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.