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The Mathematical Engines
Math G
May 2,
2001
As a young man in London during the early nineteenth century,Charles Babbage sat laboring over column after column of numbers, finding the job exceedingly tedious, long and prone to numerous errors.Babbage wished to God these calculations could be done by a steam engine. What he envisioned was an engine that could automate tasks which required precision and repetition, but could speed up the process and reduce errors. So began the inspirational steam that was to motivate Charles Babbage to bring to life his dream of a machine that could perform mechanical calculations.This machine is called the Difference Engine.
Charles Babbage was born in London on December 26, 1791.As a child he had a great curiosity about how things worked.During childhood, he suffered from fevers and was sent to a school in the Devon countryside to help improve his health.It is in Devon that Babbage attributes his learning to idleness which helped to lead him into his great mathematical skills.As a young boy,he was passionately fond of algebra and would rise everyday at 3:00 a.m. to study it.He read every book he could find on the subject.He loved numbers, percents and orders and spent much of his own time in studying mathematics.By 1812, Babbage and his friend, astronomer John Herschel, founded the Analytical Society.The Society published two books on the calculus of differentials.In 1816, Babbage applied for a job as a math professor at a college outside of London.He was told he did not get the job because he lacked influence with the board of directors.He set up a workshop and besides working on mathematical topics, he began to dabble in chemistry and mechanics.He also became a member of the Royal Society, which was Englands major scientific institution.
In 1828, Babbage was elected as Lucasian Professor of Mathematics in Cambridge, England.He did not want to accept the position as he feared it would intrude on his work with the Difference Engine.He eventually decided to accept the title and was Lucasian Professor for ten years.In 1834, Babbage helped to found The Statistical Society of London which processed and analyzed information about the British economy.
In 1821,Charles and his friend John Herschel were asked by the Astronomical Society of London to help improve their tables of the Nautical Almanac ( a publication of star positions for use at sea).It was during his work on the Nautical Almanac that Babbage began to get his idea for the Difference Engine. The design and construction of it were to occupy Babbage for the next ten years.During the nineteenth century,mathematical calculations were done by hand, thus increasing the percentage of human error.Mathematical tables were also computed by hand and were often used in such professions as navigation and government use (for determining annuity payments).Calculating the formulas was usually performed by clerks.Each calculation was then performed twice, each by a different clerk to help eliminate errors.But if both clerks made the same error, it would not be immediately apparent, but it was better than having one clerk do all the arithmetic.Babbage felt there was less chance of error if each clerk could check the others mathematical calculations. He considered how such monotonous and tedious calculations could be done by a machine instead of by hand.Charles Babbage was a genius at abstract mechanical design and had a vision of automatic computing.
In 1822, Babbage constructed a small-scale version of the Difference Engine.He engaged the help of an engineer named Joseph Clement in the actual construction of the machine. For eight years the two men traded the machine back and forth as Clement build the parts while Babbage conducted experiments on their functionality.The basic design of the Difference Engine was to automate a process of calculating a table of logarithms. Its application was the method of finite differences.It consisted of several vertical columns across the front of the machine. Each column held several rotating wheels divided into ten parts, numbered with the digits 0 to 9.On each column, the most significant digit was at the top while the least significant digit was at the bottom.The column to the farthest right was the table number, the next column to the left was the first difference and so on leftwards through the orders of difference (thus the name Difference Engine as the tables on the engine were calculated by the method of differences).
To calculate values of a function
of a variable, the variable will take the values 0, 1, 2, 3 and will be
represented by x.An example is the function f = 5x + 9.
|
x |
f |
Difference |
|
0 |
9 |
|
|
1 |
14 |
5 |
|
2 |
19 |
5 |
|
3 |
24 |
5 |
A second example is the function f = x2 + 4
|
x |
f |
Difference
1 |
Difference
2 |
|
0 |
4 |
|
|
|
1 |
5 |
1 |
|
|
2 |
8 |
3 |
2 |
|
3 |
13 |
5 |
2 |
These methods of difference tables
involve addition when calculating the next value of a function.This is easier
to calculate than multiplication and shows how the result depends on the
previous value.Example:The answer 14 depended on the previous result of 9, giving
the difference of 5, thus the function result of 19 depended on the previous
result of 15, again giving a difference of 5.The difference of 5 is considered
to be simple first order while the difference of 2 is the derived second order
and so on.By calculating each value progressively, a constant pattern of
difference begins to emerge (such as in the example of f = 5x + 9, the
difference being 5).It is this process that Babbage saw as an easy way to
automate by machine.
