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If O  be the origin and BC any arc measured in the "forward" direction on any turn of the spiral, let two circles be drawn (1) with center O and radius OB, meeting OC in C', and (2) with center O and radius OC, meeting OB produced in B'.  Then if E denote the area bounded by the larger circular arc B'C, the line B'B, and the spiral BC, while F denotes the area bounded by the smaller arc BC', the line CC' and the spiral BC,E : F = [OB + 2/3 (OC - OB) : [OB + 1/3(OC - OB].

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Ibn Sina's Method of Constructing Parabolic Points This animation depicts Ibn Sina's method of constructing parabolic points from concentric tangent circles (ie circles that meet at a common side point), by finding their intercepts with an arbitrary line oriented orthogonally to the line of the circles' center points. In this 2D animation you can see a very clear correspondence of a 3D cone tilted by 45 degrees. I should try to visualize this in a future animation. (I actually tried but Processing's orthometric mode didn't yield the results I was looking for) Primary Source: http://quadrivium.info/MathInt/Notes/Apollonius.pdf Ibn Sina: http://en.wikipedia.org/wiki/Ibrahim_ibn_Sinan

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