On 1/28/2014 2:36 PM, Neighbor wrote: > You are mentioning a straight-line with numbers on it, and since you can always find the half of a number, being this number on the straight-line, between two numbers on the straight-line there will be another, but I want you to explain me the nature of a straight-line in Euclid's geometry, not putting numbers on the straight-line, and not saying that what for numbers is true is also true for straight-lines. Just now I want to deal only with straight-lines and points on them, as Euclid does in the Elements, without putting numbers on them. > > > If I accept that between two points there is always another, then there is the aforementioned problem of division of a straight-line in half, which I cannot understand if I admit that between two points there's always another. Therefore, unless you show me how we can solve the aforementioned problem with the admission that between two points there is another, I cannot help thinking that two consecutive points exist. > > I gather we're actually discussing segments, so - if my memory serves me correctly, a segment is commonly bisected by constructing the perpendicular determined by the intersection of arcs of equal radii from the end-points of the original segment. so: if AB is thus bisected to AC and CB, the common end point of AC & CB is C.