# Difference engine

A Difference Engine is a mechanical special-purpose computer designed to tabulate polynomial functions. Since logarithmic and trigonometric functions can be approximated by polynomials, such a machine is more general than it appears at first.

History

The first of these devices was conceived in 1786 by J. H. Mueller. It was never built. Difference engines were forgotten and then reinvented in 1822 by Charles Babbage. A prototype able to tabulate quadratic polynomials was completed in 1832. This machine used the decimal numbers system and was powered by cranking a handle. The British government first financed the project but then later cut off support. Babbage went on to design his much more general analytical engine but later returned and produced an improved design (his "Difference Engine No. 2") between 1847 and 1849. Inspired by Babbage's difference engine plans, George and Edward Scheutz of Stockholm built several difference engines in 1834 - 1858; one was sold to the British government. Based on Babbage's original plans, the London Science Museum constructed a working Difference Engine No. 2 from 1989 to 1991. In 2000, the printer originally designed by Babbage was also completed. Both worked flawlessly.

Method of Differences

The principle of a difference engine is Newton's method of differences. It may be illustrated with a small example. Consider the quadratic polynomial

p(x) = 2 x2 - 3 x + 2

and suppose we want to tabulate the values p(0), p(0.1), p(0.2), p(0.3), p(0.4) etc. The table below is constructed as follows: the first column contains the values of the polynomial, the second column contains the differences of the two left neighbors in the first column, and the third row contains the differences of the two neighbors in the second column:

 p(0)=2.0 2.0-1.72=0.28 p(0.1)=1.72 0.28-0.24=0.04 1.72-1.48=0.24 p(0.2)=1.48 0.24-0.20=0.04 1.48-1.28=0.20 p(0.3)=1.28 0.20-0.16=0.04 1.28-1.12=0.16 p(0.4)=1.12

Notice how the values in the third row are constant. This is no coincidence. In fact, if you start with any polynomial of degree n, the column number n + 1 will always be constant. This crucial fact makes the method work, as we will see next.

We constructed this table from the left to the right, but now we can continue it from the right to the left in order to compute more values of our polynomial.

To calculate p(0.5) we use the values from the lowest diagonal. We start with the rightmost column value of 0.04. Then we continue the second column by subtracting 0.04 from 0.16 to get 0.12. Next we continue the first column by taking its previous value, 1.12 and subtracting the 0.12 from the second column. Thus p(0.5) is 1.12-0.12 = 1.0. In order to compute p(0.6), we iterate the same algorithm on the p(0.5) values: take 0.04 from the third column, subtract that from the second column's value 0.12 to get 0.08, then subtract that from the first column's value 1.0 to get 0.92, which is p(0.6).

This process may be continued ad infinitum. The values of the polynomial are produced without ever having to multiply. A difference engine only needs to be able to subtract. From one loop to the next, it needs to store 2 numbers in our case (the last elements in the first and second columns); if we wanted to tabulate polynomials of degree n, we'd need enough storage to hold n numbers.

Babbage's difference engine No. 2, finally built in 1991, could hold 7 numbers of 31 decimal digits each and could thus tabulate 7th degree polynomials to that precision. The best machines from Scheutz were able to store 4 numbers with 15 digits each.

The novel

The cyberpunk novelists William Gibson and Bruce Sterling co-authored a steampunk novel of alternative history entitled The Difference Engine in which Babbage's difference and analytical engines became available to Victorian society. The novel explores the consequences and implications of the early introduction of computational technology.