Figure 18. Galileo’s own diagram of parabolic motion. The ball comes from the right and then starts to fall. Incidentally, this diagram illustrates how conventions get established and become rigid – a modern version of it would certainly show the ball coming from the left and falling off on the right.) The uniformity of the horizontal inertial motion is shown by the equality of the intervals be, cd, de, ..., while the odd-numbers rule is reflected in the increasing vertical descents bo, og, gl, ....
We have only to include the water tank and assistant in the snapshots, and everything is changed. Galileo can tell us where the ball will land because he can now deduce its speed. There are some important lessons we can learn from this. First, it is water, not time, that flows. Speed is not distance divided by time but distance divided by some real change elsewhere in the world. What we call time will never be understood unless this fact is grasped. Second, we must ask what change is allowed as a measure of time. Galileo measured the water carefully and made sure that it escaped steadily from the tank – otherwise his measure of time would surely have been useless. But the innocent word ‘steadily’ itself presupposes a measure of time. Where does that come from? It looks as if we can get into an unending search all too easily. No sooner do we present some measure that is supposed to be uniform than we are challenged to prove that it is uniform.
It is an indication of how slowly basic issues are resolved – and how easily they are put aside – that Newton highlighted the issue of the ultimate source of time nearly two hundred years before serious attempts were made to find it. Even then, the attempts remained rather rudimentary and few scientists became aware of them. It is interesting that Galileo had already anticipated the first useful attempt. This was actually forced upon him by the brevity of free fall in the ball experiment: it was all over much too quickly for the water-clock to be of any use. (It came into its own when Galileo rolled balls down very gentle inclines.) To analyse the parabola, he found a handy substitute. He noted that if the horizontal motion of the falling ball does persist unchanged, then the horizontal distance traversed becomes a direct measure of time. He therefore used the horizontal motion as a clock to time the vertical motion. His famous law of free fall was then coded in the shape of the parabola. Its defining property is that the distance down from the apex (where the ball falls off the table) increases as the square of the horizontal distance from the axis. But this measures time.
Thus, time is hidden in the picture. The horizontal distance measures time. It would be nice if one could say ‘the horizontal distance is time’. This is the goal I am working towards: time will become a distance through which things have moved. Then we shall truly see time as it flows, because time will be seen for what it is – the change of things. However, there are many different motions in the universe. Are they all equally suitable for measuring time? A second question is this: what causal connections are at work here? Galileo measured time by the flow of water, but it is hard to believe that a little water flowing out of a tank in the corner of his studio directly caused the balls to trace those beautiful parabolas through the air. If time derives from motion and change – and it is quite certain that all time measures do – what motion or change, in the last resort, is telling the ball which parabola to trace? The first question is more readily answered.
Nearly two thousand years ago, astronomers knew that some motions are better than others as measures of time. This they discovered experimentally. For the early astronomers, there were two obvious and, on the face of it, equally good candidates for telling time. Both were up in the sky and both had impressive credentials. The stars made the first clock, the Sun the second.
The stars remain fixed relative to each other and define sidereal time. Any star can be chosen as the ‘hand’ of the stellar clock: one merely has to note when it is due south. The stellar clock then ticks whenever that star is due south (i.e. when it crosses the meridian). Fractions of the ‘tick unit’ are measured by its distance from the meridian. A mere glance at the night sky could tell the ancient astronomers the time to within a quarter of an hour. With some care, times could be told to a minute. There is something wonderful about this great clock in the sky. It was a unique gift to the astronomers. The discoveries that culminated with Kepler’s laws of planetary motion, and many more made until well into the twentieth century, are unthinkable without it. No other phenomenon in nature could match it for convenience and accuracy. In millennia it has lost a few hours.
But there is a rival – the Sun. It defines solar time. This is the clock by which humanity and all other animals have always lived. The principle is the same: it is noon when the sun crosses the meridian. You don’t even have to be an astronomer to tell the time by this clock; a sundial will do.
Merely describing the clocks shows that speed is not distance divided by time, but distance divided by some other real change, most conveniently another distance. Roger Bannister ran one mile in four minutes; normal mortals can usually walk four miles in one hour. What does that mean? It means that as you or I walk four miles, the sun moves 15° across the sky. But this is not quite the complete story of speed and time, because there is a subtle difference between the two clocks in the sky – they do not march in perfect step. One and the same motion will have a different speed depending on which clock is used. One difference between the clocks is triviaclass="underline" the solar day is longer than the sidereal. The Sun, tracking eastwards round the ecliptic, takes on average four minutes longer to return to the meridian than the stars do. This difference, being constant, is no problem. However, there are also two variable differences (Box 6).
BOX 6 The Equation of Time
The first difference between sidereal and solar time arises from one of the three laws discovered by Kepler that describe the motion of the planets. The Sun’s apparent motion round the ecliptic is, of course, the reflection of the Earth’s motion. But, as Kepler demonstrated with his second law, that motion is not uniform. For this reason, the Sun’s daily eastward motion varies slightly during the year from its average. The differences build up to about ten minutes at some times of the year.
The second difference arises because the ecliptic is north of the celestial equator in the (northern) summer and south in the winter. The Sun’s motion is nearly uniform round the ecliptic. However, it is purely eastward in high summer and deep winter, but between, especially near the equinoxes, there is a north-south component and the eastward motion is slower. This can lead to an accumulated difference of up to seven minutes.
The effects peak at different times, and the net effect is represented by an asymmetric curve called the equation of time (it ‘equalizes’ the times). In November the Sun is ahead of the stars by 16 minutes, but three months later it lags by 14 minutes. This is why the evenings get dark rather early in November, but get light equally early in January. The stars, not the sun, set civil time.