FOR THE HUMAN MIND the absolute continuity of motion is inconceivable. The laws of motion of any kind only become comprehensible to man when he examines units of this motion, arbitrarily selected. But at the same time it is from this arbitrary division of continuous motion into discontinuous units that a great number of human errors proceeds.

We all know the so-called sophism of the ancients, proving that Achilles would never overtake the tortoise, though Achilles walked ten times as fast as the tortoise. As soon as Achilles passes over the space separating him from the tortoise, the tortoise advances one-tenth of that space: Achilles passes over that tenth, but the tortoise has advanced a hundredth, and so on to infinity. This problem seemed to the ancients insoluble. The irrationality of the conclusion (that Achilles will never overtake the tortoise) arises from the arbitrary assumption of disconnected units of motion, when the motion both of Achilles and the tortoise was continuous.

By taking smaller and smaller units of motion we merely approach the solution of the problem, but we never attain it. It is only by assuming an infinitely small magnitude, and a progression rising from it up to a tenth, and taking the sum of that geometrical progression, that we can arrive at the solution of the problem. A new branch of mathematics, dealing with infinitely small quantities, gives now in other more complex problems of dynamics solutions of problems that seemed insoluble.

This new branch of mathematics, unknown to the ancients, by assuming infinitely small quantities, that is, such as secure the chief condition of motion (absolute continuity), corrects the inevitable error which the human intellect cannot but make, when it considers disconnected units of motion instead of continuous motion.

In the investigation of the laws of historical motion precisely the same mistake arises.

The progress of humanity, arising from an innumerable multitude of individual wills, is continuous in its motion.

The discovery of the laws of this motion is the aim of history. But in order to arrive at the laws of the continuous motion due to the sum of all these individual wills, the human mind assumes arbitrary, disconnected units. The first proceeding of the historian is taking an arbitrary series of continuous events to examine it apart from others, while in reality there is not, and cannot be, a beginning to any event, but one event flows without any break in continuity from another. The second proceeding is to examine the action of a single person, a sovereign, or a general, as though it were equivalent to the sum of many individual wills, though the sum of individual wills never finds expression in the action of a single historical personage.

Historical science as it advances is continually taking smaller and smaller units for analysis, and in this way strives to approximate the truth. But however small the units of which history takes cognisance, we feel that the assumption of a unit, disconnected from another, the assumption of a beginning of any phenomenon, and the assumption that the individual wills of all men find expression in the actions of a single historical personage are false in themselves.

Every conclusion of history can, without the slightest effort on the part of the critic, be dissipated like dust, leaving no trace, simply through criticism selecting, as the object of its analysis, a greater or smaller disconnected unit, which it has a perfect right to do, seeing that the unit of history is always selected arbitrarily.

Only by assuming an infinitely small unit for observation—a differential of history—that is, the homogeneous tendencies of men, and arriving at the integral calculus (that is, taking the sum of those infinitesimal quantities), can we hope to arrive at the laws of history.

The first fifteen years of the nineteenth century present the spectacle of an extraordinary movement of millions of men. Men leave their habitual pursuits; rush from one side of Europe to the other; plunder, slaughter one another, triumph and despair; and the whole current of life is transformed and presents a quickened activity, first moving at a growing speed, and then slowly slackening again. What was the cause of that activity, or from what laws did it arise? asks the human intellect.

The historians, in reply to that inquiry, lay before us the sayings and doings of some dozens of men in one of the buildings of the city of Paris, summing up those doings and sayings by one word—revolution. Then they give us a detailed biography of Napoleon, and of certain persons favourably or hostilely disposed to him; talk of the influence of some of these persons upon others; and then say that this it is to which that activity is due, and these are its laws.

But the human intellect not only refuses to believe in that explanation, but flatly declares that the method of explanation is not a correct one, because in this explanation a smaller phenomenon is taken as the cause of a greater phenomenon. The sum of men's individual wills produced both the revolution and Napoleon; and only the sum of those wills endured them and then destroyed them.

