/h4\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e158.\u003c/b\u003e\u003ca name=\"N158\" id=\"N158\"\u003e\u003c/a\u003e\r\nWe have seen, in discussing the axiom of Free Mobility,\r\nthat all position is relative, that is, a position exists\r\nonly by virtue of relations\u003ca name=\"FNanchor_164_164\" id=\"FNanchor_164_164\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_164_164\" class=\"fnanchor\"\u003e[164]\u003c/a\u003e. It follows that, if positions can\r\nbe defined at all, they must be uniquely and exhaustively\r\ndefined by some finite number of such relations. If Geometry\r\nis to be possible, it must happen that, after enough relations\r\nhave been given to determine a point uniquely, its relations\r\nto any fresh known point are deducible from the relations\r\nalready given. Hence we obtain, as an \u003cem\u003eà priori\u003c/em\u003e condition of\r\nGeometry, logically indispensable to its existence, the axiom\r\nthat \u003cem\u003eSpace must have a finite integral number of Dimensions\u003c/em\u003e.\r\nFor every relation required in the definition of a point constitutes\r\na dimension, and a fraction of a relation is meaningless.\r\nThe number of relations required must be finite, since an\r\ninfinite number of dimensions would be practically impossible\r\nto determine. If we remember our axiom of Free Mobility,\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_162\" id=\"Page_162\"\u003e[162]\u003c/a\u003e\u003c/span\u003eand remember also that space is a continuum, we may state\r\nour axiom, for metrical Geometry, in the form given by Helmholtz\r\n(v. \u003ca href=\"#N25\"\u003eChap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e § 25\u003c/a\u003e): \"In a space of n dimensions, the\r\nposition of every point is uniquely determined by the measurement\r\nof n continuous independent variables (coordinates).\u003ca name=\"FNanchor_165_165\" id=\"FNanchor_165_165\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_165_165\" class=\"fnanchor\"\u003e[165]\u003c/a\u003e\"\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e159.\u003c/b\u003e\u003ca name=\"N159\" id=\"N159\"\u003e\u003c/a\u003e\r\nSo much, then, is \u003cem\u003eà priori\u003c/em\u003e necessary to metrical\r\nGeometry. The restriction of the dimensions to three seems,\r\non the contrary, to be wholly the work of experience\u003ca name=\"FNanchor_166_166\" id=\"FNanchor_166_166\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_166_166\" class=\"fnanchor\"\u003e[166]\u003c/a\u003e. This\r\nrestriction cannot be logically necessary, for as soon as we have\r\nformulated any analytical system, it appears wholly arbitrary.\r\nWhy, we are driven to ask, cannot we add a fourth coordinate\r\nto our \u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e, \u003cem\u003ez\u003c/em\u003e, or give a geometrical meaning to \u003cem\u003ex\u003csup\u003e4\u003c/sup\u003e\u003c/em\u003e? In this\r\nmore special form, we are tempted to regard the axiom of\r\ndimensions, like the number of inhabitants of a town, as a\r\npurely statistical fact, with no greater necessity than such facts\r\nhave.\u003c/p\u003e\r\n\r\n\u003cp\u003eGeometry affords intrinsic evidence of the truth of my\r\ndivision of the axiom of dimensions into an \u003cem\u003eà priori\u003c/em\u003e and\r\nempirical portion. For while the extension of the number\r\nof dimensions to four, or to \u003cem\u003en\u003c/em\u003e, alters nothing in plane and\r\nsolid Geometry, but only adds a new branch which interferes\r\nin no way with the old, \u003cem\u003esome\u003c/em\u003e definite number of dimensions\r\nis assumed in all Geometries, nor is it possible to conceive of\r\na Geometry which should be free from this assumption\u003ca name=\"FNanchor_167_167\" id=\"FNanchor_167_167\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_167_167\" class=\"fnanchor\"\u003e[167]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e160.\u003c/b\u003e\u003ca name=\"N160\" id=\"N160\"\u003e\u003c/a\u003e\r\nLet us, since the point seems of some interest, repeat\r\nour proof of the apriority of this axiom from a slightly different\r\npoint of view. We will begin, this time, from the most abstract\r\nconception of space, such as we find in Riemann\u0027s dissertation,\r\nor in Erdmann\u0027s extents. We have here, an ordered\r\nmanifold, infinitely divisible and allowing of Free Mobility\u003ca name=\"FNanchor_168_168\" id=\"FNanchor_168_168\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_168_168\" class=\"fnanchor\"\u003e[168]\u003c/a\u003e.\r\nFree Mobility involves, as we saw, the power of passing continuously\r\nfrom any one point to any other, by any course which\r\nmay seem pleasant to us; it involves, also, that, in such a\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_163\" id=\"Page_163\"\u003e[163]\u003c/a\u003e\u003c/span\u003ecourse, no changes occur except changes of mere position, \u003cem\u003ei.e.\u003c/em\u003e,\r\npositions do not differ from one another in any qualitative\r\nway. (This absence of qualitative difference is the distinguishing\r\nmark of space as opposed to other manifolds, such as the\r\ncolour- and tone-systems: in these, every element has a definite\r\nqualitative sensational value, whereas in space, the sensational\r\nvalue of a position depends wholly on its spatial relation to\r\nour own body, and is thus not intrinsic, but relative.) From\r\nthe absence of qualitative differences among positions, it follows\r\nlogically that positions exist only by virtue of other positions;\r\none position differs from another just because they are two,\r\nnot because of anything intrinsic in either. Position is thus\r\ndefined simply and solely by relation to other positions. Any\r\nposition, therefore, is completely defined when, and only when,\r\nenough such relations have been given to enable us to determine\r\nits relation to any new position, this new position\r\nbeing defined by the same number of relations. Now, in order\r\nthat such definition may be at all possible, a finite number\r\nof relations must suffice. But every such relation constitutes\r\na dimension. Therefore, if Geometry is to be possible, it is\r\n\u003cem\u003eà priori\u003c/em\u003e necessary that space should have a finite integral\r\nnumber of dimensions.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e161.\u003c/b\u003e\u003ca name=\"N161\" id=\"N161\"\u003e\u003c/a\u003e\r\nThe limitation of the dimensions to three is, as we\r\nhave seen, empirical; nevertheless, it is not liable to the inaccuracy\r\nand uncertainty which usually belong to empirical\r\nknowledge. For the alternatives which logic leaves to sense\r\nare discrete\u0026mdash;if the dimensions are not three, they must be\r\ntwo or four or some other number\u0026mdash;so that \u003cem\u003esmall\u003c/em\u003e errors are\r\nout of the question\u003ca name=\"FNanchor_169_169\" id=\"FNanchor_169_169\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_169_169\" class=\"fnanchor\"\u003e[169]\u003c/a\u003e. Hence the final certainty of the axiom\r\nof three dimensions, though in part due to experience, is of\r\nquite a different order from that of (say) the law of Gravitation.\r\nIn the latter, a small inaccuracy might exist and remain undetected;\r\nin the former, an error would have to be so considerable\r\nas to be utterly impossible to overlook. It follows that\r\nthe certainty of our whole axiom, that the number of dimensions\r\nis three, is almost as great as that of the \u003cem\u003eà priori\u003c/em\u003e element,\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_164\" id=\"Page_164\"\u003e[164]\u003c/a\u003e\u003c/span\u003e\r\nsince this element leaves to sense a definite disjunction of\r\ndiscrete possibilities.\u003c/p\u003e\r\n\r\n\r\n\u003ch4\u003eIII. \u003cem\u003eThe Axiom of Distance.\u003c/em\u003e\u003c/h4\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e162.\u003c/b\u003e\u003ca name=\"N162\" id=\"N162\"\u003e\u003c/a\u003e\r\nWe have already seen, in discussing projective Geometry,\r\nthat two points must determine a unique curve, the\r\nstraight line. In metrical Geometry, the corresponding axiom\r\nis, that two points must determine a unique spatial quantity, distance.\r\nI propose to prove, in what follows, (1) that if distance,\r\nas a quantity completely determined by two points, did not exist,\r\nspatial magnitude would not be measurable; (2) that distance\r\ncan only be determined by two points, if there is an actual\r\ncurve in space determined by those two points; (3) that the\r\nexistence of such a curve can be deduced from the conception\r\nof a form of externality, and (4) that the application of quantity\r\nto such a curve necessarily leads to a certain magnitude, namely\r\ndistance, uniquely determined by any two points which determine\r\nthe curve. The conclusion will be, if these propositions\r\ncan be successfully maintained, that the axiom of distance is\r\n\u003cem\u003eà priori\u003c/em\u003e in the same double sense as the axiom of Free Mobility,\r\n\u003cem\u003ei.e.\u003c/em\u003e it is presupposed in the possibility of measurement, and\r\nit is necessarily true of any possible form of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e163.\u003c/b\u003e\u003ca name=\"N163\" id=\"N163\"\u003e\u003c/a\u003e\r\n(1) The possibility of spatial measurement allows\r\nus to infer the existence of a magnitude uniquely determined\r\nby any two points. The proof of this depends on the axiom\r\nof Free Mobility, or its equivalent, the homogeneity of space.\r\nWe have seen that these are involved in the possibility of\r\nspatial measurement; we may employ them, therefore, in any\r\nargument as to the conditions of this possibility.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow to begin with, two points must, if Geometry is to\r\nbe possible, have \u003cem\u003esome\u003c/em\u003e relation to each other, for we have seen\r\nthat such relations alone constitute position or localization.\r\nBut if two points have a relation to each other, this must be\r\nan intrinsic relation. For it follows, from the axiom of Free\r\nMobility, that two points, forming a figure congruent with the\r\ngiven pair, can be constructed in any part of space. If this\r\nwere not possible, we have seen that metrical Geometry\r\ncould not exist. But both the figures may be regarded as\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_165\" id=\"Page_165\"\u003e[165]\u003c/a\u003e\u003c/span\u003e\r\ncomposed of two points and their relation; if the two figures\r\nare congruent, therefore, it follows that the relation is quantitatively\r\nthe same for both figures, since congruence is the\r\ntest of spatial equality. Hence the two points have a quantitative\r\nrelation, which is such that they can traverse all space\r\nin a combined motion without in any way altering that relation.\r\nBut in such a general motion, any external relation\r\nof the two points, any relation involving other points or figures\r\nin space, must be altered\u003ca name=\"FNanchor_170_170\" id=\"FNanchor_170_170\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_170_170\" class=\"fnanchor\"\u003e[170]\u003c/a\u003e. Hence the relation between the\r\ntwo points, being unaltered, must be an intrinsic relation, a\r\nrelation involving no other point or figure in space; and this\r\nintrinsic relation we call distance\u003ca name=\"FNanchor_171_171\" id=\"FNanchor_171_171\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_171_171\" class=\"fnanchor\"\u003e[171]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e164.\u003c/b\u003e\u003ca name=\"N164\" id=\"N164\"\u003e\u003c/a\u003e\r\nIt might be objected, to the above argument, that it\r\ninvolves a \u003ci lang=\"la\" xml:lang=\"la\"\u003epetitio principii\u003c/i\u003e. For it has been assumed that\r\nthe two points and their relation form a figure, to which other\r\nfigures can be congruent. Now if two points have no intrinsic\r\nrelation, it would seem that they cannot form such a figure.\r\nThe argument, therefore, apparently assumes what it had to\r\nprove. Why, it may be asked, should not three points be\r\nrequired, before we obtain any relation, which Free Mobility\r\nallows us to construct afresh in other parts of space?\u003c/p\u003e\r\n\r\n\u003cp\u003eThe answer to this, as to the corresponding question in the\r\nfirst section of this chapter, lies, I think, in the passivity of\r\nspace, or the mutual independence of its parts. For it follows,\r\nfrom this independence, that any figure, or any assemblage\r\nof points, may be discussed without reference to other figures\r\nor points. This principle is the basis of infinite divisibility, of\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_166\" id=\"Page_166\"\u003e[166]\u003c/a\u003e\u003c/span\u003ethe use of quantity in Geometry, and of all possibility of\r\nisolating particular figures for discussion. It follows that two\r\npoints cannot be dependent, as to their relation, on any other\r\npoints or figures, for if they were so dependent, we should have\r\nto suppose some action of such points or figures on the two\r\npoints considered, which would contradict the mutual independence\r\nof different positions. To illustrate by an example:\r\nthe relation of two given points does not depend on the other\r\npoints of the straight line on which the given points lie. For\r\nonly through their relation, \u003cem\u003ei.e.\u003c/em\u003e through the straight line which\r\nthey determine, can the other points of the straight line be\r\nknown to have any peculiar connection with the given pair.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e165.\u003c/b\u003e\u003ca name=\"N165\" id=\"N165\"\u003e\u003c/a\u003e\r\nBut why, it may be asked, should there be only one such\r\nrelation between two points? Why not several? The\r\nanswer to this lies in the fact that points are wholly constituted\r\nby relations, and have no intrinsic nature of their own\u003ca name=\"FNanchor_172_172\" id=\"FNanchor_172_172\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_172_172\" class=\"fnanchor\"\u003e[172]\u003c/a\u003e. A\r\npoint is defined by its relations to other points, and when once\r\nthe relations necessary for definition have been given, no fresh\r\nrelations to the points used in definition are possible, since the\r\npoint defined has no qualities from which such relations could\r\nflow. Now one relation to any one other point is as good for\r\ndefinition as more would be, since however many we had, they\r\nwould all remain unaltered in a combined motion of both\r\npoints. Hence there can only be one relation determined by\r\nany two points.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e166.\u003c/b\u003e\u003ca name=\"N166\" id=\"N166\"\u003e\u003c/a\u003e\r\n(2) We have thus established our first proposition\u0026mdash;two\r\npoints have one and only one relation uniquely determined\r\nby those two points. This relation we call their distance\r\napart. It remains to consider the conditions of the measurement\r\nof distance, \u003cem\u003ei.e.\u003c/em\u003e, how far a unique value for distance involves\r\na curve uniquely determined by the two points.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, some curve joining the two points is\r\ninvolved in the above notion of a combined motion of the two\r\npoints, or of two other points forming a figure congruent with\r\nthe first two. For without some such curve, the two point-pairs\r\ncannot be known as congruent, nor can we have any test by\r\nwhich to discover when a point-pair is moving as a single\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_167\" id=\"Page_167\"\u003e[167]\u003c/a\u003e\u003c/span\u003e\r\nfigure\u003ca name=\"FNanchor_173_173\" id=\"FNanchor_173_173\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_173_173\" class=\"fnanchor\"\u003e[173]\u003c/a\u003e. Distance must be measured, therefore, by some line\r\nwhich joins the two points. But need this be a line which the\r\ntwo points completely determine?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e167.\u003c/b\u003e\u003ca name=\"N167\" id=\"N167\"\u003e\u003c/a\u003e\r\nWe are accustomed to the definition of the straight\r\nline as the \u003cem\u003eshortest\u003c/em\u003e distance between two points, which implies\r\nthat distance might equally well be measured by curved lines.\r\nThis implication I believe to be false, for the following reasons.\r\nWhen we speak of the length of a curve, we can give a meaning\r\nto our words only by supposing the curve divided into infinitesimal\r\nrectilinear arcs, whose sum gives the length of an equivalent\r\nstraight line; thus unless we presuppose the straight line,\r\nwe have no means of comparing the lengths of different curves,\r\nand can therefore never discover the applicability of our definition.\r\nIt might be thought, perhaps, that some other line, say\r\na circle, might be used as the basis of measurement. But in\r\norder to estimate in this way the length of any curve other\r\nthan a circle, we should have to divide the curve into infinitesimal\r\ncircular arcs. Now two successive points do not\r\ndetermine a circle, so that an arc of two points would have an\r\nindeterminate length. It is true that, if we exclude infinitesimal\r\nradii for the measuring circles, the lengths of the infinitesimal\r\narcs would be determinate, even if the circles\r\nvaried, but that is only because all the small circular arcs\r\nthrough two consecutive points coincide with the straight line\r\nthrough those two points. Thus, even with the help of the\r\narbitrary restriction to a finite radius, all that happens is that\r\nwe are brought back to the straight line. If, to mend matters,\r\nwe take three consecutive points of our curve, and reckon\r\ndistance by the arc of the circle of curvature, the notion of\r\ndistance loses its fundamental property of being a relation between\r\n\u003cem\u003etwo\u003c/em\u003e points. For two consecutive points of the arc could\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_168\" id=\"Page_168\"\u003e[168]\u003c/a\u003e\u003c/span\u003e\r\nnot then be said to have any corresponding distance apart\u0026mdash;three\r\npoints would be necessary before the notion of distance\r\nbecame applicable. Thus the circle is not a possible basis for\r\nmeasurement, and similar objections apply, of course, with\r\nincreased force, to any other curve. All this argument is\r\ndesigned to show, in detail, the logical impossibility of measuring\r\ndistance by any curve not completely defined by the two\r\npoints whose distance apart is required. If in the above we\r\nhad taken distance as measured by circles of \u003cem\u003egiven radius\u003c/em\u003e, we\r\nshould have introduced into its definition a relation to other\r\npoints besides the two whose distance was to be measured,\r\nwhich we saw to be a logical fallacy. Moreover, how are we to\r\nknow that all the circles have equal radii, until we have an\r\nindependent measure of distance?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e168.\u003c/b\u003e\u003ca name=\"N168\" id=\"N168\"\u003e\u003c/a\u003e\r\nA straight line, then, is not the \u003cem\u003eshortest\u003c/em\u003e distance, but\r\nis simply \u003cem\u003ethe\u003c/em\u003e distance between two points\u0026mdash;so far, this conclusion\r\nhas stood firm. But suppose we had two or more\r\ncurves through two points, and that all these curves were\r\ncongruent \u003ci lang=\"la\" xml:lang=\"la\"\u003einter se\u003c/i\u003e. We should then say, in accordance with\r\nthe definition of spatial equality, that the lengths of all these\r\ncurves were equal. Now it might happen that, although no\r\none of the curves was uniquely determined by the two end-points,\r\nyet the common length of all the curves was so determined.