**DIVINE PROPORTIONS: Rational Trigonometry to Universal Geometry**, by N J Wildberger, appeared recently in [1]. Here [2] is the wikipedia reference to the subject.In the sample chapter available for review [3] the author gives a definition of the terms

*spread*and

*quadrance*. In simple words, the

*spread*is an expression of the separation of two lines ( spread = (sin(angle))**2 ) which "replaces" the angle of classic trigonometry in rational trigonometric calculations, while the

*quadrance*is the square of a distance.Mathematically inclined minds may add more useful applications in this page. Here is a first take on how to find the

*spread*between two lines, using tcl.

# Find the spread based on the coordinates of three points P0, P1, P2, which define lines |P0,P1| and |P0,P2| # # P1 # / # / # /| # / | # P0/ | # \ | spread S0 # \ | # \| # \ # \ # P2 # # use point coordinates to find quadrance proc Qc {x1 y1 x2 y2} { return [expr {pow($x2-$x1,2)+pow($y2-$y1,2)}] } # # use side length to find quadrance proc Qs sd { return [expr {pow($sd,2)}] } # find spread given coordinates proc Sc {x0 y0 x1 y1 x2 y2} { # find the quadrances of each side of triangle P0-P1-P2 formed by the points P0,P1,P2 set q0 [Qc $x2 $y2 $x1 $y1] set q1 [Qc $x0 $y0 $x1 $y1] set q2 [Qc $x0 $y0 $x2 $y2] # use the Cross law to find S0 return [expr {1-pow($q1+$q2-$q0,2)/(4.0*$q1*$q2)}] } # # find spread given sides proc Ss {sd1 sd2 sd3} { set q1 [Qs $sd1] set q2 [Qs $sd2] set q3 [Qs $sd3] # use the Cross law to find S1 set res [expr {1-pow($q2+$q3-$q1,2)/(4.0*$q2*$q3)}] if {$res>=0 && $res <=1.0} { return $res } else { error "This triangle cannot exist!" } }

**Examples:**Assume three points P0(0,0), P1(5,0) and P2(10,10), which define a horizontal line |P0,P1| and a slanted line |P0,P2|. The angle between the two lines is 45deg . The spread S0 will be:

% Sc 0 0 5 0 10 10 0.5Other examples:

% Sc 0 0 50 0 100 75 0.36 % Sc 0 0 5 0 10 20 0.8Assuming a triangle with sides 4, 5 and 6 units, find the spread opposite to side measuring 4 units (part of an example appearing in the sample chapter of the book mentioned above):

% Ss 4 5 6 0.4375Assuming a triangle with sides 3, 4 and 5 units, find the spread opposite to side measuring 4 units:

% Ss 3 4 5 0.36and for the other sides:

% Ss 4 5 3 0.64 % Ss 5 3 4 1.0This last one says that the spread opposite to side 5 corresponds to a right angle.More on this subject as soon as the book arrives and is read :-).