“But whenever there have been wars, there have been great military leaders; whenever there have been revolutions in states, there have been great men,” says history. “Whenever there have been great military leaders there have, indeed, been wars,” replies the human reason; “but that does not prove that the generals were the cause of the wars, and that the factors leading to warfare can be found in the personal activity of one man.”

Whenever, looking at my watch, I see the hand has reached the figure x, I hear the bells beginning to ring in the church close by. But from the fact that the watch hand points to ten whenever the bells begin to ring, I have not the right to infer that the position of the hands of my watch is the cause of the vibration of the bells.

Whenever I see a steam-engine move, I hear the whistle, I see the valve open and the wheels turn; but I have no right to conclude from that that the whistle and the turning of the wheels are the causes of the steam-engine's moving.

The peasants say that in the late spring a cold wind blows because the oak-buds are opening, and, as a fact, a cold wind does blow every spring when the oak is coming out. But though the cause of a cold wind's blowing just when the oaks are coming out is unknown to me, I cannot agree with the peasants that the cause of the cold wind is the opening of the oak-buds, because the force of the wind is altogether outside the influence of the buds. I see in this simply such a coincidence of events as is common in every phenomenon of life, and I see that however long and minutely I might examine the watch hand, the valve, and the wheel of the steam-engine and the oak-bud, I shall not discover the cause of the bells ringing, of the steam-engine moving, and of the spring wind. To do that I must completely change my point of observation and study the laws of the motion of steam, of the bells, and of the wind. History must do the same. And efforts have already been made in this direction.

For the investigation of the laws of history, we must completely change the subject of observations, must let kings and ministers and generals alone, and study the homogeneous, infinitesimal elements by which masses are led. No one can say how far it has been given to man to advance in that direction in understanding of the laws of history. But it is obvious that only in that direction lies any possibility of discovering historical laws; and that the human intellect has hitherto not devoted to that method of research one millionth part of the energy that historians have put into the description of the doings of various kings, ministers, and generals, and the exposition of their own views on those doings.

人类的智慧理解不了运动的绝对连续性。人类只有在审视随意抽取的任一运动的细分单元时，方可逐步理解该运动的规律。但随即由于随意划分连续性的运动为间断性的单元，从而产生出人类的大部分迷误。

尽人皆知一条古代的辩术，讲的是阿奇里斯①总赶不上他前面的乌龟，尽管他走得比乌龟快十倍；因为每当他走完他与乌龟之间的距离时，乌龟又超前爬了这段距离的十分之一了；阿奇里斯走过这十分之一，乌龟则又超前爬了百分之一了，以此类推，直到无休无止。这道算式是一道古老的难以解决的算题。答案之荒谬（即阿奇里斯永远赶不上乌龟），仅仅是由于轻率地假定运动的不连贯单元的存在，而无论阿奇里斯或乌龟的运动，都是连续进行的。

①阿奇里斯是荷马《伊利亚特》中的英雄。

把运动的单元愈分愈细，我们只能接近问题的答案，却永远得不出答案。只有假设出无穷小数和由无穷小数产生的十分之一以下的级数，再求出这一几何级数的总量，我们才能得出问题的答案。数学的一个新的分支在解决了处理无穷小数的技术后，现已能在其他更为复杂的运动问题上求得对以前似乎解决不了的那些问题的答案。

古代人所不明了的这一新的数学分支，在研究运动问题时，因假设出无穷小数，使运动的主要条件（绝对连续性）得以复原，从而纠正人类的智慧以个别的运动单元代替对连续运动进行研究时不能不犯的错误。