\r\nIn this case, what would hinder us from calling this\r\ncommon length the distance apart, although no unique figure\r\nin space corresponded to it? This is the case contemplated by\r\nspherical Geometry, where, as on a sphere, antipodes can be\r\njoined by an infinite number of geodesics, all of which are of\r\nequal length. The difficulty supposed is, therefore, not a\r\npurely imaginary one, but one which modern Geometry forces\r\nus to face. I shall consequently discuss it at some length.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e169.\u003c/b\u003e\u003ca name=\"N169\" id=\"N169\"\u003e\u003c/a\u003e\r\nTo begin with, I must point out that my axiom is\r\nnot quite equivalent to Euclid\u0027s. Euclid\u0027s axiom states that\r\ntwo straight lines cannot enclose a space, \u003cem\u003ei.e.\u003c/em\u003e, cannot have\r\nmore than one common point. Now if every two points,\r\nwithout exception, determine a unique straight line, it follows,\r\nof course, that two different straight lines can have only one\r\npoint in common\u0026mdash;so far, the two axioms are equivalent. But\r\nit may happen, as in spherical space, that two points \u003cem\u003ein general\u003c/em\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_169\" id=\"Page_169\"\u003e[169]\u003c/a\u003e\u003c/span\u003e\r\ndetermine a unique straight line, but fail to do so when they\r\nhave to each other the special relation of being antipodes. In\r\nsuch a system every pair of straight lines in the same plane\r\nmeet in two points, which are each other\u0027s antipodes; but two\r\npoints, \u003cem\u003ein general\u003c/em\u003e, still determine a unique straight line. We\r\nare still able, therefore, to obtain distances from unique straight\r\nlines, except in limiting cases; and in such cases, we can take\r\nany point intermediate between the two antipodes, join it by\r\nthe \u003cem\u003esame\u003c/em\u003e straight line to both antipodes, and measure its\r\ndistance from those antipodes in the usual way. The sum of\r\nthese distances then gives a unique value for the distance\r\nbetween the antipodes.\u003c/p\u003e\r\n\r\n\u003cp\u003eThus even in spherical space, we are greatly assisted by the\r\naxiom of the straight line; all linear measurement is effected\r\nby it, and exceptional cases can be treated, through its help, by\r\nthe usual methods for limits. Spherical space, therefore, is not\r\nso adverse as it at first appeared to be to the \u003cem\u003eà priori\u003c/em\u003e necessity\r\nof the axiom. Nevertheless we have, so far, not attacked the\r\nkernel of the objection which spherical space suggested. To\r\nthis attack it is now our duty to proceed.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e170.\u003c/b\u003e\u003ca name=\"N170\" id=\"N170\"\u003e\u003c/a\u003e\r\nIt will be remembered that, in our \u003cem\u003eà priori\u003c/em\u003e proof\r\nthat two points must have one definite relation, we held it\r\nimpossible for those two points to have, to the rest of space,\r\nany relation which would be unaltered by motion. Now in\r\nspherical space, in the particular case where the two points are\r\nantipodes, they \u003cem\u003ehave\u003c/em\u003e a relation, unaltered by motion, to the rest\r\nof space\u0026mdash;the relation, namely, that their distance is half the\r\ncircumference of the universe. In our former discussion, we\r\nassumed that any relation to outside space must be a relation\r\nof position\u0026mdash;and a relation of position must be altered by\r\nmotion. But with a finite space, in which we have absolute\r\nmagnitude, another relation becomes possible, namely, a relation\r\nof magnitude. Antipodal points, accordingly, like coincident\r\npoints, no longer determine a unique straight line. And\r\nit is instructive to observe that there is, in consequence, an\r\nambiguity in the expression for distance, like the ordinary\r\nambiguity in angular measurement. If 1/\u003cem\u003ek\u003csup\u003e2\u003c/sup\u003e\u003c/em\u003e be the space constant,\r\nand \u003cem\u003ed\u003c/em\u003e be one value for the distance between two points,\r\n2\u003cem\u003eπkn\u003c/em\u003e ± \u003cem\u003ed\u003c/em\u003e, where \u003cem\u003en\u003c/em\u003e is any integer, is an equally good value.\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_170\" id=\"Page_170\"\u003e[170]\u003c/a\u003e\u003c/span\u003e\r\nDistance is, in short, a periodic function like angle. Thus such\r\na state of things rather confirms than destroys my contention,\r\nthat distance depends on a curve uniquely determined by two\r\npoints. For as soon as we drop this unique determination, we\r\nsee ambiguities creeping into our expression for distance.\r\nDistance still has a set of \u003cem\u003ediscrete\u003c/em\u003e values, corresponding to the\r\nfact that, given one point, the straight line is uniquely determined\r\nfor all other points but one, the antipodal point. It is\r\ntempting to go on, and say: If through \u003cem\u003eevery\u003c/em\u003e pair of points there\r\nwere an infinite number of the curves used in measuring distance,\r\ndistance would be able, for the same pair of points, to take, not\r\nonly a discrete series, but an infinite \u003cem\u003econtinuous\u003c/em\u003e series of values.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e171.\u003c/b\u003e\u003ca name=\"N171\" id=\"N171\"\u003e\u003c/a\u003e\r\nThis, however, is mere speculation. I come now to\r\nthe \u003ci lang=\"fr\" xml:lang=\"fr\"\u003epièce de résistance\u003c/i\u003e of my argument. The ambiguity in\r\nspherical space arose, as we saw, from a relation of \u003cem\u003emagnitude\u003c/em\u003e\r\nto the rest of space\u0026mdash;such a relation being unaltered by a\r\nmotion of the two points, and therefore falling outside our\r\nintroductory reasoning. But what is this relation of magnitude?\r\nSimply a relation of the \u003cem\u003edistance\u003c/em\u003e between the two\r\npoints to a \u003cem\u003edistance\u003c/em\u003e given in the nature of the space in question.\r\nIt follows that such a relation \u003cem\u003epresupposes\u003c/em\u003e a measure of distance,\r\nand need not, therefore, be contemplated in any argument\r\nwhich deals with the \u003cem\u003eà priori\u003c/em\u003e requisites for the possibility of\r\ndefinite distances\u003ca name=\"FNanchor_174_174\" id=\"FNanchor_174_174\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_174_174\" class=\"fnanchor\"\u003e[174]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e172.\u003c/b\u003e\u003ca name=\"N172\" id=\"N172\"\u003e\u003c/a\u003e\r\nI have now shown, I hope conclusively, that spherical\r\nspace affords no objection to the apriority of my axiom. Any\r\ntwo points have one relation, their distance, which is independent\r\nof the rest of space, and this relation requires, as its\r\nmeasure, a curve uniquely determined by those two points. I\r\nmight have taken the bull by the horns, and said: Two points\r\n\u003cem\u003ecan\u003c/em\u003e have no relation but what is given by lines which join\r\nthem, and therefore, if they have a relation independent of the\r\nrest of space, there must be one line joining them which they\r\ncompletely determine. Thus James says\u003ca name=\"FNanchor_175_175\" id=\"FNanchor_175_175\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_175_175\" class=\"fnanchor\"\u003e[175]\u003c/a\u003e:\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_171\" id=\"Page_171\"\u003e[171]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\"Just as, in the field of quantity, the relation between two\r\nnumbers is another number, so in the field of space the relations\r\nare facts of the same order with the facts they relate….\r\nWhen we speak of the relation of direction of two points\r\ntowards each other, we mean simply the sensation of the line\r\nthat joins the two points together. \u003cem\u003eThe line is the relation….\u003c/em\u003e\r\nThe relation of position between the top and bottom points of\r\na vertical line is that line, and nothing else.\"\u003c/p\u003e\r\n\r\n\u003cp\u003eIf I had been willing to use this doctrine at the beginning,\r\nI might have avoided all discussion. A unique relation between\r\ntwo points \u003cem\u003emust\u003c/em\u003e in this case, involve a unique line between\r\nthem. But it seemed better to avoid a doctrine not universally\r\naccepted, the more so as I was approaching the question from\r\nthe logical, not the psychological, side. After disposing of the\r\nobjections, however, it is interesting to find this confirmation\r\nof the above theory from so different a standpoint. Indeed, I\r\nbelieve James\u0027s doctrine could be proved to be a logical necessity,\r\nas well as a psychological fact. For what sort of thing\r\ncan a spatial relation between two distinct points be? It must\r\nbe something spatial, and it must, since points are wholly\r\nconstituted by their relations, be something at least as real and\r\ntangible as the points it relates. There seems nothing which\r\ncan satisfy these requirements, except a line joining them.\r\nHence, once more, a unique relation must involve a unique\r\nline. That is, linear magnitude is logically impossible, unless\r\nspace allows of curves uniquely determined by any two of their\r\npoints.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e173.\u003c/b\u003e\u003ca name=\"N173\" id=\"N173\"\u003e\u003c/a\u003e\r\n(3) But farther, the existence of curves uniquely\r\ndetermined by two points can be deduced from the nature of\r\nany form of externality\u003ca name=\"FNanchor_176_176\" id=\"FNanchor_176_176\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_176_176\" class=\"fnanchor\"\u003e[176]\u003c/a\u003e. For we saw, in discussing Free\r\nMobility, that this axiom, together with homogeneity and the\r\nrelativity of position, can be so deduced, and we saw in the\r\nbeginning of our discussion on distance, that the existence of a\r\nunique relation between two points could be deduced from the\r\nhomogeneity of space. Since position is relative, we may say,\r\nany two points must have \u003cem\u003esome\u003c/em\u003e relation to each other: since\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_172\" id=\"Page_172\"\u003e[172]\u003c/a\u003e\u003c/span\u003e\r\nour form of externality is homogeneous, this relation can be\r\nkept unchanged while the two points move in the form, \u003cem\u003ei.e.\u003c/em\u003e,\r\nchange their relations to other points; hence their relation to\r\neach other is an intrinsic relation, independent of their relations\r\nto other points. But since our form \u003cem\u003eis\u003c/em\u003e merely a complex\r\nof relations, a relation of externality must appear in the form,\r\nwith the same evidence as anything else in the form; thus if\r\nthe form be intuitive or sensational, the relation must be\r\nimmediately presented, and not a mere inference. Hence the\r\nintrinsic relation between two points must be a unique figure\r\nin our form, \u003cem\u003ei.e.\u003c/em\u003e in spatial terms, the straight line joining the\r\ntwo points.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e174.\u003c/b\u003e\u003ca name=\"N174\" id=\"N174\"\u003e\u003c/a\u003e\r\n(4) Finally, we have to prove that the existence of\r\nsuch a curve necessarily leads, when quantity is applied to the\r\nrelation between two points, to a unique magnitude, which\r\nthose two points completely determine. With this, we shall be\r\nbrought back to distance, from which we started, and shall\r\ncomplete the circle of our argument.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe saw, in \u003ca href=\"#N119\"\u003esection A § 119\u003c/a\u003e, that the figure formed by two\r\npoints is projectively indistinguishable from that formed by any\r\ntwo other points in the same straight line; the figure, in both\r\ncases, is, from the projective standpoint, simply the straight\r\nline on which the two points lie. The difference of relation, in\r\nthe two cases, is not qualitative, since projective Geometry\r\ncannot deal with it; nevertheless, there is some difference of\r\nrelation. For instance, if one point be kept fixed, while the\r\nother moves, there is obviously some change of relation. This\r\nchange, since all parts of the straight line are qualitatively\r\nalike, must be a change of quantity. If two points, therefore,\r\ndetermine a unique figure, there must exist, for the distinction\r\nbetween the various other points of this figure, a unique\r\nquantitative relation between the two determining points, and\r\ntherefore, since these points are arbitrary, between only two\r\npoints. This relation is \u003cem\u003edistance\u003c/em\u003e, with which our argument\r\nbegan, and to which it at least returns.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e175.\u003c/b\u003e\u003ca name=\"N175\" id=\"N175\"\u003e\u003c/a\u003e\r\nTo sum up: If points are defined simply by relations\r\nto other points, \u003cem\u003ei.e.\u003c/em\u003e, if all position is relative, \u003cem\u003eevery point must\r\nhave to every other point one, and only one, relation independent\r\nof the rest of space. This relation is the distance between the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_173\" id=\"Page_173\"\u003e[173]\u003c/a\u003e\u003c/span\u003e\r\ntwo points.\u003c/em\u003e Now a relation between two points can only be\r\ndefined by a line joining them\u0026mdash;nay further, it may be contended\r\nthat a relation can only \u003cem\u003ebe\u003c/em\u003e a line joining them. Hence\r\na unique relation involves a unique line, \u003cem\u003ei.e.\u003c/em\u003e, a line determined\r\nby any two of its points. Only in a space which admits of\r\nsuch a line is linear magnitude a logically possible conception.\r\nBut when once we have established the possibility, \u003cem\u003ein general\u003c/em\u003e,\r\nof drawing such lines, and therefore of measuring linear magnitudes,\r\nwe may find that a certain magnitude has a peculiar\r\nrelation to the constitution of space. The straight line may\r\nturn out to be of finite length, and in this case its length will\r\ngive a certain peculiar magnitude, the space-constant. Two\r\nantipodal points, that is, points which bisect the entire\r\nstraight line, will then have a relation of magnitude which,\r\nthough unaltered by motion, is rendered peculiar by a certain\r\nconstant relation to the rest of space. This peculiarity presupposes\r\na measure of linear magnitude in general, and cannot,\r\ntherefore, upset the apriority of the axiom of the straight line.\r\nBut it destroys, for points having the peculiar antipodal relation\r\nto each other, the argument which proved that the relation\r\nbetween two points could not, since it was unchanged by\r\nmotion, have reference to the rest of space. Thus it is intelligible\r\nthat, for such special points, the axiom breaks down, and\r\nan infinite number of straight lines are possible between them;\r\nbut unless we had started with assuming the general validity\r\nof the axiom, we could never have reached a position in which\r\nantipodal points could have been known to be peculiar, or,\r\nindeed, a position which would have enabled us to give any\r\nquantitative definition whatever of particular points.\u003c/p\u003e\r\n\r\n\u003cp\u003eDistance and the straight line, as relations uniquely determined\r\nby two points, are thus \u003cem\u003eà priori\u003c/em\u003e necessary to metrical\r\nGeometry. But further, they are properties which must belong\r\nto any form of externality. Since their necessity for Geometry\r\nwas deduced from homogeneity and the relativity of position,\r\nand since these are necessary properties of any form of externality,\r\nthe same argument proves both conclusions. We thus\r\nobtain, as in the case of Free Mobility, a double apriority:\r\nThe axiom of Distance, and its implication, the axiom of\r\nthe Straight Line, are, on the one hand, presupposed in the\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_174\" id=\"Page_174\"\u003e[174]\u003c/a\u003e\u003c/span\u003e\r\npossibility of spatial magnitude, and cannot, therefore, be contradicted\r\nby any experience resulting from the measurement\r\nof space; while they are consequences, on the other hand, of\r\nthe necessary properties of any form of externality which is to\r\nrender possible experience of an external world.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e176.\u003c/b\u003e\u003ca name=\"N176\" id=\"N176\"\u003e\u003c/a\u003e\r\nIn connection with the straight line, it will be convenient\r\nto discuss the conditions of a metrical coordinate\r\nsystem. The projective coordinate system, as we have seen,\r\naims only at a convenient nomenclature for different points,\r\nand can be set up without introducing the notion of spatial\r\nquantity. But a metrical coordinate system does much more\r\nthan this. It defines every point quantitatively, by its quantitative\r\nspatial relations to a certain coordinate figure. Only\r\nwhen the system of coordinates is thus metrical, \u003cem\u003ei.e.\u003c/em\u003e, when\r\nevery coordinate represents some spatial magnitude, which is\r\nitself a relation of the point defined to some other point or\r\nfigure\u0026mdash;can operations with coordinates lead to a metrical\r\nresult. When, as in projective Geometry, the coordinates are\r\nnot spatial magnitudes, no amount of transformation can give\r\na metrical result. I wish to prove, here, that a metrical coordinate\r\nsystem necessarily involves the straight line, and cannot,\r\nwithout a logical fallacy, be set up on any other basis. The\r\nprojective system of coordinates, as we saw, is entirely based on\r\nthe straight line; but the metrical system is more important,\r\nsince its quantities embody actual information as to spatial\r\nmagnitudes, which, in projective Geometry, is not the case.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first place, a point\u0027s metrical coordinates constitute a\r\ncomplete quantitative definition of it; now a point can only be\r\ndefined, as we have seen, by its relations to other points, and\r\nthese relations can only be defined by means of the straight\r\nline. Consequently, any metrical system of coordinates must\r\ninvolve the straight line, as the basis of its definitions of points.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis \u003cem\u003eà priori\u003c/em\u003e argument, however, though I believe it to be\r\nquite sound, is not likely to carry conviction to any one persuaded\r\nof the opposite. Let us, therefore, examine metrical\r\ncoordinate systems in detail, and show, in each case, their\r\ndependence on the straight line.\u003c/p\u003e\r\n\r\n\u003cp\u003eWe have already seen that the notion of distance is impossible\r\nwithout the straight line. We cannot, therefore, define\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_175\" id=\"Page_175\"\u003e[175]\u003c/a\u003e\u003c/span\u003e\r\nour coordinates in any of the ordinary ways, as the distances\r\nfrom three planes, lines, points, spheres, or what not. Polar\r\ncoordinates are impossible, since,\u0026mdash;waiving the straightness of\r\nthe radius vector\u0026mdash;the length of the radius vector becomes\r\nunmeaning. Triangular coordinates involve not only angles,\r\nwhich must in the limit be rectilinear, but straight lines, or at\r\nany rate some well-defined curves. Now curves can only be\r\nmetrically defined in two ways: \u003cem\u003eEither\u003c/em\u003e by relation to the\r\nstraight line, as, \u003cem\u003ee.g.\u003c/em\u003e, by the curvature at any point, \u003cem\u003eor\u003c/em\u003e by\r\npurely analytical equations, which presuppose an intelligible\r\nsystem of metrical coordinates. What methods remain for\r\nassigning these arbitrary values to different points? Nay,\r\nhow are we to get any estimate of the difference\u0026mdash;to avoid\r\nthe more special notion of distance\u0026mdash;between two points?