Also see

Babbage Difference Engine #2

- How to Initialize the Machine -

Including Calculating Initial Values

Please note:
Most explanations of how to compute initial values for the Babbage Difference Engine use simple difference tables based on the first n values of a series. This works well for polynomials with integer coefficients of power less than n. see 7. Why Simple Differences does not work well with Trig & Log tables and most other real world functions. 8. The Interpolating Polynomial then the Method of Differences using the Interpolating Polynomial works much better in these cases. !! |

Background

- The computational part of the Babbage Difference Engine #2 was built by (and is in) the British National Museum of Science and Industry in 1991. The printing part of this engine is now complete as per BBC News (April 13, 2000)

An idea of the power (speed) of Babbage's Difference Engine

Babbage's engine as built in the 1990s prints one evaluation of a 7th order polynomial each four seconds.

- Fifteen evaluations per minute.from Tim Coslett

"To evaluate seven terms of a power series took 15 minutes on the Harvard [Mark I] device of which 3 minutes was set-up time, whereas it will take at least 15 minutes to set up ENIAC and about 1 second to do the computing."

- September 2, 1944 letter to Colonel Paul Gillon, assistant director BRL from Lieutenant Herman Goldstine, U.S. Army liaison officer for Project PX (ENIAC)It seems that the Babbage machine (mechanical, designed about 1848) is about as fast as the ENIAC (electronic, designed and built about 100 years later) on this particular problem.

The Goal of this documentis to provide you with theknowledgeandcomputational tools(including free downloadable extended precision UBASIC and an On-Line Manual) (easier to use) so that you caninitialize and emulate the Babbage Difference Engine #2to:

evaluate polynomials(up to 7th order) with integer coefficients (for fun)generate trigonometric and logarithmic tables(as Babbage originally intended)without studying Numerical Analysis and getting headaches!

* Please note, this JavaScript emulator fails to run properly on Internet Explorer 7 and up.

Disclaimer- I (the author of this piece) am neither a mathematician nor historian (I are a enjineer!)

- The methods shown below may not be optimum, but no one has suggested better to me.
- Babbage wrote much, but there were no details that I found of how he was going to determine

- the polynomial coefficients to set into his proposed machine to start it computing non-trivial computations such as logarithms or trig functions,
- determine ranges over which they would provide the desired accuracy, nor other vital details.
- I have examined collections of his works - such as the 11 volume
"The Works of Charles Babbage"Edited by Martin Campbell-Kelly, published in London by William Pickering, 1989. Even Volume 2"The Difference Engine and Table Making"only gave grade school examples of integer powers of 2 and similar trivial situations.- Possibly he figured that any competent mathematician of the era would regard the determination of coefficients as obvious or child's play. Or maybe he did not care to disturb "the rest of us" with interesting complexities.

The Goal of the Babbage Difference Enginewas to provideaccurate trig and log tables(for navigation, engineering & astronomical computations). Babbage designed this version of difference engine during 1847-1849, using technical improvements discovered while he was designing his Analytical Engine.One can think of Babbage's Difference Engine as a means of providing an interpolation between widely spaced known points. To use the available power (seventh degree polynomial) of the engine, eight known points are required. Two of the points near the end points of the function to be approximated, and six points roughly evenly spaced in-between the end points.

This

web site also includestwo manuals kindly provided byBritish National Museum of Science and Industry

- Babbage Difference Engine #2, Technical Description with 145 pages of text by
Doron Swadeand22 of Babbage's original drawings- The 12 page Babbage Difference Engine #2, Instruction Manual by
National Museum of Science and Industryengineers who worked on the Difference Engine #2An article "Babbage's Expectations for his Engines" is also provided.

Please note: This paper does not discuss the Babbage Analytical Enginethat contains many of the ideas of the modern computer, and was described by "Ada King, Countess of Lovelace" (please see name discussion here) and others. The Analytical Engine development came after the construction of the Difference Engine was abandoned.The design of Difference Engine II incorporated improvements found during Babbage's design work on his Analytical Engine - yielding a reduction of parts count from 25,000 to 8,000.

For the Babbage Analytical Engine, here is Charles Babbage -- Scientist and Philosopher and an interesting web site.

