### “Notes on Note G”

[ **Part 1** | Part 2 | Part 3 ]

[**Note:** WordPress sometimes has problems including program code in posts, so I’ve made the code used in these articles available on GitHub – [link]]

I recently watched the BBC4 documentary *Calculating Ada: The Countess of Computing* (Hannah Fry’s 10-part radio series *Computing Britain* is also an interesting listen)[¹]. And although I know that Ada Lovelace is credited with publishing the first computer program, I hadn’t ever seen the form that this program took before. It’s not hard to track down – on Ada’s *Wikipedia* page there is an image of the diagram ([image] – “Diagram for the computation by the Engine of the Numbers of Bernoulli”).

The diagram itself was part of a note (*Note G*) that was attached to a translation of a French transcript (by an Italian military engineer named Luigi Frederico Menabrea – who would later become the Italian Prime Minister) of a seminar given by Charles Babbage at the University of Turin in 1840. It was the first and only time Babbage gave a public lecture on the *Analytical Engine*. Ada was commissioned to translate the paper into English. After nine months of working she published her translation, along with a set of notes (which were more than twice as long as the original paper). This was the first (and, for a hundred years, the only) text on computer programming. It is in one of the notes that the diagram that is considered to be the worlds first published computer program appeared. (The full text of Ada’s paper is available here [link], along with a large version of the diagram [link]).

Here it is:

“Diagram for the computation by the Engine of the Numbers of Bernoulli”

It’s not what we think of now as a computer program – after all it pre-dates computer programming languages. Instead it essentially the description of a sequence of operations performed by the *Analytical Engine*. So it is more like a trace of the program execution than the source code of the program itself, and it only includes the actual arithmetic operations that the engine performs. The control flow of the execution is dealt with in the attached text.

Of course the Analytical Engine was never completed, so the actual form of the instructions is not completely clear. However, I’m not trying to construct copy of the punched cards necessary to run Ada’s program. I’m attempting to transliterate her algorithm into a modern programming language in order to understand it, and the Analytical Engine, better.

Like Ada I have found that my own notes have become somewhat longer than I anticipated.

First I’ll explain what the program does. The purpose of the program is to compute *Bernoulli Numbers*. Ada’s program is concerned with calculating the odd-numbered Bernoulli Numbers[²], and is based on the following equation (given in Note G) for calculating B[2*n* − 1], given the already computed values of B[1], B[3], …, B[2*n* − 3]:

A[0] + A[1]B[1] + A[3]B[3] + … + B[2*n* − 1] = 0

where the values of the A[] coefficients are defined as follows:

A[0] = (−1/2) (2*n* − 1)/(2*n* + 1)

A[1] = 2*n*/2 = *n*

A[3] = 2*n*(2*n* − 1)(2*n* − 2) / (2.3.4)

A[5] = 2*n*(2*n* − 1)(2*n* − 2)(2*n* − 3)(2*n* − 4) / (2.3.4.5.6)

…

For *k* > 1, each A[*k*] is derived from A[*k* − 2] by multiplying the next two decreasing terms into the numerator, and the next two increasing terms into the denominator. (Note that for each value of *n* the sequence of A[] coefficients are different. We might more properly think of them as A[*n*, *k*]).

Ada’s program works by starting at *n*=1, the equation for deriving B[1] is:

A[0] + B[1] = 0

and:

A[0] = (−1/2) (2*n* − 1)/(2*n* + 1) = −1/6

So:

B[1] = 1/6

We then consider *n*=2, using the previously computed value for B[1] we have:

A[0] + A[1]B[1] + B[3] = 0

and:

A[0] = (−1/2) (2*n* − 1)/(2*n* + 1) = −3/10

A[1] = 2

So:

B[3] = −1/30

We then consider *n*=3, and use the previously computed value for B[1] and B[3] to calculate B[5]. And the procedure can be repeated as many times as we like (or until we run out of space to store the results – the proposed design for the Analytical Engine would have had room for 1,000 variables, each containing 50 decimal digits). Each time we use the results from all the previous runs to calculate the next number in the sequence.

