Hence, the rectangle and the difference of the lines MR and PS being given, each is given; hence MR is given; but MR = OM − ON; therefore OM − ON is given; and we have proved that the rectangle OM.ON = R2; therefore OM and ON are each given. In like manner, OP and OQ are each given.
Again,
Hence, since OQ and ON are each given, ρ6 and ρ7 are each given; therefore we can draw these chords, and we have the arc A6A7 between their extremities given; that is, the seventeenth part of the circumference of a circle. Hence the problem is solved.
The foregoing analysis is due to Ampere: see Catalan, Théorèmes et Problèmes de Géométrie Elémentaire. We have abridged and simplified Ampere’s solution.
NOTE D.
to find two mean proportionals between two given lines.
The problem to find two mean proportionals is one of the most celebrated in Geometry on account of the importance which the ancients attached to it. It cannot be solved by the line and circle, but is very easy by Conic Sections. The following is a mechanical construction by the Ruler and Compass.