This paper is organized into the following sections

- A short introduction
- Eight pages of an 1822 Babbage letter describing the error problem and his solution
- How the French (and most others?) calculated math tables in the early 1800s
- Examples of formulas for trig and log values
- Method of Finite Differences saves labor when making tables
- A simple example of the Method of Finite Differences

- also see "Specimen Tables, The Swedish Calculating Machine" by Scheutz (Google books)- Why Simple Differences does not work well with Trig & Log tables
- The Interpolating Polynomial

A polynomial of limited degree that approximates the desired function- Forming the Interpolating Polynomial
- Finding the Differences using the Interpolating Polynomial
- Scaling to permit dealing with fractions (or approximations to fractions like 1/3)
- Representing negative numbers, (10's complement)
- Correcting the initialization for the Babbage odd-even parallel pipelined operation
- Setting up the engine, Babbage Difference Engine #2 Instruction Manual
- Some computed example results
- A good machine test case
a JavaScript non-pipelined emulation

another simulation starting (alpha version) by Nathan Lisgo @nlisgo (https://twitter.com/#!/nlisgo) and Paul Martin.- A (mercifully) short discussion on computing errors
- Babbage Overflow-Underflow detection
- Formatting the output of the Babbage printer section.
- Some Babbage websites
- Some Babbage on line books other sites
- Some Babbage books
- Serious comments about Babbage's design.
- SetUp Script for setting coefficients in the for Babbage Engine #2, Second Edition ;-))

## 1. A short introduction

The Babbage Difference Engine is basically a fancy adding machinethat can do unexpected (amazing) things with polynomials. An example of a polynomial is

y = a * x ^{7}+ b * x^{6}+ c * x^{5}+ d * x^{4}+ e * x^{3}+ f * x^{2}+ g * x + hAssume a polynomial of degree 7 (above) or less, with integer coefficients (a, b, c, ... are integers). If you set up a Difference Engine correctly, (insert the correct things to add) the Difference Engine will evaluate the polynomial (give y) for successive integer values of x until the machine overflows (numbers get larger than the machine can handle).

This is interesting, but even more interesting is the fact that if you form a polynomial that approximates (or interpolates) a function (say the sine function), the Difference Engine will evaluate the (sine) function to arbitrary accuracy (such as 15 digits). This is useful in making those boring trig tables so necessary before electronic calculators which calculates functions only as needed.

The British needed accurate trig and other tables for navigation of their sea going merchant and naval vessels. They were very interested in such developments and machines in the 1600s through the 1800s.

Babbage had investigated errors in navigational and astronomical tables, and realized that

bothcorrectcomputationandprintingwere needed. So the second part of his Difference Engine was a type-setting machine, which reduced the probability of human error to a minimum.He was awarded a gold metal by the British Astronomical Society for his paper "Observations on the Application of Machinery to the Computation of Mathematical Tables." in 1821.

Charles Babbagewas an interesting character;very gifted, but he was a very poor project engineerand politician. This paper is not about him or his problems and failure in producing his Difference Engine, but is abouthow to set up a Difference Engine # 2 so that it can produce the desired tables.

## 2. Eight pages of an 1822 Babbage letter describing the error problem and his solution

## 3. How the French (and most others?) calculated math tables in the early 1800s

## 4. Examples of formulas for trig and log values

Examples of series expansion

sin x = x - x ^{3}/(3!) + x^{5}/(5!) - x^{7}/(7!) + ...cos x = 1 - x

^{2}/(2!) - x^{4}/(4!) - x^{6}/(6!) + ...e

^{x}= 1 + x + - x^{2}/(2!) - x^{3}/(3!) - x^{4}/(4!) + ...Log

_{e}x = (x - 1) - (1/2)*(x - 1)^{2}+ (1/3)*(x - 1)^{3}- ...

where (2 > x > 0)For more examples, and a good discussion see this Wikipedia section.

Half angle formulas, starting at known points such as sin 30 degrees = 0.5, sin 45 degrees = 0.5*SQRT(3)

Nicolaus Copernicus used half angle formulas to compute his tables to 10 minutes of angle. The user interpolated for intermediate angles.

sin 0.5x = +-SQRT((1-cos x)/2)

cos 0.5x = +- SQRT((1+cos x)/2)

- Table in "On the Shoulders of Giants" by Steven Hawking, page 44 ...More on half angle formulas.

## 5. Method of Finite Differences saves labor when making tables

The above formulas CAN be used to compute each entry in the tables of sin, cos, ...

