Chapter 5

This being clear, we must go on to consider the questions which remain. First, is there an infinite body, as the majority of the ancient philosophers thought, or is this an impossibility? The decision of this question, either way, is not unimportant, but rather all-important, to our search for the truth. It is this problem which has practically always been the source of the differences of those who have written about nature as a whole. So it has been and so it must be; since the least initial deviation¹ from the truth is multiplied later a thousandfold. Admit, for instance, the existence of a minimum magnitude, and you will find that the minimum which you have introduced, small as it is, causes the greatest truths of mathematics to totter². The reason is that a principle is great rather in power than in extent; hence that which was small at the start turns out a giant at the end. Now the conception of the infinite possesses this power of principles, and indeed in the sphere of quantity possesses it in a higher degree than any other conception; so that it is in no way absurd or unreasonable³ that the assumption that an infinite body exists should be of peculiar⁴ moment to our inquiry⁵. The infinite, then, we must now discuss, opening the whole matter from the beginning.

Every body is necessarily to be classed either as simple or as composite; the infinite body, therefore, will be either simple or composite.

But it is clear, further, that if the simple bodies are finite, the composite must also be finite, since that which is composed of bodies finite both in number and in magnitude is itself finite in respect of number and magnitude: its quantity is in fact the same as that of the bodies which compose it. What remains⁶ for us to consider, then, is whether any of the simple bodies can be infinite in magnitude, or whether this is impossible. Let us try the primary body first, and then go on to consider the others.

The body which moves in a circle must necessarily be finite in every respect, for the following reasons. (1) If the body so moving is infinite, the radii⁷ drawn⁸ from the centre will be infinite. But the space between infinite radii is infinite: and by the space between the radii I mean the area outside which no magnitude which is in contact with the two lines can be conceived as falling. This, I say, will be infinite: first, because in the case of finite radii it is always finite; and secondly¹⁰, because in it one can always go on to a width greater than any given width; thus the reasoning which forces us to believe in infinite number, because there is no maximum, applies also to the space between the radii. Now the infinite cannot be traversed, and if the body is infinite the interval¹¹ between the radii is necessarily infinite: circular motion therefore is an impossibility. Yet our eyes tell us that the heavens revolve¹² in a circle, and by argument also we have determined¹³ that there is something to which circular movement belongs.

(2) Again, if from a finite time a finite time be subtracted, what remains must be finite and have a beginning. And if the time of a journey has a beginning, there must be a beginning also of the movement, and consequently also of the distance traversed. This applies universally. Take a line, ACE⁹, infinite in one direction, E, and another line, BB, infinite in both directions. Let ACE describe a circle, revolving¹⁴ upon C as centre. In its movement it will cut BB continuously for a certain time. This will be a finite time, since the total time is finite in which the heavens complete their circular orbit, and consequently the time subtracted from it, during which the one line in its motion cuts the other, is also finite. Therefore there will be a point at which ACE began for the first time to cut BB. This, however, is impossible. The infinite, then, cannot revolve in a circle; nor could the world, if it were infinite.

(3) That the infinite cannot move may also be shown as follows. Let A be a finite line moving past the finite line, B. Of necessity A will pass clear of B and B of A at the same moment; for each overlaps¹⁵ the other to precisely¹⁶ the same extent. Now if the two were both moving, and moving in contrary directions, they would pass clear of one another more rapidly; if one were still and the other moving past it, less rapidly; provided that the speed of the latter were the same in both cases. This, however, is clear: that it is impossible to traverse an infinite line in a finite time. Infinite time, then, would be required. (This we demonstrated above in the discussion of movement.) And it makes no difference whether a finite is passing by an infinite or an infinite by a finite. For when A is passing B, then B overlaps A and it makes no difference whether B is moved or unmoved, except that, if both move, they pass clear of one another more quickly. It is, however, quite possible that a moving line should in certain cases pass one which is stationary¹⁷ quicker than it passes one moving in an opposite direction. One has only to imagine the movement to be slow where both move and much faster where one is stationary. To suppose one line stationary, then, makes no difficulty for our argument, since it is quite possible for A to pass B at a slower rate when both are moving than when only one is. If, therefore, the time which the finite moving line takes to pass the other is infinite, then necessarily the time occupied by the motion of the infinite past the finite is also infinite. For the infinite to move at all is thus absolutely impossible; since the very smallest movement conceivable must take an infinity¹⁸ of time. Moreover the heavens certainly revolve, and they complete their circular orbit in a finite time; so that they pass round the whole extent of any line within their orbit, such as the finite line AB. The revolving body, therefore, cannot be infinite.

(4) Again, as a line which has a limit cannot be infinite, or, if it is infinite, is so only in length, so a surface cannot be infinite in that respect in which it has a limit; or, indeed, if it is completely determinate, in any respect whatever. Whether it be a square or a circle or a sphere, it cannot be infinite, any more than a foot-rule can. There is then no such thing as an infinite sphere or square or circle, and where there is no circle there can be no circular movement, and similarly where there is no infinite at all there can be no infinite movement; and from this it follows that, an infinite circle being itself an impossibility, there can be no circular motion of an infinite body.

(5) Again, take a centre C, an infinite line, AB, another infinite line at right angles to it, E, and a moving radius¹⁹, CD. CD will never cease contact with E, but the position will always be something like CE, CD cutting E at F. The infinite line, therefore, refuses to complete the circle.

