chapter 12 Periodicity in nature

PERIODICITY IN NATURE

THE WHOLE life of Nature is dominated by the existence of periodic events, that is, by the existence of successive events so analogous to each other that, without any straining of language, they may be termed recurrences of the same event. The rotation of earth produces the successive days. It is true that each day is different from the preceding days, however abstractly we define the meaning of a day, so as to exclude casual phenomena. But with a sufficiently abstract definition of a day, the distinction in properties between two days becomes faint and remote from practical interest; and each day may then be conceived as a recurrence of the phenomenon of one rotation of the earth. Again the path of the earth round the sun leads to the yearly recurrence of the seasons, and imposes another periodicity on all the operations of nature. Another less fundamental periodicity is provided by the phases of the moon. In modern civilized life, with its artificial light, these phases are of slight importance, but in ancient times, in climates where the days are burning and the skies clear, human life was apparently largely influenced by the existence of moonlight. Accordingly our divisions into weeks and months, with their religious associations, have spread over the European races from Syria and Mesopotamia, though independent observances following the moon's phases are found amongst most nations. It is, however, through the tides, and not through its phases of light and darkness, that the moon's periodicity has chiefly influenced the history of the earth.

大自然的整个生命是由周期性事件的存在主导的，也就是说，主导着一系列彼此极为相似的连续事件，以至于在语言上毫不勉强地，它们可以被称为同一事件的重复。地球的自转产生了连续的日夜。虽然我们如何抽象地定义一天，排除偶然现象，每一天与前一天是不同的，但通过一个足够抽象的定义，两个日期之间的属性区别变得微弱且与实际兴趣无关；此时，每一天可以被看作是地球自转现象的重复。再者，地球绕太阳的轨迹导致了季节的年度循环，并为自然界的所有运作带来了另一个周期性。月亮的周期性则提供了另一个较为次要的周期性。在现代文明生活中，人工光源使得这些月相的重要性微乎其微，但在古代，尤其在白天气温酷热、天空晴朗的气候条件下，月光的存在显然在很大程度上影响了人类的生活。因此，我们对周和月的划分及其宗教意义，已从叙利亚和美索不达米亚传播到欧洲各族群，尽管大多数民族中也有独立的习俗遵循月相变化。然而，月亮的周期性主要通过潮汐影响了地球的历史，而不是通过光明和黑暗的变化。 Our bodies life is essentially periodic. It is dominated by the beating of the heart, and the recurrence of breathing. The presupposition of periodicity is indeed fundamental to our very conception of life. We cannot imagine a course of nature in which, as events progressed, we should be unable to say: 'This has happened before.' The whole conception of experience as a guide to conduct would be absent. Men would always find themselves in new situations possessing no substratum of identity with anything in past history. The very means of measuring time as a quantity would be absent. Events might still be recognized as occurring in a series, so that some were earlier and others later. But we now go beyond this bare recognition. We can not only say that three events, A, B, C, occurred in this order, so that A came before B, and B before C; but also we can say that the length of time between the occurrences of A and B was twice as long as that between B and C. Now, quantity of time is essentially dependent on observing the number of natural recurrences which have intervened. We may say that the length of time between A and B was so many days, or so many months, or so many years, according to the type of recurrence to which we wish to appeal. Indeed, at the beginning of civilization, these three modes of measuring time were really distinct. It has been one of the first tasks of science among civilized or semi-civilized nations, to fuse them into one coherent measure. The full extent of this task must be grasped. It is necessary to determine, not merely what number of days (e.g. 365.25...) go to some one year, but also previously to determine that the same number of days do go to the successive years. We can imagine a world in which periodicities exist, but such that no two are coherent. In some years there might be 200 days and in others 350. The determination of the broad general consistency of the more important periodicities was the first step in natural science. This consistency arises from no abstract intuitive law of thought; it is merely an observed fact of nature guaranteed by experience. Indeed, so far is it from being a necessary law, that it is not even exactly true. There are divergencies in every case. For some instances there divergencies are easily observed and are therefore immediately apparent. In other cases it requires the most refined observations and astronomical accuracy to make them apparent. Broadly speaking, all recurrences depending on living beings, such as the beatings of the heart, are subject in comparison with other recurrences to rapid variations. The great stable obvious recurrences —— stable in the sense of mutually agreeing with great accuracy—— are those depending on the motion of the earth as a whole, and on similar motions of the heavenly bodies.

