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Chapter 3 The Notion of Space
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Chapter 3
The Notion of Space
1. Introduction
In the articles I have heretofore devoted1 to space I have above all emphasized the problems raised by non-Euclidean geometry, while leaving almost completely aside other questions more difficult of approach, such as those which pertain2 to the number of dimensions. All the geometries I considered had thus a common basis, that tridimensional continuum which was the same for all and which differentiated3 itself only by the figures one drew in it or when one aspired4 to measure it.
In this continuum, primitively5 amorphous7, we may imagine a network of lines and surfaces, we may then convene8 to regard the meshes9 of this net as equal to one another, and it is only after this convention that this continuum, become measurable, becomes Euclidean or non-Euclidean space. From this amorphous continuum can therefore arise indifferently one or the other of the two spaces, just as on a blank sheet of paper may be traced indifferently a straight or a circle.
In space we know rectilinear triangles the sum of whose angles is equal to two right angles; but equally we know curvilinear triangles the sum of whose angles is less than two right angles. The existence of the one sort is not more doubtful than that of the other. To give the name of straights to the sides of the first is to adopt Euclidean geometry; to give the name of straights to the sides of the latter is to adopt the non-Euclidean geometry. So that to ask what geometry it is proper to adopt is to ask, to what line is it proper to give the name straight?
It is evident that experiment can not settle such a question; one would not ask, for instance, experiment to decide whether I should call AB or CD a straight. On the other hand, neither can I say that I have not the right to give the name of straights to the sides of non-Euclidean triangles because they are not in
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1
devoted
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| adj.忠诚的,忠实的,热心的,献身于...的 | |
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pertain
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| v.(to)附属,从属;关于;有关;适合,相称 | |
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3
differentiated
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| 区分,区别,辨别( differentiate的过去式和过去分词 ); 区别对待; 表明…间的差别,构成…间差别的特征 | |
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aspired
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| v.渴望,追求( aspire的过去式和过去分词 ) | |
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primitively
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| 最初地,自学而成地 | |
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primitive
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| adj.原始的;简单的;n.原(始)人,原始事物 | |
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amorphous
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| adj.无定形的 | |
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convene
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| v.集合,召集,召唤,聚集,集合 | |
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meshes
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| 网孔( mesh的名词复数 ); 网状物; 陷阱; 困境 | |
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conformity
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| n.一致,遵从,顺从 | |
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syllable
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| n.音节;vt.分音节 | |
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radius
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| n.半径,半径范围;有效航程,范围,界限 | |
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deforming
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| 使变形,使残废,丑化( deform的现在分词 ) | |
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deformation
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| n.形状损坏;变形;畸形 | |
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unreasonable
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| adj.不讲道理的,不合情理的,过度的 | |
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dilated
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| adj.加宽的,扩大的v.(使某物)扩大,膨胀,张大( dilate的过去式和过去分词 ) | |
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simultaneously
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| adv.同时发生地,同时进行地 | |
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apprising
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| v.告知,通知( apprise的现在分词 );评价 | |
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coordinate
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| adj.同等的,协调的;n.同等者;vt.协作,协调 | |
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coordinates
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| n.相配之衣物;坐标( coordinate的名词复数 );(颜色协调的)配套服装;[复数]女套服;同等重要的人(或物)v.使协调,使调和( coordinate的第三人称单数 );协调;协同;成为同等 | |
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inversely
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| adj.相反的 | |
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precisely
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| adv.恰好,正好,精确地,细致地 | |
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unwilling
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| adj.不情愿的 | |
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qualitative
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| adj.性质上的,质的,定性的 | |
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relatively
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| adv.比较...地,相对地 | |
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expatiate
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| v.细说,详述 | |
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spoke
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| n.(车轮的)辐条;轮辐;破坏某人的计划;阻挠某人的行动 v.讲,谈(speak的过去式);说;演说;从某种观点来说 | |
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Fahrenheit
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| n./adj.华氏温度;华氏温度计(的) | |
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exempt
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| adj.免除的;v.使免除;n.免税者,被免除义务者 | |
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quantitative
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| adj.数量的,定量的 | |
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ascertain
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| vt.发现,确定,查明,弄清 | |
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ascertainment
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| n.探查,发现,确认 | |
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peculiarity
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| n.独特性,特色;特殊的东西;怪癖 | |
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sinuous
