The modern student of physics, when asked what is meant by velocity, answers from his prompt memory, "Velocity is space divided by time." The answer is quite characteristic of the present age of science. "Take that to the calculators", was the contemptuous reply which Faraday made when somebody proposed a question for him to solve which called for no discernment of any hidden principle but was merely one of quantitative determination. The practical aspect, the quantitative aspect, the material aspect, — that is what the world is now chiefly concerned with. But to come to a right way of looking at things is an exercise in which this age does not show much talent; it has not quite been able to realise the value of so doing. Perhaps in its future development science will become a little more metaphysical and a little less materialistic. Surely the purely physical mania has fairly had its turn by this time; it has long been in possession, and might now well give place to something better. It is time for people to recognize that what is abstract and invisible is not therefore unreal, but on the contrary a degree more real and substantial than what is outward and palpable.
The teaching given to a science student whence springs the reply cited above is of a simple kind, and such as may be explained to all comers in a few words. Velocity, so termed in technical phraseology, is the same thing as speed, and is said to be the relation between space or distance traversed by the moving object and the time occupied in so doing. The relation between one mile of space or distance and one minute of time is accordingly the velocity of an express train going, as they say, "a mile a minute". Sixty miles an hour would be just the same ratio otherwise expressed. But the express train making ahead at full speed is not the only type of motion and velocity. There the speed is regular, uniform, and unvarying; at least it is so as far as we can perceive. But the motion of a stone dropped from the roof of a house has quite a different character; in this case the motion is not uniform and unvarying, and the only element of regularity in the movement is the way in which it becomes continually faster and faster. This being so, to ask "What was the speed or velocity of that falling stone?" would be an unreasonable question unless some particular instant were denned to which the question should apply. It would, however, be quite an intelligent question if one asked, "At what velocity was it moving at the instant when it struck the ground?"
Now let us reflect a little upon the foregoing considerations. An instant is to time exactly what a geometrical point is to space; indeed an instant is often called a point of time. Like the geometrical point, it has "no parts or magnitude"; all notion of how long is entirely foreign to it. How is it, then, that we can speak of the velocity of a falling stone at that instant when it touches the ground? Assuredly, at any instant, no actual motion whatever takes place; no space is passed through, neither is any time occupied. The difficulty before us is this. The scientist declares that velocity is "the space divided by the time", and yet here is a case in which we are forced to recognize velocity though neither space nor time (in that sense) enters into the question at all. This is what Dr. O. Wendell Holmes called "sticking a fact" into the lecturer; and it is a very sharp-pointed fact too. It shows that amid the enlightenment of this age (to the wise it is notorious as the Dark Age) there exists some want of reflection among scientists on the subject of velocity and motion; it shows that the philosophy of the modern scientist is of a sort that does not go to the bottom of things.
There are some persons, generally of the number of the learned whose heads are "replete with thoughts of other men", who have great difficulty in grasping this idea of an absolute instant, simple as it is in itself. These people give one a great deal of trouble in discussion; they insist on regarding an instant as an "infinitesimally short period " of time. It is as bad as if they told the geometer that his mathematical point was not an element of no magnitude, but an element of infinitesimal magnitude. But in truth a geometrical point is absolutely devoid of magnitude, and similarly an instant is not a "period" of any sort or description. To sum up this parenthesis, an instant is not anything during which either motion or any other change can occur. "During an instant" would be a self-contradictory phrase; an instant does not endure.
Let us now pause to review the position and examine the conclusions with which we are confronted. From the case of the falling stone it is made evident that a moving object has a velocity at an instant (when such elements as distance traversed and time occupied can have no existence); and also, in this example at least, it is found that velocity cannot be conceived of at all except as existing at this or that instant. For the velocity of the stone changes within the smallest fraction of a second; whatever it is at one moment, it will not be that at any succeeding moment. What, then, is to be the next step in our reasoning? If it has been established that velocity does exist at an instant, shall we imagine that it has a different character in the case of the express train maintaining an even speed? Or would it not be much reasonable to hold that velocity was the same sort of thing in all moving objects, whether their movement was uniform or accelerated? Surely nobody can hesitate to accept this latter view together with its consequence, viz., that velocity is not "the space divided by the time", but has an existence where these two elements are altogether excluded. In other words, velocity is an inherent condition of the moving object itself, and is not in any sense a dependency of motion. Indeed, this s borne out by the use of language; for we discuss the velocity of a bullet (not that of a bullet's motion.)
Here perhaps some more subtle representative of the age will tell us that he would never make the assertion that velocity was identically the ratio of the space traversed to the time occupied; he would prefer to say that velocity was measured by this ratio. That certainly would be an accurate statement. But it leaves an empty gap; because now we have no prescribed answer (for the student to learn by heart) upon the question, What is velocity in itself? Do examiners never ask the question, "What is velocity?" Or do professors never explain how such a question should be answered? It is very odd if they do not, because velocity is such an elementary topic; and it is the boast of the really able professor, as opposed to the charlatan, that he thoroughly understands the very roots of his subject, and lays the foundations of knowledge in his pupils so that the vast superstructure shall not totter.
Another scientist might say, perhaps, that velocity was a quality, attribute, or property of motion; for motion may be quick or slow.
Would anyone have the courage to say that velocity was the principle of motion, the cause of motion? Will anyone dare to say that velocity is something internal and hidden, of which motion is the outward and visible sign? If present science does not say so, peradventure future science, more metaphysical than its predecessor, will have the boldness.
Physical science, emphatically physical and non-metaphysical, cares too little for that instant in which no change ensues, but in which some thing is and tends. Paradoxical as the statement may seem, that instant is a better realisation of eternity than the most gigantic sweep's-brush of centuries jointed together within the imagination. But apart from this, it is the right aim of science to pass from effects to the recognition of their hidden causes; and the scientist who aspires to a higher wisdom should make a study of an instant, to find in it what is causal. A wonderful theme, in truth, is that instant, planted in the midst of time and yet itself no portion of time, a zero containing in itself the principle and cause of what passes in time. Assuredly we have not yet come to an end of man's store of meditable matter.