To explain the implementation of the algorithm here is my own implementation of it using modern *Python* idioms. This program is constructed using the same basic structure as Ada’s program.

from fractions import Fraction
# my implementation of the algorithm
def bernoulli():
# results
Bs = list()
# start at n=1
n = 1
# calculate the sequence
while True:
# result = A0
r = -Fraction(2 * n - 1, 2 * n + 1) / 2
# A1 = n
A = n
# for each B[k] already determined calculate the corresponding A[k]
for (i, B) in enumerate(Bs):
if i > 0:
# multiply in the 2 additional terms
j = 2 * i - 1
A *= Fraction(2 * n - j, 2 + j)
j += 1
A *= Fraction(2 * n - j, 2 + j)
# add A[k] * B[k] into the result
r += A * B
# the computed bernoulli number is -r
B = -r
# return the number
yield B
# add it to the result list
Bs.append(B)
# increase n
n += 1
# run N iterations of the algorithm
N = 10
# allow N to be specified as a command line argument
import sys
if len(sys.argv) > 1:
N = int(sys.argv[1])
k = 1 # ada's numbering (normal numbering is k + 1)
K = 2 * N + 1 # max k
for r in bernoulli():
print("B[{k}] = {r}".format(k=k, r=r))
k += 2
if k == K: break

**[Program 1 - ada1.py]**

Before we examine a more direct implementation of Ada’s diagram, first some notes on my own implementation:

- (Line 1) I’m using Python’s
`Fraction` library to generate the Bernoulli sequence. This makes it easy to check the computation against existing lists (which give them in rational form), and provides exact answers. The Analytical Engine would have computed the numbers as fixed point decimal fractions. (So small errors would accumulate in the final decimal places, and as these results are fed back into further computations the errors would compound. It would be best to use more decimal places that required in the results so that suitably accurate truncated results can be generated).
- (Line 4) The
`bernoulli()` function will generate successive Bernoulli numbers. I’ve implemented it as Python co-routine, so results are returned with the `yield` operator as they are determined. It’s up to the calling code to consume as many results as it requires. The diagram given by Ada describes the generation of a single Bernoulli number. And while her program itself sets up some of the data required for the next run, it would also require some additional modification of the cards before it is run again to compute the next number in the sequence.
- (Line 18) I’ve simplified the calculation of A1 from (2
*n*/2) to just (*n*).
- (Line 41) We can specify how many numbers in the sequence we want the program to compute (the default is 10). They are printed out according to Ada’s numbering scheme.

The program is certainly non-trivial. It involves two loops (one nested in the other) and conditional execution, as well as the storing of the computed results to use in subsequent calculations.

The diagram published by Ada shows the computation of B[7], the fourth number in the sequence. The table consists of the following columns:

1. Line number.

2. Operation.

3. Operands.

4. Assignment of results.

5-6. Comments.

7-9. Parameters.

10-19. Variables.

20-23. Results.

Only columns 2, 3 and 4 are instructions to the computer, the rest are essentially supporting documentation to explain the program to the reader.

There are a few apparent errors (the first programming bugs) in the originally published diagram:

- At line 4 the operation is given as v5 ÷ v4, although the comments and text make it clear that this should be v4 ÷ v5.
- At line 23 the formula for A3 has two 3’s in the denominator, instead of a 3 and a 4 (as in the previous line). This is just an error in the comment and would not affect execution of the program.
- At line 25 it looks like v6 and v7 are reset to their initial values (of zero). In fact it’s v7 and v13 that need to be reset for the algorithm to work correctly. In the accompanying text Ada refers to resetting v6, v7,
*and* v13 to zero (although it is not necessary to zero v6, as the first thing the program does (operation 1) is to assign 2*n* to v4, v5, and v6).

These are all “typos” rather than “thinkos”, and may not even have been present in Ada’s original notes.

