Between Centers Mishap

I think that you are right. In any event, it is fairly easy to check. mount the square between center and check runout with a dial indicator. It should be zero. Carrying it one step further, if you don't change the position of the indicator on the carriage, the readings should be identical. If they're not, it would indicate that you would cut a taper. This is essentially the same as the Edge Technology Tailstock Alignment Tool. If you check their websote, they have a manual describing use.

I actually just picked up this tailstock alignment tool because I wanted to figure out how bad I was. I initially just did a facing cut with the cutter slightly below center (leaving the 'nip'), then moved my tailstock up against the part and did it by eye. Apparently I was out by 7/10ths! Not sure I could do that again :)
 
I actually just picked up this tailstock alignment tool because I wanted to figure out how bad I was. I initially just did a facing cut with the cutter slightly below center (leaving the 'nip'), then moved my tailstock up against the part and did it by eye. Apparently I was out by 7/10ths! Not sure I could do that again :)
Like they say,"even a broken clock is right twice a day". :)

I had a machining project about fifteen years ago where I had to set my z axis zero by eye. I used a 50X microscope to do it and figured that I could repeatably hit within a thousandth lowering a sharp point to the edge of a reference surface but not much better.
 
Here is our alignment tool.

Cut only first time.

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