\r\nThe very notion of a point has become illusory. When we\r\nhave a coordinate system, we may define a point by its three\r\ncoordinates; in the absence of such a system, we may define\r\nthe notion of point \u003cem\u003ein general\u003c/em\u003e as the intersection of three surfaces\r\nor of two curves. Here we take surfaces and curves\r\nas notions which intuition makes plain, but if we wish them to\r\ngive us a precise numerical definition of \u003cem\u003eparticular\u003c/em\u003e points, we\r\nmust specify the kind of surface or curve to be used. Now\r\nthis, as we have seen, is only possible when we presuppose\r\neither the straight line, or a coordinate system. It follows that\r\nevery coordinate system presupposes the straight line, and is\r\nlogically impossible without it.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e177.\u003c/b\u003e\u003ca name=\"N177\" id=\"N177\"\u003e\u003c/a\u003e\r\nThe above three axioms, we have seen, are \u003cem\u003eà priori\u003c/em\u003e\r\nnecessary to metrical Geometry. No others can be necessary,\r\nsince metrical systems, logically as unassailable as Euclid\u0027s,\r\nand dealing with spaces equally homogeneous and equally relational,\r\nhave been constructed by the metageometers, without\r\nthe help of any other axioms. The remaining axioms of Euclidean\r\nGeometry\u0026mdash;the axiom of parallels, the axiom that the\r\nnumber of dimensions is three, and Euclid\u0027s form of the axiom\r\nof the straight line (two straight lines cannot enclose a space)\u0026mdash;are\r\nnot essential to the possibility of metrical Geometry,\r\n\u003cem\u003ei.e.\u003c/em\u003e, are not deducible from the fact that a science of spatial\r\nmagnitudes is possible. They are rather to be regarded as\r\nempirical laws, obtained, like the empirical laws of other\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_176\" id=\"Page_176\"\u003e[176]\u003c/a\u003e\u003c/span\u003e\r\nsciences, by actual investigation of the given subject-matter\u0026mdash;in\r\nthis instance, experienced space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e178.\u003c/b\u003e\u003ca name=\"N178\" id=\"N178\"\u003e\u003c/a\u003e\r\nIn summing up the distinctive argument of this\r\nSection, we may give it a more general form, and discuss\r\nthe conditions of measurement in any continuous manifold,\r\n\u003cem\u003ei.e.\u003c/em\u003e, the qualities necessary to the manifold, in order that\r\nquantities in it may be determinable, not only as to the more\r\nor less, but as to the precise \u003cem\u003ehow much\u003c/em\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eMeasurement, we may say, is the application of number\r\nto continua, or, if we prefer it, the transformation of mere\r\nquantity into number of units. Using \u003cem\u003equantity\u003c/em\u003e to denote\r\nthe vague more or less, and \u003cem\u003emagnitude\u003c/em\u003e to denote the precise\r\nnumber of units, the problem of measurement may be defined\r\nas the transformation of quantity into magnitude.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow a number, to begin with, is a whole consisting of\r\nsmaller units, all of these units being qualitatively alike.\r\nIn order, therefore, that a continuous quantity may be expressible\r\nas a number, it must, on the one hand, be itself\r\na whole, and must, on the other hand, be divisible into\r\nqualitatively similar parts. In the aspect of a whole, the\r\nquantity is \u003cem\u003eintensive\u003c/em\u003e; in the aspect of an aggregate of parts,\r\nit is \u003cem\u003eextensive\u003c/em\u003e. A purely intensive quantity, therefore, is not\r\nnumerable\u0026mdash;a purely extensive quantity, if any such could be\r\nimagined, would not be a single quantity at all, since it would\r\nhave to consist of wholly unsynthesized particulars. A measurable\r\nquantity, therefore, is a whole divisible into similar\r\nparts. But a continuous quantity, if divisible at all, must be\r\n\u003cem\u003einfinitely\u003c/em\u003e divisible. For otherwise the points at which it could\r\nbe divided would form natural barriers, and so destroy its\r\ncontinuity. But further, it is not sufficient that there should\r\nbe a possibility of division into mutually external parts; while\r\nthe parts, to be perceptible as parts, must be mutually external,\r\nthey must also, to be knowable as \u003cem\u003eequal\u003c/em\u003e parts, be\r\ncapable of overcoming their mutual externality. For this, as\r\nwe have seen, we require superposition, which involves Free\r\nMobility and homogeneity\u0026mdash;the absence of Free Mobility in\r\ntime, where all other requisites of measurement are fulfilled,\r\nrenders direct measurement of time impossible. Hence infinite\r\ndivisibility, free mobility, and homogeneity are necessary for\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_177\" id=\"Page_177\"\u003e[177]\u003c/a\u003e\u003c/span\u003e\r\nthe possibility of measurement in \u003cem\u003eany\u003c/em\u003e continuous manifold,\r\nand these, as we have seen, are equivalent to our three axioms.\r\nThese axioms are necessary, therefore, not only for spatial\r\nmeasurement, but for all measurement. The only manifold\r\ngiven in experience, in which these conditions are satisfied, is\r\nspace. All other exact measurement\u0026mdash;as could be proved, I\r\nbelieve, for every separate case\u0026mdash;is effected, as we saw in the\r\ncase of time, by reduction to a spatial correlative. This explains\r\nthe paramount importance, to exact science, of the\r\nmechanical view of nature, which reduces all phenomena to\r\nmotions in time and space. For number is, of all conceptions,\r\nthe easiest to operate with, and science seeks everywhere for\r\nan opportunity to apply it, but finds this opportunity only by\r\nmeans of spatial equivalents to phenomena\u003ca name=\"FNanchor_177_177\" id=\"FNanchor_177_177\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_177_177\" class=\"fnanchor\"\u003e[177]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e179.\u003c/b\u003e\u003ca name=\"N179\" id=\"N179\"\u003e\u003c/a\u003e\r\nWe have now seen in what the \u003cem\u003eà priori\u003c/em\u003e element of\r\nGeometry consists. This \u003cem\u003eà priori\u003c/em\u003e element may be defined as\r\nthe axioms common to Euclidean and non-Euclidean spaces,\r\nas the axioms deducible from the conception of a form of\r\nexternality, or\u0026mdash;in metrical Geometry\u0026mdash;as the axioms required\r\nfor the possibility of measurement. It remains to discuss, in\r\na final chapter, some questions of a more general philosophic\r\nnature, in which we shall have to desert the firm ground of\r\nmathematics and enter on speculations which I put forward very\r\ntentatively, and with little faith in their ultimate validity. The\r\nchief questions for this final chapter will be two: (1) How is\r\nsuch \u003cem\u003eà priori\u003c/em\u003e and purely logical necessity possible, as applied\r\nto an actually given subject-matter like space? (2) How\r\ncan we remove the contradictions which have haunted us in\r\nthis chapter, arising out of the relativity, infinite divisibility,\r\nand unbounded extension of space? These two questions are\r\nforced upon us by the present chapter, but as they open some\r\nof the fundamental problems of philosophy, it would be rash\r\nto expect a conclusive or wholly satisfactory answer. A few\r\nhints and suggestions may be hoped for, but a complete solution\r\ncould only be obtained from a complete philosophy, of which\r\nthe prospects are far too slender to encourage a confident\r\nframe of mind.\u003c/p\u003e\r\n\r\n\r\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_116_116\" id=\"Footnote_116_116\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_116_116\"\u003e\u003cspan class=\"label\"\u003e[116]\u003c/span\u003e\u003c/a\u003e See infra, Axiom of Distance, in Sec. \u003cspan class=\"fs70\"\u003eB.\u003c/span\u003e of this Chapter.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_117_117\" id=\"Footnote_117_117\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_117_117\"\u003e\u003cspan class=\"label\"\u003e[117]\u003c/span\u003e\u003c/a\u003e Thus on a cylinder, two geodesics, \u003cem\u003ee.g.\u003c/em\u003e a generator and a helix, may have\r\nany number of intersections\u0026mdash;a very important difference from the plane.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_118_118\" id=\"Footnote_118_118\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_118_118\"\u003e\u003cspan class=\"label\"\u003e[118]\u003c/span\u003e\u003c/a\u003e Cf. Cremona, \u003ccite class=\"fsnormal\"\u003eProjective Geometry\u003c/cite\u003e (Clarendon Press, 2nd ed. 1893) p. 50:\r\n\"Most of the propositions in Euclid\u0027s Elements are metrical, and it is not easy\r\nto find among them an example of a purely descriptive theorem.\"\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_119_119\" id=\"Footnote_119_119\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_119_119\"\u003e\u003cspan class=\"label\"\u003e[119]\u003c/span\u003e\u003c/a\u003e Op. cit. p. 226.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_120_120\" id=\"Footnote_120_120\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_120_120\"\u003e\u003cspan class=\"label\"\u003e[120]\u003c/span\u003e\u003c/a\u003e Some ground for this choice will appear when we come to metrical\r\nGeometry.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_121_121\" id=\"Footnote_121_121\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_121_121\"\u003e\u003cspan class=\"label\"\u003e[121]\u003c/span\u003e\u003c/a\u003e The straight line \u003cem\u003eσa\u003c/em\u003e denotes the straight line common to the planes \u003cem\u003eσ\u003c/em\u003e and\r\n\u003cem\u003ea\u003c/em\u003e, the point \u003cem\u003eσa\u003c/em\u003e denotes the point common to the plane \u003cem\u003eσ\u003c/em\u003e and the straight line\r\n\u003cem\u003ea\u003c/em\u003e, and similarly for the rest of the notation.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_122_122\" id=\"Footnote_122_122\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_122_122\"\u003e\u003cspan class=\"label\"\u003e[122]\u003c/span\u003e\u003c/a\u003e Cremona (op. cit. Chap. \u003cspan class=\"fs70\"\u003eIX.\u003c/span\u003e p. 50) defines anharmonic ratio as a metrical\r\nproperty which is unaltered by projection. This, however, destroys the logical\r\nindependence of projective Geometry, which can only be maintained by a purely\r\ndescriptive definition.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_123_123\" id=\"Footnote_123_123\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_123_123\"\u003e\u003cspan class=\"label\"\u003e[123]\u003c/span\u003e\u003c/a\u003e There is no corresponding property of \u003cem\u003ethree\u003c/em\u003e points on a line, because they\r\ncan be projectively transformed into any other three points on the same line.\r\nSee \u003ca href=\"#N120\"\u003e§ 120.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_124_124\" id=\"Footnote_124_124\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_124_124\"\u003e\u003cspan class=\"label\"\u003e[124]\u003c/span\u003e\u003c/a\u003e Due to v. Staudt\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Geometrie der Lage.\"\u003c/cite\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_125_125\" id=\"Footnote_125_125\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_125_125\"\u003e\u003cspan class=\"label\"\u003e[125]\u003c/span\u003e\u003c/a\u003e See Cremona, op. cit. Chapter \u003cspan class=\"fs70\"\u003eVIII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_126_126\" id=\"Footnote_126_126\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_126_126\"\u003e\u003cspan class=\"label\"\u003e[126]\u003c/span\u003e\u003c/a\u003e The corresponding definitions, for the two-dimensional manifold of lines\r\nthrough a point, follow by the principle of duality.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_127_127\" id=\"Footnote_127_127\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_127_127\"\u003e\u003cspan class=\"label\"\u003e[127]\u003c/span\u003e\u003c/a\u003e It is important to observe that this definition of the Point introduces\r\nmetrical ideas. Without metrical ideas, we saw, nothing appears to give the\r\nPoint precedence of the straight line, or indeed to distinguish it conceptually\r\nfrom the straight line. A reference to quantity is therefore inevitable in\r\ndefining the Point, if the definition is to be geometrical. A non-metrical\r\ndefinition would have to be also non-geometrical. See \u003ca href=\"#N196\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e §§ 196\u0026ndash;199.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_128_128\" id=\"Footnote_128_128\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_128_128\"\u003e\u003cspan class=\"label\"\u003e[128]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N163\"\u003e§§ 163\u0026ndash;175.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_129_129\" id=\"Footnote_129_129\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_129_129\"\u003e\u003cspan class=\"label\"\u003e[129]\u003c/span\u003e\u003c/a\u003e On this axiom, however, compare \u003ca href=\"#N131\"\u003e§ 131.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_130_130\" id=\"Footnote_130_130\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_130_130\"\u003e\u003cspan class=\"label\"\u003e[130]\u003c/span\u003e\u003c/a\u003e For the proof of this proposition, see \u003ca href=\"#SB\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Sec. \u003cspan class=\"fs70\"\u003eB\u003c/span\u003e\u003c/a\u003e, Axiom of\r\nDimensions.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_131_131\" id=\"Footnote_131_131\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_131_131\"\u003e\u003cspan class=\"label\"\u003e[131]\u003c/span\u003e\u003c/a\u003e The straight line and plane, in all discussions of general Geometry, are not\r\nnecessarily Euclidean. They are simply figures determined, in general, by two\r\nand by three points respectively; whether they conform to the axiom of parallels\r\nand to Euclid\u0027s form of the axiom of the straight line, is not to be considered in\r\nthe general definition.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_132_132\" id=\"Footnote_132_132\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_132_132\"\u003e\u003cspan class=\"label\"\u003e[132]\u003c/span\u003e\u003c/a\u003e That projective Geometry must have existential import, I shall attempt to\r\nprove in Chapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_133_133\" id=\"Footnote_133_133\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_133_133\"\u003e\u003cspan class=\"label\"\u003e[133]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eLogic\u003c/cite\u003e, Book \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Chapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_134_134\" id=\"Footnote_134_134\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_134_134\"\u003e\u003cspan class=\"label\"\u003e[134]\u003c/span\u003e\u003c/a\u003e Cf. Bradley\u0027s \u003ccite class=\"fsnormal\"\u003eLogic\u003c/cite\u003e, p. 63. It will be seen that the sense in which I have\r\nspoken of space as a principle of differentiation is not the sense of a \"principle\r\nof individuation\" which Bradley objects to.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_135_135\" id=\"Footnote_135_135\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_135_135\"\u003e\u003cspan class=\"label\"\u003e[135]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N186\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e §§ 186\u0026ndash;191.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_136_136\" id=\"Footnote_136_136\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_136_136\"\u003e\u003cspan class=\"label\"\u003e[136]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N201\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e § 201 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_137_137\" id=\"Footnote_137_137\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_137_137\"\u003e\u003cspan class=\"label\"\u003e[137]\u003c/span\u003e\u003c/a\u003e It is important to observe, however, that this way of regarding spatial\r\nrelations is metrical; from the projective standpoint, the relation between two\r\npoints is the whole unbounded straight line on which they lie, and need not be\r\nregarded as divisible into parts or as built up of points.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_138_138\" id=\"Footnote_138_138\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_138_138\"\u003e\u003cspan class=\"label\"\u003e[138]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N207\"\u003e§§ 207\u003c/a\u003e, \u003ca href=\"#N208\"\u003e208\u003c/a\u003e. Cf. Hegel, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eNaturphilosophie\u003c/cite\u003e, § 254.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_139_139\" id=\"Footnote_139_139\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_139_139\"\u003e\u003cspan class=\"label\"\u003e[139]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N196\"\u003eChap. \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e §§ 196\u0026ndash;199.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_140_140\" id=\"Footnote_140_140\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_140_140\"\u003e\u003cspan class=\"label\"\u003e[140]\u003c/span\u003e\u003c/a\u003e See a forthcoming article on \u003ccite class=\"fsnormal\"\u003e\"The relations of number and quantity\"\u003c/cite\u003e by\r\nthe present writer in \u003ccite class=\"fsnormal\"\u003eMind\u003c/cite\u003e, July, 1897.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_141_141\" id=\"Footnote_141_141\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_141_141\"\u003e\u003cspan class=\"label\"\u003e[141]\u003c/span\u003e\u003c/a\u003e Logic, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eVII.\u003c/span\u003e p. 211.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_142_142\" id=\"Footnote_142_142\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_142_142\"\u003e\u003cspan class=\"label\"\u003e[142]\u003c/span\u003e\u003c/a\u003e Real, as opposed to logical, diversity is throughout intended. Diverse\r\naspects may coexist in a thing at one time and place, but two diverse real\r\nthings cannot so coexist.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_143_143\" id=\"Footnote_143_143\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_143_143\"\u003e\u003cspan class=\"label\"\u003e[143]\u003c/span\u003e\u003c/a\u003e On the insufficiency of time alone, see \u003ca href=\"#N191\"\u003eChapter \u003cspan class=\"fs70\"\u003eIV.\u003c/span\u003e § 191.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_144_144\" id=\"Footnote_144_144\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_144_144\"\u003e\u003cspan class=\"label\"\u003e[144]\u003c/span\u003e\u003c/a\u003e Geometrically, the axiom of the plane is, not that three points determine\r\na figure at all, which follows from the axiom of the straight line, but that the\r\nstraight line joining two casual points of the plane lies wholly in the plane.\r\nThis axiom requires a projective method of constructing the plane, \u003cem\u003ei.e.\u003c/em\u003e of\r\nfinding all the triads of points which determine the same projective figure as\r\nthe given triad. The required construction will be obtained if we can find any\r\nprojective figure determined by three points, and any projective method\r\nof reaching other points which determine the same figure.