Here is my Python program with the `bernoulli()` function replaced with a modified version that reflects a more direct transliteration of Ada’s description:

from collections import defaultdict
from fractions import Fraction
# ada lovelace's algorithm in Python
def bernoulli():
# initially all varibles in the store are at 0
v = defaultdict(int)
# program start
# initialise the variables
v[1] = Fraction(1) # constant
v[2] = Fraction(2) # constant
v[3] = Fraction(1) # n = 1
# outer loop (compute B[2n - 1])
while True:
# pseudo-block to permit "break"
while True:
# 0: set index register
i = 1
# 1: v4 = v5 = v6 = 2n
v[4] = v[5] = v[6] = v[2] * v[3]
# 2: v4 = 2n - 1
v[4] = v[4] - v[1]
# 3: v5 = 2n + 1
v[5] = v[5] + v[1]
# 4: v11 = (2n - 1)/(2n + 1) (the diagram seems to say v[5] / v[4]) [FIX]
v[11] = v[4] / v[5]
# 5: v11 = (1/2) (2n - 1)/(2n + 1)
v[11] = v[11] / v[2]
# 6: v13 = -(1/2) (2n - 1)/(2n + 1) = A0
v[13] = v[13] - v[11]
# 7: v10 = n - 1
v[10] = v[3] - v[1]
# branch if zero to operation 24
if v[10] == 0: break
# 8: v7 = 2
v[7] = v[2] + v[7]
# 9: v11 = (2n)/2 = A1 [why not just set v[11] = v[3] instead of 8 & 9 ?]
v[11] = v[6] / v[7]
# 10: v12 = A1 * B1
v[12] = v[20 + i] * v[11]
i += 1
# 11: v13 = A0 + A1 * B1
v[13] = v[12] + v[13]
# 12: v10 = n - 2
v[10] = v[10] - v[1]
# branch if zero to operation 24
if v[10] == 0: break
# for each computed result, B = B3 [1], B5 [2], ...
while True:
# 13: v6 = 2n - 1 [1], 2n - 3 [2], ...
v[6] = v[6] - v[1]
# 14: v7 = 3 [1], 5 [2], ...
v[7] = v[1] + v[7]
# 15: v8 = (2n - 1)/3 [1], (2n - 3)/5 [2], ...
v[8] = v[6] / v[7]
# 16: v11 = (2n)/2 * (2n - 1)/3 [1], (2n)/2 * (2n - 1)/3 * (2n - 3)/5 [2], ...
v[11] = v[8] * v[11]
# 17: v6 = 2n - 2 [1], 2n - 4 [2], ...
v[6] = v[6] - v[1]
# 18: v7 = 4 [1], 6 [2], ...
v[7] = v[1] + v[7]
# 19: v9 = (2n - 2)/4 [1], (2n - 4)/6
v[9] = v[6] / v[7]
# 20: v11 = (2n)/2 * (2n - 1)/3 * (2n - 2)/4 = A3 [1], (2n)/2 * (2n - 1)/3 * (2n - 2)/4 * (2n - 3)/5 * (2n - 4)/6 = A5 [2], ...
v[11] = v[9] * v[11]
# 21: v12 = A3 * B3 [1], A5 * B5 [2], ...
v[12] = v[20 + i] * v[11]
i += 1
# 22: v13 = A0 + A1 * B1 + A3 * B3 [1], A0 + A1 * B1 + A3 * B3 + A5 * B5 [2], ...
v[13] = v[12] + v[13]
# 23: v10 = n - 3 [1], n - 4 [2], ...
v[10] = v[10] - v[1]
# branch if non-zero to operation 13
if v[10] == 0: break
# terminate the pseudo-block
break
# 24: result (-v13) is copied into the results
r = v[20 + i] = v[20 + i] - v[13]
# the result could be printed here
# we pass it back to the calling routine
yield r
# 25: increase n, and reset working variables
v[3] = v[1] + v[3]
v[7] = v[13] = 0
# branch to operation 0
# run N iterations of the algorithm
N = 10
# allow N to be specified as a command line argument
import sys
if len(sys.argv) > 1:
N = int(sys.argv[1])
k = 1 # ada's numbering (normal numbering is k + 1)
K = 2 * N + 1 # max k
for r in bernoulli():
print("B[{k}] = {r}".format(k=k, r=r))
k += 2
if k == K: break

**[Program 2 - ada2.py]**

Ada’s diagram shows how the program would run when *n*=4, and it not a direct indication of the punched cards that would be fed to Analytical Engine. Of course as the Analytical Engine was never completed it’s actual instruction set is not something we can know for certain. But the way it is described as working is as follows:

- An operation card indicates the operation that the engine is to perform. (One of +, −, ×, ÷).
- A variable card indicates which variable is transferred from the store (memory) to the first Ingress Axis of the mill (CPU). This provides the first operand.
- A second variable card indicates which variable is transferred from the store to the second Ingress Axis if the mill. This provides the second operand.
- The operation is then performed, the result appears in the Egress Axis of the mill.
- A variable card indicates which variable in the store the contents of the Egress Axis are transferred to.

