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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Sun Apr 30, 2006 7:31 am Post subject: (Math) Calculators: Constructing Difference Engines 


Science News Online
Week of April 29, 2006; Vol. 169, No. 17
Constructing Difference Engines
Ivars Peterson
When I was in college, several decades ago, I relied heavily on a computational device called a slide rule and on a little, paperbound handbook of mathematical and scientific tables. I occasionally enjoyed simply browsing the handbook's many tables, formulas, and other data. The table of random digits, by itself, presented intriguing mysteries.
I also wondered at times about the sources of all this information. Mathematicians and many others have relied on tables of logarithms, trigonometric functions, and other mathematical formulas for a very long time. Carl Friedrich Gauss (1777–1855), for one, pondered such tables in formulating ideas about the distribution of prime numbers ( see Gauss' Prime Tables at http://blog.sciencenews.org/20.....les_1.html ).
Computing mathematical tables itself has a long history, often tied closely to astronomical and navigational requirements. In the 18th and 19th centuries, in particular, there arose a great need for large tables of numerical values of a wide variety of mathematical functions.
Newton's method of differences is the basis of one technique for computing values of polynomial functions. Consider, for example, the quadratic polynomial p(x) = x^2 – 4x + 3. Suppose that you want to compute the values of p(x) when x = 0, 0.1, 0.2, 0.3, 0.4, and so on.
In the table below, the first column contains the value of the polynomial, the second column contains the differences of two neighbors in the first column, and the third column contains the differences of two neighbors in the second column.
p(0) = 3
############# 3 – 2.61 = 0.39
p(0.1) = 2.61 ################## 0.39 – 0.37 = 0.02
############# 2.61 – 2.24 = 0.37
p(0.2) = 2.24 ################## 0.37 – 0.35 = 0.02
############# 2.24 – 1.89 = 0.35
p(0.3) = 1.89 ################## 0.35 – 0.33 = 0.02
############# 1.89 – 1.56 = 0.33
p(0.4) = 1.56
Note that the values in the third column are constant. In general, if you start with any polynomial of degree n, the number in column n + 1 will always be constant.
With such a start, you can readily compute p(0.5). Starting from the right with the value 0.02, you subtract it from the value in the second column, 0.33 – 0.02, to get 0.31. Subtracting this result from the value for p(0.4), 1.56 – 0.31, gives the value p(0.5) = 1.25.
So, it's possible to generate successive values of a polynomial without having to multiply.
This process can be automated in a mechanical device. In the 18th century, J.H. Mueller designed such a mechanism—a difference machine—but it was never built. Charles Babbage (1791–1871) revived the idea in 1821 when he proposed a design for a decimalbased difference engine operated by a hand crank. He designed a second, improved version between 1847 and 1849. However, Babbage made little progress in building either machine.
In 1985, the Science Museum in London launched a project to build a complete, working version of Babbage's second difference engine. This model was completed and working in November 1991, 1 month before the anniversary of Babbage's birth. See http://www.sciencemuseum.org.u...../index.asp .
Babbage's second difference engine could hold seven numbers of 31 decimal digits each, allowing it to tabulate seventhdegree polynomials to high precision.
The challenge of reconstructing Babbage's difference engines has also attracted others. In 2003, Tim Robinson constructed smallscale, working models of both of Babbage's difference engines entirely from standard Meccano parts ( see http://www.meccano.us/difference_engines/ ).
Andrew Carol, a software developer for Apple, recently built a working difference engine out of Lego pieces ( see http://acarol.woz.org/LegoDifferenceEngine.html ).
Although none of these models matches the complexity and intricacy of Babbage's original designs, they are still mechanical calculators that do the same job that Babbage's machines were supposed to do. They're remarkable tributes to mechanical ingenuity—and patience.
