
Re: Euclid's Elements Book 1 Proposition 1  something Euclid missed?
Posted:
Feb 16, 2014 5:42 PM


On Sunday, February 16, 2014 10:09:02 PM UTC+2, Ross A. Finlayson wrote: > On 2/14/2014 10:24 PM, John Gabriel wrote: > > > On Saturday, February 15, 2014 6:04:10 AM UTC+2, Ross A. Finlayson wrote: > > >> On 2/14/2014 6:00 PM, Ken Quirici wrote: > > >> > > >>> According to Wikipedia, Book 1 Proposition 1 (constructing equilateral triangle on given line segment) > > >> > > >>> is missing a premise  that if two different circles share a radius (that is, a line segment connecting > > >> > > >>> their centers is a radius for both circles). > > >> > > >>> > > >> > > >>> Could somebody provide a proof of this premise? > > >> > > >>> > > >> > > >>> Regards, > > >> > > >>> > > >> > > >>> Ken > > >> > > >>> > > >> > > >> > > >> > > >> > > >> > > >> > > >> > > >> > > >> > > >> (proximal, proximity) > > >> > > >> > > >> > > >> The circle fills space uniformly with the least perimeter for > > >> > > >> area so they are the same. It is halfway then for the > > >> > > >> equilateral, because the angles are equiangular, in the > > >> > > >> evolution of angles, they are equal. > > >> > > >> > > >> > > >> Sorry, this is a rather poor geometric construction, I'd be > > >> > > >> most surprised to hear of something like that in Euclid. > > >> > > >> > > >> > > >> Then, it would because the radii at a distance from the line > > >> > > >> with the parallel line, with the adjustable compass from > > >> > > >> drawing the parallel lines, would form the triangles with that > > >> > > >> height that form the equilateral triangle (that the side is > > >> > > >> equal, to find then the parallel line at the height of the > > >> > > >> equilateral triangle. This is finding the center from > > >> > > >> shifting that up then finding the midpoint of that, bisecting > > >> > > >> the segment with the compass and edge, then halving and > > >> > > >> doubling, for the projection of the triangle that is of the > > >> > > >> parallel lines, to the orthogonal parallel distance that is of > > >> > > >> the third side of the equilateral triangle, through that. > > >> > > >> > > >> > > >> Or, having three compasses. > > > > > > What absolute rubbish. I can tell you don't understand the Elements at all! > > > > > > > About Euclid, no that actually is a reasonable construction > > of the equilateral triangle. > > > > Here, the question is "define the area of a triangle". > > > > It is a question for Euclid.
Proposition has NOTHING to do with area of a triangle.