HOWEVER, the cost in terms of computation was excessive in the days of hand calculations. Your modern hand calculator (and personal computer) calculate the instances of these functions using formulas similar to the above formulas carefully modified to require a minimum of terms to attain the required results.But for tables, not much used these happy electronic days, other methods were used to save arithmetic operation,

The "

Method of Differences", dear to the hearts of Numerical Analyst's,saves an enormous amount of arithmetic, especially multiplications and divisionswhch are much more labor intensive than additions.You will note below that to compute the next value (x = x + 1) for a polynomial, the only processing required is a number of additions equal to the degree of the polynomial. An example, a

7th order polynomial (shown above) requires just 7 additions for each successive value of x.

## 6. A simple example of the Method of Finite Differences

- also see "Specimen Tables, The Swedish Calculating Machine" by Scheutz (Google books)

Boldednumbers below are difference values obtained by the process

Method of Finite Differences for y = x^{3}The symbol -> indicates subtract the number just above from the number just below and place the result to the right.

x (cycle) y = x ^{3}1st dif 2nd dif 3rd dif 4th dif 0 0

-> 1

1 1 -> 6

-> 7 -> 6

2 8 -> 12 -> 0

-> 19 -> 6

3 27 -> 18 -> 0

-> 37 -> 6

4 64 -> 24

-> 61

5 125

The difference values obtained by the above process are 0, 1, 6, 6, and 0

Using above difference values to calculate y = x^{3}The difference values obtained by the above process (

0, 1, 6, 6, and 0)

are placed into the Difference Columns of a "difference engine".note: the noun "Difference Column" has been abreviated to "DC" in the table below.

adding DC1 to DC0, then DC2 to DC1, ...

cycle

(x)DC0

(y=x^{3})DC1 DC2 DC3 DC4 0

01 6 6 0 1

17 12 6 0 2

819 18 6 0 3

2737 24 6 0 4

6461 30 6 0 5

12591 36 6 0 6

216127 42 6 0 7

343169 48 6 0 Boldednumbers above are the results obtained by the process.

An example of the above, and the BASIC program, is provided here

Summaryof the above difference method. A polynomial of limited degree (number of terms, power of exponents, ...) can be evaluated exactly to many values of x by using the method of differences. In the method of differences, you evaluate a few values of x, do the differences, and by simple addition, and can evaluate a large number of values.

## 7. Why Simple Differences does not work well with Trig & Log tables -

The above simple system works perfectly with a polynomial of a limited number of terms with integer coefficients.Unfortunately, trig and log functions are not low order polynomials with integer coefficients. In fact, trig and log functions are NOT polynomials, but irrational, transcendental functions.

The method of differences works exactly for the number of terms used to compute the differences, but begins to fail on succeeding cycles. The Taylor series for sin:

sin x = x - x3/(3!) + x5/(5!) - x7/(7!) + ... has just 4 derivatives in the range to the seventh power. Using the method of finite differences for the first 8 increments provides perfect results (duplicates the inputs) for the 8 iterations, but begins to deviate significantly after a number of iterations.

The differences depend upon the relative range of the difference sample versus the range of the desired calculations. In a

MATHEMATICA exampleand aUBASIC exampleusing the sin function, 0 to 7 degrees were used, and the desired range was 45 degrees. The error at the far end of the range (45 degrees) approaches 0.0001 percent.This is just a table of 45 values - trivial - and the errors are almost intolerable already.

HOWEVER - in a more useful sin table, with a value every 1 minute, things go dramatically wrong using this simple method. See this

UBASIC exampleof a sin table with an output every 1 minute of angle.

By the way, you notice that I make a big deal about extended precision. there is an example of using normal IEEE floating point (about 15 digit precision). You may notice that similar code to the example in the above paragraph yields an error of 54% at a mere 12 degrees. To summarize, if you are forming trig and log tables, using the method that works so well with polynomials of 7th degree or less fails. The first 8 values have 0 error, then the rest of the values have increasingly bad errors.

In the

next section, we will examine a method where few if any values are EXACTLY correct, butall values are within known (and hopefully acceptable) error limits.

