On 2/17/2014 3:20 PM, David Hartley wrote: > In message <email@example.com>, Ken > Quirici <firstname.lastname@example.org> writes >>> Consider a system consisting of a single straight line and the points >>> >>> thereon. It satisfies all Euclid's explicit postulates (and Pasch's >>> >>> axiom) but circles have only two points in their circumferences and >>> >>> circles with a common radius do not intersect. There are no equilateral >>> >>> triangles. >>> >>> >> >> there's problem in that his definitions include (at least in Heath's >> translation) surface - that which has length and breadth only. >> >> Does this vitiate your statement? >> > No, the axioms don't imply that a surface exists. > > Of course the problem of non-standard models like the one I've suggested > is trivial to fix. Just add axioms to say that there is at least one > line and a point not on that line, (and a plane which does not contain > the line if you want 3-D). I assume, from the discussions I've seen on > the web, there are models which satisfy this but still have circles > which don't intersect even though they "should". Further Googling would > probably turn up an example but for now it's more fun trying to find one > on my own. > > > .... >> Thanks. I'm hoping Moise's book on Elementary Geometry from an >> Advanced Standpoint will provide a sufficiently rigorous approach. >> Have you read it? If so, is it sufficiently rigorous? > > No I haven't read it. I've never studied Euclid's axioms before. >
Is the circle, an infinity-sided "polygon"? Where vertices divide edges, the "graph" of the circle has only one edge (or with one vertex and another that is disconnected, but maintained).
How about whether it is going one way or the other direction, for how much is going through, then simply reading out that?
Then for this the fixed point is still to the principal component, here as is maintained a 1-D line of reals, R.
Then, for example, it wouldn't work for just the positive or negative, only positive and negative (and zero, for convenience, and as so, where it is not: where that is so).
Here the translation of rate, positive or negative, is measured in the spinning of a circle. Then, it is the actual surface or curve of that, what the spinning circle rights out from its points as they fall to the next.
Here this can be constructed with a spinning top and string.