Pythagoras' Choice

The usual way of viewing the "metric" of spacetime is in the 
"Pythagorean" form  
                       s^2 = x^2 - t^2

whereas this relation can also be expressed in the hyperbolic 
form s^2 = uw.  This raises some interesting questions about how 
"inevitable" are the basic forms of mathematics.  Certainly the 
"sum of squares" is likely to emerge from just about any historical 
circumstance, because of its close connection to orthogonality, 
and how naturally it generalizes to more dimensions.  Also, it 
arises in many non-geometric contexts, such as the fact that the 
standard deviations of sums of random variables exhibit a "sum 
of squares" relation.

However, I still wonder if some other civilization (e.g., the Mayan)
might have formulated things differently.  Consider a general
"right" triangle with the hypotenuse drawn as the base

                    /|\
                  /  |   \
              a /    |      \ b
              /      |c        \
            /________|____________\
                 d          e 

Pythagoras chose to observe the characteristic relation between the
three edge lengths 
                       a^2 + b^2  =  (d+e)^2

whereas it's conceivable that someone might focus instead on the
characteristic relation between the altitude and the partition of
the base
                            c^2  =  d e

Both of these equations capture the "rightness" of the triangle, but
they emphasize different aspects of it.  Would Western mathematics
have been hindered if Pythagoras (and, later, Euclid) had chosen the
hyperbolic formulation of "rightness" instead of the circular?  Was
it inevitable that a "sum of squares" viewpoint would prevail?  It's
interesting that in recent times our views of nature have often
found expression as hyperbolic relations (e.g., xp=h).