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The Mathematical Engines
Math G
May 2,2001
As a young man in London during theearly nineteenth century,Charles Babbage sat laboring over column after columnof numbers, finding the job exceedingly tedious, long and prone to numerouserrors.Babbage wished to God these calculations could be done by a steamengine. What he envisioned was an engine that could automate tasks whichrequired precision and repetition, but could speed up the process and reduceerrors. So began the inspirational steam that was to motivate Charles Babbageto bring to life his dream of a machine that could perform mechanicalcalculations.This machine is called the Difference Engine.
Charles Babbage was born in Londonon December 26, 1791.As a child he had a great curiosity about how thingsworked.During childhood, he suffered from fevers and was sent to a school inthe Devon countryside to help improve his health.It is in Devon that Babbageattributes his learning to idleness which helped to lead him into his greatmathematical skills.As a young boy,he was passionately fond of algebra andwould rise everyday at 3:00 a.m. to study it.He read every book he could findon the subject.He loved numbers, percents and orders and spent much of his owntime in studying mathematics.By 1812, Babbage and his friend, astronomer JohnHerschel, founded the Analytical Society.The Society published two books on thecalculus of differentials.In 1816, Babbage applied for a job as a mathprofessor at a college outside of London.He was told he did not get the jobbecause he lacked influence with the board of directors.He set up a workshopand besides working on mathematical topics, he began to dabble in chemistry andmechanics.He also became a member of the Royal Society, which was Englandsmajor scientific institution.
In 1828, Babbage was elected asLucasian Professor of Mathematics in Cambridge, England.He did not want toaccept the position as he feared it would intrude on his work with theDifference Engine.He eventually decided to accept the title and was LucasianProfessor for ten years.In 1834, Babbage helped to found The StatisticalSociety of London which processed and analyzed information about the Britisheconomy.
In 1821,Charles and his friend JohnHerschel were asked by the Astronomical Society of London to help improve theirtables of the Nautical Almanac ( a publication of star positions for use atsea).It was during his work on the Nautical Almanac that Babbage began to gethis idea for the Difference Engine. The design and construction of it were tooccupy Babbage for the next ten years.During the nineteenthcentury,mathematical calculations were done by hand, thus increasing thepercentage of human error.Mathematical tables were also computed by hand andwere often used in such professions as navigation and government use (fordetermining annuity payments).Calculating the formulas was usually performed byclerks.Each calculation was then performed twice, each by a different clerk tohelp eliminate errors.But if both clerks made the same error, it would not beimmediately apparent, but it was better than having one clerk do all thearithmetic.Babbage felt there was less chance of error if each clerk couldcheck the others mathematical calculations. He considered how such monotonousand tedious calculations could be done by a machine instead of by hand.CharlesBabbage was a genius at abstract mechanical design and had a vision ofautomatic computing.
In 1822, Babbage constructed asmall-scale version of the Difference Engine.He engaged the help of an engineernamed Joseph Clement in the actual construction of the machine. For eight yearsthe two men traded the machine back and forth as Clement build the parts whileBabbage conducted experiments on their functionality.The basic design of theDifference Engine was to automate a process of calculating a table oflogarithms. Its application was the method of finite differences.It consistedof several vertical columns across the front of the machine. Each column heldseveral rotating wheels divided into ten parts, numbered with the digits 0 to9.On each column, the most significant digit was at the top while the least significantdigit 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 leftwardsthrough the orders of difference (thus the name Difference Engine as the tableson the engine were calculated by the method of differences).
To calculate values of a functionof a variable, the variable will take the values 0, 1, 2, 3 and will berepresented 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 tablesinvolve addition when calculating the next value of a function.This is easierto calculate than multiplication and shows how the result depends on theprevious value.Example:The answer 14 depended on the previous result of 9, givingthe difference of 5, thus the function result of 19 depended on the previousresult of 15, again giving a difference of 5.The difference of 5 is consideredto be simple first order while the difference of 2 is the derived second orderand so on.By calculating each value progressively, a constant pattern ofdifference begins to emerge (such as in the example of f = 5x + 9, thedifference being 5).It is this process that Babbage saw as an easy way toautomate 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 themachine.For the machine to be automated, the starting wheel has to be aconstant.The following is an example of a table that shows the squares ofintegers 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 afterthe 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 mighttake until the fourth or fifth difference to find a constant value.Babbagedevised a system of mechanical notation that showed how the parts of theDifference Engine moved.The mechanical notation was a table of numbers, linesand symbols to describe the machines actions. He published his notation in thePhilosophical Transactions of the Royal Society in 1826.
By 1828, Charles Babbage had spentsix thousand British pounds of his own money on the construction of theDifference Engine.By this time, the governmentwho had shown initial interest inthe project,had only reimbursed him for fifteen hundred pounds.His own engineerthreatened to quit if he wasnt paid.Thousands of parts had been made for themachine but his workers had not assembled any.Babbage and his engineer, JosephClement 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 nofurther work was done on the machine.It wasnt until 1991 that the ScienceMuseum of London built a replica of the Difference Engine.The machine consistedof four thousand parts, weighed three tons and required modern computer-aideddesigns to produce.
