II The System of Co-ordinates

 On the basis of the physical interpretation¹ of distance which has been indicated, we are also in a position to establish the distance between two points on a rigid² body by means of measurements. For this purpose we require a “distance” (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line joining them according to the rules of geometry; then, starting from A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length.1

 

Every description of the scene of an event or of the position of an object in space is based on the specification³ of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification “Times Square, New York,”2 I arrive at the following result. The earth is the rigid body to which the specification of place refers; “Times Square, New York,” is a well-defined point, to which a name has been assigned, and with which the event coincides in space.3

 

This primitive⁴ method of place specification deals only with places on the surface of rigid bodies, and is dependent on the existence of points on this surface which are distinguishable from each other. But we can free ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering⁵ over Times Square, then we can determine its position relative to the surface of the earth by erecting⁶ a pole perpendicularly⁸ on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement⁹ of the conception of position has been developed.

 

(a) We imagine the rigid body, to which the place specification is referred, supplemented in such a manner that the object whose position we require is reached by the completed rigid body.

(b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference.

We speak of the height of the cloud even when the pole which reaches the cloud has not been erected¹⁰. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud.

From this consideration we see that it will be advantageous¹¹ if, in the description of position, it should be possible by means of numerical measures to make ourselves independent of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained¹² by the application of the Cartesian system of co-ordinates.

 

This consists of three plane surfaces perpendicular⁷ to each other and rigidly¹³ attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined¹⁴ (for the main part) by the specification of the lengths of the three perpendiculars¹⁵ or co-ordinates left-parenthesis x comma y comma z right-parenthesis which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry.

 

In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications¹⁶ of position must always be sought in accordance with the above considerations.4

 

We thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for “distances;” the “distance” being represented physically¹⁷ by means of the convention of two marks on a rigid body.

 

1 Here we have assumed that there is nothing left over i.e. that the measurement gives a whole number. This difficulty is got over by the use of divided measuring-rods, the introduction of which does not demand any fundamentally new method. ↑

 

2 Einstein used “Potsdamer Platz, Berlin” in the original text. In the authorised translation this was supplemented with “Trafalgar Square, London”. We have changed this to “Times Square, New York”, as this is the best known location to English speakers in the present day. ↑

 

3 It is not necessary here to investigate further the significance of the expression “coincidence in space.” This conception is sufficiently¹⁸ obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice. ↑

 

4 A refinement and modification¹⁹ of these views does not become necessary until we come to deal with the general theory of relativity, treated in the second part of this book. 

1 interpretation   

n.解释，说明，描述；艺术处理

参考例句：

• His statement admits of one interpretation only.他的话只有一种解释。
• Analysis and interpretation is a very personal thing.分析与说明是个很主观的事情。

2 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。

3 specification   

n.详述；[常pl.]规格，说明书，规范

参考例句：

• I want to know his specification of details.我想知道他对细节的详述。
• Examination confirmed that the quality of the products was up to specification.经检查，产品质量合格。

4 primitive   

adj.原始的；简单的；n.原(始)人，原始事物

参考例句：

• It is a primitive instinct to flee a place of danger.逃离危险的地方是一种原始本能。
• His book describes the march of the civilization of a primitive society.他的著作描述了一个原始社会的开化过程。

5 hovering   

鸟( hover的现在分词 )； 靠近（某事物）； （人）徘徊； 犹豫

参考例句：

• The helicopter was hovering about 100 metres above the pad. 直升机在离发射台一百米的上空盘旋。
• I'm hovering between the concert and the play tonight. 我犹豫不决今晚是听音乐会还是看戏。

6 erecting   

v.使直立，竖起( erect的现在分词 )；建立

参考例句：

• Nations can restrict their foreign trade by erecting barriers to exports as well as imports. 象设置进口壁垒那样，各国可以通过设置出口壁垒来限制对外贸易。 来自辞典例句
• Could you tell me the specific lift-slab procedure for erecting buildings? 能否告之用升板法安装楼房的具体程序？ 来自互联网

7 perpendicular   

adj.垂直的，直立的；n.垂直线，垂直的位置

参考例句：

• The two lines of bones are set perpendicular to one another.这两排骨头相互垂直。
• The wall is out of the perpendicular.这墙有些倾斜。

8 perpendicularly   

adv. 垂直地, 笔直地, 纵向地

参考例句：

• Fray's forehead was wrinkled both perpendicularly and crosswise. 弗雷的前额上纹路纵横。
• Automatic resquaring feature insures nozzle is perpendicularly to the part being cut. 自动垂直功能，可以确保刀头回到与工件完全垂直的位置去切割。

9 refinement   

n.文雅；高尚；精美；精制；精炼

参考例句：

• Sally is a woman of great refinement and beauty. 莎莉是个温文尔雅又很漂亮的女士。
• Good manners and correct speech are marks of refinement.彬彬有礼和谈吐得体是文雅的标志。

10 ERECTED   

adj. 直立的，竖立的，笔直的 vt. 使 ... 直立，建立

参考例句：

• A monument to him was erected in St Paul's Cathedral. 在圣保罗大教堂为他修了一座纪念碑。
• A monument was erected to the memory of that great scientist. 树立了一块纪念碑纪念那位伟大的科学家。

11 advantageous   

adj.有利的；有帮助的

参考例句：

• Injections of vitamin C are obviously advantageous.注射维生素C显然是有利的。
• You're in a very advantageous position.你处于非常有利的地位。

12 attained   

（通常经过努力）实现( attain的过去式和过去分词 )； 达到； 获得； 达到（某年龄、水平、状况）

参考例句：

• She has attained the degree of Master of Arts. 她已获得文学硕士学位。
• Lu Hsun attained a high position in the republic of letters. 鲁迅在文坛上获得崇高的地位。

13 rigidly   

adv.刻板地，僵化地

参考例句：

• Life today is rigidly compartmentalized into work and leisure. 当今的生活被严格划分为工作和休闲两部分。
• The curriculum is rigidly prescribed from an early age. 自儿童时起即已开始有严格的课程设置。

14 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。

15 perpendiculars   

n.垂直的，成直角的( perpendicular的名词复数 )；直立的

参考例句：

• It is normal if two hands are in a line and perpendiculars to the floor. 标准姿势就是两只手在一条直线上与地面成直角。 来自互联网

16 specifications   

n.规格；载明；详述；（产品等的）说明书；说明书( specification的名词复数 )；详细的计划书；载明；详述

参考例句：

• Our work must answer the specifications laid down. 我们的工作应符合所定的规范。 来自《简明英汉词典》
• This sketch does not conform with the specifications. 图文不符。 来自《现代汉英综合大词典》

17 physically   

adj.物质上，体格上，身体上，按自然规律

参考例句：

• He was out of sorts physically,as well as disordered mentally.他浑身不舒服，心绪也很乱。
• Every time I think about it I feel physically sick.一想起那件事我就感到极恶心。

18 sufficiently   

adv.足够地，充分地

参考例句：

• It turned out he had not insured the house sufficiently.原来他没有给房屋投足保险。
• The new policy was sufficiently elastic to accommodate both views.新政策充分灵活地适用两种观点。

19 modification   

n.修改，改进，缓和，减轻

参考例句：

• The law,in its present form,is unjust;it needs modification.现行的法律是不公正的，它需要修改。
• The design requires considerable modification.这个设计需要作大的修改。