(6) Again, if the heaven is infinite and moves in a circle, we shall have to admit that in a finite time it has traversed the infinite. For suppose the fixed²⁰ heaven infinite, and that which moves within it equal to it. It results that when the infinite body has completed its revolution, it has traversed an infinite equal to itself in a finite time. But that we know to be impossible.

(7) It can also be shown, conversely, that if the time of revolution is finite, the area traversed must also be finite; but the area traversed was equal to itself; therefore, it is itself finite.

We have now shown that the body which moves in a circle is not endless or infinite, but has its limit.

1 deviation   

n.背离，偏离；偏差，偏向；离题

参考例句：

• Deviation from this rule are very rare.很少有违反这条规则的。
• Any deviation from the party's faith is seen as betrayal.任何对党的信仰的偏离被视作背叛。

2 totter   

v.蹒跚， 摇摇欲坠；n.蹒跚的步子

参考例句：

• He tottered to the fridge,got a beer and slumped at the table.他踉跄地走到冰箱前，拿出一瓶啤酒，一屁股坐在桌边。
• The property market is tottering.房地产市场摇摇欲坠。

3 unreasonable   

adj.不讲道理的，不合情理的，过度的

参考例句：

• I know that they made the most unreasonable demands on you.我知道他们对你提出了最不合理的要求。
• They spend an unreasonable amount of money on clothes.他们花在衣服上的钱太多了。

4 peculiar   

adj.古怪的，异常的；特殊的，特有的

参考例句：

• He walks in a peculiar fashion.他走路的样子很奇特。
• He looked at me with a very peculiar expression.他用一种很奇怪的表情看着我。

5 inquiry   

n.打听，询问，调查，查问

参考例句：

• Many parents have been pressing for an inquiry into the problem.许多家长迫切要求调查这个问题。
• The field of inquiry has narrowed down to five persons.调查的范围已经缩小到只剩5个人了。

6 remains   

n.剩余物，残留物；遗体，遗迹

参考例句：

• He ate the remains of food hungrily.他狼吞虎咽地吃剩余的食物。
• The remains of the meal were fed to the dog.残羹剩饭喂狗了。

7 radii   

n.半径；半径（距离）( radius的名词复数 )；用半径度量的圆形面积；半径范围；桡骨

参考例句：

• Hence, the damage radii can not be determined from overpressure alone. 因此，破坏半径不能单单由超压力大小来决定。 来自辞典例句
• It is now necessary to introduce a sign convention for radii of curvature. 现在必须介绍曲率半径的正负号规则。 来自辞典例句

8 drawn   

v.拖，拉，拔出；adj.憔悴的，紧张的

参考例句：

• All the characters in the story are drawn from life.故事中的所有人物都取材于生活。
• Her gaze was drawn irresistibly to the scene outside.她的目光禁不住被外面的风景所吸引。

9 ace   

n.A牌；发球得分；佼佼者；adj.杰出的

参考例句：

• A good negotiator always has more than one ace in the hole.谈判高手总有数张王牌在手。
• He is an ace mechanic.He can repair any cars.他是一流的机械师,什么车都会修。

10 secondly   

adv.第二，其次

参考例句：

• Secondly,use your own head and present your point of view.第二，动脑筋提出自己的见解。
• Secondly it is necessary to define the applied load.其次，需要确定所作用的载荷。

11 interval   

n.间隔，间距；幕间休息，中场休息

参考例句：

• The interval between the two trees measures 40 feet.这两棵树的间隔是40英尺。
• There was a long interval before he anwsered the telephone.隔了好久他才回了电话。

12 revolve   

vi.(使)旋转；循环出现

参考例句：

• The planets revolve around the sun.行星绕着太阳运转。
• The wheels began to revolve slowly.车轮开始慢慢转动。

13 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。

14 revolving   

adj.旋转的，轮转式的；循环的v.（使）旋转( revolve的现在分词 )；细想

参考例句：

• The theatre has a revolving stage. 剧院有一个旋转舞台。
• The company became a revolving-door workplace. 这家公司成了工作的中转站。

15 overlaps   

v.部分重叠( overlap的第三人称单数 )；（物体）部份重叠；交叠；（时间上）部份重叠

参考例句：

• The style in these two books largely overlaps. 这两本书的文体有许多处是一致的。 来自《现代英汉综合大词典》
• The new office overlaps the functions of the one already in existence. 新机构的职能与那个现存机构的职能部分重叠。 来自辞典例句

16 precisely   

adv.恰好，正好，精确地，细致地

参考例句：

• It's precisely that sort of slick sales-talk that I mistrust.我不相信的正是那种油腔滑调的推销宣传。
• The man adjusted very precisely.那个人调得很准。

17 stationary   

adj.固定的，静止不动的

参考例句：

• A stationary object is easy to be aimed at.一个静止不动的物体是容易瞄准的。
• Wait until the bus is stationary before you get off.你要等公共汽车停稳了再下车。

18 infinity   

n.无限，无穷，大量

参考例句：

• It is impossible to count up to infinity.不可能数到无穷大。
• Theoretically,a line can extend into infinity.从理论上来说直线可以无限地延伸。

19 radius   

n.半径，半径范围；有效航程，范围，界限

参考例句：

• He has visited every shop within a radius of two miles.周围两英里以内的店铺他都去过。
• We are measuring the radius of the circle.我们正在测量圆的半径。

20 fixed   

adj.固定的，不变的，准备好的；(计算机)固定的

参考例句：

• Have you two fixed on a date for the wedding yet?你们俩选定婚期了吗？
• Once the aim is fixed,we should not change it arbitrarily.目标一旦确定，我们就不应该随意改变。