我们的生命本质上是周期性的。它受到心跳的节奏和呼吸的周期性支配。周期性的前提实际上是我们对生命本质的基本认知。我们无法想象在一个自然过程当中，随着事件的进展，我们无法说“这曾经发生过”。体验作为行为指南的整体概念将会缺失。人们将总是发现自己处于全新的境地，与过去历史中的任何事物没有相同的底层身份。甚至连作为量度时间的手段都会缺失。事件仍然可以被认为是发生在一系列当中的，因此有些发生在前，有些发生在后。但我们现在超越了这一基本的识别。我们不仅可以说三个事件 A、B、C 是按顺序发生的，A 发生在 B 之前，B 发生在 C 之前；我们还可以说 A 和 B 之间发生的时间是 B 和 C 之间时间的两倍。现在，时间的量度本质上依赖于观察其间发生的自然周期的次数。我们可以说，A 和 B 之间的时间长度是几天，几个月，或几年，具体取决于我们希望参照的周期类型。事实上，在文明的初期，这三种时间度量方式实际上是截然不同的。在文明或半文明的民族中，科学的首要任务之一，就是将它们融为一个连贯的度量。必须理解这一任务的完整范围。我们不仅仅需要确定多少天（例如 365.25 天）构成一年，还需要事先确定相同数量的天数适用于接下来的每一年。我们可以想象一个周期性存在的世界，但其中没有两个周期是相互一致的。在某些年份，可能有 200 天，而在其他年份，可能有 350 天。确定更重要周期的广泛一致性是自然科学的第一步。这种一致性并非源自任何抽象的直观思维法则；它仅仅是通过经验得到验证的自然事实。事实上，它远不是一条必要的法则，甚至不完全准确。每一种情况都会有偏差。在某些情况下，这些偏差容易被观察到，因此立即显现出来。而在其他情况下，则需要极其精密的观察和天文精度才能使这些偏差显现出来。广义上讲，所有依赖于生物体的周期性，比如心跳，都会相对于其他周期性呈现快速变化。那些稳定且明显的周期性——在精确度上相互一致的——是依赖于地球整体运动和天体类似运动的。

We therefore assume that these astronomical recurrences mark out equal intervals of time. But how are we to deal with their discrepancies which the refined observations of astronomy detect? Apparently we are reduced to the arbitrary assumption that one or other of these sets of phenomena marks out equal times——e.g. that either all days are of equal length, or that all years are of equal length. This is not so: some assumptions must be made, but the assumption which underlies the whole procedure of the astronomers in determining the measure of time is that the laws of motion are exactly verified. Before explaining how this is done, it is interesting to observe that this relegation of the determination of the measure of time to the astronomers arises (as has been said) from the stable consistency of the recurrences with which they deal. If such a superior consistency had been noted among the recurrences characteristic of the human body, we should naturally have looked to the doctors of medicine for the regulation of our clocks.

因此，我们假定这些天文循环标志着相等的时间间隔。但对于天文学精密观测所发现的这些循环的不一致性，我们该如何处理呢？显然，我们被迫做出一个武断的假设，即这些现象中的某一类能够标志相等的时间，例如，要么所有的日长都相等，要么所有的年长都相等。事实并非如此：某些假设确实必须被提出，但天文学家在确定时间度量的整个过程中所依据的基本假设是运动定律被严格验证。在解释这一点之前，有一个有趣的现象值得注意：将时间度量的确定交由天文学家处理（如前所述），是因为他们所研究的循环具有稳定的一致性。如果在人类身体特征性的循环中发现了这种高度的一致性，那么我们很可能会自然而然地依赖医学博士来调节我们的时钟。