## 8. The Interpolating Polynomial

A polynomial of limited degree that approximates the desired functionFortunately there is a good "work around" to the fact that some of the most desirable functions are not polynomials. You can form a polynomial that approximates the function (example sine) to high precision (say 15 digits) over the range of interest (say 0 to 45 degrees) and accept that most values in that range will not be "perfect". But the values can be within the allowed error tolerance, say 15 decimal digits. If you print the table accurate to 15 decimal digits, everyone will be satisfied - actually happy.

This page (750 KB) from "Applied Numerical Methods" by Carnahan shows that "least squares curve fit" is not needed when exact points are available.

The higher the order of the polynomial, the closer you can "fit" the polynomial to the desired function. Unfortunately, the higher the order of the polynomial, the more expensive the computing machine. In these days of wonderfully cheap computing capability, you could choose a 100th degrees polynomial with no visible impact on your budget. However, Babbage was making his machine from very expensive precision machinery, and he had to compromise. His compromise, which was still too expensive to attract investment money - was a 7th degree polynomial design.

The same cost driven compromise caused Babbage to restrict the precision of the Difference Engine to 31 decimal places. This seems like an excessive precision - BUT - a wide variety of error sources, which accumulate in the calculations, are caused by a finite limit to the computing accuracy.

ANDseveral digit positions of the machine are need to count operations to provide the "argument" column of the printed table, such as from abovey = x^{3}example.

x^{ }y = x^{3}0 0 1 1 2 8 3 27 4 64 5 125 6 216 7 343 Fortunately, you can reduce the errors caused by the missing higher order terms by adjusting the available (7th order) terms by closely fitting a lower order (7th order) polynomial as closely as practical to the function over the range of interest. (And if the errors are too great, you can reduce the range and reduce the errors.) Curve fitting the approximating polynomial to the actual function helps correct for the missing higher order terms of the actual function.

The individual values of the approximating polynomial, while not usually perfect, can closely approximate the function over the entire desired range. I will show that by using just a 7th order "interpolating polynomial" the sin function can be computed in 1 minute steps from 0 to 23 degrees (1860 steps) with a maximum error of 10

^{-12}. With this kind of accuracy, if every thing else is perfect, a navigator can determine his position within an inch relative to the earth's surface. That ought to be good enough, even for the British Navy. ;-))

## 9. Forming the Interpolating Polynomial

Now, how to make a polynomial from the sine trig function.

We now have the coefficients, accurate to about 31 decimal places, of the sine function over the range of 0 to 23 degrees.

- Pick a degree of polynomial. In our case, since we are already fighting for accuracy, pick the Babbage machine maximum, a 7th degree polynomial.
- Pick a range, say 0 degrees to 23 degrees (about half of the desired 0 - 45 degrees - (we know from previous experience that a seventh order polynomial in the range of 0 - 45 degrees will not be as accurate as we desire.
- select at least the polynomial degree plus one (eight) points rather evenly spaced along the selected range to be evaluated. In the example below, we picked the following degrees: 0, 3, 6, 9, 12, 15, 18, 21, 24.

Of course, Nicolaus Copernicus and Isaac Newton did not have handy calculators. However they could get a nice selection of numbers using symmetry formulas here The "half angle" and "sum of angles" formulas are of particular interest. For instance, the sine (and cosine) of 45 degrees is 1/(square root of 2) exactly so using half angle formulas the angles

45, 22.5, 11.25, 5.625, 2.8125, ...

are available using the half angle formula

(sin(a/2) = SquareRoot((1-cos(a))/2))And using the sum of angles formula

(sin(a+b) = sin(a)cos(b) + cos(a)sin(b))

yields values for

2.8125, 5.625, 8.4375, 11.25, 14.0625, 16.875, 19.6875, 22.5 degrees.

And of course, the sine of 0 degrees is 0.0- Calculate the sine values of these points to at least 31 digits of precision. A handy method of doing this is to use the "Calculator" in Windows.
To save error prone manual copying, there is a "copy" available in the "Edit" button that copies to the "clipboard" and you can copy the clipboard value to an editor or some curve-fit programs.

