Euclid's Elements - The Application of Areas

[Hunus]

In Heath's commentary on Euclid's Elements he stresses the importance of the application of areas (Book I Proposition 44) with, "The marvellous ingenuity of the solution is indeed worth of the 'godlike men of old'...".

The proposition, "To a given straight line to apply, in a given rectilineal angle, a parallelogram equal to a given triangle" seemed to me to be proven with, "Let the parallelogram BEFG be constructed equal to the triangle C, in the angle EBG which is equal to the rectilineal angle D [I.42]", but he goes on to employ I.43 (complements of a parallelogram about its diameter are equal to one another) to be able to change the dimensions of the area, but to hold the area constant.

Now I understand that that is important in and of itself, but I don't see why using [I.42] isn't 'applying' the area to the straight line.

 

[Hunus]

I figured out what was confusing me. My misconception was that I believed AB to be an infinite straight line with A and B only denoting the different directions of the line that had not yet been fixed -- in which case the proof would have ended after the construction of the parallelogram BEFG -- but AB is a finite segment and BE is only a production of AB beyond B.

So you have to apply the area to AB itself not AB produced -- which is why Euclid employed I.43.