I mean to say "a line without breadth with points on it" and in the Euclid's Elements this particular kind of lines (i.e. straight-lines) is defined as one which lies evenly with points on itself and a point is that of which there is no part, and my question is: Are there infinite points or not on them? If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing, but if we admit that a point has only breadth (a very small amount but not 0), then the sum of infinite points will also have breadth, and since they are infinite the straight-line they made up is also with infinite breadth (i.e. an infinite straight-line). Therefore, if we suppose that straight-lines have infinite points on them, then all the straight-lines will be either infinite or nothing (depending how we define a point). On the other hand as you know, there are also finite straight lines, how can we admit their existence if they are made up of infinite points? Now, if we suppose that straight-lines have a finite number of points on them, and a point is defined as having only breadth, the smallest possible and not having parts, then a finite number of such points will make a finite straight-line and an infinite number of them, a infinite straight-line. As you can see this is the only case which seems to be true, but if so, the divisibility to infinity of a finite straight-line would also be impossible, for dividing this we can arrive to the single point which have no parts, as an atom defined by Democritus. Maybe I've been talking nonsense, 'cause all these concepts puzzle me, and if you can show something better, please, I should very much like to know what you think.