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Chapter 4 Space and its Three Dimensions
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Chapter 4
Space and its Three Dimensions
1. The Group of Displacements1
Let us sum up briefly3 the results obtained. We proposed to investigate what was meant in saying that space has three dimensions and we have asked first what is a physical continuum and when it may be said to have n dimensions. If we consider different systems of impressions and compare them with one another, we often recognize that two of these systems of impressions are indistinguishable (which is ordinarily expressed in saying that they are too close to one another, and that our senses are too crude, for us to distinguish them) and we ascertain4 besides that two of these systems can sometimes be discriminated5 from one another though indistinguishable from a third system. In that case we say the manifold of these systems of impressions forms a physical continuum C. And each of these systems is called an element of the continuum C.
How many dimensions has this continuum? Take first two elements A and B of C, and suppose there exists a series Σ of elements, all belonging to the continuum C, of such a sort that A and B are the two extreme terms of this series and that each term of the series is indistinguishable from the preceding. If such a series Σ can be found, we say that A and B are joined to one another; and if any two elements of C are joined to one another, we say that C is all of one piece.
Now take on the continuum C a certain number of elements in a way altogether arbitrary. The aggregate6 of these elements will be called a cut. Among the various series Σ which join A to B, we shall distinguish those of which an element is indistinguishable from one of the elements of the cut (we shall say that these are they which cut the cut) and those of which all the elements are distinguishable from all those of the cut. If all the series Σ which join A to B cut the cut, we shall say that A and B are separated by the cut, and that the cut divides C. If we can not find on C two elements which are separated by the cut, we shall say that the cut does not divide C.
These definitions laid down, if the continuum C can be divided by cuts which do not themselves form a continuum, this continuum C has only one dimension; in the contrary case it has several. If a cut forming a continuum of 1 dimension suffices to divide C, C will have 2 dimensions; if a cut forming a continuum of 2 dimensions suffices, C will have 3 dimensions, etc. Thanks to these definitions, we can always recognize how many dimensions any physical continuum has. It only
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1
displacements
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| n.取代( displacement的名词复数 );替代;移位;免职 | |
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displacement
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| n.移置,取代,位移,排水量 | |
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briefly
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| adv.简单地,简短地 | |
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ascertain
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| vt.发现,确定,查明,弄清 | |
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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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aggregate
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| adj.总计的,集合的;n.总数;v.合计;集合 | |
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remains
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| n.剩余物,残留物;遗体,遗迹 | |
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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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postulated
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| v.假定,假设( postulate的过去式和过去分词 ) | |
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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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fixed
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| adj.固定的,不变的,准备好的;(计算机)固定的 | |
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detour
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| n.绕行的路,迂回路;v.迂回,绕道 | |
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fibers
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| 光纤( fiber的名词复数 ); (织物的)质地; 纤维,纤维物质 | |
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fiber
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| n.纤维,纤维质 | |
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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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qualitatively
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| 质量上 | |
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purely
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| adv.纯粹地,完全地 | |
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luminous
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| adj.发光的,发亮的;光明的;明白易懂的;有启发的 | |
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simultaneously
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| adv.同时发生地,同时进行地 | |
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considerably
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| adv.极大地;相当大地;在很大程度上 | |
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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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enunciated
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| v.(清晰地)发音( enunciate的过去式和过去分词 );确切地说明 | |
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budge
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| v.移动一点儿;改变立场 | |
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afterward
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| adv.后来;以后 | |
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budged
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| v.(使)稍微移动( budge的过去式和过去分词 );(使)改变主意,(使)让步 | |
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inverse
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| adj.相反的,倒转的,反转的;n.相反之物;v.倒转 | |
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compensate
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| vt.补偿,赔偿;酬报 vi.弥补;补偿;抵消 | |
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anticipation
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| n.预期,预料,期望 | |
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contented
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| adj.满意的,安心的,知足的 | |
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criteria
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| n.标准 | |
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analyze
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| vt.分析,解析 (=analyse) | |
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analyzed
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| v.分析( analyze的过去式和过去分词 );分解;解释;对…进行心理分析 | |
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engender
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| v.产生,引起 | |
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interval
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| n.间隔,间距;幕间休息,中场休息 | |
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inverses
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| vt.使倒转(inverse的第三人称单数形式) | |
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budging
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| v.(使)稍微移动( budge的现在分词 );(使)改变主意,(使)让步 | |
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distinguished
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| adj.卓越的,杰出的,著名的 | |
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remarkable
