On Saturday, February 15, 2014 1:23:27 AM UTC-5, John Gabriel wrote: > On Saturday, February 15, 2014 4:00:41 AM UTC+2, 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 > > > > The Greek implies that the circles so drawn are equal, for otherwise the construction such that the sides of the equilateral triangle are equal would not be possible. It follows logically from the context. There is no missing premise. In the construction you must use circles with equal radii for otherwise you cannot construct an equalitareal triangle. This proposition is used to construct angles such as 60 degrees and 30 degrees.
Thanks for all replies.
The premise that I think is missing is that the two circles must intersect - Euclid uses this premise to find the vertex of the equilateral triangle whose base is the common radius of the two circles - the vertex being either one of the two intersecting points of the circles.
Imagine you start with the two centers, and the length of the radius. Now start drawing the two circumferences. It's clear that they eventually intersect, but how do you demonstrate this rigorously - that is, just using Euclid's Definitions, Postulates, and Common Notions? It's obvious, but so is that you can draw a circle given a center and radius, but Euclid felt it necessary to include that in his 5 Postulates. So even if it's obvious that the two circles intersect, it still needs to be somehow demonstrated OR assumed.
Try something else. Start with one circle and its center, already 'constructed' as the Postulate asserts it is possible to do. Now take your compass and put it on the circumference of this circle, and its 'pencil' at the center of the first circle. Now start swinging the compass. It eventually hits the first circle's circumference. Obviously. But WHY.
I'm beginning to think that Euclid takes a huge number of 'obvious' facts for granted. I'm beginning to think Euclid's elements can only REALLY be made rigorous by making the apparently simple geometry a subset of R^2 real analysis - the whole kit-and-caboodle of calculus, continuity, &c.