\r\n\u003c/p\u003e\r\n\r\n\u003cdiv class=\"figcenter\"\u003e\r\n\u003cimg src=\"https://chrisdeasy.com/wp-content/uploads/gutenberg-an-essay-on-the-foundations-of-geometry-imagep145.jpg\" width=\"300\" alt=\"\" /\u003e\r\n\u003c/div\u003e\r\n\r\n\u003cp\u003e\r\nLet \u003cem\u003eO\u003c/em\u003e, \u003cem\u003eP\u003c/em\u003e, \u003cem\u003eQ\u003c/em\u003e be the three points whose projective relation is required.\r\nThen we have given us the three straight lines \u003cem\u003ePQ\u003c/em\u003e, \u003cem\u003eQO\u003c/em\u003e, \u003cem\u003eOP\u003c/em\u003e. Metrically, the\r\nrelation between these points is made up of the area, and the magnitude of the\r\nsides and angles, of the triangle \u003cem\u003eOPQ\u003c/em\u003e, just as the relation between two points\r\nis distance. But projectively, the figure is unchanged when \u003cem\u003eP\u003c/em\u003e and \u003cem\u003eQ\u003c/em\u003e travel\r\nalong \u003cem\u003eOP\u003c/em\u003e and \u003cem\u003eOQ\u003c/em\u003e, or when \u003cem\u003eOP\u003c/em\u003e and \u003cem\u003eOQ\u003c/em\u003e turn about \u003cem\u003eO\u003c/em\u003e in such a way as still to\r\nmeet \u003cem\u003ePQ\u003c/em\u003e. This is a result of the general principle of projective equivalence\r\nenunciated above (\u003ca href=\"#N108\"\u003e§§ 108\u003c/a\u003e, \u003ca href=\"#N109\"\u003e109\u003c/a\u003e). Hence the projective relation between \u003cem\u003eO\u003c/em\u003e, \u003cem\u003eP\u003c/em\u003e, \u003cem\u003eQ\u003c/em\u003e\r\nis the same as that between \u003cem\u003eO\u003c/em\u003e, \u003cem\u003ep\u003c/em\u003e, \u003cem\u003eq\u003c/em\u003e or \u003cem\u003eO\u003c/em\u003e, \u003cem\u003eP′\u003c/em\u003e, \u003cem\u003eQ′\u003c/em\u003e; that is, \u003cem\u003ep\u003c/em\u003e, \u003cem\u003eq\u003c/em\u003e and \u003cem\u003eP′\u003c/em\u003e, \u003cem\u003eQ′\u003c/em\u003e lie in\r\nthe plane \u003cem\u003eOPQ\u003c/em\u003e. In this way, any number of points on the plane may be\r\nobtained, and by repeating the construction with fresh triads, every point of the\r\nplane can be reached. We have to prove that, when the plane is so constructed,\r\nthe straight line joining any two points of the plane lies wholly in the plane.\r\n\u003c/p\u003e\r\n\u003cp\u003e\r\nIt is evident, from the manner of construction, that any point of \u003cem\u003ePQ\u003c/em\u003e, \u003cem\u003eOP\u003c/em\u003e,\r\n\u003cem\u003eOQ\u003c/em\u003e, \u003cem\u003eOP′\u003c/em\u003e or \u003cem\u003eOQ′\u003c/em\u003e lies in the plane. If we can prove that any point of \u003cem\u003epq\u003c/em\u003e lies in\r\nthe plane, we shall have proved all that is required, since \u003cem\u003epq\u003c/em\u003e may be transformed,\r\nby successive repetitions of the same construction, into any straight line\r\njoining two points of the plane. But we have seen that the same plane is\r\ndetermined by \u003cem\u003eO\u003c/em\u003e, \u003cem\u003ep\u003c/em\u003e, \u003cem\u003eq\u003c/em\u003e and by \u003cem\u003eO\u003c/em\u003e, \u003cem\u003eP\u003c/em\u003e, \u003cem\u003eQ\u003c/em\u003e. The straight lines \u003cem\u003ePQ\u003c/em\u003e, \u003cem\u003epq\u003c/em\u003e have, therefore,\r\nthe same relation to the plane. But \u003cem\u003ePQ\u003c/em\u003e lies wholly in the plane; therefore \u003cem\u003epq\u003c/em\u003e\r\nalso lies wholly in the plane. Hence our axiom is proved.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_145_145\" id=\"Footnote_145_145\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_145_145\"\u003e\u003cspan class=\"label\"\u003e[145]\u003c/span\u003e\u003c/a\u003e A detailed proof has been given above, Chap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e 3rd period. It is to\r\nbe observed that any reference to infinitely distant elements involves metrical\r\nideas.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_146_146\" id=\"Footnote_146_146\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_146_146\"\u003e\u003cspan class=\"label\"\u003e[146]\u003c/span\u003e\u003c/a\u003e Cf. \u003ca href=\"#N115\"\u003eSection A, §§ 115\u0026ndash;117.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_147_147\" id=\"Footnote_147_147\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_147_147\"\u003e\u003cspan class=\"label\"\u003e[147]\u003c/span\u003e\u003c/a\u003e Contrast Erdmann, op. cit. p. 138.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_148_148\" id=\"Footnote_148_148\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_148_148\"\u003e\u003cspan class=\"label\"\u003e[148]\u003c/span\u003e\u003c/a\u003e Cf. Erdmann, op. cit. p. 164.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_149_149\" id=\"Footnote_149_149\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_149_149\"\u003e\u003cspan class=\"label\"\u003e[149]\u003c/span\u003e\u003c/a\u003e Strictly speaking, this method is only applicable where the two magnitudes\r\nare commensurable. But if we take infinite divisibility rigidly, the units can\r\ntheoretically be taken so small as to obtain any required degree of approximation.\r\nThe difficulty is the universal one of applying to continua the essentially\r\ndiscrete conception of number.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_150_150\" id=\"Footnote_150_150\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_150_150\"\u003e\u003cspan class=\"label\"\u003e[150]\u003c/span\u003e\u003c/a\u003e Cf. Erdmann, op. cit. p. 50.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_151_151\" id=\"Footnote_151_151\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_151_151\"\u003e\u003cspan class=\"label\"\u003e[151]\u003c/span\u003e\u003c/a\u003e Also called the axiom of congruence. I have taken congruence to be the\r\n\u003cem\u003edefinition\u003c/em\u003e of spatial equality by superposition, and shall therefore generally speak\r\nof the \u003cem\u003eaxiom\u003c/em\u003e as Free Mobility.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_152_152\" id=\"Footnote_152_152\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_152_152\"\u003e\u003cspan class=\"label\"\u003e[152]\u003c/span\u003e\u003c/a\u003e For the sense in which these figures are to be regarded as material,\r\nsee criticism of Helmholtz, \u003ca href=\"#N69\"\u003eChapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e §§ 69 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_153_153\" id=\"Footnote_153_153\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_153_153\"\u003e\u003cspan class=\"label\"\u003e[153]\u003c/span\u003e\u003c/a\u003e Op. cit. p. 60.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_154_154\" id=\"Footnote_154_154\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_154_154\"\u003e\u003cspan class=\"label\"\u003e[154]\u003c/span\u003e\u003c/a\u003e The view of Helmholtz and Erdmann, that mechanical experience suffices\r\nhere, though geometrical experience fails us, has been discussed above,\r\n\u003ca href=\"#N73\"\u003eChapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e §§ 73\u003c/a\u003e, \u003ca href=\"#N82\"\u003e82.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_155_155\" id=\"Footnote_155_155\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_155_155\"\u003e\u003cspan class=\"label\"\u003e[155]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N81\"\u003eChapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e § 81.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_156_156\" id=\"Footnote_156_156\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_156_156\"\u003e\u003cspan class=\"label\"\u003e[156]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N72\"\u003eChapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e § 72.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_157_157\" id=\"Footnote_157_157\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_157_157\"\u003e\u003cspan class=\"label\"\u003e[157]\u003c/span\u003e\u003c/a\u003e\r\nContrast \u003cins class=\"corr\" title=\"Transcriber\u0027s Note\u0026mdash;Original text: \u0027Delboeuf\u0027\"\u003eDelbœuf\u003c/ins\u003e, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003eL\u0027ancienne et les nouvelles géométries\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Rev. Phil.\r\n1894, Vol. xxxvii. p. 354.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_158_158\" id=\"Footnote_158_158\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_158_158\"\u003e\u003cspan class=\"label\"\u003e[158]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eProlegomena\u003c/cite\u003e, § 13. See Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 518\u0026ndash;532 esp.\r\npp. 521\u0026ndash;2. The above was Kant\u0027s whole purpose in 1768, but only part of his\r\npurpose in the Prolegomena, where the intuitive nature of space was also to be\r\nproved.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_159_159\" id=\"Footnote_159_159\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_159_159\"\u003e\u003cspan class=\"label\"\u003e[159]\u003c/span\u003e\u003c/a\u003e On the subject of time measurement, cf. Bosanquet\u0027s \u003ccite class=\"fsnormal\"\u003eLogic\u003c/cite\u003e, Vol. i.\r\npp. 178\u0026ndash;183. Since time, in the above account, is measured by motion, its\r\nmeasurement presupposes that of spatial magnitudes.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_160_160\" id=\"Footnote_160_160\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_160_160\"\u003e\u003cspan class=\"label\"\u003e[160]\u003c/span\u003e\u003c/a\u003e Cf. Stumpf. \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eUrsprung der Raumvorstellung\u003c/cite\u003e, p. 68.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_161_161\" id=\"Footnote_161_161\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_161_161\"\u003e\u003cspan class=\"label\"\u003e[161]\u003c/span\u003e\u003c/a\u003e As is Helmholtz\u0027s other axiom, that the possibility of superposition\r\nis independent of the course pursued in bringing it about.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_162_162\" id=\"Footnote_162_162\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_162_162\"\u003e\u003cspan class=\"label\"\u003e[162]\u003c/span\u003e\u003c/a\u003e Cf. \u003ca href=\"#N129\"\u003e§§ 129\u003c/a\u003e, \u003ca href=\"#N130\"\u003e130.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_163_163\" id=\"Footnote_163_163\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_163_163\"\u003e\u003cspan class=\"label\"\u003e[163]\u003c/span\u003e\u003c/a\u003e This deduction is practically the same as that in Sec. A, but I have stated\r\nit here with more special reference to space and to metrical Geometry.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_164_164\" id=\"Footnote_164_164\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_164_164\"\u003e\u003cspan class=\"label\"\u003e[164]\u003c/span\u003e\u003c/a\u003e The question: \"Relations to what?\" is a question involving many\r\ndifficulties. It will be touched on later in this chapter, and answered, as far as\r\npossible, in the fourth chapter. For the present, in spite of the glaring circle\r\ninvolved, I shall take the relations as relations to other positions.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_165_165\" id=\"Footnote_165_165\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_165_165\"\u003e\u003cspan class=\"label\"\u003e[165]\u003c/span\u003e\u003c/a\u003e Wiss. Abh. Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 614.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_166_166\" id=\"Footnote_166_166\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_166_166\"\u003e\u003cspan class=\"label\"\u003e[166]\u003c/span\u003e\u003c/a\u003e Cp. Grassmann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eAusdehnungslehre von 1844\u003c/cite\u003e, 2nd ed. p. \u003cspan class=\"fs70\"\u003eXXIII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_167_167\" id=\"Footnote_167_167\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_167_167\"\u003e\u003cspan class=\"label\"\u003e[167]\u003c/span\u003e\u003c/a\u003e\r\n\u003cins class=\"corr\" title=\"Transcriber\u0027s Note\u0026mdash;Original text: \u0027Delboeuf\u0027\"\u003eDelbœuf\u003c/ins\u003e, it is true, speaks of Geometries with \u003cem\u003em\u003c/em\u003e/\u003cem\u003en\u003c/em\u003e dimensions, but gives\r\nno reference (Rev. Phil. T. xxxvi. p. 450).\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_168_168\" id=\"Footnote_168_168\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_168_168\"\u003e\u003cspan class=\"label\"\u003e[168]\u003c/span\u003e\u003c/a\u003e In criticizing Erdmann, it will be remembered, we saw that Free Mobility\r\nis a necessary property of his extents, though he does not regard it as such.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_169_169\" id=\"Footnote_169_169\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_169_169\"\u003e\u003cspan class=\"label\"\u003e[169]\u003c/span\u003e\u003c/a\u003e Cf. Riemann, \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eHypothesen welche der Geometrie zu Grunde liegen,\r\nGesammelte Werke,\u003c/cite\u003e p. 266; also Erdmann, op. cit. p. 154.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_170_170\" id=\"Footnote_170_170\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_170_170\"\u003e\u003cspan class=\"label\"\u003e[170]\u003c/span\u003e\u003c/a\u003e This is subject, in spherical space, to the modification pointed out below,\r\nin dealing with the exception to the axiom of the straight line. See\r\n\u003ca href=\"#N168\"\u003e§§ 168\u0026ndash;171.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_171_171\" id=\"Footnote_171_171\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_171_171\"\u003e\u003cspan class=\"label\"\u003e[171]\u003c/span\u003e\u003c/a\u003e In speaking of distance at once as a quantity and as an intrinsic\r\nrelation, I am anxious to guard against an apparent inconsistency. I have\r\nspoken of the judgment of quantity, throughout, as one of comparison; how,\r\nthen, can a quantity be intrinsic? The reply is that, although measurement\r\nand the judgment of quantity express the result of comparison, yet the terms\r\ncompared must exist before the comparison; in this case, the terms compared\r\nin measuring distances, \u003cem\u003ei.e.\u003c/em\u003e in comparing them \u003ci lang=\"la\" xml:lang=\"la\"\u003einter se\u003c/i\u003e, are intrinsic relations\r\nbetween points. Thus, although the \u003cem\u003emeasurement\u003c/em\u003e of distance involves a\r\nreference to other distances, and its expression as a magnitude requires such a\r\nreference, yet its existence does not depend on any external reference, but\r\nexclusively on the two points whose distance it is.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_172_172\" id=\"Footnote_172_172\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_172_172\"\u003e\u003cspan class=\"label\"\u003e[172]\u003c/span\u003e\u003c/a\u003e See the end of the argument on Free Mobility, \u003ca href=\"#N155\"\u003e§ 155 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_173_173\" id=\"Footnote_173_173\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_173_173\"\u003e\u003cspan class=\"label\"\u003e[173]\u003c/span\u003e\u003c/a\u003e In Frischauf\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003e\"Absolute Geometrie nach Johann Bolyai,\"\u003c/cite\u003e Anhang, there\r\nis a series of definitions, starting from the sphere, as the locus of congruent\r\npoint-pairs when one point of the pair is fixed, and hence obtaining the circle\r\nand the straight line. From the above it follows, that the sphere so defined\r\nalready involves a curve between the points of the point-pair, by which\r\nvarious point-pairs can be known as congruent; and it will appear, as we\r\nproceed, that this curve must be a straight line. Frischauf\u0027s definition by\r\nmeans of the sphere involves, therefore, a vicious circle, since the sphere\r\npresupposes the straight line, as the test of congruent point-pairs.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_174_174\" id=\"Footnote_174_174\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_174_174\"\u003e\u003cspan class=\"label\"\u003e[174]\u003c/span\u003e\u003c/a\u003e Nor in any argument which, like those of projective Geometry, avoids the\r\nnotion of magnitude or distance altogether. It follows that the propositions of\r\nprojective Geometry apply, without reserve, to spherical space, since the\r\nexception to the axiom of the straight line arises only on metrical ground.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_175_175\" id=\"Footnote_175_175\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_175_175\"\u003e\u003cspan class=\"label\"\u003e[175]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003ePsychology\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 149\u0026ndash;150.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_176_176\" id=\"Footnote_176_176\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_176_176\"\u003e\u003cspan class=\"label\"\u003e[176]\u003c/span\u003e\u003c/a\u003e This step in the argument has been put very briefly, since it is a mere\r\nrepetition of the corresponding argument in Section A, and is inserted here only\r\nfor the sake of logical completeness. See \u003ca href=\"#N137\"\u003e§ 137 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_177_177\" id=\"Footnote_177_177\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_177_177\"\u003e\u003cspan class=\"label\"\u003e[177]\u003c/span\u003e\u003c/a\u003e Cf. Hannequin, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003eEssai critique sur l\u0027hypothèse des atomes\u003c/cite\u003e, Paris, 1895,\r\npassim.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\r\n\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_178\" id=\"Page_178\"\u003e[178]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\u003cp class=\"p4\" /\u003e\r\n\r\n\u003ch2\u003e\u003ca name=\"CHAP_IV\" id=\"CHAP_IV\"\u003e\u003c/a\u003eCHAPTER IV.\u003cbr /\u003e\r\n\u003cbr /\u003e\r\n\u003cspan class=\"fs60\"\u003ePHILOSOPHICAL CONSEQUENCES.\u003c/span\u003e\u003c/h2\u003e\r\n\r\n\r\n\u003cp\u003e\u003cb\u003e180.\u003c/b\u003e\u003ca name=\"N180\" id=\"N180\"\u003e\u003c/a\u003e\r\nIn the present chapter, we have to discuss two questions\r\nwhich, though scarcely geometrical, are of fundamental\r\nimportance to the theory of Geometry propounded above. The\r\nfirst of these questions is this: What relation can a purely\r\nlogical and deductive proof, like that from the nature of a form\r\nof externality, bear to an experienced subject-matter such as\r\nspace? You have merely framed a general conception, I may\r\nbe told, containing space as a particular species, and you have\r\nthen shown, what should have been obvious from the beginning,\r\nthat this general conception contained some of the attributes\r\nof space. But what ground does this give for regarding these\r\nattributes as \u003cem\u003eà priori\u003c/em\u003e? The conception Mammal has some of\r\nthe attributes of a horse; but are these attributes therefore \u003cem\u003eà\r\npriori\u003c/em\u003e adjectives of the horse? The answer to this obvious\r\nobjection is so difficult, and involves so much general philosophy,\r\nthat I have kept it for a final chapter, in order not to\r\ninterrupt the argument on specially geometrical topics.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e181.\u003c/b\u003e\u003ca name=\"N181\" id=\"N181\"\u003e\u003c/a\u003e\r\nI have already indicated, in general terms, the ground\r\nfor regarding as \u003cem\u003eà priori\u003c/em\u003e the properties of any form of externality.