在历史运动规律的探讨中也完全是这样。

人类的运动由不计其数的人们的随意行为所产生，是持续不断地进行着的。

了解这一运动的规律，是历史学的目的。但为了理解人们的随意行为的总和所构成的连续运动的规律，人类的智慧便假设出了随意可以截取而互不连贯的单元来。史学的第一个步骤，在于任意抽取一系列连续发生的事件，将其逐个分开来加以研究，这就没有也不可能有任何事件的开端，永远是一个事件不间断地从另一事件涌现出来。第二步骤在于把个人的、帝王的、统帅的行动，作为人们的无意识行为的总和来加以研究，而个别历史人物的行动却又永远反映不出人类无意识行为的总和。

历史科学在本身的运作中，经常划分小而又小的单元以供研究，以此接近对真理的认识。但无论史学划分出怎样的细小单元，我们感觉到，假设出彼此脱节的单元，假设有某种现象的·开·端，假设所有人的随意行为会在个别历史人物的行动中反映出来，其本身便是虚妄。

史学的任何结论，无须评论界劳神，便会土崩瓦解，不著痕迹，只须论者对一或大或小的前后不连贯的单元加以考察就行了；评论界总有权利这样做，任何一个历史单元不也是任意截取的吗？

只有采取无限小的观察单位——历史的微分，即人们的共同倾向，并运用积分法（即得出这些无限小的总和），我们才有希望了解历史的规律。

十九世纪最初的十五年，欧洲出现了一次数百万人的不寻常的运动。人们抛弃他们的日常职业，从欧洲的一边到另一边去抢动和撕杀，凯歌胜利和绝望呻吟，因而整个生活的进程在几年间变化不定，表现为一种先高涨而后衰落的激烈运动。这一运动的原因何在，它是按照什么规律运行的呢？——人类的智慧要问个明白。

历史学家回答这一问题时，向我们叙述巴黎城内一座大楼里的几十个人的言行，称这些言行为革命；然后出版拿破仑的，以及同情或敌视他的人物的详细传记，讲述其中一些人对另一些人的影响，说出：这就是这一运动发生的原因，这就是它的规律。

但是，人类的智慧不仅不肯相信这种解释，还干脆说，这种解释方法就不可信以为真，因为这种解释是把最微弱的现象视为最有力的论据。人们无意识行为的总和造成了革命，也造就了拿破仑，也只是这些无意识行为的总和，才容忍了，尔后又消灭了前后两者。

“但无论何时，有战伐必有征服者；无论何时，国家有变，必出伟人。”历史如是说。事实上，每当征服者出现，便爆发战争，这是人类智慧的回答，但这并不证明征服者便是战争的原因，且在个别人物的个人行动中能找出战争的规律。每当我看看钟，看到钟的指针走到十，便会听见邻近的教堂敲起钟声，但是，从指针走到十点祈祷钟声便敲响这一点出发，我无权下结论说，指针的位置是教堂的钟运动的原因。

每当我看到火车头起动，便听到汽笛声，看到阀门打开，车轮转动；但我无权由此得出结论：汽笛声和车轮转动是机车运动的实质原因。

农民说，暮春刮寒风，是因为橡树的芽苞绽开了，而事实上，每年春天当橡树抽芽时，都刮冷风。但是，虽然我不知道橡树抽芽时刮冷风的原因，我亦不同意农民的看法，认为橡树抽芽是刮冷风的原因，因为芽苞影响不到风力。我只看到日常生命现象中一些条件的偶合，我清楚，无论我多么仔细地观察时钟的指针，机车的阀门和车轮及橡树芽，我依然不会明白祈祷钟声，机车运动和倒春寒的原因。要明白其究竟，我必须完全改变观察点，去研究蒸汽、教堂大钟及风力的运动规律。史学也应如此。而且有人做了这方面的尝试。

为了研究历史规律，我们应该完全改变观察目标，敞开帝王大臣将军们，转而研究民众所遵循的同一类型的无穷小的因素。谁也无法说出，用这一方法，人类能获得对历史规律的几许了解；但是显而易见，这条途径有获取历史规律的机会；且这条途径使人类智慧付出的努力，还不及史学家用来描述帝王将相的行动，和据此行动发挥其想象所费精力的百万分之一。