Could Pythagoras Have Derived the Formulas for a Sum of Two Rotations?

  In the last post we derived formulas for rotations from trigonometric formulas but we may have gotten a little ahead of ourselves. So we need to ask if Pythagoras could have derived the same formulas using geometry. Let's suppose we have two right triangles with rational coordinates (x,y) and (u,v) respectively. To add the rotations we can construct the second triangle, ΔOBE, on the hypotenuse of the first triangle, ΔOAP.

So we need to determine the coordinates of the point B on the unit circle. The major difficulty is to find the sides of triangle ΔEFB. To do so we construct a copy of ΔEFB at the origin, namely, triangle ΔOGD. Line BE is perpendicular to line OA so to find the direction of BE from the origin we need to construct a copy of triangle ΔOAP based on the vertical axis, namely, ΔOCQ. We can then mark off point D a distance v away from the origin. Using the proportions for the similar triangles in triangles ΔOAP and ΔOCQ we can deduce the formulas for OH, EH, GO, and OR. The new coordinates are just sums and differences of these lengths.

So one would not need trigonometry to deduce the formulas for the addition of the two triangles and Pythagoras should have been able to to this. The remaining question is whether he was motivated to do so or not.