|
Squares |
|
|
|
|
|
|
Sequence |
1st. Difference |
2nd Difference |
3rd Difference |
|
|
1 |
|
|
|
|
|
4 |
3 |
|
|
|
|
9 |
5 |
2 |
|
|
|
16 |
7 |
2 |
0 |
|
|
25 |
9 |
2 |
0 |
|
|
36 |
11 |
2 |
0 |
|
|
49 |
13 |
2 |
0 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cubes |
|
|
|
|
|
|
Sequence |
1st Difference |
2nd Difference |
3rd Difference |
4th Difference |
|
1 |
|
|
|
|
|
8 |
7 |
|
|
|
|
27 |
19 |
12 |
|
|
|
64 |
37 |
18 |
6 |
|
|
125 |
61 |
24 |
6 |
0 |
|
216 |
91 |
30 |
6 |
0 |
|
343 |
127 |
36 |
6 |
0 |
To make the Difference Engine work, the operator has to specify the initial differences to be entered into the machine.For the machine to be automated, the starting wheel has to be a constant.The following is an example of a table that shows the squares of integers with the second difference constant at 2 and in the cubes of integers, the third difference becomes constant at 6.Note that the next difference after the constant 1 is zero.For squares, the engine would need three sets of wheels, for cubes, four.More wheels would be needed to calculate numbers that might take until the fourth or fifth difference to find a constant value.Babbage devised a system of mechanical notation that showed how the parts of the Difference Engine moved.The mechanical notation was a table of numbers, lines and symbols to describe the machines actions. He published his notation in the Philosophical Transactions of the Royal Society in 1826.
By 1828, Charles Babbage had spent six thousand British pounds of his own money on the construction of the Difference Engine.By this time, the governmentwho had shown initial interest in the project,had only reimbursed him for fifteen hundred pounds.His own engineer threatened to quit if he wasnt paid.Thousands of parts had been made for the machine but his workers had not assembled any.Babbage and his engineer, Joseph Clement soon parted ways due to money issues on the Difference Engine project, but not before assembly of the engine was completed (without the printing section) in 1832.Unfortunately, all funding for the engine began to diminish and no further work was done on the machine.It wasnt until 1991 that the Science Museum of London built a replica of the Difference Engine.The machine consisted of four thousand parts, weighed three tons and required modern computer-aided designs to produce.
In between his inventions, Babbage
became interested in the calculation tables widely used during the nineteenth
century. His goal was to devise a table completely free of errors and to update
tables that had been in use for some two hundred years.The table he worked on
was a logarithm table.Logarithm is an algebraic term that involves
exponentiation (multiplying a number by itself some number of times).Before the
invention of calculators, multiplying large numbers was an extremely difficult
and tedious task, but with the use of logarithm tables the job was made much
easier. In logarithms, numbers are multiplied by adding their exponents.An
example is na x nb
= na+b .n represents 10 so 102 x 103= 102+3= 105.Multiplying ten five times itself may be quite a large number if done by hand, but the use of logarithms helped to simplify the equation.By using a logarithm table, the chore of multiplying 105 became much easier by finding the number and its corresponding logarithm.Heres an example of what a logarithm table looked like:
|
Number |
Logarithm |
|
2 |
0.30103 |
|
3 |
0.47712 |
|
6 |
0.77815 |
Using logarithms instead of numbers is a process that substitutes addition for multiplication, subtraction for division and multiplication for exponentiation.These are the rules for logarithms:
Log (A x B) = log (A) + log (B)
Log (A ¬½B) = log(A) ? log (B)
Log (Ab) = log (A) x B
After Babbages work on both the Difference Engine and the logarithm tables, he began to throw himself full force into making a better machine that could solve more complicated problems.It would do a lot more than adding and subtracting of fixed numbers.It would solve equations.In 1834, Babbage began work on this next project which he called the Analytical Engine. Many people today call this machine the worlds first computer.He began to design this machine using his own money but knew he didnt have the capability to actually build one. The contrast between Babbages Difference and Analytical Engines were that instead of entering a new constant by hand, he developed a way for the differences to be done mechanically. To achieve this, Babbage used a punch card system, which were cards with holes in them to represent numbers.The cards became known as operation cards and were similar to the punched cards used in the Jacquard Loom to weave intricate patterns of cloth.It was Lady Ada Lovelace, daughter of the English poet Lord Byron, who coined the phrase the Analytical Engine weaves algebraic patterns just as the Jacquard loom weaves flowers and leaves.
