why parabolic?

A parabola is a geometric curve that gives the greatest strength to weight ratio in a spanning beam.  It is defined as the curve a plane section makes passing through a cone parallel to the cone’s axis.  They can be wide or narrow, depending on the angle of the cone.  

I believe that braces should be shaped parabolically both in length and cross section to best flow with and transmit the vibrations of the strings.  This is not just my opinion.   I have inspected a number of “vintage” Martins.  They all had parabolically shaped braces on the back, as well as parabolic finger braces, and half-parabolas on the arms of the X brace.  Only the X brace legs and the two belly braces were scalloped. 

Once luthiers started copying Martin bracing, they seem to have concentrated only on scalloping braces, ignoring the parabolic shaping that so contributed to the sound of vintage days.