\r\nThis ground is transcendental, \u003cem\u003ei.e.\u003c/em\u003e it is to be found in\r\nthe conditions required for the possibility of experience. The\r\nform of externality, like Riemann\u0027s manifolds, is a general class-conception,\r\nincluding time as well as Euclidean and non-Euclidean\r\nspaces. It is not motived, however, like the manifolds,\r\nby a \u003cem\u003equantitative\u003c/em\u003e resemblance to space, but by the fact that\r\nit fulfils, if it has more than one dimension, all those functions\r\nwhich, in our actual world, are fulfilled by space. But a form\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_179\" id=\"Page_179\"\u003e[179]\u003c/a\u003e\u003c/span\u003e\r\nof externality, in order to accomplish this, must be, not a mere\r\nconception, but an actually experienced intuition. Hence the\r\nconception of such a form is the general \u003cem\u003econception\u003c/em\u003e, containing\r\nunder it every logically possible \u003cem\u003eintuition\u003c/em\u003e which can fulfil the\r\nfunction actually fulfilled by space. And this function is, to\r\nrender possible experience of diverse but interrelated things.\r\nSome form in sense-perception, then, whose conception is\r\nincluded under our form of externality, is \u003cem\u003eà priori\u003c/em\u003e necessary to\r\nexperience of diversity in relation, and without experience of this,\r\nwe should, as modern logic shows, have no experience at all.\r\nThis still leaves untouched the relation of the \u003cem\u003eà priori\u003c/em\u003e to the\r\nsubjective: the form of externality is necessary to experience,\r\nbut is not, \u003cem\u003eon that account\u003c/em\u003e, to be declared purely subjective. Of\r\ncourse, necessity for experience can only arise from the nature\r\nof the mind which experiences; but it does not follow that the\r\nnecessary conditions could be fulfilled, unless the objective\r\nworld had certain properties. The \u003cem\u003eground\u003c/em\u003e of necessity, we may\r\nsafely say, arises from the mind; but it by no means follows\r\nthat the \u003cem\u003etruth\u003c/em\u003e of what is necessary depends only on the constitution\r\nof the mind. Where this is not the case, our conclusion,\r\nwhen a piece of knowledge has been declared \u003cem\u003eà priori\u003c/em\u003e,\r\ncan only be: Owing to the constitution of the \u003cem\u003emind\u003c/em\u003e, experience\r\nwill be impossible unless the \u003cem\u003eworld\u003c/em\u003e accepts certain adjectives.\u003c/p\u003e\r\n\r\n\u003cp\u003eSuch, in outline, will be the argument of the first half of\r\nthis chapter, and such will be the justification for regarding\r\nas \u003cem\u003eà priori\u003c/em\u003e those axioms of Geometry, which were deduced\r\nabove from the conception of a form of externality. For these\r\naxioms, and these only, are necessarily true of any world in\r\nwhich experience is possible.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e182\u003ca name=\"FNanchor_178_178\" id=\"FNanchor_178_178\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_178_178\" class=\"fnanchor\"\u003e[178]\u003c/a\u003e.\u003c/b\u003e\u003ca name=\"N182\" id=\"N182\"\u003e\u003c/a\u003e\r\nThe view suggested has, obviously, much in common\r\nwith that of the Transcendental Aesthetic. Indeed the whole\r\nof it, I believe, can be obtained by a certain limitation and\r\ninterpretation of Kant\u0027s classic arguments. But as it differs,\r\nin many important points, from the conclusions aimed at by\r\nKant, and as the agreement may easily seem greater than it is,\r\nI will begin by a brief comparison, and endeavour, by reference\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_180\" id=\"Page_180\"\u003e[180]\u003c/a\u003e\u003c/span\u003e\r\nto authoritative criticisms, to establish the legitimacy of my\r\ndivergence from him.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e183.\u003c/b\u003e\u003ca name=\"N183\" id=\"N183\"\u003e\u003c/a\u003e\r\nIn the first place, the psychological element is much\r\nlarger in Kant\u0027s thesis than in mine. I shall contend, it is true,\r\nthat a form of externality, if it is to do its work, must not be\r\na mere conception or a mere inference, but must be a given\r\nelement in sense-perception\u0026mdash;not, of course, originally given in\r\nisolation, but discoverable, through analysis, by attention to\r\nthe object of sense-perception\u003ca name=\"FNanchor_179_179\" id=\"FNanchor_179_179\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_179_179\" class=\"fnanchor\"\u003e[179]\u003c/a\u003e. But Kant contended, not only\r\nthat this element is given, but also that it is subjective. Space,\r\nfor him, is, on the one hand, not conceptual, but on the other\r\nhand, not sensational. It forms, for him, no part of the data of\r\nsense, but is added by a subjective intuition, which he regards\r\nas not only logically, but psychologically, prior to objects in\r\nspace\u003ca name=\"FNanchor_180_180\" id=\"FNanchor_180_180\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_180_180\" class=\"fnanchor\"\u003e[180]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis part of Kant\u0027s argument is wholly irrelevant for us.\r\nWhether a form of externality be given in sense, or in a pure\r\nintuition, is for us unimportant, since we neglect the question\r\nas to the connection of the \u003cem\u003eà priori\u003c/em\u003e and the subjective; while\r\nthe temporal priority of space to objects in it has been generally\r\nrecognized as irrelevant to Epistemology, and has often\r\nbeen regarded as forming no part of Kant\u0027s thesis\u003ca name=\"FNanchor_181_181\" id=\"FNanchor_181_181\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_181_181\" class=\"fnanchor\"\u003e[181]\u003c/a\u003e. If we call\r\nintuitional whatever is given in sense-perception, then we may\r\ncontend that a form of externality must be intuitional; but\r\nwhether it is a pure intuition, in Kant\u0027s sense, or not, is\r\nirrelevant to us, as is its priority to the objects in it.\u003c/p\u003e\r\n\r\n\u003cp\u003eThat the non-sensational nature of space is no essential part\r\nof Kant\u0027s \u003cem\u003elogical\u003c/em\u003e teaching, appears from an examination of his\r\nargument. He has made, in the introduction, the purely\r\nlogical distinction of matter and form, but has given to this\r\ndistinction, in the very moment of suggesting it, a psychological\r\nimplication. This he does by the assertion that the\r\nform, in which the matter of sensations is ordered, cannot\r\nitself be sensational. From this assumption it follows, of course,\r\nthat space cannot be sensational. But the assumption is\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_181\" id=\"Page_181\"\u003e[181]\u003c/a\u003e\u003c/span\u003etotally unsupported by argument, being set forth, apparently,\r\nas a self-evident axiom; it has been severely criticized by\r\nStumpf\u003ca name=\"FNanchor_182_182\" id=\"FNanchor_182_182\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_182_182\" class=\"fnanchor\"\u003e[182]\u003c/a\u003e and others\u003ca name=\"FNanchor_183_183\" id=\"FNanchor_183_183\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_183_183\" class=\"fnanchor\"\u003e[183]\u003c/a\u003e, and has been described by Vaihinger as a\r\nfatal \u003ci lang=\"la\" xml:lang=\"la\"\u003epetitio principii\u003c/i\u003e\u003ca name=\"FNanchor_184_184\" id=\"FNanchor_184_184\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_184_184\" class=\"fnanchor\"\u003e[184]\u003c/a\u003e; it is irrelevant to the logical argument,\r\nwhen this argument is separated, as we have separated it, from\r\nall connection with psychological subjectivity; and finally, it\r\nleaves us a prey to psychological theories of space, which have\r\nseemed, of late, but little favourable to the pure Kantian\r\ndoctrine.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e184.\u003c/b\u003e\u003ca name=\"N184\" id=\"N184\"\u003e\u003c/a\u003e\r\nWe have a right, therefore, in an epistemological\r\ninquiry, to neglect Kant\u0027s psychological teaching\u0026mdash;in so far,\r\nat any rate, as it distinguishes spatial intuition from sensation\u0026mdash;and\r\nattend rather to the logical aspect alone. That part of\r\nhis psychological teaching, which maintains that space is not a\r\nmere conception, is, with certain limitations, sufficiently evident\r\nas applied to actual space; but for us, it must be transformed\r\ninto a much more difficult thesis, namely, that \u003cem\u003eno\u003c/em\u003e form of\r\nexternality, which renders experience of diversity in relation\r\npossible, can be merely conceptual. This question, to which we\r\nmust return later, is no longer psychological, but belongs wholly\r\nto Epistemology.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e185.\u003c/b\u003e\u003ca name=\"N185\" id=\"N185\"\u003e\u003c/a\u003e\r\nWhat, then, remains the kernel, for our purposes, of\r\nKant\u0027s first argument for the apriority of space? His argument,\r\nin the form in which he gave it, is concerned with the\r\neccentric projection of sensations. In order that I may refer\r\nsensations, he says, to something outside myself, I must already\r\nhave the subjective space-form in the mind. In this shape, as\r\nVaihinger points out (\u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 69, 165), the argument\r\nrests on a \u003ci lang=\"la\" xml:lang=\"la\"\u003epetitio principii\u003c/i\u003e, for only if sensations are\r\nnecessarily non-spatial does their projection demand a subjective\r\nspace-form. But, further, is the logical apriority of space\r\nconcerned with the externality of things to ourselves?\u003c/p\u003e\r\n\r\n\u003cp\u003eSpace \u003cem\u003eseems\u003c/em\u003e to perform two functions: on the one hand,\r\nit reveals things, by the eccentric projection of sensations,\r\nas external to the self, while, on the other hand, it reveals\r\nsimultaneously presented things as mutually external. These\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_182\" id=\"Page_182\"\u003e[182]\u003c/a\u003e\u003c/span\u003etwo functions, though often treated as coordinate and almost\r\nequivalent\u003ca name=\"FNanchor_185_185\" id=\"FNanchor_185_185\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_185_185\" class=\"fnanchor\"\u003e[185]\u003c/a\u003e, seem to me widely different. Before we discuss\r\nthe apriority of space, we must carefully distinguish, I think,\r\nbetween these two functions, and decide which of them we are\r\nto argue about.\u003c/p\u003e\r\n\r\n\u003cp\u003eNow externality to the Self, it would seem, must necessarily\r\nraise the whole question of the nature and limits of the Ego,\r\nand what is more, it cannot be derived from spatial presentation,\r\nunless we give the Self a definite position in space. But things\r\nacquire a position in space only when they can appear in sense-perception;\r\nwe are forced, therefore, if we adopt this view of\r\nthe function of space, to regard the Self as a phenomenon\r\npresented to sense-perception. But this reduces externality to\r\nthe Self to externality to the body. The body, however, is a\r\npresented object like any other, and externality of objects to it\r\nis, therefore, a special case of the mutual externality of presented\r\nthings. Hence we cannot regard space as giving, primarily at\r\nany rate, externality to the Self, but only the mutual externality\r\nof the things presented to sense-perception\u003ca name=\"FNanchor_186_186\" id=\"FNanchor_186_186\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_186_186\" class=\"fnanchor\"\u003e[186]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e186.\u003c/b\u003e\u003ca name=\"N186\" id=\"N186\"\u003e\u003c/a\u003e\r\nThis, then, is the kind of externality we are to expect\r\nfrom space, and our question must be: Would the existence of\r\ndiverse but interrelated things be unknowable, if there were\r\nnot, in sense-perception, some form of externality? This is the\r\ncrucial question, on which turns the apriority of our form, and\r\nhence of the necessary axioms of Geometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e187.\u003c/b\u003e\u003ca name=\"N187\" id=\"N187\"\u003e\u003c/a\u003e\r\nThe converse argument to mine, the argument from\r\nthe spatio-temporal element in perception to a world of interrelated\r\nbut diverse things, is developed at length in Bradley\u0027s\r\nLogic. It is put briefly in the following sentence (p. 44, note):\r\n\"If space and time are continuous, and if all appearance must\r\noccupy some time or space\u0026mdash;and it is not hard to support both\r\nthese \u003cem\u003etheses\u003c/em\u003e\u0026mdash;we can at once proceed to the conclusion, no mere\r\nparticular exists. Every phenomenon will exist in more times\r\nor spaces than one; and against that diversity will be itself an\r\nuniversal\u003ca name=\"FNanchor_187_187\" id=\"FNanchor_187_187\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_187_187\" class=\"fnanchor\"\u003e[187]\u003c/a\u003e.\" The importance of this fact appears, when we\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_183\" id=\"Page_183\"\u003e[183]\u003c/a\u003e\u003c/span\u003econsider that, if any \u003cem\u003emere\u003c/em\u003e particular existed, all judgment and\r\ninference as to that particular would be impossible, since all\r\njudgment and inference necessarily operate by means of universal.\r\nBut all reality is constructed from the \u003cem\u003eThis\u003c/em\u003e of immediate\r\npresentation, from which judgment and inference necessarily\r\nspring. Owing, however, to the continuity and relativity of\r\nspace and time, no \u003cem\u003eThis\u003c/em\u003e can be regarded either as simple or as\r\nself-subsistent. Every \u003cem\u003eThis\u003c/em\u003e, on the one hand, can be analyzed\r\ninto \u003cem\u003eThises\u003c/em\u003e, and on the other hand, is found to be necessarily\r\nrelated to other things, outside the limits of the given object\r\nof sense-perception. This function of space and time is presupposed\r\nin the following statement from Bosanquet\u0027s Logic\r\n(Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e pp. 77\u0026ndash;78): \"Reality is given for me \u003cem\u003ein\u003c/em\u003e present\r\nsensuous perception, and \u003cem\u003ein\u003c/em\u003e the immediate feeling of my own\r\nsentient existence that goes with it. The real world, as a\r\ndefinite organized system, is \u003cem\u003efor me\u003c/em\u003e an extension of this present\r\nsensation and self feeling by means of judgment, and it is the\r\nessence of judgment to effect and sustain such an extension….\r\nThe subject in every judgment of Perception is some given spot\r\nor point in sensuous contact with the percipient self. But, as\r\nall reality is continuous, the subject is not \u003cem\u003emerely\u003c/em\u003e this given\r\nspot or point.\"\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e188.\u003c/b\u003e\u003ca name=\"N188\" id=\"N188\"\u003e\u003c/a\u003e\r\nThis doctrine of Bradley and Bosanquet is the\r\nconverse of the epistemological doctrine I have to advocate.\r\nOwing to the continuity and relativity of space and time, they\r\nsay, we are able to construct a systematic world, by judgment\r\nand inference, out of that fragmentary and yet necessarily\r\ncomplex existence which is given in sense-perception. My\r\ncontention is, conversely, that since all knowledge is necessarily\r\nderived by an extension of the \u003cem\u003eThis\u003c/em\u003e of sense-perception, and\r\nsince such extension is only possible if the \u003cem\u003eThis\u003c/em\u003e has that\r\nfragmentary and yet complex character conferred by a form of\r\nexternality, therefore some form of externality, given with the\r\n\u003cem\u003eThis\u003c/em\u003e, is essential to all knowledge, and is thus logically \u003cem\u003eà priori\u003c/em\u003e.\r\nBradley\u0027s argument, if sound, already proves this contention;\r\nfor while, on the one hand, he uses no properties of space and\r\ntime but those which belong to every form of externality, he\r\nproves, on the other hand, that judgment and inference require\r\nthe \u003cem\u003eThis\u003c/em\u003e to be neither single nor self-subsistent. But I will\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_184\" id=\"Page_184\"\u003e[184]\u003c/a\u003e\u003c/span\u003e\r\nendeavour, since the point is of fundamental importance, to\r\nreproduce the proof, in a form more suited than Bradley\u0027s to\r\nthe epistemological question.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e189.\u003c/b\u003e\u003ca name=\"N189\" id=\"N189\"\u003e\u003c/a\u003e\r\nThe essence of my contention is that, if experience is\r\nto be possible, every sensational \u003cem\u003eThis\u003c/em\u003e must, when attended to,\r\nbe found, on the one hand, resolvable into \u003cem\u003eThises\u003c/em\u003e, and on the\r\nother hand dependent, for some of its adjectives, on external\r\nreference. The second of these theses follows from the first,\r\nfor if we take one of the \u003cem\u003eThises\u003c/em\u003e contained in the first \u003cem\u003eThis\u003c/em\u003e, we\r\nget a new \u003cem\u003eThis\u003c/em\u003e necessarily related to the other \u003cem\u003eThises\u003c/em\u003e which\r\nmake up the original \u003cem\u003eThis\u003c/em\u003e. I may, therefore, confine myself to\r\nthe first proposition, which affirms that the object of perception\r\nmust contain a diversity, not only of conceptual content, but of\r\nexistence, and that this can only be known if sense-perception\r\ncontains, as an element, some form of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003eMy premiss, in this argument, is that all knowledge involves\r\na recognition of diversity in relation, or, if we prefer it, of\r\nidentity in difference. This premiss I accept from Logic, as\r\nresulting from the analysis of judgment and inference. To\r\nprove such a premiss, would require a treatise on Logic; I\r\nmust refer the reader, therefore, to the works of Bradley and\r\nBosanquet on the subject. It follows at once, from my premiss,\r\nthat knowledge would be impossible, unless the object of\r\nattention could be complex, \u003cem\u003ei.e.\u003c/em\u003e not a \u003cem\u003emere\u003c/em\u003e particular. Now\r\ncould the mental object\u0026mdash;\u003cem\u003ei.e.\u003c/em\u003e, in this connection, the object of\r\na cognition\u0026mdash;be complex, if the object of immediate perception\r\nwere always simple?\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e190.\u003c/b\u003e\u003ca name=\"N190\" id=\"N190\"\u003e\u003c/a\u003e\r\nWe might be inclined, at first sight, to answer this\r\nquestion affirmatively. But several difficulties, I think, would\r\nprevent such an answer. In the first place, knowledge must\r\nstart from perception. Hence, either we could have no knowledge\r\nexcept of our present perception, or else we must be able\r\nto contrast and compare it with some other perception. Now\r\nin the first case, since the present perception, by hypothesis, is\r\na mere particular, knowledge of it is impossible, according to\r\nour premiss. But in the second case, the other perception,\r\nwith which we compare our first, must have occurred at some\r\nother time, and with time, we have at once a form of externality.