degrees value of sin ------- ------------ 0, 0 3, 0.052335956 2429438327 2211862960 90784 6, 0.104528463 2676534713 9983415480 2498 9, 0.156434465 0402308690 1010531946 7167 12, 0.207911690 8177593371 0174228440 5125 15, 0.258819045 1025207623 4889883762 4048 18, 0.309016994 3749474241 0229341718 2819 21, 0.358367949 5453002734 8413778941 3467 24, 0.406736643 0758002077 5398599034 1498- Use these values to determine the coefficients of a polynomial using Gaussian Reduction or some other curve fitting technique. Ideally, you will have an extended precision mathematical package such as MATHEMATICA or MAPLE to do this quite automatically (ah, this modern world :-)
Here is an example, written in MATHEMATICA (a wonderful and expensive extended precision mathematical tool - I have a very old student version). The extended precision is needed as IEEE floating point used by many low cost or free tools, such as Euler is accurate to "only" 15 decimal digits. Since the Babbage machine is capable of up to 31 digit precision, errors due to the limited precision of the floating point other packages introduces serious simulation errors.

- 1.011225786 6802907976 5083139397 *10^{12}+ 0.017453292 5216557274 7695492508 *x^{1}- 9.650085084 7235102648 2968837056 *10^{13}*x^{2}- 8.860959019 0158230245 8402902990 *10^{7}*x^{3}- 3.559895737 1823585126 5053546540 *10^{14}*x^{4}+ 1.349880364 7432610642 7099346582 *10^{11}*x^{5}- 1.182215638 0312276038 2796654565 *10^{16}*x^{6}- 9.563667920 8157451630 8224691444 *10^{17}*x^{7}Unfortunately, the above packages are not free - even student versions are sold for about $140.

A handy

15 decimal digit accuracy curve fit packageis found at the web site of John Kennedy, retired mathematics instructor from Santa Monica College.Make a new directory, then access John Kennedy's web site, click on "Downloadable Math Software", then download "JK32W2.ZIP" to your new directory. (I am assuming that you have access to PKZIP or a similar expansion routine.) Unzip the file, and Start polyfit.exe. This starts a handy Windows program that can calculate "Least Squares Fit" of data to the limits of IEEE floating point.

Coefficient Term -------------------- ----- -9.28351019821882E-16 constant 0.0174532925208996 x -7.57903133381811E-13 xFor your convenience, I have included John Kennedy's POLYFIT directory here.^{2}-8.86095929472524E-07 x^{3}-3.36840732998678E-14 x^{4}1.34987376001792E-11 x^{5}-1.17367893545279E-16 x^{6}-9.56350174721563E-17 x^{7}

You should be able to click on the .pdf and .txt files to see the contents.

To run PolyFit.exe , copy it to a directory, and click on it there -

## 10. Finding the Differences using the Interpolating Polynomial

Using the above coefficients, do the following

- select the interpolation interval or steps - say 1 minute intervals
- using the above coefficients, calculate the values for the first 8 intervals, i.e. 0 deg 0 min, 0 deg 1 min, 0 deg 2 min, ...
- figure how to scale the values in the machine, i.e., multiply the values so that the final result will not overflow the machine, but retaining as many digits of accuracy in the machine as possible
- using the values obtained, do the method of differences (above)
- insert the results of the method of differences into the machine (
we will worry about pipeline corrections later)- operate the machine until values for the entire range are obtained
Yielding the table 0-2 degrees 8 K bytes, table 0-23 degrees 88 K bytes, and a table with errors

## 11. Scaling to permit dealing with fractions

(or approximations to fractions like 1/3) Scaling is multiplying or dividing a variable by some constant so that you can make it fit into your computing machine. It is a little like the U.S. government budget. We humans divide all the numbers by say a million or a billion so we can think of them in our rather limited minds. Computers are also limited in the size and precision of the numbers they can handle naturally.

(Humans can write and add numbers as large as they wish, limited only by the size of the sheet of paper and the time you wish to spend. You can write down 2 one-hundred digit numbers just fine, and add them with no problem. However, a one-hundred digit number in meaningless to most of us - just

HUGE!)Most modern machines, for reasons of economy and speed, work naturally with a few sizes of numbers. A 2 byte word can represent numbers within the range of about +- 32,000, about 4 decimal digits of precision. A 4 byte word can represent numbers with a range of about +- 2,000,000,000, about 9 decimal digits of precision. The most usual "scientific" floating point has about 15 decimal digits of precision, and an automatic method of sliding the decimal point around in a handy fashion.

All of the above have a handy way of detecting "overflow" or "underflow" where the correct answer after an arithmetic operation cannot fit into the defined number range.

