THE human intellect cannot grasp the idea of absolutely uninterrupted motion. Man can begin to understand the laws of any kind of motion only when he takes into consideration arbitrarily selected units of such motion. But at the same time from this arbitrary division of unbroken motion into measurable units flows the greater part of human errors.

Take, for instance, the so-called "sophism" of the ancients, to prove that Achilles would never overtake a tortoise which had the start of him, even though Achilles ran ten times more swiftly than the tortoise. As soon as Achilles had passed over the distance between them, the tortoise would have advanced one-tenth of that distance ; Achilles runs that tenth, the tortoise advances a hundredth, and so on ad infinitum.

This problem seemed to the ancients unsolvable. The fallacy of the reasoning that Achilles would never overtake the tortoise arose from this: simply, that intermitted units of motions were arbitrarily taken for granted, whereas the motion of Achilles and -the tortoise were continuous.

By assuming ever smaller and smaller units of motion, we only approach the settlement of this question, we never really attain to it. Only by assuming infinitesimal quantities, and the progression up to one-tenth, and by taking the sum of this geometrical progression, can we attain the solution of the question. The new branch of Vol, v, .i . i

mathematics which is the science of reckoning with in finitesimals enables us to deal with still more complicated problems of motion, and solves problems which to the ancients seemed unanswerable.

This new branch of mathematics, which was unknown to the ancients, and applies so admirably to the problems of motion, by admitting infinitesimally small quantities, . that is, those by which the principal condition of motion is reestablished,.namely, absolute continuity, in itself corrects the inevitable error which the human mind is bound to make when it contemplates the separate units of motion instead of continuous motion.

In searching for the laws of historical movements precisely the same things must be observed. The progress of humanity, arising from an infinite collection of human wills, is continuous.

The apprehension of the laws of this onward march is the aim of history.

But in order to discover the laws of continuous motion in the sum of all the volitions of men, human reason assumes arbitrary and separate units. History first studies an arbitrary series of uninterrupted events, and contemplates it separate from the others, albeit there is and can be no beginning of an event, but every event is the direct outgrowth of its predecessor.

Secondly, history studies the deeds of a single man, a tsar, a colonel, as representing the sum of men's volitions, when in reality the sum of men's volitions is never expressed in the activities of any one historical personage.

The science of history is constantly taking ever smaller and smaller units for study, and in this way strives to reach the truth. But, however small the . units which history takes, we feel that the assumption of any unit separate from another, the,assumption of a beginning of any phenomenon whatever, and the assumption that the volitions of all men are expressed in the actions of any historical character, must be false per se.

Every deduction of history falls to pieces, like powder, without the slightest effort on the part of criticism, leav-

ing nothing behind it, simply in consequence of the fact that the criticism chooses as the object of its observation a more or less interrupted unit; and it has always the right to do this, since every historical unit is always arbitrary.

Only by assuming the infinitesimal unit for our observation,. as the differential of history, .in other words, the homogeneous tendencies of men, and by attaining the art of integrating (calculating the sum of these infinitesimal differentials), can we expect to attain to the laws of history.

The first fifteen years of the nineteenth century in Europe exhibit an extraordinary movement of millions of men. Men abandon their ordinary vocations, rush from one end of Europe to the other, rob, slaughter one another; they are filled with triumph and with despair, and the whole course of their lives is for a number of years changed, and undergoes a powerful movement, which at first goes on increasing and then slackens.

"What is the cause of this movement, or by what laws did it take place?" asks the human mind.

The historians, replying to this question, bring to our notice certain acts and speeches of certain dozens of men, in one of the buildings of the city of Paris, and call these acts and speeches "the Revolution"; then they give a circumstantial account of Napoleon, and of certain sympathizers and enemies of his, tell about the influence which certain of these individuals had upon the others, and they say : .

" This was the cause of this movement, and here are its laws."

But the human mind not only refuses to put credence in this explanation, but declares, up and down, that this manner of explanation is fallacious, for the reason that, according to it, a feeble phenomenon is taken as the cause of a mighty one. The sum of human volitions produced both the Revolution and Napoleon, and only the sum of these volitions sustained them and destroyed them.

" But in every case where there have been conquests there have been conquerors; in every case where there have been revolutions in a kingdom there have been great men," says history.

" Indeed, in every case where conquerors have appeared, there have been wars," replies human reason ; but this does not prove that the conquerors were the cause of the wars, or that it is possible to discover the laws of war in the personal activity of a single man.

In every case when I, looking at my watch, observe that the hand points at ten, I hear the bells ringing in the neighboring church ; but from the fact that in every case when the hand reaches ten o'clock, the ringing of the bells begins, I have no right to draw the conclusion that the position of the hands is the cause of the motion in the bells.

Every time when I observe an engine in motion, I hear the sound of the whistle, I see the valves open and the wheels in motion; but from this I have no right to conclude that the whistle and the movement of the wheels are the cause of the movement of the engine.

The peasants say that in late spring the cold wind blows because the oak tree is budding, and it is a fact that every spring a cold wind blows when the oaks are in bloom. But, although the cause of the cold wind blowing during the blossoming-time of the oaks is unknown to me, I am unable to agree with the peasants in attributing the cause of the cold winds to the burgeoning buds on the oaks, for the reason that the force of the wind is wholly outside the influence of the oak buds. I see only a coincidence of conditions, which is found in all the phenomena of life, and I see that, no matter how carefully I may contemplate the hands of the watch, the valves and wheels of the engine, and the oak buds, I shall never learn the cause that makes the church-bell chime, the engine to move, and the wind to blow in the spring. To discover this, I must entirely change my point of view, and study the laws that regulate steam, bells, and the wind.

History must do the same thing.

And experiments in this have already been made.

For, studying the laws of history, we must absolutely change the objects of our observation, leave kings, ministers, and generals out of the account, and select for study the homogeneous, infinitesimal elements that regulate the masses. No one can say how far it is given to man to attain by this path an understanding of the laws of history; but evidently on this path only is there any possibility of grasping the laws of history, and the human intellect has not, so far, devoted to this method the one-millionth part of the energies that have been expended by historians in the description of the deeds of individual tsars, colonels, and ministers, and in the elucidation of their combinations, resulting from these deeds.