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| adj.蜿蜒的,迂回的 | |
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whatsoever
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| adv.(用于否定句中以加强语气)任何;pron.无论什么 | |
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purely
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| adv.纯粹地,完全地 | |
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distinguished
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| adj.卓越的,杰出的,著名的 | |
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apropos
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| adv.恰好地;adj.恰当的;关于 | |
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verities
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| n.真实( verity的名词复数 );事实;真理;真实的陈述 | |
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futile
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| adj.无效的,无用的,无希望的 | |
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experimentation
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| n.实验,试验,实验法 | |
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derive
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| v.取得;导出;引申;来自;源自;出自 | |
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obviated
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| v.避免,消除(贫困、不方便等)( obviate的过去式和过去分词 ) | |
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exterior
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| adj.外部的,外在的;表面的 | |
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nebula
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| n.星云,喷雾剂 | |
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attain
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| vt.达到,获得,完成 | |
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receded
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| v.逐渐远离( recede的过去式和过去分词 );向后倾斜;自原处后退或避开别人的注视;尤指问题 | |
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discriminate
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| v.区别,辨别,区分;有区别地对待 | |
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discriminated
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| 分别,辨别,区分( discriminate的过去式和过去分词 ); 歧视,有差别地对待 | |
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sufficiently
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| adv.足够地,充分地 | |
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postulated
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| v.假定,假设( postulate的过去式和过去分词 ) | |
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remains
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| n.剩余物,残留物;遗体,遗迹 | |
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solely
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| adv.仅仅,唯一地 | |
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justify
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| vt.证明…正当(或有理),为…辩护 | |
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essentially
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| adv.本质上,实质上,基本上 | |
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susceptible
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| adj.过敏的,敏感的;易动感情的,易受感动的 | |
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engendered
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| v.产生(某形势或状况),造成,引起( engender的过去式和过去分词 ) | |
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expounded
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| 论述,详细讲解( expound的过去式和过去分词 ) | |
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tempted
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| v.怂恿(某人)干不正当的事;冒…的险(tempt的过去分词) | |
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entirely
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| ad.全部地,完整地;完全地,彻底地 | |
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fictitious
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| adj.虚构的,假设的;空头的 | |
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displacement
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| n.移置,取代,位移,排水量 | |
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displacements
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| n.取代( displacement的名词复数 );替代;移位;免职 | |
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aggregate
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| adj.总计的,集合的;n.总数;v.合计;集合 | |
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revolves
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| v.(使)旋转( revolve的第三人称单数 );细想 | |
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spherical
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| adj.球形的;球面的 | |
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axis
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| n.轴,轴线,中心线;坐标轴,基准线 | |
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fiber
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| n.纤维,纤维质 | |
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qualitatively
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| 质量上 | |
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analogous
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| adj.相似的;类似的 | |
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tactile
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| adj.触觉的,有触觉的,能触知的 | |
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olfactory
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| adj.嗅觉的 | |
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varied
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| adj.多样的,多变化的 | |
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rotation
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| n.旋转;循环,轮流 | |
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ascertained
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| v.弄清,确定,查明( ascertain的过去式和过去分词 ) | |
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affected
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| adj.不自然的,假装的 | |
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systematically
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| adv.有系统地 | |
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artifice
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| n.妙计,高明的手段;狡诈,诡计 | |
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analyze
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| vt.分析,解析 (=analyse) | |
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mingling
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| adj.混合的 | |
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situated
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| adj.坐落在...的,处于某种境地的 | |
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physiology
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| n.生理学,生理机能 | |
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