The

Babbage Difference Engine #2represents numbers as 31 decimal digits, greater precision than "natural" in modern machines. But even 31 decimal digits precision cannot exactly represent the fraction one/three or 1/3 that we can approximate as 0.333333333333333333... .Let us assume the decimal point to the right of a computer number, and add

000000000001.

plus000000000003.

generating000000000004.

No problem, however if we do the division, and keep the decimal point on the right,000000000001.

divided by000000000003.

generating000000000000.

clearly incorrect - the fractional part disappeared off to the right and is lost.HOWEVER, if we multiplied the factors by say 1,000,000,000 we get

001.000000000

divided by003.000000000

generating000.333333333

Clearly a better approximation :-)This

multiplying of variables by a constantto fit the machine is one usage ofscalingand is used a great deal in Babbage and in modern machines.A major advantage of using floating point arithmetic and hardware, included in most machines in our current era of almost free logic, is to reduce the need for, and human errors in estimating, scaling.

## 12. Representing negative numbers, (10's complement)

Human notation for negative number is usually a minus sign followed by the absolute value the number. This is very convenient for humans, "no problem". :-) A human eye can scan the number field and easily detect the sign of the number, and based on the sign of the number, "do the right thing".To do the equivalent with a machine takes extra hardware and (possibly more important) time. By custom hardware representation of negative numbers is made such that you can add or subtract numbers (quickly & cheaply) and not even know or care about the sign.

In most machines, a negative number is represented as though you took the absolute value of the negative number and subtracted it from the largest value the machine can represent, and then adding the smallest positive number the machine can represent. This is called 10s (or 2s) complement.

An example is a little easier to see, we have say -3. The absolute value is

003.000000000

subtract the above from999.999999999

generating996.999999999

adding000.000000001

generates997.000000000

with the high digit not zero indicating that the number is negativeAdding +5 to the -3 above yields

005.000000000

plus997.000000000

generates002.000000000

and the high digit is zero indicating positive.10s (or 2s) complement arithmetic is a little tough for humans, but fast and simple for machines - so 10s (or 2s) complement inside, converted to the other form out side the machine for us slow humans.

Unfortunately, to set up a Babbage machine, you must set the rotors for 10s complement arithmetic - the negative numbers "look funny".

HOWEVER, the print subroutines I provide in TOOLS output 10s complement values.

## 13. The parallel "pipeline" nature of the Babbage machine and:

## ----- Correcting for printing after the adds

----- Correcting for the odd-even phasing of adds

The goal of this section is to make the output of the Babbage machine identical with the ouputs of section 6. "A simple example of the Method of Finite Differences" above.

----- Correcting for printing after the adds

Dick Guertin noted that data for the printer is taken from DC0 during the 2nd half cycle, after an add into DC0, not the 1st half cycle. Thanks DickFortunately, there is a simple solution for the above situation. This BASIC code shows how to back up the input one step so that the expected results will be output.

for I = 6 to 0 step -1 DC(i) = DC(i) - DC(i+1) next i

----- Correcting for the odd-even phasing of adds

To save cranking, and to speed operations, the Babbage machine is arranged for parallel "pipeline" operation.The names of the difference columns are taken from the conventions in section 6 above.

The

"Technical Description Manual", page 66 uses different Difference Column numbers, which we will ignore to reduce confusion.The initialization must be adjusted to allow for this even-odd, parallel, "pipeline" operation. In the notation below,

DC0+=DC1means add DC1 to DC0Normal serial operation(Babbage outputs to printer at the end) DC0+=DC1 DC1+=DC2 DC2+=DC3 DC3+=DC4 DC4+=DC5 DC5+=DC6 DC6+=DC7 Printer<- DC0 (copy DC0 to the printer)

The above could easily take 8 machine cycles HOWEVER, as discussed in the "Technical Description Manual" Babbage does as much as possible in parallel - called "pipelining"Babbage pipeline - First half cycleDC0+=DC1 DC2+=DC3 DC4+=DC5 DC6+=DC7Babbage pipeline - Second half cycleDC1+=DC2 DC3+=DC4 DC5+=DC6 Printer<- DC0 (copy DC0 to the printer)

Unfortunately, the simple difference set up used in the Normal serial operation does not work for the Babbage parallel operation - expected operations are done out of sequence, and the results are just wrong.