At stage 5 there may be multiple variable cards instructing that the result is stored in several variables, as we see in the first operation of Ada’s table. If we were to write an assembly language version of this operation it would look something like this:

`# operation 1`

MUL

LOAD v[2]

LOAD v[3]

STORE v[4]

STORE v[5]

STORE v[6]

And each of these opcodes corresponds to a card that acts as instructions to the Analytical Engine.

Ada’s accompanying text describes the need for some common programming idioms that we take for granted in modern programming languages, but are not present in the diagram (which just shows the arithmetic operations performed).

First there is the need for conditional execution. She mentions that when computing B[1] operation 6 would complete the calculation. In my program at line 47 I exit the block when v10 becomes 0. Similarly, as Ada says, operation 12 works in a similar way. Again in my program I conditionally exit the block if v10 becomes 0.

Secondly there is the need for repeated execution of a sequence of instructions (looping). This is dealt with extensively in the notes where the repeated execution of operations 13-23 is described.

Both these could be achieved with a conditional branch instruction. Although the mechanism for branching is not explicitly mentioned in the paper it is clear that the Analytical Engine implements such a function. For example, if the Analytical Engine set a flag when a result was 0 (as microprocessors often do), then there could be an instruction to skip forward or backwards a certain number of program steps and resume operation from there. The instructions were to be provided on a punched cards, so this would require the card reader to skip forward or rewind a certain number of cards.

(In the Fourmilab simulation of the Analytical Engine a “run-up lever” is used. This is set by an operation if the sign of the result of the operation being different from the sign of the first operand (or if an overflow occurs). A conditional branch can then be made based on the state of the lever. So, we could test if a variable was zero by subtracting 1 from it. The only way the sign will change is if the variable initially held the value of 0 (so the result is −1), and the lever will be set. As Ada doesn’t show these operations in her table I’m going to simplify by referring to a “branch if zero” instruction. In a similar way branching on other conditions can be implemented).

This would allow us to place an instruction after operation 7 and operation 12 of “branch if zero to operation 24”. The inner loop would be implemented by placing an instruction after operation 23 of “branch if non-zero to operation 13”. The outer loop would be implemented by placing an instruction after operation 25 of “branch always to operation 1” (or more likely to branch if *n* was less than a predefined constant value, so that a Bernoulli sequence of the specified length is generated).

The text also hints at the notion of *indirection*, where one variable (or register) can tell us which of a number of other variables we need to access. This is necessary as each time we iterate through the operations 13-23, we have computed the A[] coefficient after executing operation 20, but then in operation 21 we multiply this the corresponding number already computed in the B[] sequence. The first time round we would need B[3], stored in variable v22 (as specified in the operands for operation 21 in the diagram), but the next time round the loop we would need the B[5] value stored in v23, and so forth. Likewise when the final result (B[7]) is calculated it is stored by operation 24 into v24 and *n* is increased. The next time we get to operation 24 the final result will be B[9] and it will be stored into v25. Although the diagram itself does not indicate how the instructions of operations 21 and 24 are modified as the program is executed it is clear that it is an issue that Ada has considered.

From the way Ada describes the program it seems that way this would happen is that for each value of *n* the program is run once. Then it must be set up slightly differently for next run. Ada notes that exactly the same sequence of operation cards can be reused, but the variable cards must be modified slightly on each iteration to achieve the required results.

One way the Analytical Engine could do indirection would be to allow the index of a variable to be specified in another variable, so instead of a “LOAD v[12]” (load the Ingress Axis from variable 12 in the store), we would have something like “LOADI v[12]” (load the Ingress Axis from the variable in the store whose index is stored in v12). So if v[12] = 21, “LOADI v[12]” would be the same as “LOAD v[21]”.

