chapter 24

 Euclidean and Non-Euclidean Continuum

 

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a “neighbouring” one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing “jumps.” I am sure the reader will appreciate with sufficient clearness what I mean here by “neighbouring” and by “jumps” (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

 

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab¹. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping². We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals of which are equally long. To ensure the equality of the diagonals, we make use of a little testing-rod. To this square we add similar ones, each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such, that each side of a square belongs to two squares and each corner to four squares.

 

It is a veritable wonder that we can carry out this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and, as a consequence, the arrangement of the remaining two sides of the square is already completely determined³. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord, then this is an especial favour of the marble slab and of the little rods, about which I can only be thankfully surprised. We must experience many such surprises if the construction is to be successful.

 

If everything has really gone smoothly⁴, then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod, which has been used as a “distance” (line-interval). By choosing one corner of a square as “origin” I can characterise every other corner of a square with reference to this origin by means of two numbers. I only need state how many rods I must pass over when, starting from the origin, I proceed towards the “right” and then “upwards,” in order to arrive at the corner of the square under consideration. These two numbers are then the “Cartesian co-ordinates” of this corner with reference to the “Cartesian co-ordinate system” which is determined by the arrangement of little rods.

 

By making use of the following modification⁵ of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods “expand” by in amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery⁶, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder⁷ during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.

 

With reference to our little rods—defined as unit lengths—the marble slab is no longer a Euclidean continuum, and we are also no longer in the position of defining Cartesian co-ordinates directly with their aid, since the above construction can no longer be carried out. But since there are other things which are not influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a “Euclidean continuum.” This can be done in a satisfactory manner by making a more subtle stipulation⁸ about the measurement or the comparison of lengths.

 

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behaviour of our rods in experiments analogous⁹ to the one described above, then our best plan would be to assign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these two points; for how else should we define the distance without our proceeding¹⁰ being in the highest measure grossly arbitrary? The method of Cartesian coordinates¹¹ must then be discarded, and replaced by another which does not assume the validity of Euclidean geometry for rigid¹² bodies.1 The reader will notice that the situation depicted¹³ here corresponds to the one brought about by the general postulate¹⁴ of relativity (Section XXIII).

 

1 Mathematicians¹⁵ have been confronted with our problem in the following form. If we are given a surface (e.g. an ellipsoid) in Euclidean three-dimensional space, then there exists for this surface a two-dimensional geometry, just as much as for a plane surface. Gauss undertook the task of treating this two-dimensional geometry from first principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If we imagine constructions to be made with rigid rods in the surface (similar to that above with the marble slab), we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed¹⁶ out the way to the method of Riemann of treating multi-dimensional, non-Euclidean continuum. Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity. 

1 slab   

n.平板，厚的切片；v.切成厚板，以平板盖上

参考例句：

• This heavy slab of oak now stood between the bomb and Hitler.这时笨重的橡木厚板就横在炸弹和希特勒之间了。
• The monument consists of two vertical pillars supporting a horizontal slab.这座纪念碑由两根垂直的柱体构成，它们共同支撑着一块平板。

2 overlapping   

adj./n.交迭（的）

参考例句：

• There is no overlapping question between the two courses. 这两门课程之间不存在重叠的问题。
• A trimetrogon strip is composed of three rows of overlapping. 三镜头摄影航线为三排重迭的象片所组成。

3 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。

4 smoothly   

adv.平滑地，顺利地，流利地，流畅地

参考例句：

• The workmen are very cooperative,so the work goes on smoothly.工人们十分合作，所以工作进展顺利。
• Just change one or two words and the sentence will read smoothly.这句话只要动一两个字就顺了。

5 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。

6 periphery   

n.(圆体的)外面；周围

参考例句：

• Geographically, the UK is on the periphery of Europe.从地理位置上讲，英国处于欧洲边缘。
• The periphery of the retina is very sensitive to motion.视网膜的外围对运动非常敏感。

7 disorder   

n.紊乱，混乱；骚动，骚乱；疾病，失调

参考例句：

• When returning back,he discovered the room to be in disorder.回家后，他发现屋子里乱七八糟。
• It contained a vast number of letters in great disorder.里面七零八落地装着许多信件。

8 stipulation   

n.契约，规定，条文；条款说明

参考例句：

• There's no stipulation as to the amount you can invest. 没有关于投资额的规定。 来自《简明英汉词典》
• The only stipulation the building society makes is that house must be insured. 建屋互助会作出的唯一规定是房屋必须保险。 来自《简明英汉词典》

9 analogous   

adj.相似的；类似的

参考例句：

• The two situations are roughly analogous.两种情況大致相似。
• The company is in a position closely analogous to that of its main rival.该公司与主要竞争对手的处境极为相似。

10 proceeding   

n.行动，进行，(pl.)会议录，学报

参考例句：

• This train is now proceeding from Paris to London.这次列车从巴黎开往伦敦。
• The work is proceeding briskly.工作很有生气地进展着。

11 coordinates   

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

参考例句：

• The town coordinates on this map are 695037. 该镇在这幅地图上的坐标是695037。 来自《简明英汉词典》
• The UN Office for the Coordination of Humanitarian Affairs, headed by the Emergency Relief Coordinator, coordinates all UN emergency relief. 联合国人道主义事务协调厅在紧急救济协调员领导下，负责协调联合国的所有紧急救济工作。 来自《简明英汉词典》

12 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。

13 depicted   

描绘，描画( depict的过去式和过去分词 )； 描述

参考例句：

• Other animals were depicted on the periphery of the group. 其他动物在群像的外围加以修饰。
• They depicted the thrilling situation to us in great detail. 他们向我们详细地描述了那激动人心的场面。

14 postulate   

n.假定，基本条件；vt.要求，假定

参考例句：

• Let's postulate that she is a cook.我们假定她是一位厨师。
• Freud postulated that we all have a death instinct as well as a life instinct.弗洛伊德曾假定我们所有人都有生存本能和死亡本能。

15 mathematicians   

数学家( mathematician的名词复数 )

参考例句：

• Do you suppose our mathematicians are unequal to that? 你以为我们的数学家做不到这一点吗？ 来自英汉文学
• Mathematicians can solve problems with two variables. 数学家们可以用两个变数来解决问题。 来自哲学部分

16 pointed   

adj.尖的，直截了当的

参考例句：

• He gave me a very sharp pointed pencil.他给我一支削得非常尖的铅笔。
• She wished to show Mrs.John Dashwood by this pointed invitation to her brother.她想通过对达茨伍德夫人提出直截了当的邀请向她的哥哥表示出来。