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Yes, τ in the first line and the expression on the right in the second line are what I understand as a way of expressing the proper time formula – only with the signs reversed – but that is just the convention used ( I believe).If you want to understand this stuff then you cannot simply give up on this part. You must pursue it until it is not over your head.

Look at the right hand side of the formula in the inertial frame. Do you understand that? Do you see the similarity with the Pythagorean theorem?

Yes, because the hypotenuse is the same length and the sum of the squares of the catheti will always equal the square of the hypotenuse.If you take the Pythagorean theorem and calculate the distance between two points and then you rotate your coordinates and recalculate you get the same result.

I'm feeling a bit stupid here...In spacetime the proper time is the equivalent of the Pythagorean theorem and a Lorentz transform (boost) is the equivalent of a rotation.

but how is part of the Pythagorean theorem is equivalent to proper time?

When we say a rotation, what exactly are we rotating and how? I think that if we take two events with different time coordinates and increase the x coordinate of one of them then the connecting line is rotated, and the slope of the connecting line changes as the relative speed increases, from 0 when they are vertical to horizontal at c (because at c time stops?)A rotation/boost does not change the length/proper-time. It is not that hard to prove, I would recommend doing it to convince yourself.

Which part of my analysis are you referring to?Note that in a rotation both the x and y coordinates are changed in such a way that the distance is preserved. Similarly here both the t and x coordinates are changed in such a way that the proper time is preserved. In your analysis above you neglected the change in t.

I thought I had used τ for proper time and t for coordinate time - should I have used t and t' instead?