This post is a continuation of analytical programming about the programming model used to program Babbage’s analytical engine. In the previous post I talked about two of the ways programming was used, the table format used to describe programs and the microcode format used internally within the engine. The third program format which is what this post is all about is the one used to feed programs to the engine. (Side note: if this looks familiar it’s because it used to be the second half of that post which you may already have read; I’ve just pulled it out into a separate post)

In some regards the card format is similar to the bytecode formats we know today. That’s one of the reasons I find it especially interesting, because it’s so relatively familiar and relatable for a modern programmer. However, in one regard is is very different from any modern programming model. Babbage felt very strongly that there was a fundamental difference between specifying which operation to perform and which variables to perform the operation on. Modern programming and mathematics has generally moved in the opposite direction, seeing operations and function as values of a different type that can nonetheless be abstracted over much the same way that numeric values can. Babbage would have disagreed strongly with this.

Because of this view a punched card program is divided into two distincts parts: one set of cards containing all the operations to perform and another set specifying the variables to perform them on. For instance, a program that multiplied three numbers, V_{1}, V_{2}, and V_{3} and stored the result in V_{4}, similar the table program from the previous post but using multiplication instead of addition, would be split from the format where operations and variables are together,

V_{1} × V_{2} → V_{4}

V_{3} × V_{4} → V_{4}

and into two separate sets of cards, one giving the operations

and one giving the column indices

In the following I’ll use these colored boxes to represent punched cards. The real punched cards didn’t look anything like this as you’ve seen from the picture at the beginning. For the variable cards I’m also omitting the part that indicates whether reads are clearing or restoring, that would have been there in the original design.

Besides specifying which operation to perform the operation cards all specified how many times the operation should be repeated; using this the operations program above could be specified even more succinctly as

To execute an instruction the engine would first read the operations card and store the number of repeats in a register, the *Operation Card Counting Apparatus*. Then it would repeatedly perform the operation, decrementing the O.C.C.A for each and reading the next operation card when it reached zero. For each operation it would read as many variable cards as were appropriate for that operation.

In the following I’ll describe each of the instructions understood by the engine. It’s only about a dozen. The number of variable cards for an operation card depends on the repeat count but I’ll show the variable cards that go with a single repetition. I’ll separate the cards representing input and output with an arrow, just to make it easier to read. It’s purely a visual aid though, in practice you had to keep track what the cards meant yourself.

## Basic Operations

The most straightforward arithmetic instructions, instruction format wise, are the multiplicative ones. We have multiplication of two columns, V_{i} and V_{j}:

*n*

_{i}

_{j}

_{h}

_{m}

_{t}

Wait, you might say, didn’t you just show multiplication using just three variable cards? Yes, and that was a simplification. In reality multiplying two 50-digit numbers can give you up to 100 digits and as a general principle the analytical wouldn’t discard digits. Instead it would store the most significant 50 digits in one column and the least significant 50 in another. In addition, unlike the difference engine the analytical engine had built-in support for fixed-precision values by allowed you to specify an implicit multiplier which would be taken into account during all operations. This could cause a single multiplication to produce up to 150 digits, hence you need three columns to be able to store the result of any multiplication, called the head, middle, and tail values. If you know your output fits within 50 digits you can use scratch columns for the most significant parts and only use the least significant 50 bits of the output, that’s up to you, but you always have to pass three output variable cards.

Symmetric to multiplication the engine also supported division:

*n*

_{i}

_{j}

_{h}

_{m}

_{t}

The input/output logistics of this operation are the same as multiplication, the result can be up to 150 digits long. The microcode for division is even more complex than multiplication, as you would probably expect. It’s an iterative approach that first approximates the result by looking just at the most significant digits of both inputs and gradually refines the result by considering more and more less significant digits. Both operations take time proportional to the number of digits of the numbers involved.

Besides arithmetic operations the engine also supported shifting values up and down in base 10, corresponding to multiplying or dividing by powers of 10. Here is the *step down* operation, what we would call right-shifting:

*n*

*a*

_{i}

_{h}

_{t}

This instruction shifts the value in V_{i} *a* steps to the right, effectively dividing the value by 10^{a}. The *n* as usual specifies how many times to repeat the operation. Like with the multiplicative operations the shift, sorry *stepping*, operations are somewhat familiar but also quite different. The step up operation takes two parameters: the number of times to repeat the operation and how far to shift the value. Since each operation card only holds one parameter and this one uses two we need to operation cards, the operation itself and a dummy card that has no function but to hold the amount to shift by.

