It looks like Philo’s Line can be drawn as below, using only a straightedge and a compass, by virtue of knowing NordaVinci’s Formula for the Area of a Right Triangle, which can be used to show that the two tangent points 1) the inscribed circle tangent point to the hypotenuse, and 2) the escribed circle tangent point to the hypotenuse, are equidistant from the endpoints of the hypotenuse on each end.

Some history . . .

http://mathshistory.st-andrews.ac.uk/HistTopics/Doubling_the_cube.html

Some more history . . .

http://mathshistory.st-andrews.ac.uk/Biographies/Philon.html

PHD Thesis on IMPOSSIBILITY RESULTS: FROM GEOMETRY TO

ANALYSIS by Davide Crippa

Davide Crippa. IMPOSSIBILITY RESULTS: FROM GEOMETRY TO ANALYSIS : A study in early modern conceptions of impossibility. History, Philosophy and Sociology of Sciences. UniveristÃ© Paris, Diderot Paris 7, 2014. English. fftel-01098493

HAL Id: tel-01098493

https://hal.archives-ouvertes.fr/tel-01098493

Submitted on 25 Dec 2014