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Encyclopedia > Difference Engine

A difference engine is a special-purpose mechanical digital calculator, 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. Part of Charles Babbages Difference Engine assembled after his death by Babbages son, using parts found in his laboratory. ... Part of Charles Babbages Difference Engine assembled after his death by Babbages son, using parts found in his laboratory. ... A calculator is a device for performing calculations. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... Logarithms to various bases: is to base e, is to base 10, and is to base 1. ... All of the trigonometric functions of an angle Î¸ can be constructed geometrically in terms of a unit circle centered at O. In mathematics, the trigonometric functions are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other applications. ...

## History

The first of these devices was conceived in 1786 by J.H. Müller but it was never built. J.H. MÃ¼ller was an engineer in the Hessian army who in 1786 conceived the idea that would later evolve into modern computers, the Difference Engine. ...

Based on Babbage's original plans, the London Science Museum constructed a working Difference Engine No. 2 from 1989 to 1991, under Doron Swade, the then Curator of Computing. This was to celebrate the 200th anniversary of Babbage's birth. In 2000, the printer which Babbage originally designed for the difference engine was also completed. The conversion of the original design drawings into drawings suitable for engineering manufacturers' use revealed some minor errors in Babbage's design (introduced by accident or perhaps as a protection against unauthorized use), which had to be corrected. Once completed, both the engine and its printer worked flawlessly, and still do. The difference engine and printer were constructed to tolerances achievable with 19th century technology, resolving a long-standing debate whether Babbage's design would actually have worked. (One of the reasons formerly advanced for the non-completion of Babbage's engines had been that engineering methods were insufficiently developed in the Victorian era.) The National Science Museum in London The Science Museum on Exhibition Road, Kensington, London, is part of the National Museum of Science and Industry. ... This article or section does not cite its references or sources. ... Alternative meaning: Nineteenth Century (periodical) (18th century &#8212; 19th century &#8212; 20th century &#8212; more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...

## Operation

The difference engine consist of a number of columns, numbered from 1 to N. Each column is able to store one decimal number. The only operator the engine can do, is add the value of column n + 1 to column n to produce the new value of n. Column N can only store a constant, column 1 displays (and possibly prints) the value of the calculation on the current iteration. 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... Look up printer in Wiktionary, the free dictionary. ... The word iteration is sometimes used in everyday English with a meaning virtually identical to repetition. ...

The engine is programmed by setting initial values to the columns. Column 1 is set to the value of the polynomial at the start of computation. column 2 is set to a value derived from the first and higher derivatives of the polynomial at the same value of X. Each of the columns from 3 to N is set to a value derived from the n − 1th and higher derivatives of the polynomial. In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...

### Timing

In the Babbage design, one iteration i.e. one full set of addition and carry operations happens once for four rotations of the columns axes. Odd and even columns alternatively perform the addition every two rotations. The sequence of operations for column n is thus: The term carry may refer to: A violation whilst dribbling in the game of basketball. ...

1. Addition from column n + 1
2. Carry propagation
3. Addition to column n − 1
4. Rest

In electronics, an adder is a device which will perform the addition, S, of two numbers. ...

## Method of differences The London Science Museum's replica difference engine, built from Babbage's design. The design has the same precision on all columns, but when calculating converging polynomials, the precision on the higher-order columns could be lower.

As the differential engine can not do multiplication, it is unable to calculate the value of a polynomial. However, if the initial value of the polynomial (and of its derivatives) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... The National Science Museum in London The Science Museum on Exhibition Road, Kensington, London, is part of the National Museum of Science and Industry. ... Look up replica in Wiktionary, the free dictionary. ... In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, a derivative is defined as the instantaneous rate of change of a function and the process of finding the derivative is called differentiation. ...

The principle of a difference engine is Newton's method of divided differences. It may be illustrated with a small example. Consider the quadratic polynomial In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is the interpolation polynomial for a given set of data points in the Newton form. ... In mathematics divided differences is a recursive division process. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ...

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 column are constant. This is no mere 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. Look up Ad infinitum in Wiktionary, the free dictionary. ...

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.

## Initial values

The initial values of columns can be calculated by first manually calculating N consecutive values of the function, and by backtracking, i.e. calculating the required differences. Backtracking is a strategy for finding solutions to constraint satisfaction problems. ...

Col 10 gets the value of the function at the start of computation f(0). Col 20 is the difference between f(1) and f(0)...

### Use of derivatives

A more general and in many cases more useful method is to calculate the initial values from the values of the derivatives of the function at the start of computation. Each value is thus represented as power series of the different derivates. The constants of the series can be calculated by first expressing a function as a Taylor series i.e. a sum of its derivatives. Setting 0 as the start of computation we get the Maclaurin series In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ... As the degree of the Taylor series rises, it approaches the correct function. ... $sum_{n=0}^{infin} frac{f^{(n)}(0)}{n!} (x)^{n}.$

Calculating the values numerically, we get the following serial representations for the initial values of the columns:

Let f,f',f'',f''',f''''... be the values of the function and its derivatives at the start of computation

• Col 10 = f
• Col 20 = f' + 1 / 2f'' + 1 / 6f''' + 1 / 24f'''' + 1 / 120f''''' + ...
• Col 30 = f'' + f''' + 14 / 24f'''' + 23 / 120f''''' + ...
• Col 40 = f''' + 36 / 24f'''' + 171 / 120f''''' + ...
• Col 50 = f'''' + 378 / 120f''''' + ... Results from FactBites:

 Difference engine - Wikipedia, the free encyclopedia (767 words) Difference engines were forgotten and then rediscovered in 1822 by Charles Babbage, who proposed it in a paper to the Royal Astronomical Society entitled "Note on the application of machinery to the computation of very big mathematical tables." 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. The difference engine and printer were constructed to tolerances achievable with 19th century technology, resolving a long-standing debate whether Babbage's design would actually have worked.
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