h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

I see a problem with the fifth postulate as stated here.
The first four seem to be independent of the number of dimensions we are talking about. The fifth, as it is stated here, requires a two dimensional space.

This is easily shown by starting with two meeting lines and a connecting line in two dimensions. Now 'disconnect' the meetingpoint and raise one of the lines a bit in the third dimension. The angles will stay practically the same, but the lines now cross without meeting each other.

Euclid probably did state it in terms of 2 dimensions. I'll have to look it up to see. I'd say the ambiguity lies in my paraphrasing of it.

No, it was implicit in Euclid's definition that he was talking about lines that were all in the one plane. So as you say, it was two dimensions only.

meeting point of two lines is two dimensional and on raising one of the lines is on the third dimension

Flyingtwinkle, I deliberately moved it to the third dimension to show the problem in the way the postulate is stated.

I wonder if there is a way to formulate this postulate so it also will be independent of the number of dimensions. I can think of a few that need terms like 'distance' or 'right angle', but I am not sure you can derive these from the first four postulates.

hi archangel nice to know we agree on some points like the third dimension a straight line is also an angle of 180 degrees so it is also two dimensional
a straight line is also an angle of 360 degrees so it is also three dimensional
a staight line if split lengthwise become two parrallel lines

Just so you know, two lines that are not on the same plane are
called "skew" lines; they do not intersect AND they aren't parralel.

On a similiar not, the two lines perpindicular to another line
with a weird parralel relationship can exist; however, the two lines are skew, not parralel.

not= note