PART I The Special Theory of Relativity I Physical Meaning of Geometrical Propositions

 In your schooldays most of you who read this book made acquaintance with the noble building of Euclid’s geometry, and you remember—perhaps with more respect than love—the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious¹ teachers. By reason of our past experience, you would certainly regard everyone with disdain² who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: “What, then, do you mean by the assertion that these propositions are true?” Let us proceed to give this question a little consideration.

 

Geometry sets out form certain conceptions such as “plane,” “point,” and “straight line,” with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue³ of these ideas, we are inclined to accept as “true.” Then, on the basis of a logical process, the justification⁴ of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct (“true”) when it has been derived⁵ in the recognised manner from the axioms. The question of “truth” of the individual geometrical propositions is thus reduced to one of the “truth” of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely⁶ without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called “straight lines,” to each of which is ascribed the property of being uniquely determined⁷ by two points situated⁸ on it. The concept “true” does not tally⁹ with the assertions of pure geometry, because by the word “true” we are eventually in the habit of designating always the correspondence with a “real” object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.

 

It is not difficult to understand why, in spite of this, we feel constrained¹⁰ to call the propositions of geometry “true.” Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly¹¹ the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity¹². The practice, for example, of seeing in a “distance” two marked positions on a practically rigid¹³ body is something which is lodged¹⁴ deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation.

 

If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1 Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately¹⁵ ask as to the “truth” of geometrical propositions interpreted in this way, since we are justified¹⁶ in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the “truth” of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses.

 

Of course the conviction of the “truth” of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the “truth” of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this “truth” is limited, and we shall consider the extent of its limitation.

 

1 It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for the present purpose. 

1 conscientious   

adj.审慎正直的，认真的，本着良心的

参考例句：

• He is a conscientious man and knows his job.他很认真负责，也很懂行。
• He is very conscientious in the performance of his duties.他非常认真地履行职责。

2 disdain   

n.鄙视，轻视；v.轻视，鄙视，不屑

参考例句：

• Some people disdain labour.有些人轻视劳动。
• A great man should disdain flatterers.伟大的人物应鄙视献媚者。

3 virtue   

n.德行，美德；贞操；优点；功效，效力

参考例句：

• He was considered to be a paragon of virtue.他被认为是品德尽善尽美的典范。
• You need to decorate your mind with virtue.你应该用德行美化心灵。

4 justification   

n.正当的理由；辩解的理由

参考例句：

• There's no justification for dividing the company into smaller units. 没有理由把公司划分成小单位。
• In the young there is a justification for this feeling. 在年轻人中有这种感觉是有理由的。

5 derived   

vi.起源；由来；衍生；导出v.得到( derive的过去式和过去分词 )；（从…中）得到获得；源于；（从…中）提取

参考例句：

• Many English words are derived from Latin and Greek. 英语很多词源出于拉丁文和希腊文。 来自《简明英汉词典》
• He derived his enthusiasm for literature from his father. 他对文学的爱好是受他父亲的影响。 来自《简明英汉词典》

6 entirely   

ad.全部地，完整地；完全地，彻底地

参考例句：

• The fire was entirely caused by their neglect of duty. 那场火灾完全是由于他们失职而引起的。
• His life was entirely given up to the educational work. 他的一生统统献给了教育工作。

7 determined   

adj.坚定的；有决心的

参考例句：

• I have determined on going to Tibet after graduation.我已决定毕业后去西藏。
• He determined to view the rooms behind the office.他决定查看一下办公室后面的房间。

8 situated   

adj.坐落在...的，处于某种境地的

参考例句：

• The village is situated at the margin of a forest.村子位于森林的边缘。
• She is awkwardly situated.她的处境困难。

9 tally   

n.计数器，记分，一致，测量；vt.计算，记录，使一致；vi.计算，记分，一致

参考例句：

• Don't forget to keep a careful tally of what you spend.别忘了仔细记下你的开支账目。
• The facts mentioned in the report tally to every detail.报告中所提到的事实都丝毫不差。

10 constrained   

adj.束缚的,节制的

参考例句：

• The evidence was so compelling that he felt constrained to accept it. 证据是那样的令人折服，他觉得不得不接受。
• I feel constrained to write and ask for your forgiveness. 我不得不写信请你原谅。

11 undoubtedly   

adv.确实地，无疑地

参考例句：

• It is undoubtedly she who has said that.这话明明是她说的。
• He is undoubtedly the pride of China.毫无疑问他是中国的骄傲。

12 unity   

n.团结，联合，统一；和睦，协调

参考例句：

• When we speak of unity,we do not mean unprincipled peace.所谓团结，并非一团和气。
• We must strengthen our unity in the face of powerful enemies.大敌当前，我们必须加强团结。

13 rigid   

adj.严格的，死板的；刚硬的，僵硬的

参考例句：

• She became as rigid as adamant.她变得如顽石般的固执。
• The examination was so rigid that nearly all aspirants were ruled out.考试很严，几乎所有的考生都被淘汰了。

14 lodged   

v.存放( lodge的过去式和过去分词 )；暂住；埋入；（权利、权威等）归属

参考例句：

• The certificate will have to be lodged at the registry. 证书必须存放在登记处。 来自《简明英汉词典》
• Our neighbours lodged a complaint against us with the police. 我们的邻居向警方控告我们。 来自《简明英汉词典》

15 legitimately   

ad.合法地；正当地，合理地

参考例句：

• The radio is legitimately owned by the company. 该电台为这家公司所合法拥有。
• She looked for nothing save what might come legitimately and without the appearance of special favour. 她要的并不是男人们的额外恩赐，而是合法正当地得到的工作。 来自英汉文学 - 嘉莉妹妹

16 justified   

a.正当的，有理的

参考例句：

• She felt fully justified in asking for her money back. 她认为有充分的理由要求退款。
• The prisoner has certainly justified his claims by his actions. 那个囚犯确实已用自己的行动表明他的要求是正当的。