(Math) Calculators: Constructing Difference Engines
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#1: (Math) Calculators: Constructing Difference Engines Author: adediosLocation: Angel C. de Dios PostPosted: Sun Apr 30, 2006 7:31 am
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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 decimal-based 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 seventh-degree polynomials to high precision.

The challenge of reconstructing Babbage's difference engines has also attracted others. In 2003, Tim Robinson constructed small-scale, 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.

*************************************************************

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

Fraction-Decimal 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://www-groups.dcs.st-and.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

#2: Scientists Unravel Mystery of Ancient Greek Machine Author: adediosLocation: Angel C. de Dios PostPosted: Thu Nov 30, 2006 2:01 pm
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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 2,000-year-old astronomical calculator built by ancient Greeks.

A new analysis of the Antikythera Mechanism [image], a clock-like machine consisting of more than 30 precise, hand-cut 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 three-dimensional X-ray scanners to reconstruct the workings of the device's gears and high-resolution surface imaging to enhance faded inscriptions on its surface.

For the full article:

http://www.livescience.com/his....._mech.html



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