of change whatever, take the twenty seconds themselves elapsing.  If time be infinitely divisible, and it must be so on intellectualist principles, they simply cannot elapse, their end cannot be reached; for no matter how much of them has already elapsed, before the remainder, however minute, can have wholly elapsed, the earlier half of it must first have elapsed.  And this ever re-arising need of making the earlier half elapse first leaves time with always something to do before the last thing is done, so that the last thing never gets done.  Expressed in bare numbers, it is like the convergent series 1/2 plus 1/4 plus 1/8..., of which the limit is one.  But this limit, simply because it is a limit, stands outside the series, the value of which approaches it indefinitely but never touches it.  If in the natural world there were no other way of getting things save by such successive addition of their logically involved fractions, no complete units or whole things would ever come into being, for the fractions’ sum would always leave a remainder.  But in point of fact nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, etc., and adding them together.  She either makes a whole egg at once or none at all, and so of all her other units.  It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come that Zeno’s paradox gives trouble.

And it gives trouble then only if the succession of steps of change be infinitely divisible.  If a bottle had to be emptied by an infinite number of successive decrements, it is mathematically impossible that the emptying should ever positively terminate.  In point of fact, however, bottles and coffee-pots empty themselves by a finite number of decrements, each of definite amount.  Either a whole drop emerges or nothing emerges from the spout.  If all change went thus drop-wise, so to speak, if real time sprouted or grew by units of duration of determinate amount, just as our perceptions of it grow by pulses, there would be no zenonian paradoxes or kantian antinomies to trouble us.  All our sensible experiences, as we get them immediately, do thus change by discrete pulses of perception, each of which keeps us saying ‘more, more, more,’ or ‘less, less, less,’ as the definite increments or diminutions make themselves felt.  The discreteness is still more obvious when, instead of old things changing, they cease, or when altogether new things come.  Fechner’s term of the ‘threshold,’ which has played such a part in the psychology of perception, is only one way of naming the quantitative discreteness in the change of all our sensible experiences.  They come to us in drops.  Time itself comes in drops.