In modern programming shift operations always discard bits, either high or low bits. As we’ve seen Babbage made sure operations never discarded bits. In this case we get two outputs: the shifted value on one column and the bits that were shifted away on the other. Again, if you really intend to discard the bits that were shifted away you can just store them in a scratch column.

Internally the engine had primitives that could shift by 1 and 2 digits in one cycle and the shift card was implemented by repeatedly shifting by 1 digit. If you wanted to shift by an even number there was a separate card that worked the same way as the single shift but repeating the 2-digit shift primitive:

*n*

*a*

_{i}

_{h}

_{t}

This instruction shifts by 2*a* but uses only *a* cycles to do it where the single step down operation would take 2*a* cycles. Symmetrically there are operations for *stepping up*, what we would call shifting left:

*n*

*a*

_{i}

_{h}

_{t}

*n*

*a*

_{i}

_{h}

_{t}

There’s a few things to note about this set of operations. First of all, the amount to shift by is always fixed by an operation card so there is no way to shift by a variable or computed amount, only by a constant. Also, since no values are ever discarded the up/down variants are complementary: stepping up by 26 gives you the same result as stepping down by 24 except that the head and tail of the results are swapped. So you would never step more than 25 in any direction because you could get the same result with fewer cycles by stepping the other way.

Now we get to addition and subtraction. You might think they were among the simpler operations, simpler than division surely, but no, the instruction format is really hairy. The thing is, you very often end up having to perform long sequences of additions and subtractions and if the format was straightforward like division you would end up constantly writing values out to columns and reading them back in. What you really want is to add and subtract a sequence of numbers into a running sum inside the engine’s mill and only once you’ve done store the result in a column. The interface for doing this changed many times and the final result is clearly a work in progress itself. The basic form looks like this:

*a*

*b*

_{i0}

_{i1}

_{ia}

_{j0}

_{j1}

_{jb}

_{o}

This instruction first adds *a* values together, then subtracts *b* values, and finally stores the result in V_{o}. The second addition card seems a bit random; it’s not clear why it’s even necessary but in any case it’s just an end marker, it doesn’t cause any numbers to be added.

You might be thinking to yourself: surely you need two columns to store the result or you may end up discarding digits? Well yes, sort of. The internal column that stores the running total before it’s transferred out has 3 extra digits of precision so you can safely add and subtract lots of numbers without overflowing. And as long as you’re sure that the final result doesn’t exceed 50 digits then you’re good. If the result *does* exceed 50 digits then the machine would stop and notify the machine’s operator, presumably by ringing a bell. What he could do to resolve the issue is unclear. But that’s how it worked.

Now, Babbage felt that this was a somewhat limited interface: you have to add first, then subtract, and then you’re done. Sometimes you want to subtract first. Other times you want to add, then subtract, then add some more, and so on. So he experimented with other approaches, like allowing as many add and subtract cards as you want, in any order, terminated by a special *F* card. Often you want just one addition or subtraction so he introduced two new cards, add-once and subtract-once. Why that’s better than a general add card with a repetition count of one is unclear but presumably they were faster. This was still a work in progress and we don’t know how addition would ultimately have worked if he’d been able to complete his design.

## Special operations

The operations I’ve covered so far are the most straightforward ones, the arithmetic operations. The remaining ones are a set of somewhat obscure arithmetic operations and the control structures. I’ll go over the obscure ones first, starting with the operations for counting significant digits.

_{1}

*a*

_{i}

_{o}

_{2}

*a*

_{i}

_{j}

_{o}

The one-input version computes the number of significant digits of a single value, the two-input version computes the sum of the number of significant digits in two inputs. The one-input operation is somewhat useful; the two-input one is more puzzling. It could be implemented using the one-input version and addition, though this would be a lot less efficient. Babbage must have had some use in mind for both but it’s not clear from his notes what it was.

