Hey, I just finished restoring a similar Jsameter and this thread helped a lot with finding additional information so I just wanted to share what I found out.
The Jsameter is an adjustable snap gauge, how it works: Depress the plunger to retract the anvil, insert the part and let go of the plunger – the springs can only push the anvil forward until it touches the part and the plunger stops and shows a reading on the scale:
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The mechanism is quite interesting, on the inside there is a very slightly tapered pin the length of the measuring range and connected to the plunger, it rides between 7 loose bearing balls so be aware when opening one of those up. The plunger is spring-loaded and pushes the bearing balls apart which rest inside a cone that is connected to the anvil – the balls move outward and push the cone forward – the anvil moves forward. Depressing the plunger lets the anvil move back/open up with the help of another spring.
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The end part of the barrel with the measuring line on it is pressed in, there are no screws and only two small ridges to pull on. I took out the scale and used parallel pliers on the flat part with the line to wiggle it out with the barrel in the vise with copper jaws. After getting it out I immediately spilled the bearing balls on the floor 
My idea of the origin of the name – there is a dial snap gauge called the Passameter, maybe this is the Passameter from J Weco – the Jsameter.
I was also able to find a patent from Wizemann showing the exact instrument in the drawings, except in the patent it is a thread gauge. The patent is from 1939: espacenet: Patent CH236608A
But the mechanism might be a little older, there is a patent from inventor Walter Wolff form 1931: espacenet: Patent DE601437C
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Kristof