\r\nBut what is more, our present perception is no\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_185\" id=\"Page_185\"\u003e[185]\u003c/a\u003e\u003c/span\u003e\r\nlonger a mere particular. For the power of comparing it with\r\nanother perception involves a point of identity between the\r\ntwo, and thus renders both complex. Moreover, time must be\r\ncontinuous, and the present, as Bradley points out, is no mere\r\npoint of time\u003ca name=\"FNanchor_188_188\" id=\"FNanchor_188_188\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_188_188\" class=\"fnanchor\"\u003e[188]\u003c/a\u003e. Thus our present perception contains the\r\ncomplexity involved in duration throughout the specious present:\r\nits mere particularity and its simplicity are lost. Its\r\nself-subsistence is also lost, for beyond the specious present, lie\r\nthe past and the future, to which our present perception thus\r\nunavoidably refers us. Time at least, therefore, is essential to\r\nthat identity in difference, which all knowledge postulates.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e191.\u003c/b\u003e\u003ca name=\"N191\" id=\"N191\"\u003e\u003c/a\u003e\r\nBut we have derived, from all this, no ground for\r\naffirming a multiplicity of real things, or a form of externality\r\nof more than one dimension, which, we saw, was necessary for\r\nthe truth of two out of our three axioms. This brings us to\r\nthe question: Have we enough, with time alone as a form of\r\nexternality, for the possibility of knowledge?\u003c/p\u003e\r\n\r\n\u003cp\u003eThis question we must, I think, answer in the negative.\r\nWith time alone, we have seen, our presented object must be\r\ncomplex, but its complexity must, if I may use such a phrase,\r\nbe merely adjectival. Without a second form of externality,\r\nonly one thing can be given at one moment\u003ca name=\"FNanchor_189_189\" id=\"FNanchor_189_189\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_189_189\" class=\"fnanchor\"\u003e[189]\u003c/a\u003e, and this one\r\nthing, therefore, must constitute the whole of our world. The\r\nobject of past perception must\u0026mdash;since our one thing has nothing\r\nexternal to it, by which it could be created or destroyed\u0026mdash;be\r\nregarded as the same thing in a different state. The complexity,\r\ntherefore, will lie only in the changing states of our\r\none thing\u0026mdash;it will be adjectival, not substantival. Moreover we\r\nhave the following dilemma: Either the one thing must be\r\nourselves, or else self-consciousness could never arise. But the\r\nchief difficulty of such a world would lie in the changes of the\r\nthing. What could cause these changes, since we should know\r\nof nothing external to our thing? It would be like a Leibnitzian\r\nmonad, without any God outside it to prearrange its changes.\r\nCausality, in such a world, could not be applied, and change\r\nwould be wholly inexplicable.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_186\" id=\"Page_186\"\u003e[186]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eHence we require also the possibility of a diversity of\r\nsimultaneously existing things, not merely of successive adjectives;\r\nand this, we have seen, cannot be given by time alone,\r\nbut only by a form of externality for simultaneous parts of one\r\npresentation. We could never, in other words, infer the\r\nexistence of diverse but interrelated things, unless the object\r\nof sense-perception could have substantival complexity, and for\r\nsuch complexity we require a form of externality other than\r\ntime. Such a form, moreover, as was shown in \u003ca href=\"#N135\"\u003eChapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e,\r\nSection A (§ 135)\u003c/a\u003e, can only fulfil its functions if it has more\r\nthan one dimension. In our actual world, this form is given\r\nby space; in any world, knowable to beings with our laws of\r\nthought, some such form, as we have now seen, must be given\r\nin sense-perception.\u003c/p\u003e\r\n\r\n\u003cp\u003eThis argument may be briefly summed up, by assuming the\r\ndoctrine of Bradley, that all knowledge is obtained by inference\r\nfrom the \u003cem\u003eThis\u003c/em\u003e of sense-perception. For, if this be so, the\r\n\u003cem\u003eThis\u003c/em\u003e\u0026mdash;in order that inference, which depends on identity in\r\ndifference, may be possible at all\u0026mdash;must itself be complex, and\r\nmust, on analysis, reveal adjectives having a reference beyond\r\nitself. But this, as was shown above, can only happen by\r\nmeans of a form of externality. This establishes the à priori\r\naxioms of Geometry, as necessarily having existential import\r\nand validity in any intelligible world.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e192.\u003c/b\u003e\u003ca name=\"N192\" id=\"N192\"\u003e\u003c/a\u003e\r\nThe above argument, I hope, has explained why I\r\nhold it possible to deduce, from a mere conception like that of\r\na form of externality, the logical apriority of certain axioms as\r\nto experienced space. The Kantian argument\u0026mdash;which was\r\ncorrect, if our reasoning has been sound, in asserting that real\r\ndiversity, in our actual world, could only be known by the help\r\nof space\u0026mdash;was only mistaken, so far as its purely logical scope\r\nextends, in overlooking the possibility of other forms of\r\nexternality, which could, if they existed, perform the same task\r\nwith equal efficiency. In so far as space differs, therefore,\r\nfrom these other conceptions of possible intuitional forms, it is\r\na mere experienced fact, while in so far as its properties are\r\nthose which all such forms must have, it is \u003cem\u003eà priori\u003c/em\u003e necessary\r\nto the possibility of experience.\u003c/p\u003e\r\n\r\n\u003cp\u003eI cannot hope, however, that no difficulty will remain, for\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_187\" id=\"Page_187\"\u003e[187]\u003c/a\u003e\u003c/span\u003e\r\nthe reader, in such a deduction, from abstract conceptions, of\r\nthe properties of an actual \u003cem\u003edatum\u003c/em\u003e in sense-perception. Let\r\nus consider, for example, such a property as impenetrability.\r\nTo suppose two things simultaneously in the same position\r\nin a form of externality, is a logical contradiction; but can we\r\nsay as much of actual space and time? Is not the impossibility,\r\nhere, a matter of experience rather than of logic? Not\r\nif the above argument has been sound, I reply. For in that\r\ncase, we infer real diversity, \u003cem\u003ei.e.\u003c/em\u003e the existence of different things,\r\nonly from difference of position in space or time. It follows,\r\nthat to suppose two things in the same point of space and\r\ntime, is still a logical contradiction: not because we have\r\nconstructed the data of sense out of logic, but because logic\r\nis dependent, as regards its application, on the nature of these\r\ndata. This instance illustrates, what I am anxious to make\r\nplain, that my argument has not attempted to construct the\r\nliving wealth of sense-perception out of \"bloodless categories,\"\r\nbut only to point out that, unless sense-perception contained\r\na certain element, these categories would be powerless to\r\ngrapple with it.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e193.\u003c/b\u003e\u003ca name=\"N193\" id=\"N193\"\u003e\u003c/a\u003e\r\nHow we are to account for the fortunate realization\r\nof these requirements\u0026mdash;whether by a pre-established harmony,\r\nby Darwinian adaptation to our environment, by the subjectivity\r\nof the necessary element in sense-perception, or by\r\na fundamental identity and unity between ourselves and the\r\nrest of reality\u0026mdash;is a further question, belonging rather to\r\nmetaphysics than to our present line of argument. The \u003cem\u003eà\r\npriori\u003c/em\u003e, we have said throughout, is that which is necessary for\r\nthe possibility of experience, and in this we have a purely\r\nlogical criterion, giving results which only Logic and Epistemology\r\ncan prove or disprove. What is subjective in experience,\r\non the contrary, is primarily a question for psychology,\r\nand should be decided on psychological grounds alone. When\r\nthese two questions have been separately answered, but not\r\ntill then, we may frame theories as to the connection of the \u003cem\u003eà\r\npriori\u003c/em\u003e and the subjective; to allow such theories to influence\r\nour decision, on either of the two previous questions, is liable,\r\nsurely, to confuse the issue, and prevent a clear discrimination\r\nbetween fundamentally different points of view.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_188\" id=\"Page_188\"\u003e[188]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e194.\u003c/b\u003e\u003ca name=\"N194\" id=\"N194\"\u003e\u003c/a\u003e\r\nI come now to the second question with which this\r\nchapter has to deal, the question, namely: What are we to\r\ndo with the contradictions which obtruded themselves in\r\nChapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e, whenever we came to a point which seemed\r\nfundamental? I shall treat this question briefly, as I have\r\nlittle to add to answers with which we are all familiar. I\r\nhave only to prove, first, that the contradictions are inevitable,\r\nand therefore form no objection to my argument; secondly,\r\nthat the first step in removing them is to restore the notion\r\nof matter, as that which, in the data of sense-perception, is\r\nlocalized and interrelated in space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e195.\u003c/b\u003e\u003ca name=\"N195\" id=\"N195\"\u003e\u003c/a\u003e\r\nThe contradictions in space are an ancient theme\u0026mdash;as\r\nancient, in fact, as Zeno\u0027s refutation of motion. They are,\r\nroughly, of two kinds, though the two kinds cannot be sharply\r\ndivided. There are the contradictions inherent in the notion\r\nof the continuum, and the contradictions which spring from\r\nthe fact that space, while it must, to be knowable, be pure\r\nrelativity, must also, it would seem, since it is immediately\r\nexperienced, be something more than mere relations. The\r\nfirst class of contradictions has been encountered more frequently\r\nin this essay, and is also, I think, the more definite,\r\nand the more important for our present purpose. I doubt,\r\nhowever, whether the two classes are really distinct; for any\r\ncontinuum, I believe, in which the elements are not data, but\r\nintellectual constructions resulting from analysis, can be shown\r\nto have the same relational and yet not wholly relational character\r\nas belongs to space.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe three following contradictions, which I shall discuss\r\nsuccessively, seem to me the most prominent in a theory\r\nof Geometry.\u003c/p\u003e\r\n\r\n\u003cp\u003e(1) Though the parts of space are intuitively distinguished,\r\nno conception is adequate to differentiate them.\r\nHence arises a vain search for elements, by which the differentiation\r\ncould be accomplished, and for a whole, of which\r\nthe parts of space are to be components. Thus we get the\r\npoint, or zero extension, as the spatial element, and an infinite\r\nregress or a vicious circle in the search for a whole.\u003c/p\u003e\r\n\r\n\u003cp\u003e(2) All positions being relative, positions can only be\r\ndefined by their relations, \u003cem\u003ei.e.\u003c/em\u003e by the straight lines or planes\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_189\" id=\"Page_189\"\u003e[189]\u003c/a\u003e\u003c/span\u003e\r\nthrough them; but straight lines and planes, being all qualitatively\r\nsimilar, can only be defined by the positions they\r\nrelate. Hence, again, we get a vicious circle.\u003c/p\u003e\r\n\r\n\u003cp\u003e(3) Spatial figures must be regarded as relations. But a\r\nrelation is necessarily indivisible, while spatial figures are\r\nnecessarily divisible \u003ci lang=\"la\" xml:lang=\"la\"\u003ead infinitum\u003c/i\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e196.\u003c/b\u003e\u003ca name=\"N196\" id=\"N196\"\u003e\u003c/a\u003e\r\n(1) \u003cem\u003ePoints.\u003c/em\u003e The antinomy of the point\u0026mdash;which\r\narises wherever a continuum is given, and elements have to be\r\nsought in it\u0026mdash;is fundamental to Geometry. It has been given,\r\nperhaps unintentionally, by Veronese as the first axiom, in the\r\nform: \"There are different points. All points are identical\"\r\n(\u003ccite\u003eop. cit.\u003c/cite\u003e p. 226). We saw, in discussing projective Geometry,\r\nthat straight lines and planes must be regarded, on the one hand\r\nas relations between points, and on the other hand as made up\r\nof points\u003ca name=\"FNanchor_190_190\" id=\"FNanchor_190_190\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_190_190\" class=\"fnanchor\"\u003e[190]\u003c/a\u003e. We saw again, in dealing with measurement, how\r\nspace must be regarded as infinitely divisible, and yet as mere\r\nrelativity. But what is divisible and consists of parts, as space\r\ndoes, must lead at last, by continued analysis, to a simple and\r\nunanalyzable part, as the unit of differentiation. For whatever\r\ncan be divided, and has parts, possesses some thinghood, and\r\nmust, therefore, contain two ultimate units, the whole namely,\r\nand the smallest element possessing thinghood. But in space\r\nthis is notoriously not the case. After hypostatizing space, as\r\nGeometry is compelled to do, the mind imperatively demands\r\nelements, and insists on having them, whether possible or not.\r\nOf this demand, all the geometrical applications of the infinitesimal\r\ncalculus are evidence\u003ca name=\"FNanchor_191_191\" id=\"FNanchor_191_191\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_191_191\" class=\"fnanchor\"\u003e[191]\u003c/a\u003e. But what sort of elements do\r\nwe thus obtain? Analysis, being unable to find any earlier\r\nhalting-place, finds its elements in points, that is, in zero quanta\r\nof space. Such a conception is a palpable contradiction, only\r\nrendered tolerable by its necessity and familiarity. A point\r\nmust be spatial, otherwise it would not fulfil the function of a\r\nspatial element; but again it must contain no space, for any\r\nfinite extension is capable of further analysis. Points can\r\nnever be given in intuition, which has no concern with the\r\ninfinitesimal: they are a purely conceptual construction, arising\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_190\" id=\"Page_190\"\u003e[190]\u003c/a\u003e\u003c/span\u003eout of the need of terms between which spatial relations can\r\nhold. If space be more than relativity, spatial relations must\r\ninvolve spatial relata; but no relata appear, until we have\r\nanalyzed our spatial data down to nothing. The contradictory\r\nnotion of the point, as a thing in space without spatial\r\nmagnitude, is the only outcome of our search for spatial\r\nrelata. This \u003ci lang=\"la\" xml:lang=\"la\"\u003ereductio ad absurdum\u003c/i\u003e surely suffices, by itself, to\r\nprove the essential relativity of space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e197.\u003c/b\u003e\u003ca name=\"N197\" id=\"N197\"\u003e\u003c/a\u003e\r\nThus Geometry is forced, since it wishes to regard\r\nspace as independent, to hypostatize its abstractions, and\r\ntherefore to invent a self-contradictory notion as the spatial\r\nelement. A similar absurdity appears, even more obviously, in\r\nthe notion of a whole of space. The antinomy may, therefore,\r\nbe stated thus: Space, as we have seen throughout, must, if\r\nknowledge of it is to be possible, be mere relativity; but it\r\nmust also, if \u003cem\u003eindependent\u003c/em\u003e knowledge of it, such as Geometry\r\nseeks, is to be possible, be something more than mere relativity,\r\nsince it is divisible and has parts. But we saw, in \u003ca href=\"#N133\"\u003eChap. \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e,\r\nSection A (§ 133)\u003c/a\u003e that knowledge of a form of externality must\r\nbe logically independent of the particular matter filling the\r\nform. How then are we to extricate ourselves from this\r\ndilemma?\u003c/p\u003e\r\n\r\n\u003cp\u003eThe only way, I think, is, not to make Geometry dependent\r\non Physics, which we have seen to be erroneous\u003ca name=\"FNanchor_192_192\" id=\"FNanchor_192_192\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_192_192\" class=\"fnanchor\"\u003e[192]\u003c/a\u003e, but to give\r\nevery geometrical proposition a certain reference to matter in\r\ngeneral. And at this point an important distinction must be\r\nmade. We have hitherto spoken of space as relational, and\r\nof spatial figures as relations. But space, it would seem, is\r\nrather relativity than relations\u0026mdash;itself not a relation, it gives\r\nthe bare possibility of relations between diverse things\u003ca name=\"FNanchor_193_193\" id=\"FNanchor_193_193\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_193_193\" class=\"fnanchor\"\u003e[193]\u003c/a\u003e. As\r\napplied to a spatial figure, which can only arise by a differentiation\r\nof space, and hence by the introduction of some\r\ndifferentiating matter, the word relation is, perhaps, less\r\nmisleading than any other; as applied to empty undifferentiated\r\nspace, it seems by no means an accurate description.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut a bare possibility cannot exist, or be given in sense-perception!\r\nWhat becomes, then, of the arguments of the\r\n\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_191\" id=\"Page_191\"\u003e[191]\u003c/a\u003e\u003c/span\u003efirst part of this chapter? I reply, it is not empty space, but\r\nspatial figures, which sense-perception reveals, and spatial\r\nfigures, as we have just seen, involve a differentiation of space,\r\nand therefore a reference to the matter which is in space. It\r\nis spatial figures, also, and not empty space, with which\r\nGeometry has to deal. The antinomy discussed above arises\r\nthen\u0026mdash;so it would seem\u0026mdash;from the attempt to deal with empty\r\nspace, rather than with spatial figures and the matter to which\r\nthey necessarily refer.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e198.\u003c/b\u003e\u003ca name=\"N198\" id=\"N198\"\u003e\u003c/a\u003e\r\nLet us see whether, by this change, we can overcome\r\nthe antinomy of the point. Spatial figures, we shall now say,\r\nare relations between the matter which differentiates empty\r\nspace. Their divisibility, which seemed to contradict their\r\nrelational character, may be explained in two ways: first, as\r\nholding of the figures considered as parts of empty space, which\r\nis itself not a relation; second, as denoting the possibility\r\nof continuous change in the relation expressed by the spatial\r\nfigure. These two ways are, at bottom, the same; for empty\r\nspace is a possibility of relations, and the figure, when viewed\r\nin connection with empty space, thus becomes a \u003cem\u003epossible\u003c/em\u003e relation,\r\nwith which other possible relations may be contrasted or\r\ncompared. But the second way of regarding divisibility is the\r\nbetter way, since it introduces a reference to the matter which\r\ndifferentiates empty space, without which, spatial figures, and\r\ntherefore Geometry, could not exist. It is empty space, then\u0026mdash;so\r\nwe must conclude\u0026mdash;which gives rise to the antinomy in\r\nquestion; for empty space is a bare possibility of relations,\r\nundifferentiated and homogeneous, and thus wholly destitute\r\nof parts or of thinghood. To speak of parts of a possibility is\r\nnonsense; the parts and differentiations arise only through a\r\nreference to the matter which is differentiated in space.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e199.\u003c/b\u003e\u003ca name=\"N199\" id=\"N199\"\u003e\u003c/a\u003e\r\nBut what nature must we ascribe to this matter,\r\nwhich is to be involved in all geometrical propositions? In\r\ncriticizing Helmholtz (\u003ca href=\"#N73\"\u003eChap. