Difference Engine History Method Of Differences See Also

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.

1 History

The first of these devices was conceived in 1786 by J. H. Mueller . It was never built.

Difference engines were forgotten and then rediscovered in 1822 by Charles Babbage. 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, Per Georg Scheutz built several difference engines from 1855 onwards; one was sold to the British government in 1859. Martin WibergMartin Wiberg ( September 4, 1826 December 29 1905) was born in Viby, Scania enrolled at Lund University in 1845 and became a doctor of philosophy in 1850. He is known as a computer pioneer for inventing a machine the size of a sewing machine, that could improved the construction even further but used his device only for producing and publishing printed logarithmic tables.

Based on Babbage's original plans, the London Science Museum constructed a working Difference Engine No. 2 from 19891989 is a common year starting on Sunday (click on link for calendar). Events January January 7 Akihito becomes Emperor of Japan following the death of Hirohito. The Heisei period begins January 8 the Kegworth Air Disaster A British Midland Boeing 737 cra to 19911991 like 2002, is a palindromic year. It also has the same calendar as 2002, including Easter on March 31. It is a common year starting on Tuesday. Events January January 2 Sharon Pratt Dixon is sworn in as mayor of Washington, DC becoming the first blac. In 2000This page is about the year 2000. See 2000 AD for the UK comic book, Number 2000 for other uses. 2000 is a leap year starting on Saturday (see link for calendar), and also the International Year for a Culture of Peace''. Events Y2K passes without the seri, the printerA computer printer is a computer peripheral device that produces a hard copy (permanent human-readable text and/or graphics, usually on paper) from data stored in a computer connected to it. Printing mode The data received by a printer may be: # a string originally designed by Babbage was also completed. Both worked flawlessly. The difference engine and printer were constructed to tolerances achievable with 19th centuryAlternative meaning: Nineteenth Century (periodical ( 18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801- 1900. Events The Little Ice Age ended technology, resolving a long-standing debate whether Babbage's design would actually have worked.

2 Method of differences

The principle of a difference engine is NewtonKneller's portrait of 1689. Sir Isaac Newton ( December 25, 1642 March 20, 1727 by the Julian calendar then in use; or January 4, 1643 March 31, 1727 by the Gregorian calendar) was an English physicist, mathematician, astronomer, philosopher, and alchemis's method of differences. It may be illustrated with a small example. Consider the quadratic polynomial

p(x) = 2x² − 3x + 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 column 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.

3 See also

4 External links

Mathematical tools

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