To correct the parameters set into the Babbage machine, you MUST Subtract various differences. Perform the following

.in exactly this orderUsing the above recipe, inserting these values into the machine (see section below), and adding paper and ink, thefor I = 6 to 0 step -1 DC(i) = DC(i) - DC(i+1) next i DC6=DC6-DC7 ' back off iteration 3, 1st half cycle DC5=DC5-DC6 ' back off iteration 2, 2nd half cycle DC6=DC6-DC7 ' , 1st half cycle DC4=DC4-DC5 DC3=DC3-DC4 ' back off iteration 1, 2nd half cycle DC5=DC5-DC6 DC2=DC2-DC3 ' , 1st half cycle DC4=DC4-DC5 DC6=DC6-DC7Babbage Difference Engine #2is ready to compute and print your polynomial.A code example and output is here.

## 14. Setting up the engine, Babbage Difference Engine #2 Instruction Manual

## 15. Some computed example results

- A 31 digit difference engine in UBASIC
- sin 1-23 degrees, 1 minute increments, polynomial steps 3 degrees
- (max error about 1.5*10
^{-10}%- sin 0-45 degrees, 1 minute increments, polynomial steps 6 degrees
- (max error about 3.7*10
^{-7}%- A 31 digit difference engine in MATHEMATICA
- code - NOT pipelined, see below
- output, 129 K bytes
- A much improved version of this web page by Scott Centoni
- Documentation of Method of Differences, series, errors in Mathmatica .nb format - 409 K bytes
- Documentation of Method of Differences, series, errors - in .pdf format - 563 K bytes
- tabular result and error of sin(0) to sin(22.5 degrees) step 1 minute - in .txt format - 70 K bytes
- from Tim Robinson web site - a Mathematica notebook - .nb [of logarithms 1.0 to 1.6] with the setup and its derivation. This is set up so as to reproduce the first twelve pages of Babbage's own 7 figure log table from a single setup (about 6000 values). .pdf version

## 16. A good machine test case y = K * x

^{7}through the range -70 < x < 70

UBASIC code and results- serial operation, non Babbage

parallel operation, Babbage

MATHEMATICA code and results

## 17.

a JavaScript non-pipelined emulation

## 18. The Computing Errors

The following discusses errors caused by the nature of the numbers representable in a digital (not analog, like slide rule) computing mechanism.

- limited length (number of digits) and "overflow" - fractions (and other non integer numbers) might not be represented exactly

(I think) that each itteration will cause a maximum error of 1 count in the low order digit, 10^010 itterations will cause a maximum error of 1 count in the next higher digit, 10^1 100 itterations will cause a maximum error of 1 count in the next higher digit, 10^2

and so forth, a creeping increasing error.

on an 8 axis machine such as the Babbage Difference Engine, the error is 8 times greater -hence the large number of digits used per axis.

## 19. Babbage Overflow-Underflow detection

There was no overflow or underflow detection. [ Underflow here defined as exceeding the negative range of the machine. ]The automatic sensing/warning of these conditions had to wait for possibly the Manchester Baby - about a hundred and fifty years later.

Even Atanosoff's ABC in 1940 did not include overflow checking.

## 20. Formatting the output of the Babbage printer section.

From Doron Swade September 2002, in-charge of building the Babbage Difference Engine - British Museum of Science and Industry

Negative numbers are dealt with as complements as you correctly surmise. There is no provision for normalising for conventional signing before printing. It is most likely that the stereotypes would have been left intact and the results printed in complement form with appropriate instructions in a customary Preface on use of the tables and the techniques for inverting the complements. There is no provision in the printer apparatus on DE2 to print signs and I cannot immediately see how the machine could be modified to do so.>The printing section does not seem to discuss >whether negative numbers are printed in > a) 10s complement > b) signed absolute > >This is certainly not a problem in most trig and log generation, > but is to me an interesting general question. > >Could you answer this matter of printing sign representation.With the printer and stereotyping apparatus coupled to the calculating section as depicted in Babbage's drawings all the apparatus can do is produce an inscriptional and impressed record of the last column of results in the calculating section. This 31-digit result (printed to 30 places) is the outcome of repeated addition using finite differences. It is strictly decimal i.e. the carry mechanism works strictly modulo 10. To convert to sexagessimal notation would require a completely new set of 248 figure wheels with carry fingers differently positioned. The timing cycle would also be affected. In short, the answer is, 'no': the machine is strictly decimal and converting to an alternative modulus would require redesign and rebuild. (In contrast, the Scheutz Difference Engine was easily modified for sexagessimal use).>Is there an option to print say > a) degrees, decimal fraction minutes > with correct roll over from 59.xxx minutes ? > b) degrees, minutes, seconds ? > c) some other conventional representation, > other than say GradsThe printing and stereotyping apparatus is now complete and working. There were many modifications necessary and it was a more difficult job than the calculating section. It is now likely that I will be spending time next year completing the documentation i.e. updating and completing the Technical Description you have to include the printer/stereotyping build.