In considering how the laws of motion come into the matter, note that two inconsistent modes of measuring time will yield different variations of velocity to the same body. For example, suppose we define an hour as one twenty-fourth of a day, and take the case of a train running uniformly for two hours at the rate of twenty miles per hour. Now take a grossly inconsistent measure of time, and suppose that it makes the first hour to be twice as long as the second hour. Then, according to this other measure of duration, the time of the train's run is divided into two parts, during each of which it has traversed the same distance, namely, twenty miles; but the duration of the first part is twice as long as that of the second part. Hence the velocity of the train has not been uniform, and on the average the velocity during the second period is twice that during the first period. Thus the question as to whether the train has been running uniformly or not entirely depends on the standard of time which we adopt.

在考虑运动定律如何与此相关时，注意到两种不一致的时间测量方式会导致同一物体产生不同的速度变化。例如，假设我们将一小时定义为一天的二十四分之一，并以每小时二十英里的速度均匀行驶两小时的火车为例。现在假设我们采用一种严重不一致的时间度量方式，假设这种度量方式使得第一小时是第二小时的两倍长。那么，根据这种其他的时间度量方式，火车的行驶时间被分为两部分，在每一部分中，火车行驶的距离相同，即二十英里；但第一部分的持续时间是第二部分的两倍。因此，火车的速度并不均匀，平均而言，第二部分的速度是第一部分的两倍。由此可见，火车是否均匀行驶完全取决于我们采用的时间标准。

Now, for all ordinary purposes of life on the earth, the various astronomical recurrences may be looked on as absolutely consistent; and, furthermore assuming their consistency, and thereby assuming the velocities and changes of velocities possessed by bodies, we find that the laws of motion, which have been considered above, are almost exactly verified. But only exactly when we come to some of the astronomical phenomena. We find, however, that by assuming slightly different velocities for the rotations and motions of the planets and stars, the laws would be exactly verified. This assumption is then made; and we have, in fact thereby, adopted a measure of time, which is indeed defined by reference to the astronomical phenomena, but not so as to be consistent with the uniformity or any one of them. But the broad fact remains that the uniform flow of time on which so much is based, is itself dependent on the observation of periodic events.

现在，对于地球上的所有日常生活目的，各种天文循环可以被视为绝对一致的；而且，进一步假设它们的一致性，从而假设物体的速度和速度变化，我们发现，之前讨论的运动定律几乎完全得到验证。但在某些天文现象中，只有几乎完全验证。当我们假设行星和恒星的自转和运动具有稍微不同的速度时，定律将得到完全验证。于是做出了这一假设；事实上，我们因此采用了一种时间度量方式，这种度量方式确实是通过参考天文现象来定义的，但并非与任何一个天文现象的均匀性一致。然而，广义的事实依然是，许多事情所依赖的时间的均匀流动本身是依赖于周期性事件的观察的。

Even phenomena, which on the surface seem casual and exceptional, or, on the other hand, maintain themselves with a uniform persistency, may be due to the remote influence of periodicity. Take, for example, the principle of resonance. Resonance arises when two sets of connected circumstances have the same periodicities. It is a dynamical law that the small vibrations of all bodies when left to themselves take place in definite times characteristic of the body. Thus a pendulum with a small swing always vibrates in some definite time, characteristic of its shape and distribution of weight and length. A more complicated body may have many ways of vibrating; but each of its mode of vibration will have its own peculiar 'period'. Those periods of vibration of a body are called its 'free' periods. Thus a pendulum has but one period of vibration, while a suspension bridge will have many. We get a musical instrument, like a violin string, when the periods of vibration are all simple submultiples of the longest; i.e. if seconds be the longest period, the others are , and so on, where any of these smaller periods may be absent. Now, suppose we excite the vibrations of a body by a cause which is itself periodic; then, if the period of the cause is very nearly that of one of the periods of the body, that mode of vibration of the body is very violently excited; even although the magnitude of the exciting cause is small. This phenomenon is called "resonance". The general reason is easy to understand. Any one wanting to upset a rocking stone will push 'in tune' with the oscillations of the stone, so as always to secure a favourable moment for a push. If the pushed are out of tune, some increase the oscillations, but others check them. But when they are in tune, after a time all the pushes are favourable. The word 'resonance' comes from considerations of sound: but the phenomenon extends far beyond the region of sound. The laws of absorption and emission of light depend on it, the 'tuning' of receivers for wireless telegraphy, the comparative importance of the influences of planets on each other's motion, the danger to a suspension bridge as troops march over it in step, and the excessive vibration of some ships, under the rhythmical beat of their machinery at certain speeds. This coincidence of periodicities may produce steady phenomena when there is a constant association of the two periodic events, or it may produce violent and sudden outbursts when the association is fortuitous and temporary.

