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Well, for 1, consider spherical geometry. In Euclidean geometry, there can be only one line between any two points. But consider the lines on a sphere between t
by jhbadger 1y ago
Well, for 1, consider spherical geometry. In Euclidean geometry, there can be only one line between any two points. But consider the lines on a sphere between the antipodes (like lines of longitude on Earth) -- there are infinite lines of longitude that connect the North and South Poles, not just one.
- thaumasiotes 1y agoThat's postulate 5, parallel lines meeting. Postulate 1 says that a line can be drawn between any two points. Drawing more than one line between a certain pair of points doesn't contradict that. If you can draw two lines, you can draw one line.
- jhbadger 1y agoNo. Postulate 1 says that for any two points A and B, there is a line AB connecting them. While you could argue that "infinite lines" are a superset of "a line" it is pretty obvious that Euclid meant one and only one line (which is true in Euclidean geometry)
- thaumasiotes 1y agoYou don't seem comfortable with math. You might observe that the way Euclid uses postulate 1 is to provide geometric constructions of shapes. If you want to argue about what he meant, what he meant was "you have a straightedge". Similarly, postulate 3 says "you have a compass", and postulate 2 says... "you have a straightedge". You've described drawing several parallel lines that meet at two opposite points of a sphere. Postulate 1 has no problem with that - all of those lines are straight. Postulate 5 has a problem with it, because they meet.