Babbage's Difference Engine: Difference Table Plotter

Polynomial f(n) = an³+bn²+cn+d

a (cubic) 1

b (quadratic) 0

c (linear) 0

d (constant) 2

Crank

Crank speed 1.0 turns/s

Registers

Turns made (n)

Multiplications used

How it works

Charles Babbage's Difference Engine (design finalised 1849) computes values of a polynomial using only repeated addition — no multiplication or division gearing at all, once it is set up. It exploits the method of finite differences: for a polynomial f(n) of degree d, the d-th forward difference is a constant.

Δf(n)  = f(n+1) − f(n)
Δ²f(n) = Δf(n+1) − Δf(n)
Δ³f(n) = Δ²f(n+1) − Δ²f(n)   ← constant, for a cubic f

Once the starting register values f(0), Δf(0), Δ²f(0), Δ³f(0) are known, every later value is produced by the "table of differences" update, applied from the bottom register upward on each turn of the crank:

f(n+1)   = f(n)   + Δf(n)
Δf(n+1)  = Δf(n)  + Δ²f(n)
Δ²f(n+1) = Δ²f(n) + Δ³f(n)
Δ³f(n+1) = Δ³f(n)            (unchanged — it is the constant)

This 2D version renders the same register arithmetic as a live cascade diagram — four stacked value boxes with the addition pulse flowing upward from Δ³ into f(n) on every crank turn — plus a rolling plot of f(n) traced point by point above it, exactly like the paper printouts the real engine's stereotype printer produced.

a, b, c, d sliders — set the coefficients of f(n); the engine re-derives the four starting register values from them and resets to n = 0.
Turn crank — applies one step of the addition chain above and appends the new point to the plot.
Auto-crank — turns the crank continuously at the chosen rate.
"Multiplications used" stays at 0 forever — a polynomial table of any length is produced from a handful of additions per row, which is why it was mechanically buildable with 1840s precision engineering.

Babbage's actual Difference Engine No. 2 carried 31-digit registers across 7 orders of difference; this model uses the same 4-register, 3rd-difference arithmetic, enough to reproduce the same principle exactly.

f(n) — plotted curve Δ¹ register Δ² register Δ³ register (constant)