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| adj.显著的,异常的,非凡的,值得注意的 | |
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behold
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| v.看,注视,看到 | |
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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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rigidly
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| adv.刻板地,僵化地 | |
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rigid
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| adj.严格的,死板的;刚硬的,僵硬的 | |
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conformity
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| n.一致,遵从,顺从 | |
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inversely
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| adj.相反的 | |
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previously
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| adv.以前,先前(地) | |
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authorizes
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| 授权,批准,委托( authorize的名词复数 ) | |
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invert
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| vt.使反转,使颠倒,使转化 | |
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abridged
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| 削减的,删节的 | |
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disappearance
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| n.消失,消散,失踪 | |
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exterior
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| adj.外部的,外在的;表面的 | |
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legitimate
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| adj.合法的,合理的,合乎逻辑的;v.使合法 | |
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analogous
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| adj.相似的;类似的 | |
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justify
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| vt.证明…正当(或有理),为…辩护 | |
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phenomena
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| n.现象 | |
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physiology
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| n.生理学,生理机能 | |
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physicist
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| n.物理学家,研究物理学的人 | |
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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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constrained
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| adj.束缚的,节制的 | |
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virtue
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| n.德行,美德;贞操;优点;功效,效力 | |
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allured
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| 诱引,吸引( allure的过去式和过去分词 ) | |
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simplicity
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| n.简单,简易;朴素;直率,单纯 | |
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repelled
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| v.击退( repel的过去式和过去分词 );使厌恶;排斥;推开 | |
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habitual
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| adj.习惯性的;通常的,惯常的 | |
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invincible
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| adj.不可征服的,难以制服的 | |
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prominence
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| n.突出;显著;杰出;重要 | |
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imposing
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| adj.使人难忘的,壮丽的,堂皇的,雄伟的 | |
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thereby
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| adv.因此,从而 | |
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refractory
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| adj.倔强的,难驾驭的 | |
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defense
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| n.防御,保卫;[pl.]防务工事;辩护,答辩 | |
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entangled
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| adj.卷入的;陷入的;被缠住的;缠在一起的v.使某人(某物/自己)缠绕,纠缠于(某物中),使某人(自己)陷入(困难或复杂的环境中)( entangle的过去式和过去分词 ) | |
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cultivation
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| n.耕作,培养,栽培(法),养成 | |
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scythe
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| n. 长柄的大镰刀,战车镰; v. 以大镰刀割 | |
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precisely
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| adv.恰好,正好,精确地,细致地 | |
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chamber
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| n.房间,寝室;会议厅;议院;会所 | |
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amorphous
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| adj.无定形的 | |
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quantitative
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| adj.数量的,定量的 | |
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anterior
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| adj.较早的;在前的 | |
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physiologist
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| n.生理学家 | |
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physiologists
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| n.生理学者( physiologist的名词复数 );生理学( physiology的名词复数 );生理机能 | |
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sufficiently
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| adv.足够地,充分地 | |
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orientation
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| n.方向,目标;熟悉,适应,情况介绍 | |
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adverse
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| adj.不利的;有害的;敌对的,不友好的 | |
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physiologically
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| ad.生理上,在生理学上 | |
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extremities
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| n.端点( extremity的名词复数 );尽头;手和足;极窘迫的境地 | |
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irreproachable
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| adj.不可指责的,无过失的 | |
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manifestations
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| n.表示,显示(manifestation的复数形式) | |
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destined
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| adj.命中注定的;(for)以…为目的地的 | |
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vertical
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| adj.垂直的,顶点的,纵向的;n.垂直物,垂直的位置 | |
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components
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| (机器、设备等的)构成要素,零件,成分; 成分( component的名词复数 ); [物理化学]组分; [数学]分量; (混合物的)组成部分 | |
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acceleration
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| n.加速,加速度 | |
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rotation
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| n.旋转;循环,轮流 | |
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integration
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| n.一体化,联合,结合 | |
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rotations
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| 旋转( rotation的名词复数 ); 转动; 轮流; 轮换 | |
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