special relativity - Definition of the Spacetime Interval

The spacetime interval is defined as follows:

$$\Delta s^2 = -(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$$

or in tensor notation:

$$\Delta s^2 = \eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$$

When I first studied introductory special relativity, I didn't even pay much attention to this quantity -- it was mostly time dilation, length contraction, and fancy paradoxes.

However, it has caught my attention now. The book I'm reading simply defines the quantity, and claims that it's invariant.

Now, just from tensor analysis and ignoring special relativity, $\eta_{\mu\nu} \Delta x^\mu \Delta x^\nu$ looks like a contracted product of a doubly covariant tensor with two contravariant tensors, mathematically proving it's an invariant. Great!

But, what I do not understand is why is the spacetime invariant defined the way it is? Why is it $-(c\Delta t)^2$, and not $(c\Delta t)^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ ?

I want the physical motivation behind this formula.