Carry on (the) Arithmometer

There are a number of devices that go by the name "Arithmometer," including the original one which went into production in 1851. They aren't computers, to be sure, and are best described as mechanical calculators, but I like them because they're fun to play with, they don't need a power source (other than a stylus and your fingers), and — as the manual for this Tasco Pocket Arithmometer says — if they're operated correctly the machine will always give you the correct answer immediately.

The Tasco Pocket Arithmometer here has an unusual lineage that appears to start in 1913 in Germany with the Trick Rechenmaschine (Trick Calculating Machine), though "trick" may not be the image you'd want in mind to suggest accurate accounting. The Trick itself is an evolution of the "slide adder," using rows of sliding numbers to calculate sums; its primary innovations were a semi-automatic carry feature and a sliding subtraction plate, making it probably the prototype of this class of "shepherd's crook" mechanical calculators. In the 1920s the design ended up Stateside and was manufactured by the Gray Arithmometer Company in Ithaca, NY. These early units had a dull nickel finish, and some are labeled Morse Chain Co., who were at one time the primary distributor. Somewhere around 1940 to 1945 manufacture of the product was taken over by Tasco, apparently unrelated to the modern manufacturer of crappy telescopes, who produced it in almost exactly the same form except for a new name plate and a chrome finish. They seem to have been manufactured at least through the 1950s, and came with a manual, a metal stylus that could poke out an eye and a little leatherette snap case.

The T-P-A does addition and (with the use of the sliding subtraction plate) subtraction. Since multiplication is merely repeated addition, it can do that too, and division via repeated subtraction with some additional effort, though these are plainly not its strong suits. I'll talk specifically about addition here, being the T-P-A's primary function.

Numbers are dialed in by pulling the top tab, which clears all the columns, and then by using a stylus in the provided holes. The metal between the holes is either white or red. When the colour above the number is white, we pull the stylus down; when it is red, we pull it up. In this animation, we demonstrate twelve plus five by dialing 2 and 10, and then pulling down 5 to yield 17:

The "hook" at the top (or at the bottom, when the subtraction plate is engaged by sliding it up) is the automatic carry system. When adding, the stylus gets pulled up and over the hook to add one to the next column. Here is 12 plus 9, dialing in 2 and 10, then pulling up and over the 9 to yield 21:

Although fascinating and intuitive when you know the trick, the manual assumes a lot of someone not already familiar with this style of calculator. In particular, while the automatic carry is innovative, it does not cascade. The reason for this can be seen when we try to dial in 999:

The left-most 9 (in the hundreds place) has a block in the right, but the stylus can still move into the hook. However, the other two nines in the tens and units places are "double blocked": two blocks completely prevent the stylus from advancing into the hook. What the manual doesn't make clear is that you have to cascade the carry yourself if you have multiple positions that are double blocked. Let's consider how to add 1 to 999.

1. We attempt to add 1 to the rightmost column. This column is red, so we put the stylus in the one-slot and pull it all the way up (yielding zero), but this column is double blocked so we can't move into the hook to automatically transfer the carry.
2. We thus attempt to add 1 to the next column as well, but it's double blocked also, so we end up zeroing that column too and continuing to move the carry to the next column to the left.
3. This last column is single blocked, so now we can finally pull the one up and over the hook, setting it to zero and carrying the 1 automatically. There's nothing more to add, so we're done.

By cascading the carry over multiple columns, the machine reads 1-0-0-0, which is the answer.

Similarly, consider adding 11 to 999. This is nearly the same, except having added the 1 and propagated the carry left, we then just add 10. The columns now read 1-0-1-0, which is the answer.

Finally, let's consider the case of 999+999=1998.

1. We attempt to add 9 to the rightmost column, but it is double blocked. Instead, we put the stylus in the nine-slot and pull it up, setting this column to 8, and carry the 1 to the left.
2. We attempt to add 1 to the next column, but it is double blocked, so we put the stylus in the one-slot and pull it up, setting this column to zero, and carry the 1 to the left.
3. We attempt to add 1 to the next column, and it is single blocked, so we put the stylus in the one-slot and pull it up and over the hook, setting this column to zero and carrying 1 to the left.

At this point the dials read 1-0-0-8, but you've probably figured out the next two steps already:

4. We add 90 (1-0-9-8).
5. We add 900 (1-9-9-8).

Demonstrated below:

This is the answer, correctly and immediately.