Wouldn't it make more sense to go off on a normal instead?

Go off on a tandem; go off at a tangent.

OK, why not go off at a normal then?

Go off at half cock.

It makes most sense to remain calm and focussed

Rather than to go off on one?

Go off on a tangent: there is normally a tangential relation involved.

Go off on a normal: maybe there is no tangential relation involved

Go off on one: normally a relation involved

This post speaks for itself really

I wouldn't have expected anything less although the original question still stands as it occurred to me today that this expression doesn't make sense when looking at the textbook definition of the word tangent.

Quick refresher course.

Consider the flight of a slingshot. Before it's released, it is moving in a circle, constrained by the sling; when it's released, it maintains that velocity - along the tangent of the circle at the point it's released.

Imagine the sling being swung anticlockwise. When the sling is east, the velocity[1] is north; when the sling is north, the velocity is west, and so on.

The projectile is off at a tangent...

[1] yes, I know: velocity has a speed and a direction. We care only about the direction here.

barnacle; not just there for the bad things in life.

Understood and what I'm getting at it is that the direction doesn't change when it (or anything else on any curved path) leaves. The tangent is in the same direction [1] as the object is travelling at that instant.

The turn of phrase is used to imply a change of direction but the tangent is exactly the same direction.

If someone/thing goes off on a tangent then they are just carrying on in the same direction. It's everyone/thing else that's changed [2]!



[1] Let's leave gravity out of this.
[2] Assuming we're considering a curve not a straight line.

Perhaps it's because the original subject of the discussion gets left behind at the most rapid possible rate?

No, that's just general chat, oh hang on...

I always wondered when people refer to cement / concrete as having "gone off" when it has cured. I would have used the word "set."

Sorry if I've gone off on a tangent. I think I may have simply gone off.

or set, can you still move?

Look, just because you've found x don't start getting over-confident.