In between his inventions, Babbagebecame interested in the calculation tables widely used during the nineteenthcentury. His goal was to devise a table completely free of errors and to updatetables that had been in use for some two hundred years.The table he worked onwas a logarithm table.Logarithm is an algebraic term that involvesexponentiation (multiplying a number by itself some number of times).Before theinvention of calculators, multiplying large numbers was an extremely difficultand tedious task, but with the use of logarithm tables the job was made mucheasier. In logarithms, numbers are multiplied by adding their exponents.Anexample is na x nb
= na+b .nrepresents 10 so 102 x 103= 102+3= 105.Multiplyingten five times itself may be quite a large number if done by hand, but the useof logarithms helped to simplify the equation.By using a logarithm table, thechore of multiplying 105 became much easier by finding the numberand its corresponding logarithm.Heres an example of what a logarithm tablelooked like:
| Number | Logarithm |
| 2 | 0.30103 |
| 3 | 0.47712 |
| 6 | 0.77815 |
Using logarithms instead of numbersis a process that substitutes addition for multiplication, subtraction fordivision and multiplication for exponentiation.These are the rules forlogarithms:
Log (A x B) = log (A) + log (B)
Log (A ΒΈB) = log(A) ? log (B)
Log (Ab) = log (A) x B
After Babbages work on both theDifference Engine and the logarithm tables, he began to throw himself fullforce into making a better machine that could solve more complicatedproblems.It would do a lot more than adding and subtracting of fixed numbers.Itwould solve equations.In 1834, Babbage began work on this next project which hecalled the Analytical Engine. Many people today call this machine the worldsfirst computer.He began to design this machine using his own money but knew hedidnt have the capability to actually build one. The contrast between BabbagesDifference and Analytical Engines were that instead of entering a new constantby hand, he developed a way for the differences to be donemechanically. To achieve this, Babbage used a punch card system, whichwere cards with holes in them to represent numbers.The cards became known asoperation cards and were similar to the punched cards used in the Jacquard Loomto weave intricate patterns of cloth.It was Lady Ada Lovelace, daughter of theEnglish poet Lord Byron, who coined the phrase the Analytical Engine weavesalgebraic patterns just as the Jacquard loom weaves flowers and leaves.
Ada Lovelace was introduced tomathematics by her mother Lady Byron at a very young age.It is said she spentmore time in mathematics than in raising her own three children.She first metCharles Babbage in 1833 and established a lifelong friendship. He was impressedwith her energy and eagerness to learn and encouraged her to pursue hermathematical interests.In 1843, Ada made a significant contribution to thepublics awareness of Babbages Analytical Engine.In 1842, an Italianmathematician, Luigi Menabrea published a twenty-four page description inFrench 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 amazingcapabilities of the Analytical Engine. Ada emphasized how the Analytical Enginecould compute trigonometric functions containing variables and how it couldcompute Bernoulli numbers.Because of her keen insight on the abilities of theAnalytical Engine,Babbage referred to her as the Enchantress of Numbers.Theimportance of her notes stressed the ability of the Analytical Engine to beprogrammed with general information supplied by the operation cards (orsoftware design as we know it today).Thus, Ada is known today as the firstcomputer programmer. In 1980, the U.S. Department of Defense developed auniversal computer language that is named in her honor (ADA).
Babbages Analytical Engine, ifbuilt, would have been quite large. It was to be steam powered (thus the termsteam engine) and was equivalent in size to the computers of the 1960s and1970s.It was a decimal system machine rather than binary and it could handleall four arithmetic functions.The machine was designed to stop and ring a bellif it required further data to complete its calculations.The Analytical Enginehad 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 onlysolve equations by having the correct sequence of operations.The engine waseasily capable of carrying out any calculations Babbage could ever require ofit.Logarithm tables could be calculated along with the rational roots of somefunctions.
Babbage worked diligently on theAnalytical Engine for a period of twenty years.Unfortunately, the AnalyticalEngine was never built while Babbage lived.His dreams of building a steamengine were not fully realized partly because of finances, but also because themachinery and technology was beyond the engineering capability that waspossibleat that time. The Analytical Engine remained on paper in the form of Babbagesnotes.It wasnt until 1937 when a Harvard physicist, Howard Aiken conceived of aprogrammable electromechanical calculating machine.It was finished in 1944 andwas called the Mark I Computer.Like Babbages Difference Engine, the Mark I wasdesigned and used for calculating and printing mathematical tables.It is thismachine that earned the title Babbages Dream Comes True.
Babbages work on such machines asthe Difference Engine and the Analytical Engine pioneered the trail of a newcomputer era.His achievements, along with Ada Lovelaces programming work,earned him the reputation as being the grandfather of computing.
UnversityPress, Inc.,1998.
Press,1985