## 21. Babbage related web sites

- A great description of the principles of operation of the Babbage Difference Engine using animated .gif images, other good info.
- My trip report on visit to Babbage Engine
- The MacTutor History of Mathematics archive - Babbage recommended -
- Nanotechnology
- Some links to information on Charles Babbage
- Robinson's Difference Engine #1 from standard Meccano parts.
- Robinson's Difference Engine #2 from standard Meccano parts.
- IEEE Annals of the History of Computing Unfortunately, you have to be (or know) an IEEE member to access the site. from Dan Lasley

## 22.Babbage books on-line

other sites

- Babbage, Charles: The Economy of Machinery and Manufactures
- Babbage, Charles: Reflections on the Decline of Science in England, and on Some of Its Causes

## 23. A few of the books on Babbage

- The Difference Engine: Charles Babbage and the Quest to Build the First Computer by Doron Swade
- Charles Babbage and His Calculating Engines (Paperback)
- Glory and Failure: The Difference Engines of Johann Müller, Charles Babbage, and Georg and Edvard Sheutz (History of Computing) by Michael Lindgren (I haven't examined this book yet.)

## 24. Serious comments about Babbage's design.

## 25. SetUp Script for setting coefficients in the Babbage Engine #2, Second Edition

The following is a check off script for the Babbage Engine currently in the Computer History Museum. It is destined for Nathan Myhrvold's living room? to go with the T-Rex and other trinkets in mid 2009. The author is probably Tim Robinson. It was in use mid June 2008.

Babbage Engine Calculation Setup Checklist- Rev 4This his checklist must be completed anytime the setup procedure is carried out on the Engine. All steps must be checked off by the checker as the step is performed. Under no circumstances may steps be changed or omitted. Doing so could result in physical damage to the Engine. Refer to the User Manual for a full description of each step.

Special setup for printer test.

Step Check Description 1 Cycle the Engine to the zero (Full Cycle) point 2 Lift the release lever to disengage the sector drive (jiggle lifting levers if necessary) 3 Lift one of the lifting levers 4 Lock in raised position 5 Raise and lock the second lifting lever 6 Advance the Engine to the 10-unit point 7 Engage the four manual odd setting locks 8 Check all four odd setting locks are secured engaged, all even locks unengaged 9 Advance the Engine to the 35-unit point 10 Engage the four manual even setting locks 11 Physically check all 210 carry levers are reset. If any found warned or carried STOP IMMEDIATFLY and seek assistance. (Disabled levers are OK) 12 Complete the cycle by advancing the Engine to zero. If setting all zeros, go to step 26 13 Advance the Engine to the 20-unit (Set Odd) point 14 Disengage the manual lock on the 7th difference column 15 Determine which wheels are to be used for the main calculation and which for index 16 Enter the 7th difference value on the figure wheels 17 Reengage the manual lock 18 Repeat steps 14 to 17 for the other three odd difference columns, remembering the index on D1 19 Advance the Engine to the 45 unit (Set Even) point 20 Disengage the manual lock on the 6th difference column 21 Enter the 6th difference value on the figure wheels 22 Reengage the manual lock 23 Repeat steps 20 to 22 for the other two even difference columns 24 Repeat steps 20 to 22 for the results column 25 Advance the Engine to the zero point 26 Lower the release lever and reengage the sector drive 27 Retract the locking plungers and lower the sector levers 28 Disengage all eight manual locks and secure in the disengaged position 29 Check all eight manual locks are disengaged 30 If required, rewind the stereotype trays

If you have comments or suggestions, Send e-mail to Ed Thelen

Doran Swade talk Stanford

Last update Jan, 2016

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