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Questions to explore further this topic:
Who is Charles Babbage?
http://www.sciencemuseum.org.u...../index.asp
http://www.sciencemuseum.org.u...../page2.asp
http://www.cbi.umn.edu/exhibits/cb.html
What is the "tables crisis"
http://www.sciencemuseum.org.u...../page1.asp
What are mathematical tables?
http://en.wikipedia.org/wiki/Mathematical_table
http://www.sosmath.com/tables/tables.html
Number Notation
http://www.math2.org/math/general/numnotation.htm
Addition Table
http://www.math2.org/math/general/addtable.htm
Multiplication Table
http://www.math2.org/math/gene.....ytable.htm
FractionDecimal Conversion
http://www.math2.org/math/gene.....fradec.htm
Trigonometric Table
http://www.math2.org/math/trig/tables.htm
Logarithmic Table (base 10)
http://www.sosmath.com/tables/.....table.html
A History of Mathematical Tables
http://functions.wolfram.com/About/history.html
http://faculty.oxy.edu/jquinn/.....931AD.html
Are there ancient mathematical tables?
http://wwwgroups.dcs.stand.a.....atics.html
http://it.stlawu.edu/%7Edmelvi...../Recip.pdf
What is a slide rule?
http://www.hpmuseum.org/sliderul.htm
How does one use a slide rule?
http://www.hpmuseum.org/srinst.htm
What are circular slide rules?
http://www.hpmuseum.org/srcirc.htm
Loga
http://www.hpmuseum.org/loga.jpg
Atlas
http://www.hpmuseum.org/atlas.jpg
Fowler
http://www.hpmuseum.org/fowler.jpg
Lietz
http://www.hpmuseum.org/lietz.jpg
What are cylindrical slide rules?
http://www.hpmuseum.org/srcyl.htm
Fuller calculator
http://www.hpmuseum.org/big/fullinbx.jpg
Otis King calculator
http://www.hpmuseum.org/big/otisking.jpg
Images of various slide rules
http://www.hpmuseum.org/srmore.htm
What is the difference engine?
http://www.sciencemuseum.org.u...../page3.asp
What is the difference engine no.2?
http://www.sciencemuseum.org.u...../page4.asp
What is an analytical engine?
http://www.sciencemuseum.org.u...../page5.asp
http://www.fourmilab.ch/babbage/contents.html
What is an abacus?
http://www.oldcalculatormuseum.com/articles.html
http://www.hitmill.com/compute.....bacus.html
What are calculators?
http://en.wikipedia.org/wiki/Calculator
A Calculator Museum
http://www.oldcalculatormuseum.com/articles.html
History of Calculating Machines
http://www.pbs.org/nerds/timeline/index.html
http://www.webcom.com/calc/
http://www.maxmon.com/timeline.htm
A Collection of Online Calculators
http://www.coolmath.com/calculators/index.html
http://www.martindalecenter.com/Calculators.html
Who is Seymour Cray?
http://www.cbi.umn.edu/exhibits/cray/index.html
A Course on the Development of Algorithms
http://www.csc.liv.ac.uk/~ped/.....ntent.html
GAMES
http://www.pbs.org/nerds/game.html
http://www.k111.k12.il.us/KING/math.htm
http://resources.kaboose.com/games/math2.html
Last edited by adedios on Sat Jan 27, 2007 3:08 pm; edited 1 time in total 

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adedios SuperPoster
Joined: 06 Jul 2005 Posts: 5060 Location: Angel C. de Dios

Posted: Thu Nov 30, 2006 2:01 pm Post subject: Scientists Unravel Mystery of Ancient Greek Machine 


Scientists Unravel Mystery of Ancient Greek Machine
By Ker Than
LiveScience Staff Writer
posted: 29 November 2006
01:05 pm ET
Scientists have finally demystified the incredible workings of a 2,000yearold astronomical calculator built by ancient Greeks.
A new analysis of the Antikythera Mechanism [image], a clocklike machine consisting of more than 30 precise, handcut bronze gears, show it to be more advanced than previously thought—so much so that nothing comparable was built for another thousand years.
"This device is just extraordinary, the only thing of its kind," said study leader Mike Edmunds of Cardiff University in the UK. "The design is beautiful, the astronomy is exactly right…In terms of historical and scarcity value, I have to regard this mechanism as being more valuable than the Mona Lisa."
The researchers used threedimensional Xray scanners to reconstruct the workings of the device's gears and highresolution surface imaging to enhance faded inscriptions on its surface.
For the full article:
http://www.livescience.com/his....._mech.html 

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