即使是看似偶然和特殊的现象，或者另一方面，表现出均匀持续性的现象，也可能是周期性影响的远程结果。例如，考虑共鸣原理。共鸣发生在两组相互关联的情境具有相同的周期性时。一个力学法则是，所有物体的微小振动，在不受外部干扰时，都会在特定的时间内发生，这段时间是该物体的特征，取决于其形状、质量分布和长度。因此，一个小摆幅的钟摆总是以一个确定的时间振动，这个时间是由它的形状、质量分布和长度所决定的。一个更复杂的物体可能有多种振动方式；但每种振动模式都有其独特的“周期”。这些物体的振动周期称为它的“自由”周期。因此，钟摆只有一个振动周期，而悬索桥则有多个振动周期。当我们得到像小提琴弦这样的乐器时，所有的振动周期都是最长周期的简单子倍数；也就是说，如果 秒是最长的周期，其他周期是 ，依此类推，其中任何较小的周期可能会缺失。现在，假设我们通过一个周期性原因来激发物体的振动；如果激发的周期与物体的某个周期非常接近，那么该振动模式会被非常剧烈地激发；即使激发原因的幅度很小。这个现象叫做“共鸣”。这个原理很容易理解。任何想要推动摇摆石块的人，会根据石块的振荡周期来“调和”推动，这样他每次施力时都能抓住一个有利时刻。如果推动力与振荡不一致，有些推动力会增加振荡，而其他则会抑制振荡。但当它们调和时，经过一段时间，所有的推动力都会是有利的。虽然“共鸣”一词源于声音，但这一现象远远超出了声音的范畴。光的吸收和发射定律依赖于它，无线电报接收器的“调谐”、行星对彼此运动影响的重要性比较、部队行进时悬索桥所面临的危险，以及某些船只在其机器节奏性震动下在特定速度下产生的过度振动。周期性的一致性可能在两种周期事件持续相关时产生稳定的现象，或者在偶然和短暂的关联时产生剧烈和突如其来的爆发。

Again, the characteristic and constant periods of vibration mentioned above are the underlying causes of what appear to us as steady excitements of our senses. We work for hours in a steady light, or we listen to a steady unvarying sound. But, if modern science be correct, this steadiness has no counterpart in nature. The steady light is due to the impact on the eye of a countless number of periodic waves in a vibrating ether, and the steady sound to similar waves in a vibrating air. It is not our purpose here to explain the theory of light or the theory of sound. We have said enough to make it evident that one of the first step necessary to make mathematics a fit instrument for the investigation of Nature is that it should be able to express the essential periodicity of things. If we have grasped this, we can understand the importance of the mathematical conceptions which we have next to consider, namely, periodic functions.

再者，上述提到的特征性和恒定的振动周期是我们感知到的稳定感官刺激的根本原因。我们在稳定的光线下工作数小时，或者我们听到不变的声音。但如果现代科学是正确的，这种稳定性在自然界中是没有对应物的。稳定的光是由于无数周期性波动的冲击作用在眼睛上，而稳定的声音则是由于类似的波动作用在振动的空气中。我们在这里并不打算解释光的理论或声音的理论。我们已经说得足够清楚，足以使人显而易见，数学作为研究自然的有效工具，首先需要能够表达事物的基本周期性。如果我们已经理解这一点，就能理解接下来我们需要考虑的数学概念的重要性，即周期函数。