Part 4

The Definition of Time

by H. Poincare Icon

One will say that all this is unimportant. Our instruments of measurement are imperfect. But it suffices that we can conceive a perfect instrument.

This ideal can not be reached. But it is enough to have conceived it and so to have put rigor into the definition of the unit of time.

The problem is that there is no rigor in the definition.

We use the pendulum to measure time. But what postulate do we implicitly admit?

It is that the duration of two identical phenomena is the same; or, if you prefer, that the same causes take the same time to produce the same effects.

At first sight, this is a good definition of the equality of two durations. But is it impossible that experiment may some day contradict our postulate?

Suppose that at a certain place in the world the phenomenon α happens. It causes at the end of a certain time the effect α'. At another place in the world very far away from the first, happens the phenomenon β. It which causes the effect β'.

The phenomena α and β are simultaneous, as are also the effects α' and β'.

Later, the phenomenon α is reproduced under approximately the same conditions as before. Simultaneously, the phenomenon β is also reproduced at a very distant place in the world and almost under the same circumstances.

The effects α' and β' also take place.

Let us suppose that the effect α happens perceptibly before the effect β.

If experience made us witness such a sight, our postulate would be contradicted. Experience would tell us that the first duration αα’ is equal to the first duration ββ' and that the second duration αα’ is less than the second duration ββ'.

On the other hand, our postulate would require that the two durations αα' should be equal to each other, as likewise the two durations ββ'.

The equality and the inequality deduced from experience would be incompatible with the two equalities deduced from the postulate.

Can we affirm that the hypotheses I have just made are absurd?

They are in no wise contrary to the principle of contradiction. Doubtless they could not happen without the principle of sufficient reason seeming violated. But to justify a definition so fundamental I should prefer some other guarantee.

In physical reality, one cause does not produce a given effect, but a multitude of distinct causes contribute to produce it, without our having any means of discriminating the part of each of them.

Physicists seek to make this distinction. But they make it only approximately, and, however they progress, they never will make it except approximately.

It is approximately true that the motion of the pendulum is due solely to the earth’s attraction; but in all rigor every attraction, even of Sirius, acts on the pendulum.

The causes which have produced a certain effect will only be reproduced approximately.

  • It follows that we should modify our postulate and our definition.

Instead of saying: ‘The same causes take the same time to produce the same effects,’ we should say: ‘Causes almost identical take almost the same time to produce almost the same effects.’

Our definition therefore is no longer anything but approximate.

Besides, as M. Calinon very justly remarks in a recent memoir:[1]

One of the circumstances of any phenomenon is the velocity of the earth's rotation; if this velocity of rotation varies, it constitutes in the reproduction of this phenomenon a circumstance which no longer remains the same. But to suppose this velocity of rotation constant is to suppose that we know how to measure time.

Our definition is therefore not yet satisfactory. It is certainly not that which the astronomers of whom I spoke above implicitly adopt, when they affirm that the terrestrial rotation is slowing down.

What meaning according to them has this affirmation? We can only understand it by analyzing the proofs they give of their proposition. They say first that the friction of the tides producing heat must destroy vis viva. They invoke therefore the principle of vis viva, or of the conservation of energy.

They say next that the secular acceleration of the moon, calculated according to Newton’s law, would be less than that deduced from observations unless the correction relative to the slowing down of the terrestrial rotation were made. They invoke therefore Newton’s law.

In other words, they say that time should be defined in a way that Newton’s law and that of vis viva may be verified.

Newton’s law is an experimental truth; as such it is only approximate, which shows that we still have only a definition by approximation.

If another way of measuring time is adopted, the experiments on which Newton’s law is founded would nonetheless have the same meaning.

  • Only the enunciation of the law would be different, because it would be translated into another language.
  • It would be much less simple.

So that the definition implicitly adopted by the astronomers may be summed up thus: “Time should be so defined that the equations of mechanics may be as simple as possible.”

In other words, there is not one way of measuring time more true than another; that which is generally adopted is only more convenient.

Of 2 watches, we have no right to say that the one goes true, the other wrong; we can only say that it is advantageous to conform to the indications of the first.

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