But variables in the store hold 50 decimal digits and indices into the store are only 3 decimal digits, so it is a bit odd. Instead we might consider that the Analytical Engine could be equipped with a few special purpose three decimal digit index registers, these could then be used to provide an additional offset in a load instruction.

For the purposes of Ada’s program it would be necessary to only have a single additional index register. We could imagine a that “LOADI v[20]” would correspond to “LOAD v[20 + *i*]” where *i* represents the value in the index register. So if *i* = 1 the instruction “LOADI v[20]” would be the same as “LOAD v[21]”. And there would be a corresponding “STOREI” instruction. To increase the offset we would need an “INC” instruction, which would increment the number in the index register by 1. This would be sufficient to implement Ada’s program.

But perhaps this approach is too influenced by modern computing. A different approach would be to allow the LOAD and STORE instructions to get a variable from an alternate stack of cards. “LOAD alt” and “STORE alt”. That way the alternate stack of cards could be filled with the addresses of v21 to (say) v30. And in operation 10 the address that the first LOAD card uses isn’t punched on the card itself, instead it is read from the alternate stack (it’s the first card in the stack so it would be v21), and the stack moves to the next card. When we reach operation 21 we would again use a “LOAD alt” instruction to get the index of the variable from the alternate stack. The first time round the loop it will be v22, then v23 and so on. When the loop ends, and we have processed all the previously calculated results the next card in the alternate stack will be the correct address to store the newly computed number. So operation 24 uses “STORE alt” and the number is stored.

Finally at the end of the diagram as well as looping back to the beginning we also rewind the alternate card deck and we are ready to go again without needing to alter any of the cards. After 10 iterations the first 10 Bernoulli numbers will have been stored in the variables referenced by the alternate stack, and the program can halt.

To produce the transliteration of Ada’s program that can run without intervention I have adopted the addition of an *index register* to allow indirection, although, as far as I know, this is not historically accurate.

Finally Ada notes that the process described in her diagram produces results of the correct magnitude, but not the correct sign. If there is no simple way to change the sign of a value, this can easily be remedied by subtracting the result (in v[13]) from the variable that the result is to be stored in (as it will initially be 0), and then storing it back into the result variable. Consequently I’ve adjusted operation 24 from “v[24] = v[13] + v[24]” to be “v[24] = v[24] − v[13]”, the sign is then correctly set for value to be reused by the algorithm.

Writing an equivalent pseudo-assembly language version of my transliteration would look something like this:

# initialise the variables
SET v[1] 1 # constant 1
SET v[2] 2 # constant 2
SET v[3] 1 # n = 1
# operation 0: initialise the index register
SET i 1
# operation 1
MUL
LOAD v[2]
LOAD v[3]
STORE v[4]
STORE v[5]
STORE v[6]
# operation 2
SUB
LOAD v[4]
LOAD v[1]
STORE v[4]
# operation 3
ADD
LOAD v[5]
LOAD v[1]
STORE v[5]
# operation 4
DIV
LOAD v[4]
LOAD v[5]
STORE v[11]
# operation 5
DIV
LOAD v[11]
LOAD v[2]
STORE v[11]
# operation 6
SUB
LOAD v[13]
LOAD v[11]
STORE v[13]
# operation 7
SUB
LOAD v[3]
LOAD v[1]
STORE v[10]
# branch if zero
BRANCH ZERO (operation 24)
# operation 8
ADD
LOAD v[2]
LOAD v[7]
STORE v[7]
# operation 9
DIV
LOAD v[6]
LOAD v[7]
STORE v[11]
# operation 10
MUL
LOAD v[20 + i] # indirection
LOAD v[11]
STORE v[12]
INC i # index register is incremented
# operation 11
ADD
LOAD v[12]
LOAD v[13]
STORE v[13]
# operation 12
SUB
LOAD v[10]
LOAD v[1]
STORE v[10]
# branch if zero
BRANCH ZERO (operation 24)
# operation 13
SUB
LOAD v[6]
LOAD v[1]
STORE v[6]
# operation 14
ADD
LOAD v[1]
LOAD v[7]
STORE v[7]
# operation 15
DIV
LOAD v[6]
LOAD v[7]
STORE v[8]
# operation 16
MUL
LOAD v[8]
LOAD v[11]
STORE v[11]
# operation 17
SUB
LOAD v[6]
LOAD v[1]
STORE v[6]
# operation 18
ADD
LOAD v[1]
LOAD v[7]
STORE v[7]
# operation 19
DIV
LOAD v[6]
LOAD v[7]
STORE v[9]
# operation 20
MUL
LOAD v[9]
LOAD v[11]
STORE v[11]
# operation 21
MUL
LOAD v[20 + i] # indirection
LOAD v[11]
STORE v[12]
INC i # increment index register
# operation 22
ADD
LOAD v[12]
LOAD v[13]
STORE v[13]
# operation 23
SUB
LOAD v[10]
LOAD v[1]
STORE v[10]
# branch if non-zero
BRANCH NON-ZERO (operation 13)
# operation 24
SUB
LOAD v[20 + i] # indirection
LOAD v[13]
STORE v[20 + i] # indirection
# the result can be output at this point
PRINT
# operation 25
ADD
LOAD v[1]
LOAD v[3]
STORE v[3]
# reset working variables
SET v[7] 0
SET v[13] 0
# repeat
# (or if v[3] < N)
BRANCH ALWAYS (operation 0)
HALT