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e § 73\u003c/a\u003e), it may be remembered,\r\nwe decided that Geometry refers to a peculiar and abstract\r\nkind of matter, which is not regarded as possessing any causal\r\nqualities, as exerting or as subject to the action of forces. And\r\nthis is the matter, I think, which we require for the needs of\r\nthe moment. Not that we affirm, of course, that actual matter\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_192\" id=\"Page_192\"\u003e[192]\u003c/a\u003e\u003c/span\u003e\r\ncan be destitute of the properties with which Physics is cognizant,\r\nbut that we abstract from these properties, as being\r\nirrelevant to Geometry. All that we require, for our immediate\r\npurpose, is a subject of that diversity which space renders\r\npossible, or terms for those relations by which empty space, if\r\nspace is to be studied at all, must be differentiated. But how\r\nmust a matter, which is to fulfil this function, be regarded?\u003c/p\u003e\r\n\r\n\u003cp\u003eEmpty space, we have said, is a possibility of diversity in\r\nrelation, but spatial figures, with which Geometry necessarily\r\ndeals, are the actual relations rendered possible by empty space.\r\nOur matter, therefore, must supply the terms for these relations.\r\nIt must be differentiated, since such differentiation, as we have\r\nseen, is the special work of space. We must find, therefore,\r\nin our matter, that unit of differentiation, or atom\u003ca name=\"FNanchor_194_194\" id=\"FNanchor_194_194\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_194_194\" class=\"fnanchor\"\u003e[194]\u003c/a\u003e, which in\r\nspace we could not find. This atom must be simple, \u003cem\u003ei.e.\u003c/em\u003e it\r\nmust contain no real diversity; it must be a \u003cem\u003eThis\u003c/em\u003e not resolvable\r\ninto \u003cem\u003eThises\u003c/em\u003e. Being simple, it can contain no relations within\r\nitself, and consequently, since spatial figures are mere relations,\r\nit cannot appear as a spatial figure; for every spatial figure\r\ninvolves some diversity of matter. But our atom must have\r\nspatial relations with other atoms, since to supply terms for\r\nthese relations is its only function. It is also capable of having\r\nthese relations, since it is differentiated from other atoms.\r\nHence we obtain an unextended term for spatial relations,\r\nprecisely of the kind we require. So long as we sought this\r\nterm without reference to anything more than space, the self-contradictory\r\nnotion of the point was the only outcome of our\r\nsearch; but now that we allow a reference to the matter differentiated\r\nby space, we find at once the term which was needed,\r\nnamely, a non-spatial simple element, with spatial relations to\r\nother elements. To Geometry such a term will appear, owing\r\nto its spatial relations, as a point; but the contradiction of the\r\npoint, as we now see, is a result only of the undue abstraction\r\nwith which Geometry deals.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e200.\u003c/b\u003e\u003ca name=\"N200\" id=\"N200\"\u003e\u003c/a\u003e\r\n(2) \u003cem\u003eThe circle in the definition of straight lines and\r\nplanes.\u003c/em\u003e This difficulty need not long detain us, since we have\r\nalready, with the material atom, broken through the relativity\r\nwhich caused our circle. Straight lines, in the purely geometrical\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_193\" id=\"Page_193\"\u003e[193]\u003c/a\u003e\u003c/span\u003e\r\nprocedure, are defined only by points, and points only by\r\nstraight lines. But points, now, are replaced by material\r\natoms: the duality of points and lines, therefore, has disappeared,\r\nand the straight line may be defined as the spatial\r\nrelation between two unextended atoms. These atoms have\r\nspatial adjectives, derived from their relations to other atoms;\r\nbut they have no \u003cem\u003eintrinsic\u003c/em\u003e spatial adjectives, such as could\r\nbelong to them if they had extension or figure. Thus straight\r\nlines and planes are the true spatial units, and points result\r\nonly from the attempt to find, within space, those terms for\r\nspatial relations which exist only in a more than spatial\r\nmatter. Straight lines, planes and volumes are the spatial\r\nrelations between two, three or four unextended atoms, and\r\npoints are a merely convenient geometrical fiction, by which\r\npossible atoms are replaced. For, since space, as we saw, is a\r\npossibility, Geometry deals not with actually realized spatial\r\nrelations, but with the whole scheme of possible relations.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e201.\u003c/b\u003e\u003ca name=\"N201\" id=\"N201\"\u003e\u003c/a\u003e\r\n(3) \u003cem\u003eSpace is at once relational and more than relational.\u003c/em\u003e\r\nWe have already touched on the question how far\r\nspace is other than relations, but as this question is quite\r\nfundamental, as \u003cem\u003erelation\u003c/em\u003e is an ambiguous and dangerous word,\r\nas I have made constant use of the relativity of space without\r\nattempting to define a relation, it will be necessary to discuss\r\nthis antinomy at length.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e202.\u003c/b\u003e\u003ca name=\"N202\" id=\"N202\"\u003e\u003c/a\u003e\r\nNow for this discussion it is essential to distinguish\r\nclearly between empty space and spatial figures. Empty\r\nspace, as a form of externality, is not actual relations, but\r\nthe possibility of relations: if we ascribe existential import\r\nto it, as the ground, in reality, of all diversity in relation, we at\r\nonce have space as something not itself relations, though giving\r\nthe possibility of all relations. In this sense, space is to be\r\ndistinguished from spatial order. Spatial order, it may be said,\r\npresupposes space, as that in which this order is possible. Thus\r\nStumpf says\u003ca name=\"FNanchor_195_195\" id=\"FNanchor_195_195\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_195_195\" class=\"fnanchor\"\u003e[195]\u003c/a\u003e: \"There is no order or relation without a positive\r\nabsolute content, underlying it, and making it possible to order\r\nanything in this manner. Why and how should we otherwise\r\ndistinguish one order from another?… To distinguish different\r\norders from one another, we must everywhere recognize a\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_194\" id=\"Page_194\"\u003e[194]\u003c/a\u003e\u003c/span\u003e\r\nparticular absolute content, in relation to which the order takes\r\nplace. And so space, too, is not a mere order, but just that\r\nby which the spatial order, side-by-sideness (\u003ci lang=\"de\" xml:lang=\"de\"\u003eNebeneinander\u003c/i\u003e)\r\ndistinguishes itself from the rest.\"\u003c/p\u003e\r\n\r\n\u003cp\u003eMay we not, then, resolve the antinomy very simply, by a\r\nreference to this ambiguity of space? Bradley contends (Appearance\r\nand Reality, pp. 36\u0026ndash;7) that, on the one hand, space\r\nhas parts, and is therefore not mere relations, while on the\r\nother hand, when we try to say what these parts are, we find\r\nthem after all to be mere relations. But cannot the space\r\nwhich has parts be regarded as empty space, Stumpf\u0027s absolute\r\nunderlying content, which is not mere relations, while the parts,\r\nin so far as they turn out to be mere relations, are those relations\r\nwhich constitute spatial order, not empty space? If this\r\ncan be maintained, the antinomy no longer exists.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut such an explanation, though I believe it to be a first\r\nstep towards a solution, will, I fear, itself demand almost as\r\nmuch explanation as the original difficulty. For the connection\r\nof empty space with spatial order is itself a question full of\r\ndifficulty, to be answered only after much labour.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e203.\u003c/b\u003e\u003ca name=\"N203\" id=\"N203\"\u003e\u003c/a\u003e\r\nLet us consider what this empty space is. (I speak\r\nof \"empty\" space without necessarily implying the absence of\r\nmatter, but only to denote a space which is not a mere\r\norder of material things.) Stumpf regards it as given in sense;\r\nKant, in the last two arguments of his metaphysical deduction,\r\nargues that it is an intuition, not a concept, and must be\r\nknown before spatial order becomes possible. I wish to maintain,\r\non the contrary, that it is wholly conceptual; that space is\r\ngiven only as spatial order; that spatial relations, being given,\r\nappear as more than mere relations, and so become hypostatized;\r\nthat when hypostatized, the whole collection of them is\r\nregarded as contained in empty space; but that this empty\r\nspace itself, if it means more than the logical possibility of\r\nspace-relations, is an unnecessary and self-contradictory assumption.\r\nLet us begin by considering Kant\u0027s arguments on\r\nthis point.\u003c/p\u003e\r\n\r\n\u003cp\u003eLeibnitz had affirmed that space was only relations, while\r\nNewton had maintained the objective reality of absolute space.\r\nKant adopted a middle course: he asserted absolute space, but\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_195\" id=\"Page_195\"\u003e[195]\u003c/a\u003e\u003c/span\u003e\r\nregarded it as purely subjective. The assertion of absolute\r\nspace is the object of his second argument; for if space were\r\nmere relations between things, it would necessarily disappear\r\nwith the disappearance of the things in it; but this the second\r\nargument denies\u003ca name=\"FNanchor_196_196\" id=\"FNanchor_196_196\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_196_196\" class=\"fnanchor\"\u003e[196]\u003c/a\u003e. Now spatial order obviously does disappear\r\nwith matter, but absolute or empty space may be supposed to\r\nremain. It is this, then, which Kant is arguing about, and it is\r\nthis which he affirms to be a pure intuition, necessarily presupposed\r\nby spatial order\u003ca name=\"FNanchor_197_197\" id=\"FNanchor_197_197\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_197_197\" class=\"fnanchor\"\u003e[197]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e204.\u003c/b\u003e\u003ca name=\"N204\" id=\"N204\"\u003e\u003c/a\u003e\r\nBut can we agree in regarding empty space, the\r\n\"infinite given whole,\" as really given? Must we not, in spite\r\nof Kant\u0027s argument, regard it as wholly conceptual? It is not\r\nrequired, in the first place, by the argument of the first half of\r\nthis chapter, which required only that every \u003cem\u003eThis\u003c/em\u003e of sense-perception\r\nshould be resolvable into \u003cem\u003eThises\u003c/em\u003e, and thus involved\r\nonly an order among \u003cem\u003eThises\u003c/em\u003e, not anything given originally\r\nwithout reference to them at all. In the second place, Kant\u0027s\r\ntwo arguments\u003ca name=\"FNanchor_198_198\" id=\"FNanchor_198_198\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_198_198\" class=\"fnanchor\"\u003e[198]\u003c/a\u003e designed to prove that empty space is not\r\nconceptual, are inadequate to their purpose. The argument\r\nthat the parts of space are not contained \u003cem\u003eunder\u003c/em\u003e it, but \u003cem\u003ein\u003c/em\u003e it,\r\nproves certainly that space is not a general conception, of which\r\nspatial figures are the instances; but it by no means follows\r\nthat empty space is not a conception. Empty space is undifferentiated\r\n\u003cins class=\"corr\" title=\"Transcriber\u0027s Note\u0026mdash;Original text: \u0027and homogenous\u0027\"\u003eand homogeneous;\u003c/ins\u003e parts of space, or spatial\r\nfigures, arise only by reference to some differentiating matter,\r\nand thus belong rather to spatial order than to empty space.\r\nIf empty space be the pre-condition of spatial order, we cannot\r\nexpect it to be connected with spatial relations as genus with\r\nspecies. But empty space may nevertheless be a universal\r\nconception; it may be related to spatial order as the state to\r\nthe citizens. These are not instances of the state, but are\r\ncontained in it; they also, in a sense, presuppose it, for a man\r\ncan only become a citizen by being related to other citizens in\r\na state\u003ca name=\"FNanchor_199_199\" id=\"FNanchor_199_199\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_199_199\" class=\"fnanchor\"\u003e[199]\u003c/a\u003e.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_196\" id=\"Page_196\"\u003e[196]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eThe uniqueness of space, again, seems hardly a valid argument\r\nfor its intuitional nature; to regard it as an argument\r\nimplies, indeed, that all conceptions are abstracted from a series\r\nof instances\u0026mdash;a view which has been criticized in \u003ca href=\"#N77\"\u003eChapter II.\r\n(§ 77)\u003c/a\u003e, and need not be further discussed here\u003ca name=\"FNanchor_200_200\" id=\"FNanchor_200_200\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_200_200\" class=\"fnanchor\"\u003e[200]\u003c/a\u003e. There is no\r\nground, therefore, in Kant\u0027s two arguments for the intuitional\r\nnature of empty space, which can be maintained against\r\ncriticism.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e205.\u003c/b\u003e\u003ca name=\"N205\" id=\"N205\"\u003e\u003c/a\u003e\r\nAnother ground for condemning empty space is to\r\nbe found in the mathematical antinomies. For it is no solution,\r\nas Lotze points out (Metaphysik, Bk. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e Chap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e, § 106), to\r\nregard empty space as purely subjective: contradictions in a\r\nnecessary subjective intuition form as great a difficulty as in\r\nanything else. But these antinomies arise only in connection\r\nwith empty space, not with spatial order as an aggregate of\r\nrelations. For only when space is regarded as possessed of some\r\nthinghood, can a whole or a true element be demanded. This\r\nwe have seen already in connection with the Point. When\r\nspace is regarded, so far as it is valid, as only spatial order,\r\nunbounded extension and infinite divisibility both disappear.\r\nWhat is divided is not spatial relations, but matter; and if\r\nmatter, as we have seen that Geometry requires, consists of\r\nunextended atoms with spatial relations, there is no reason to\r\nregard matter either as infinitely divisible, or as consisting of\r\natoms of finite extension.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e206.\u003c/b\u003e\u003ca name=\"N206\" id=\"N206\"\u003e\u003c/a\u003e\r\nBut whence arises, on this view, the paradox that we\r\ncannot but regard space as having more or less thinghood, and\r\nas divisible \u003ci lang=\"la\" xml:lang=\"la\"\u003ead infinitum\u003c/i\u003e? This must be explained, I think, as\r\na psychological illusion, unavoidably arising from the fact that\r\nspatial relations are immediately presented. They thus have\r\na peculiar psychical quality, as immediate experiences, by which\r\nquality they can be distinguished from time-relations or any\r\nother order in which things may be arranged. To Stumpf,\r\nwhose problem is psychological, such a psychical quality would\r\nconstitute an absolute underlying content, and would fully\r\njustify his thesis; to us, however, whose problem is epistemological,\r\nit would not do so, but would leave the \u003cem\u003emeaning\u003c/em\u003e of the\r\nspatial element in sense-perception free from any implication\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_197\" id=\"Page_197\"\u003e[197]\u003c/a\u003e\u003c/span\u003e\r\nof an absolute or empty space\u003ca name=\"FNanchor_201_201\" id=\"FNanchor_201_201\"\u003e\u003c/a\u003e\u003ca href=\"#Footnote_201_201\" class=\"fnanchor\"\u003e[201]\u003c/a\u003e. May we not, then, abandon\r\nempty space, and say: Spatial order consists of \u003cem\u003efelt\u003c/em\u003e relations,\r\nand \u003cem\u003equâ\u003c/em\u003e felt has, for Psychology, an existence not wholly\r\nresolvable into relations, and unavoidably \u003cem\u003eseeming\u003c/em\u003e to be more\r\nthan mere relations. But when we examine the information,\r\nas to space, which we derive from sense-perception, we find\r\nourselves plunged in contradictions, as soon as we allow this\r\ninformation to consist of more than relations. This leaves\r\nspatial order alone in the field, and reduces empty space to a\r\nmere name for the logical possibility of spatial relations.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e207.\u003c/b\u003e\u003ca name=\"N207\" id=\"N207\"\u003e\u003c/a\u003e\r\nThe apparent divisibility of the relations which constitute\r\nspatial order, then, may be explained in two ways,\r\nthough these are at bottom equivalent. We may take the\r\nrelation as considered in connection with empty space, in which\r\ncase it becomes more than a relation; but being falsely hypostatized,\r\nit appears as a complex thing, necessarily composed of\r\nelements, which elements, however, nowhere emerge until we\r\nanalyze the pseudo-thing down to nothing, and arrive at the\r\npoint. In this sense, the divisibility of spatial relations is an\r\nunavoidable illusion. Or again, we may take the relation in\r\nconnection with the material atoms it relates. In this case,\r\nother atoms may be imagined, differently localized by different\r\nspatial relations. If they are localized on the straight line\r\njoining two of the original atoms, this straight line appears as\r\ndivided by them. But the original relation is not really\r\ndivided: all that has happened is, that two or more equivalent\r\nrelations have replaced it, as two compounded relations of\r\nfather and son may replace the equivalent relation of grandfather\r\nand grandson. These two ways of viewing the apparent\r\ndivisibility are equivalent: for empty space, in so far as it is not\r\nillusion, is a name for the aggregate of possible space-relations.\r\nTo regard a figure in empty space as divided, therefore, means,\r\nif it means anything, to regard two or more other possible relations\r\nas substituted for it, which gives the second way of viewing\r\nthe question.\u003c/p\u003e\r\n\r\n\u003cp\u003eThe same reference to matter, then, by which the antinomy\r\nof the Point was solved, solves also the antinomy as to the\r\nrelational nature of space. Space, if it is to be freed from\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_198\" id=\"Page_198\"\u003e[198]\u003c/a\u003e\u003c/span\u003e\r\ncontradictions, must be regarded exclusively as spatial order, as\r\nrelations between unextended material atoms. Empty space,\r\nwhich arises, by an inevitable illusion, out of the spatial element\r\nin sense-perception, may be regarded, if we wish to retain it, as\r\nthe bare principle of relativity, the bare logical possibility of\r\nrelations between diverse things. In this sense, empty space is\r\nwholly conceptual; spatial order alone is immediately experienced.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e208.\u003c/b\u003e\u003ca name=\"N208\" id=\"N208\"\u003e\u003c/a\u003e\r\nBut in what sense does spatial order consist of relations?\r\nWe have hitherto spoken of externality as a relation,\r\nand in a sense such a manner of speaking is justified. Externality,\r\nwhen predicated of anything, is an adjective of that thing,\r\nand implies a reference to some other thing. To this extent,\r\nthen, externality is analogous to other relations; and only to this\r\nextent, in our previous arguments, has it been regarded as a\r\nrelation. But when we take account of further qualities of\r\nrelations, externality begins to appear, not so much as a relation,\r\nbut rather as a necessary aspect or element in every\r\nrelation. And this is borne out by the necessity, for the\r\nexistence of relations, of some given form of externality.