The Speed at a Single Instant

A speedometer says 50 km/h. Ask what that means at the instant you read it and the arithmetic falls apart.

Speed is distance divided by time. At a single instant, no distance has been covered and no time has passed. Zero divided by zero is not a small number or a difficult number; it is not a number, because every number would satisfy it equally well. Zeno built an argument against the possibility of motion on exactly this, and for two thousand years nobody had a good reply.

The reply, when it came, does not divide by zero. It refuses to.

Take the average speed over the second around that instant. Then over a tenth of a second. Then a hundredth, a thousandth, a millionth. Every one of those is an honest division with a real answer. What you notice is that the answers settle: they crowd towards a particular value and stay near it, and the smaller the interval the closer they crowd. That value is the speed at the instant, and it is defined as the thing the averages approach rather than as any one of them.

This is the derivative, and three things about it are worth holding.

It is a definition, not a measurement. No instrument gets there, and no better instrument would help, because the difficulty was never practical. The quantity had to be constructed.

Geometrically it is a slope. Draw the curve of distance against time, and the derivative at a point is the steepness of the line that just touches the curve there. Steeper means faster. This is why a flat tangent marks a peak or a trough, which turns out to be the most useful fact in applied mathematics: the way to find the best of something is to find where its rate of change is zero.

And it was used for a century and a half before anybody could justify it.

Fermat was finding maxima this way before Newton was born. At the top of a curve the thing is momentarily flat, so set the rate of change to nothing and solve. It worked. It kept working. Nobody could say what the vanishing intervals were.

Bishop Berkeley wrote the objection down in 1734 and it is the best thing written against calculus. The quantities being divided are either zero or they are not. If they are not zero, the answer is approximate and the triumphant exactness is a fraud. If they are zero, the division is meaningless. He called them the ghosts of departed quantities, and he was not being unfair; he was pointing out that the most successful mathematics of the age rested on a step nobody could state.

It took until Cauchy in the 1820s to answer him, by defining the limit carefully enough that no ghost is required: not a number the interval reaches but a value you can get as close to as you like by making the interval small enough. The definition is fussy, and the fussiness is the entire content. It is what replaced a working trick with a reason.

That order of events is normal rather than embarrassing, and it is worth expecting. A technique that produces right answers gets adopted, used, and relied on, and the account of why it works arrives a generation or three later, usually from someone who was irritated by a critic.