Each program line would correspond to a punched card that would be fed to the Analytical Engine, so this is the closest we can get to what the original stack of punched cards, and hence the first program, would have looked like.

**Note:** Whilst writing up this article I have found a lot more information out about the Analytical Engine than when I started, and there is much more for me to go through. I intend to investigate the Analytical Engine further, and I will correct any glaring errors written above as I become more familiar with the subject. Please let me know of any problems you find and I will endeavour to sort them out.

— J.M.R.

Here are some useful resources that I have found about Ada and the Analytical Engine:

[¹] Apologies if you can’t access these programmes. BBC iPlayer is only available in the UK, and programmes are only available for a limited time.

[²] Ada’s indices for the Bernoulli Numbers are one less than the indices now generally used.

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It occurred to me that it is possible to write Ada’s program without the need for full indirection. The input data for a single run of the algorithm are the Bernoulli Numbers already calculated. These data are accessed in order, sequentially and each item only once. A stack of cards is sequentially accessed memory. So if an extra stack of “input data” cards were provided composed of the already calculated Bernoulli numbers, then they would provide the correct values to the program in the right order as it ran. When they were all processed the Engine would print out the next Bernoulli number on a card, which would be added to the input data stack for the next run. All that would be required would be a “LOAD input” instruction to load the register in the mill from the current input data card. After the card was read the next input data card would be available for the next “LOAD input” instruction.

Here’s the Python transliteration:

And the corresponding assembly language for the Analytical Engine:

We use “LOAD input” to access cards from the input data stream in operation 10 and operation 21. And we no longer store the result in the store in operation 24.

On each run the Engine will print a card with next Bernoulli number in the sequence on it. The control and input cards are reset and the newly printed output card is added to the end of the input stack. The Engine is then ready to run the next iteration of the algorithm with exactly the same control cards as before.

Theoretically this method would allow you to calculate as many Bernoulli numbers as you had cards to print them out onto, regardless of the number of variables in the store (as long as there are at least 14). But if the 1,000 variable Analytical Engine had ever been built you would probably run out of accuracy, or time, before you exhausted the storage available.

Clever lass, our Ada. What a pity Babbage’s plan was so overambitious and he didn’t start small and work up, like Konrad Zuse. Few people have heard of him, but he was ahead of the Americans with their ENIAC. His first model was constructed in his bedroom at his parents’ home, painstakingly cutting and filing pieces of metal to form latches. He saw from the start that binary is the way to go. It’s perhaps an interesting reflection on different national characteristics that during the war the Germans were using his computer (a later model with electromechanical relays) to help design aircraft wing sections; it didn’t occur to them to try cracking ciphers that way!

What were they intending to do with all those Bernouilli numbers? My 50-year-old copy of

Mathematical Methods for Science Studentsgives an utterly unpractical formula but makes no mention of what use they might be. I believe the terms of series expansions for tan and cot can be expressed using Bernouilli numbers. Do they have other applications?