\u003c/p\u003e\r\n\r\n\u003cp\u003eEvery relation, we may say, involves a diversity between\r\nthe related terms, but also some unity. Mere diversity does not\r\ngive a ground for that interaction, and that interdependence,\r\nwhich a relation requires. Mere unity leaves the terms identical,\r\nand thus destroys the reference of one to another required for a\r\nrelation. Mere externality, taken in abstraction, gives only the\r\nelement of diversity required for a relation, and is thus more\r\nabstract than any actual relation. But mere diversity does not\r\ngive that indivisible whole of which any actual relation must\r\nconsist, and is thus, when regarded abstractly, not subject to the\r\nrestrictions of ordinary relations.\u003c/p\u003e\r\n\r\n\u003cp\u003eBut with mere diversity, we seem to have returned to empty\r\nspace, and abandoned spatial order. Mere diversity, surely, is\r\neither complete or non-existent; degrees of diversity, or a\r\nquantitative measure of it, are nonsense. We cannot, therefore,\r\nreduce spatial order to mere diversity. Two things, if they\r\noccupy different positions in space, are necessarily diverse, but\r\nare as necessarily something more; otherwise spatial order\r\nbecomes unmeaning.\u003c/p\u003e\r\n\r\n\u003cp\u003e\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_199\" id=\"Page_199\"\u003e[199]\u003c/a\u003e\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eEmpty space, then, in the above sense of the possibility\r\nof spatial relations, contains only one aspect of a relation,\r\nnamely the aspect of diversity; but spatial order, by its reference\r\nto matter, becomes more concrete, and contains also the element\r\nof unity, arising out of the connection of the different material\r\natoms. Spatial order, then, consists of relations in the ordinary\r\nsense; its merely spatial element, however\u0026mdash;if one may make\r\nsuch a distinction\u0026mdash;the element, that is, which can be abstracted\r\nfrom matter and regarded as constituting empty space, is only\r\none aspect of a relation, but an aspect which, in the concrete,\r\nmust be inseparably bound up with the other aspect. Here,\r\nonce more, we see the ground of the contradictions in empty\r\nspace, and the reason why spatial order is free from these\r\ncontradictions.\u003c/p\u003e\r\n\r\n\r\n\u003ch4\u003e\u003cem\u003eConclusion.\u003c/em\u003e\u003c/h4\u003e\r\n\r\n\u003cp\u003e\u003cb\u003e209.\u003c/b\u003e\u003ca name=\"N209\" id=\"N209\"\u003e\u003c/a\u003e\r\nWe have now completed our review of the foundations\r\nof Geometry. It will be well, before we take leave of the\r\nsubject, briefly to review and recapitulate the results we have\r\nwon.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the first chapter, we watched the development of a branch\r\nof Mathematics designed, at first, only to establish the logical\r\nindependence of Euclid\u0027s axiom of parallels, and the possibility\r\nof a self-consistent Geometry which dispensed with it. We\r\nfound the further development of the subject entangled, for a\r\nwhile, in philosophical controversy; having shown one axiom to\r\nbe superfluous, the geometers of the second period hoped to\r\nprove the same conclusion of all the others, but failed to\r\nconstruct any system free from three fundamental axioms.\r\nBeing concerned with analytical and metrical Geometry, they\r\ntended to regard Algebra as \u003cem\u003eà priori\u003c/em\u003e, but held that those\r\nproperties of spatial magnitudes, which were not deducible\r\nfrom the laws of Algebra, must be empirical. In all this, they\r\naimed as much at discrediting Kant as at advancing Mathematics.\r\nBut with the third period, the interest in Philosophy\r\ndiminishes, the opposition to Euclid becomes less marked, and\r\nmost important of all, measurement is no longer regarded as\r\nfundamental, and space is dealt with by descriptive rather than\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_200\" id=\"Page_200\"\u003e[200]\u003c/a\u003e\u003c/span\u003e\r\nquantitative methods. But nevertheless, three axioms, substantially\r\nthe same as those retained in the second period, are\r\nstill retained by all geometers.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the second chapter, we endeavoured, by a criticism of\r\nsome geometrical philosophies, to prepare the ground for a\r\nconstructive theory of Geometry. We saw that Kant, in\r\napplying the argument of the Transcendental Aesthetic to\r\nspace, had gone too far, since its logical scope extended only\r\nto some form of externality in general. We saw that Riemann,\r\nHelmholtz and Erdmann, misled by the quantitative bias, overlooked\r\nthe qualitative substratum required by all judgments\r\nof quantity, and thus mistook the direction in which the\r\nnecessary axioms of Geometry are to be found. We rejected,\r\nalso, Helmholtz\u0027s view that Geometry depends on Physics,\r\nbecause we found that Physics must assume a knowledge of\r\nGeometry before it can become possible. But we admitted, in\r\nGeometry, a reference to matter\u0026mdash;not, however, to matter as\r\nempirically known in Physics, but to a more abstract matter,\r\nwhose sole function is to appear in space, and supply the terms\r\nfor spatial relations. We admitted, however, besides this, that\r\nall \u003cem\u003eactual\u003c/em\u003e measurement must be effected by means of \u003cem\u003eactual\u003c/em\u003e\r\nmatter, and is only empirically possible, through the empirical\r\nknowledge of approximately rigid bodies. In criticizing Lotze,\r\nwe saw that the most important sense, in which non-Euclidean\r\nspaces are possible, is a philosophical sense, namely, that they\r\nare not condemned by any \u003cem\u003eà priori\u003c/em\u003e argument as to the necessity\r\nof space for experience, and that consequently, if they are not\r\naffirmed, this must be on empirical grounds alone. Lotze\u0027s\r\nstrictures on the mathematical procedure of Metageometry we\r\nfound to be wholly due to ignorance of the subject.\u003c/p\u003e\r\n\r\n\u003cp\u003eProceeding, in the third chapter, to a constructive theory\r\nof Geometry, we saw that projective Geometry, which has no\r\nreference to quantity, is necessarily true of any form of\r\nexternality. Its three axioms\u0026mdash;homogeneity, dimensions, and\r\nthe straight line\u0026mdash;were all deduced from the conception of a\r\nform of externality, and, since some such form is necessary to\r\nexperience, were all declared \u003cem\u003eà priori\u003c/em\u003e. In metrical Geometry,\r\non the contrary, we found an empirical element, arising out of\r\nthe alternatives of Euclidean and non-Euclidean space. Three\u003cspan class=\"pagenum\"\u003e\u003ca name=\"Page_201\" id=\"Page_201\"\u003e[201]\u003c/a\u003e\u003c/span\u003e\r\n\u003cem\u003eà priori\u003c/em\u003e axioms, common to these spaces, and necessary conditions\r\nof the possibility of measurement, still remained; these\r\nwere the axiom of Free Mobility, the axiom that space has a\r\nfinite integral number of dimensions, and the axiom of distance.\r\nExcept for the new idea of motion, these were found equivalent\r\nto the projective triad, and thus necessarily true of any form\r\nof externality. But the remaining axioms of Euclid\u0026mdash;the\r\naxiom of three dimensions, the axiom that two straight lines\r\ncan never enclose a space, and the axiom of parallels\u0026mdash;were\r\nregarded as empirical laws, derived from the investigation and\r\nmeasurement of our actual space, and true only, as far as the\r\nlast two are concerned, within the limits set by errors of\r\nobservation.\u003c/p\u003e\r\n\r\n\u003cp\u003eIn the present chapter, we completed our proof of the\r\napriority of the projective and equivalent metrical axioms, by\r\nshowing the necessity, for experience, of some form of externality,\r\ngiven by sensation or intuition, and not merely inferred from\r\nother data. Without this, we said, a knowledge of diverse but\r\ninterrelated things, the corner-stone of all experience, would be\r\nimpossible. Finally, we discussed the contradictions arising\r\nout of the relativity and continuity of space, and endeavoured\r\nto overcome them by a reference to matter. This matter, we\r\nfound, must consist of unextended atoms, localized by their\r\nspatial relations, and appearing, in Geometry, as points. But\r\nthe non-spatial adjectives of matter, we contended, are irrelevant\r\nto Geometry, and its causal properties may be left out of\r\naccount. To deal with the new contradictions, involved in such\r\na notion of matter, would demand a fresh treatise, leading us,\r\nthrough Kinematics, into the domains of Dynamics and Physics.\r\nBut to discuss the special difficulties of space is all that is\r\npossible in an essay on the Foundations of Geometry.\u003c/p\u003e\r\n\r\n\u003cdiv class=\"footnotes\"\u003e\u003ch3\u003eFOOTNOTES:\u003c/h3\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_178_178\" id=\"Footnote_178_178\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_178_178\"\u003e\u003cspan class=\"label\"\u003e[178]\u003c/span\u003e\u003c/a\u003e Compare, with the following paragraphs, the admirable discussion in\r\nMr Hobhouse\u0027s \u003ccite class=\"fsnormal\"\u003eTheory of Knowledge\u003c/cite\u003e (Methuen 1896), Part \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Chapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_179_179\" id=\"Footnote_179_179\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_179_179\"\u003e\u003cspan class=\"label\"\u003e[179]\u003c/span\u003e\u003c/a\u003e I speak of sense-perception instead of sensation, so as not to prejudge the\r\nissue as to the sensational nature of space.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_180_180\" id=\"Footnote_180_180\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_180_180\"\u003e\u003cspan class=\"label\"\u003e[180]\u003c/span\u003e\u003c/a\u003e See Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 86\u0026ndash;7, 168\u0026ndash;171.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_181_181\" id=\"Footnote_181_181\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_181_181\"\u003e\u003cspan class=\"label\"\u003e[181]\u003c/span\u003e\u003c/a\u003e See Caird, \u003ccite class=\"fsnormal\"\u003eCritical Philosophy of Kant\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 287.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_182_182\" id=\"Footnote_182_182\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_182_182\"\u003e\u003cspan class=\"label\"\u003e[182]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eUrsprung der Raumvorstellung\u003c/cite\u003e, pp. 12\u0026ndash;30.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_183_183\" id=\"Footnote_183_183\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_183_183\"\u003e\u003cspan class=\"label\"\u003e[183]\u003c/span\u003e\u003c/a\u003e See the references in Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 76 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_184_184\" id=\"Footnote_184_184\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_184_184\"\u003e\u003cspan class=\"label\"\u003e[184]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 71 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_185_185\" id=\"Footnote_185_185\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_185_185\"\u003e\u003cspan class=\"label\"\u003e[185]\u003c/span\u003e\u003c/a\u003e \u003cem\u003eE.g.\u003c/em\u003e by Caird, \u003ccite\u003eop. cit.\u003c/cite\u003e Vol. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e p. 286.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_186_186\" id=\"Footnote_186_186\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_186_186\"\u003e\u003cspan class=\"label\"\u003e[186]\u003c/span\u003e\u003c/a\u003e I have no wish to deny, however, that space is essential in the subsequent\r\ndistinction of Self and not-Self.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_187_187\" id=\"Footnote_187_187\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_187_187\"\u003e\u003cspan class=\"label\"\u003e[187]\u003c/span\u003e\u003c/a\u003e See also Book I. Chap \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e passim; especially p. 51 ff. and pp. 70\u0026ndash;1.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_188_188\" id=\"Footnote_188_188\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_188_188\"\u003e\u003cspan class=\"label\"\u003e[188]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\"\u003eLogic\u003c/cite\u003e, p. 51 ff.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_189_189\" id=\"Footnote_189_189\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_189_189\"\u003e\u003cspan class=\"label\"\u003e[189]\u003c/span\u003e\u003c/a\u003e For the \u003cem\u003eThis\u003c/em\u003e, on such a hypothesis, has a purely temporal complexity, and\r\nis not resolvable into coexisting \u003cem\u003eThises\u003c/em\u003e.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_190_190\" id=\"Footnote_190_190\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_190_190\"\u003e\u003cspan class=\"label\"\u003e[190]\u003c/span\u003e\u003c/a\u003e \u003ca href=\"#N131\"\u003eChapter \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e Section A, (§ 131).\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_191_191\" id=\"Footnote_191_191\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_191_191\"\u003e\u003cspan class=\"label\"\u003e[191]\u003c/span\u003e\u003c/a\u003e Cf. Hannequin, \u003ccite class=\"fsnormal\" lang=\"fr\" xml:lang=\"fr\"\u003eEssai critique sur l\u0027hypothèse des atomes\u003c/cite\u003e, Paris 1895,\r\nChap. \u003cspan class=\"fs70\"\u003eI.\u003c/span\u003e Section \u003cspan class=\"fs70\"\u003eIII.\u003c/span\u003e; especially p. 43.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_192_192\" id=\"Footnote_192_192\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_192_192\"\u003e\u003cspan class=\"label\"\u003e[192]\u003c/span\u003e\u003c/a\u003e See \u003ca href=\"#N69\"\u003eChapter \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e § 69 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_193_193\" id=\"Footnote_193_193\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_193_193\"\u003e\u003cspan class=\"label\"\u003e[193]\u003c/span\u003e\u003c/a\u003e See third antinomy below, \u003ca href=\"#N201\"\u003e§ 201 ff.\u003c/a\u003e\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_194_194\" id=\"Footnote_194_194\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_194_194\"\u003e\u003cspan class=\"label\"\u003e[194]\u003c/span\u003e\u003c/a\u003e This atom, of course, must not be confounded with the atom of Chemistry.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_195_195\" id=\"Footnote_195_195\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_195_195\"\u003e\u003cspan class=\"label\"\u003e[195]\u003c/span\u003e\u003c/a\u003e \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eUrsprung der Raumvorstellung\u003c/cite\u003e, p. 15.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_196_196\" id=\"Footnote_196_196\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_196_196\"\u003e\u003cspan class=\"label\"\u003e[196]\u003c/span\u003e\u003c/a\u003e See Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e pp. 189\u0026ndash;190.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_197_197\" id=\"Footnote_197_197\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_197_197\"\u003e\u003cspan class=\"label\"\u003e[197]\u003c/span\u003e\u003c/a\u003e See ibid. p. 224 ff. for Kant\u0027s inconsistencies on this point.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_198_198\" id=\"Footnote_198_198\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_198_198\"\u003e\u003cspan class=\"label\"\u003e[198]\u003c/span\u003e\u003c/a\u003e The fourth and fifth in the first edition, the third and fourth in the\r\nsecond.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_199_199\" id=\"Footnote_199_199\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_199_199\"\u003e\u003cspan class=\"label\"\u003e[199]\u003c/span\u003e\u003c/a\u003e Cf. Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 218.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_200_200\" id=\"Footnote_200_200\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_200_200\"\u003e\u003cspan class=\"label\"\u003e[200]\u003c/span\u003e\u003c/a\u003e Cf. Vaihinger\u0027s \u003ccite class=\"fsnormal\" lang=\"de\" xml:lang=\"de\"\u003eCommentar\u003c/cite\u003e, \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e p. 207.\u003c/p\u003e\u003c/div\u003e\r\n\r\n\u003cdiv class=\"footnote\"\u003e\r\n\r\n\u003cp\u003e\u003ca name=\"Footnote_201_201\" id=\"Footnote_201_201\"\u003e\u003c/a\u003e\u003ca href=\"#FNanchor_201_201\"\u003e\u003cspan class=\"label\"\u003e[201]\u003c/span\u003e\u003c/a\u003e Cf. James, \u003ccite class=\"fsnormal\"\u003ePsychology\u003c/cite\u003e, Vol. \u003cspan class=\"fs70\"\u003eII.\u003c/span\u003e, p. 148 ff.\u003c/p\u003e\u003c/div\u003e\u003c/div\u003e\r\n\r\n\r\n\u003chr class=\"chap pg-brk\" /\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\u003cp class=\"pfs90\"\u003e\u003cspan class=\"antiqua\"\u003eCambridge:\u003c/span\u003e\u003c/p\u003e\r\n\r\n\u003cp class=\"pfs70\"\u003ePRINTED BY J. AND C. F. CLAY,\u003c/p\u003e\r\n\u003cp class=\"pfs70\"\u003eAT THE UNIVERSITY PRESS.\u003c/p\u003e\r\n\u003cp class=\"p6\" /\u003e\r\n\r\n\r\n\u003chr class=\"chap\" /\u003e\r\n\r\n\u003cdiv class=\"transnote\"\u003e\r\n\u003ca name=\"TN\" id=\"TN\"\u003e\u003c/a\u003e\r\n\u003cp\u003e\u003cstrong\u003eTRANSCRIBER\u0027S NOTE\u003c/strong\u003e\u003c/p\u003e\r\n\r\n\u003cp\u003eObvious typographical errors and punctuation errors have been\r\ncorrected after careful comparison with other occurrences within\r\nthe text and consultation of external sources.\u003c/p\u003e\r\n\r\n\u003cp\u003eA minus sign is represented by the n-dash character \"\u0026ndash;\".\r\nDate and number ranges also use the n-dash \"\u0026ndash;\".\u003c/p\u003e\r\n\r\n\u003cp\u003eExcept for those changes noted below, all misspellings in the text,\r\nand inconsistent or archaic usage, have been retained. For example,\r\nco-exist, coexist; every-day, everyday; connexion; assertorial;\r\napodeictic; premisses.\u003c/p\u003e\r\n\r\n\u003cp\u003e\r\n\u003ca href=\"#N82\"\u003e§ 82\u003c/a\u003e, \u0027so Erdmannn confidently\u0027 replaced by \u0027so Erdmann confidently\u0027.\u003cbr /\u003e\r\n\u003ca href=\"#N150\"\u003e§ 150\u003c/a\u003e Footnote \u003ca href=\"#Footnote_157_157\"\u003e[157]\u003c/a\u003e, \u0027Delboeuf\u0027 replaced by \u0027Delbœuf\u0027 for consistency.\u003cbr /\u003e\r\n\u003ca href=\"#N152\"\u003e§ 152\u003c/a\u003e, \u0027one subdivison must\u0027 replaced by \u0027one sub-division must\u0027.\u003cbr /\u003e\r\n\u003ca href=\"#N159\"\u003e§ 159\u003c/a\u003e Footnote \u003ca href=\"#Footnote_167_167\"\u003e[167]\u003c/a\u003e, \u0027Delboeuf\u0027 replaced by \u0027Delbœuf\u0027 for consistency.\u003cbr /\u003e\r\n\u003ca href=\"#N204\"\u003e§ 204\u003c/a\u003e, \u0027and homogenous;\u0027 replaced by \u0027and homogeneous;\u0027.\u003cbr /\u003e\r\n\u003c/p\u003e\r\n\u003c/div\u003e\r\n\u003c/article\u003e"}],"SectionSequence":["Back Link","Work Title","Deck","Author","Period","Era","Composition","Date Note","Region","Terra Avita","Terra Avita Region","Modern Country","Original Title","Language","Primary Discipline","Secondary Discipline","Tradition","Full Versions","Core Thesis","Classification","Arguments","Influence","Significance","Evidence Note","Full Text"],"Counts":{"ContextCards":3,"GeoCards":4,"DisciplineCards":2,"Links":11,"Sections":25,"Styles":3,"Scripts":1}}