# Overflow behavior in turn counters of Odhner-style arithmometers

I have a Felix M Arithmometer (shown below) that appears to follow the same design as an Odhner arithmometer.

One thing I noticed about the device is that the turn counter (on the left) does not overflow like the addition register does.

This means if I want to multiply 37 * 49 I must do the following

Procedure A:
1. enter 37
2. rotate + (9 times)
3. shift right
4. rotate + (4 times)


Performing the operation "subtractively" will result in the right answer, but won't display the correct result in the turn counter

Procedure B
1. enter 37
2. rotate - (1 time) (causing underflow)
3. shift right
4. rotate + (5 times)


OR

Procedure C
1. enter 37
2. shift right
3. rotate + (5 times)
4. shift left
5. rotate - (1 time)


If I do procedure B or C, then I see 51 in the turn counter instead of the expected 49 even though the number in the addition register is still 1813 as expected.

There's some logic to this. If I multiply 37 * 48 subtractively, then 52 appears in the turn counter, which can be though of as 5 in the tens place and -2 in the units place. So you can check your work by consulting the turn counter if you remember which digits are supposed to be negative.

I think that a register with overflow is heavier and more expensive to produce than a register without overflow, so it makes sense that the turn counter wouldn't overflow in exactly the same way as the addition register. However, that still doesn't explain why the turn counter counts up then down instead of always counting up.

Were any arithmometers produced anywhere with ordinary overflow behavior in the turn counter?

Is multiplying "subtractively" and remembering which digits are supposed to be negative when interpreting the turn counter an intended mode of operation for an Odhner arithmometer? ## 1 Answer

After a little bit of research, I think the answer is that the turn counter works the way it does to support division and division would not function as intended if the turn counter carried. "Subtractive" multiplication is not well supported because it would make division less intuitive.

This video https://www.youtube.com/watch?v=aDN4s8ElxqE shows an arithmometer of similar design and the intended algorithm for doing division problems starting at about 4:57.

The turn counter shows the quotient in a division problem. Quotients are computed by repeatedly subtracting the divisor multiplied by a power of ten until just before the divisor underflows.

The quotient is a positive number and the quotient-so-far (in the turn counter) must increase each time a multiple of the divisor is subtracted.

If the turn counter carried, then the result of a division problem would be the nine's complement of the real answer. Alternatively, if the divisor or dividend were complemented, then the quotient would be non-complemented. This is explained in detail here: https://www.youtube.com/watch?v=YXMuJco8onQ starting around 5:19.