> John K Clark explains
> : > Or you can prove that all lines are composed of the same number of points
> : > regardless of length. Draw 2 parallel lines, a short one and a long one
> : > below it, pick a point midway along the short line but above it.
>
> : > /\
> : > / \
> : > /________\
> : > / \
> : > /________________\
>
> : > Draw a line from that point to any place on the short line, then continue
> : > it until you hit the long line. You've made a one to one correspondence
> : > between all the points in the short line and all the points in the long
> : > line, so they must have an equal number of points.
>
> Wolfkin asks:
> : How did you get this? It seems obvious that because we admitted that
> : the longer line *was* longer, after you have used all of the
> : (infinity of) points on the shorter line, there will be points on the
> : longer line in between the lines you drew across.
>
> Do it the other way. Draw a line through the apex and any point on the
> longer segment. If the shorter segment contains only half as many points
> as the longer segment, then half of the time your new line will intersect
> *no* point on the shorter segment. I think Euclid would disagree.
Probably. :) Ok, then, if we do it the other way...
Why would it "intersect *no* point"? This just seems to mean that
two different lines *can* be drawn through two points. ;)
Wolfkin.