Ada Lovelace was introduced to mathematics by her mother Lady Byron at a very young age.It is said she spent more time in mathematics than in raising her own three children.She first met Charles Babbage in 1833 and established a lifelong friendship. He was impressed with her energy and eagerness to learn and encouraged her to pursue her mathematical interests.In 1843, Ada made a significant contribution to the publics awareness of Babbages Analytical Engine.In 1842, an Italian mathematician, Luigi Menabrea published a twenty-four page description in French of Babbages Analytical Engine.Ada Lovelace, encouraged by Babbage, translated Menabreas article into English adding many pages of her own notes. Adas notes gave more explanations and details expounding the amazing capabilities of the Analytical Engine. Ada emphasized how the Analytical Engine could compute trigonometric functions containing variables and how it could compute Bernoulli numbers.Because of her keen insight on the abilities of the Analytical Engine,Babbage referred to her as the Enchantress of Numbers.The importance of her notes stressed the ability of the Analytical Engine to be programmed with general information supplied by the operation cards (or software design as we know it today).Thus, Ada is known today as the first computer programmer. In 1980, the U.S. Department of Defense developed a universal computer language that is named in her honor (ADA).
Babbages Analytical Engine, if built, would have been quite large. It was to be steam powered (thus the term steam engine) and was equivalent in size to the computers of the 1960s and 1970s.It was a decimal system machine rather than binary and it could handle all four arithmetic functions.The machine was designed to stop and ring a bell if it required further data to complete its calculations.The Analytical Engine had 3 card types used in instructing the machine:
-Number:allocate constants
be done (+ - * /)
-Variable:determine which columns the results
are to be sent to.
ax + by = m
cx + dy = n
|
Step |
Operation |
Variables |
Column |
Answer |
|
1 |
* |
v4 * v3 |
v8 |
=md |
|
2 |
* |
v5 * v1 |
v9 |
=nb |
|
3 |
* |
v0 * v3 |
v10 |
=ad |
|
4 |
* |
v1 * v2 |
v11 |
=bc |
|
5 |
- |
v8 - v9 |
v12 |
=md - nb |
|
6 |
- |
v10 - v11 |
v13 |
=ad - bc |
|
7 |
/ |
v12 / v13 |
v14 |
=md - nb/ad - bc |
The Analytical Engine could only solve equations by having the correct sequence of operations.The engine was easily capable of carrying out any calculations Babbage could ever require of it.Logarithm tables could be calculated along with the rational roots of some functions.
Babbage worked diligently on the Analytical Engine for a period of twenty years.Unfortunately, the Analytical Engine was never built while Babbage lived.His dreams of building a steam engine were not fully realized partly because of finances, but also because the machinery and technology was beyond the engineering capability that waspossible at that time. The Analytical Engine remained on paper in the form of Babbages notes.It wasnt until 1937 when a Harvard physicist, Howard Aiken conceived of a programmable electromechanical calculating machine.It was finished in 1944 and was called the Mark I Computer.Like Babbages Difference Engine, the Mark I was designed and used for calculating and printing mathematical tables.It is this machine that earned the title Babbages Dream Comes True.
Babbages work on such machines as
the Difference Engine and the Analytical Engine pioneered the trail of a new
computer era.His achievements, along with Ada Lovelaces programming work,
earned him the reputation as being the grandfather of computing.
Unversity
Press, Inc.,1998.
Press,1985