These are Golden Gem portable adding machines made by the Automatic Adding Machine Company. The chrome one on the left was made around 1910 and the black one on the right, which has its original box, case, warranty card and receipt, was purchased in February 1941 for $9.16.
These are known as chain adders. By inserting a stylus and dragging the chain down, you increment the register. You can see the carry mechanism in this video, which uses a spring-loaded tooth that rides along the outside of the gear and engages with the next gear only after 9.
These are portable, but are bigger and heavier than other pocket calculators of the time and primarily sat in or on a desk using the kickstand, and were used for quick arithmetic. To add a number, insert a stylus at the corresponding digit and drag down to the bottom.
The chain only moves in one direction, so to subtract you have to use the complements method. To subtract 42 from 31342, I add 9999958.
To clear the register, you turn the knob on the bottom right side anti-clockwise until all the digits say 0.
Dragging the chain and hearing the gears click into place is very satisfying--nice tactile feedback. However, carrying a one across the whole register takes some force. It looks nice and is OK as a small desktop calculator, but subtracting w/ it makes me prefer the Addometer.
Multiplication is repeated addition, shifting left for each digit in the multiplier. Here is 12 x 12. Because of the extended complements for subtraction method, division is possible but not ideal.