*n*

*a*

_{i0}

_{i1}

_{ia}

_{o}

The analytical engine was a generalization of the difference engine but the goals were much the same: producing arithmetic tables. Hence it seems natural that the analytical engine should have “native” support for difference engine style calculations. This operation provides that support. It performs *n* iterations of the *a*-order finite difference tabulation. How finite difference tabulation works is covered in full detail in my post about the difference engine. This operation, which is naturally repetitive, may be the inspiration for having a repeat count on the other operations.

For each iteration you specify the set of columns that hold the differences and the output column. This means that for say 100 iterations you need to specify hundreds of variable cards. It makes sense to some extent, you probably want the output of each iteration to get its own column rather than override the previous value difference-engine-style. On the other hand, a modern programmer would have used a 50-element array and stored the output at successive indexes, rather than duplicate the code 50 times to store the results in what is essentially 50 global variables. The underlying issue is a broader one: the analytical engine only supported what we would call direct addressing, global variables basically. There was no such thing as writing to or reading from a column whose index was computed at runtime, so you had no data structures of any kind, not even flat arrays. Indirect addressing wouldn’t actually have been that difficult to implement but apparently it just hadn’t occurred to Babbage and the differences operation is one of the places where you see the effect.

The last few operations we don’t actually know the instruction format for. The first two are **approximate multiplication** and **division**. You’ll remember that multiplication and division take time proportional to the number of digits. If an approximate result is good enough for a computation you can use the approximate versions which works essentially the same way as their accurate counterparts except for a step before the main computation that right-shifts the operands to discard the least significant digits, and in the case of multiplication a step at the end that left-shifts the result back to get the right magnitude.

The last arithmetic operation is **double-length addition** which is similar to the addition above but for each operand takes two columns, the head and the tail of a 60-digit value, and produces a 60-digit result. It’s pretty straightforward really, the only really notable thing about it is that Babbage’s implementation has a subtle bug which may cause the output to have the wrong sign in some cases.

## Control

As you may have noticed I’ve been going from the best to the least well understood operations and now we get to the control flow operations, the least well understood of all. Babbage does mention them but spends very little time on them. His focus was on the more challenging operations like the arithmetic ones and control flow, which would be easy to implement mechanically, he more or less ignored. He knew at a high level which kind of control flow the engine should support and knew that it would be easy to implement – so why spend too much time on it at the design phase?

The two control operations we do know is branch-if-zero and branch-if-negative:

*a*

_{i}

*b*

*a*

_{i}

*b*

The first thing you’ll notice is that the variable cards are different – where normally a variable card specifies a single column, for the control operations they also specify a count. They’re also different in that the argument is not the number of times to repeat the operation because that doesn’t make sense for a branch.

They way both operations work is that first the selected column is checked for whether it’s zero or negative respectively. If the condition is true the operation card stream is moved by *a* cards and the variable card stream is moved by *b* cards. It’s not clear which direction the cards are moved but since we know programs that (at least appear to) branch backward it seems plausible that they’re backward branches. Also, looking at the table programs that were written both by Babbage and others there can be little doubt that his intention was for the engine to be what we today call Turing complete. And if the branches go forward it wouldn’t be (though the margin is too small for me to prove that formally).

Note that since the operation and variable cards are moved independently it’s quite possible to run the same operation cards with different variable cards as input as well as the other way round, by branching by unaligned amounts. It also makes programming more error prone though.

Basically it’s clear that the control operations were never fully developed and just because we only know of a limited doesn’t necessarily mean that the finished engine wouldn’t have had a full-featured set of control operations. It’s likely that he just never completed the design.

You have now seen the three different ways the analytical engine could be programmed, including the full instruction set as we know it today. Our understanding of the engine is incomplete though and it’s possible that further research into Babbage’s notes will tell us more. But even what we do know gives a clear sense of the flavor of programming the engine supported. My next post will focus on one particular program, the famous first program for computing Bernoulli numbers from Ada Lovelace’s notes on the engine. As you’ve seen the engine is rich in programming and some of it, for instance Menabrea’s small programs above, predate the Bernoulli program. The next post will explain why that program nonetheless deserves to be singled out and celebrated as the first example of what we today understand as programming.

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