OK but how do you get the sine of the angle to square without trig - just wondering.
I'm telling you it's the square of the sine so you know what it is, but that's not the way the book develops it. Use the law of cosines instead -- cos C = (a^2 + b^2 - c^2)/2ab.
So, the "spread" of angle C can be defined in terms of the "quadrance" --
S(C) = 1-((Q(a)+Q(b)-Q(c))^2)/(4Q(a)Q(b))
I think the book has an even simpler formula, but the one I just showed follows immediately from the law of cosines, and the calculations don't require tables or calculators.