Babbage's Difference Engine

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

f(n)0

Δ¹0

Δ²0

Δ³ (const)0

Turns made

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 engine's digit-wheel registers are cranked into their starting values f(0), Δf(0), Δ²f(0), Δ³f(0), 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)

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; every decimal digit wheel rotates to its new digit, exactly like the real mechanism's carry-propagating gear trains.
Auto-crank — turns the crank continuously at the chosen rate, the way a steam or hand-driven engine would run a printed table.
"Multiplications used" stays at 0 forever — that is the entire point of the design: 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 4 registers of 6 digits and up to the 3rd difference, enough to reproduce the same principle exactly.

f(n) register Δ¹ register Δ² register Δ³ register

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