We proceed to fix the positions of a series of points in this way: there are two straight lines, OX and OY, at right angles to each other, and we measure off certain steps along the line OX; these steps are OX0·5, OX1, OX1·5, OX2, and so on, the small numerals indicating the distance of each point (OX0·5, etc.) from the origin O. We then draw lines perpendicular to the X-axis through these points. We have now to calculate one-half of the square of each of these lengths OX0·5, OX1, etc., and then we mark off these calculated lengths along the perpendicular lines. The point A, for instance, is 1/2(0·5)2 from the point X0·5, B is 1/2(1)2 from X1, and so on. In this way we obtain a series of points, A, B, C, D, E, etc., and these are points on the locus of the “moving” point.
Fig. 6.
There is nothing at all about motion here. All that we have done is to measure lengths. We have made a kind of counterpoint, X-points against Y-points, but we have not even made a curve. We connect the points A, B, C, D, E, etc., by means of short, straight lines, and then we may connect together these short lines, and, if we plot a number of intermediate points between those that we have already obtained and join these, the points may be so close together that they may seem to be indistinguishable from a curve. Yet, no matter how numerous they may be, they can never be connected together so as to form a curve; we therefore draw a curved line freehand through them, and at once, in so doing, we abandon our intellectual methods, for our curve depends on our intuition of continuously changing direction. But if we think about it we shall find that we can form no clear intellectual notion of continuity and we can only measure the curvature of a line at a point in the line by drawing a tangent to the curve at this point, and then by measuring the slope of the tangent. The curve itself we obviously leave out of consideration.
We cannot conceive of the point moving along the locus OD. We can think of it only as at the places O, A, B, C, D, E, etc., but we must neglect the intervals OA, AB, BC, CD, DE, and so on, or we can divide them into smaller intervals by supposing the point to have occupied the positions f, g, i, j, between the points A and B, for instance. Yet, no matter how many these intervals may be, we can only think of the point as being at the places O, A, B, C, D, E, or at f, g, i, j, and so on. We never think of the intervals themselves, and, if all we think about is the position of the point, we do not really think of it as in motion at all. We can see it in motion, but we cannot form an intellectual concept of its motion. It is not really necessary that we should in the affairs of everyday life, but for the adequate treatment of problems involving rates of change science had to wait for the invention of the methods of the infinitesimal calculus before this disability of the human mind could be circumvented.
But the moving point occupies successively a number of different positions in space. If it is a material point that we observe to move from one place to another, we perceive that a certain interval of our duration corresponds with the change of position of the point. Duration was not used up in the occupancy of the different positions O, A, B, C, D, E, and so on, nor in that of the occupancy of the indefinitely numerous other positions in which we may place the moving point, but in the intervals themselves. We have said “duration” and not “time,” using Bergson’s term. By duration and time we understand different things.
Time is, for us, only a series of standard events which punctuate, so to speak, our experienced duration. The unit of time is the sidereal day, that is, the interval of time between two successive transits of a fixed star across the arbitrary meridian. But if we try to conceptualise this interval we find that we can do so only by breaking it up into smaller intervals, and this we do by using a pendulum of a certain length which makes a certain number of swings (86,400) during the interval between the two transits of the star. Thus we obtain a smaller interval of duration and we call this a second of time. But for many purposes this interval is too long, and we can again sub-divide it by making use of a tuning-fork which makes, say, 1000 complete vibrations in a second; in this way we obtain still smaller intervals of duration—the sigmata of the physiologists. A sigma, therefore, represents the interval between the beginning and end of one complete vibration of a certain kind of tuning-fork; a second, that between the beginning and end of one complete swing of a pendulum of a certain length, placed at certain parts of the earth’s surface; and a day, that between two successive transits of a fixed star across a selected meridian, after all the necessary corrections have been made to the observation. These actual occurrences, the positions of the prongs of the tuning-fork, or those of the bob of the pendulum, or those of the fixed star do not involve duration. We consider the meridian of Greenwich as an imaginary line drawn across the celestial sphere, and the star as a point of light, so that the actual transit is, in the limit, an occurrence which occupies only an “infinitesimal” interval of duration. So also with the pendulum and the tuning-fork; the positions of these things do not “use up” time, and even if the intervals into which we divide astronomical time are indefinitely numerous no real quantity of duration is taken up by their occurrence. We know that the interval between two successive transits of a fixed star are not really constant, that is, the astronomical day is lengthening by an incredibly small part of a second each year, but how do we know this? It is not that we can feel the increments of duration, but just that we assume that Newton’s laws of motion are true; and hence that the tidal friction due to the motions of the earth, sun, and moon must retard the period of rotation of the earth so that the intervals between two successive transits of a star must become greater.
Thus we do not conceptualise the actual intervals of duration of which we are able to mark the end-points; they are lived by us, and they are real absolute things independent of our wills. Suppose we come in from a long walk, tired and thirsty, and ask the maid to get tea ready at once. She puts the kettle on the gas stove and then sits down to read. The water takes, say, five minutes to boil. What do we mean by this?
This is what we mean:—