To draw the y axis, draw circles of radius 2 about 1 and -1. They intersect in two points on the traditional y axis, at Âħsqrt(3). A line drawn through these points becomes the y axis. Establish the integers along the y axis, just as we did for the x axis. It is now possible to reach any lattice (integer) point in the xy plane.
If you want to cut a line segment in half, draw the perpendicular bisector, as we did before. Repeat this to cut the segment into quarters, then eighths, and so on. Thus any distance n/2m can be constructed. At this point any figure can be approximated to arbitrary precision, and the engineers are satisfied, but the mathematicians press on.
The unit segment can be divided into 2n equal parts by drawing circles of radius n around both endpoints, and treating their intersection as the apex of an isosceles triangle. Mark off integral lengths along the sides of this triangle, and project these points down into the base, thus cutting the base into 2n equal parts. Now all rational coordinates are accessible.