Change and calculus

1665–1684

The invention of calculus

Isaac Newton and Gottfried Wilhelm Leibniz, independently

How fast is something going at one exact instant, when an instant has no time in it?

Newton worked out calculus in 1665–66 while Cambridge was closed by plague, but kept it mostly to himself. Leibniz found it independently and published first, in 1684, with the d and ∫ symbols we still use. Their supporters argued for decades over who came first.

First, make a guess

If you drive at a constant speed, what does your distance graph look like?

How they figured it out

In 1665–66 the plague closed Cambridge, and the 23-year-old Newton went home to Woolsthorpe. There he worked out his “fluxions” (quantities flowing in time), began his theory of gravity and experimented with light. He told almost nobody.

In Paris in the 1670s, Leibniz thought instead about infinitely small differences, dx, and on 29 October 1675 first wrote the elongated S, ∫, for “summa”. He published in 1684. Years later a Royal Society report accused Leibniz of stealing the idea; it had been written, anonymously, by Newton himself. Today both are credited, and the whole world uses Leibniz’s notation.

The equation

∫abf′(x) dx=f(b)−f(a)\int_a^b f'(x)\,dx = f(b) - f(a)

Adding up a rate of change over time gives you the total change. Speed and distance are two views of one journey.

Everyday echoYour car’s speedometer and trip counter are the two halves of calculus.

See it in real life

Filling a bath

How far the tap is open is a rate (litres per minute); how much water is in the bath is a total. The total is the rate added up over time (integration), and the rate is how fast the total changes (differentiation). Your car’s speedometer and odometer are the same pair.

Where engineers use it

  • Everywhere in engineering

    Stress in beams, current in circuits, flow in pipes and heat in engines are all rates of change.

  • Numerical simulation

    Computers integrate by adding thin slices, exactly the rectangles in the simulation.

Worked example: A steady acceleration

  1. Speed v(t)=2tv(t) = 2t m/s.
  2. Distance in 5 s: ∫052t dt=[t2]05=25\int_0^5 2t\,dt = [t^2]_0^5 = 25 m.

Where it came from

  • Infinite sums in Kerala c. 1400

    Same technique

    Madhava’s infinite series anticipate the power series at the heart of calculus, about 250 years earlier.

    Where the comparison stops: There is no evidence Newton or Leibniz knew the Kerala work; the ideas were found independently, and Madhava did not have the general machinery of derivatives and integrals.

  • Shapes become equations 1637

    Built on

    Calculus works on curves written as equations, which Descartes supplied.

What it led to