Philon's Line

      Philon of Byzantium , sometimes called Philo, was a lessor known scientist/mathematician living around 280 BC.  One of his projects in geomery was the ancient problem of making a cube with twice the volume of a given cube.  In trying to solve the problem, he encountered and solved the following problem:
Given an angle BAC and a point X, interior to the angle, find the shortest line segment through X reaching from ray AC to ray AB.  (see figure)
Philon's discovery was that the shortest segment occurred when the distance from the point X to one ray was equal to the distance from the other ray to the foot (G in the figure) of the perpendicular  to the segment from A .
   Philon's only surviving work is his treatise on mechanics.  It appears that he travelled throughout the Eastern Mediterranean studying the construction of catapults.  He may have been a militrary advisor of some sort.  He is mentioned in the writings of Heron of Alexandria.  You can find more about his life at the St Andrews University web site.