Change and calculus

1715

Taylor series

Brook Taylor (building on James Gregory and others)

How does a calculator know sin 37°?

Taylor published a general recipe for turning almost any smooth curve into a polynomial: match its value, then its slope, then its bend, and so on. A few terms are often enough near one point. This is how a pendulum clock’s maths becomes simple for small swings.

First, make a guess

As you add more terms to the polynomial for sin x…

How they figured it out

Brook Taylor published his general recipe in 1715, but he was not first: James Gregory had it in 1671, and the Kerala school had special cases centuries earlier. For decades few noticed how important it was.

Then Lagrange called it the main foundation of differential calculus. Taylor himself used it on problems like the vibrating string. Today it is how engineers simplify almost every hard equation: keep the first one or two terms and check the size of the next.

The equation

f(x)=f(a)+f′(a)(x−a)+f′′(a)2!(x−a)2+⋯f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots

Near a point, any smooth curve looks like a straight line, then a parabola, then a better and better polynomial.

Everyday echoEngineers’ favourite shortcut, sin θ ≈ θ for small angles, is the first term.

See it in real life

Why grandfather clocks swing small

For small swings, sin⁡θ≈θ\sin\theta \approx \theta, and a pendulum’s period hardly depends on how far it swings. That is why clock pendulums swing only a few degrees: big swings break the approximation and the clock runs slow.

Where engineers use it

  • Linearisation

    Engineers replace hard equations with their first-order Taylor terms near an operating point: control systems, circuits, structures.

  • Error estimates

    The first missing term tells you how wrong an approximation is.

Worked example: sin(0.1) by hand

  1. sin⁡x≈x−x3/6\sin x \approx x - x^3/6
  2. 0.1−0.001/6=0.09983330.1 - 0.001/6 = 0.0998333
  3. True value: 0.09983340.0998334.

Where it came from

  • The invention of calculus 1665–1684

    Built on

    Taylor’s recipe uses derivatives of every order.

  • Infinite sums in Kerala c. 1400

    Same technique

    Madhava’s series for sine and cosine are exactly what Taylor’s recipe gives for those functions.

    Where the comparison stops: Madhava found series for particular functions; Taylor gave a general method for any smooth function.

What it led to