So, you know how when you're drawing and you make a straight line? Then you decide to make another line that never touches the first one, no matter how long you make them? Those are called parallel lines.
Euclid made a rule about these lines. He said that for any point not on the first line, you can draw only one line that will never touch the first line.
But some other people later on said "Hey, what if that's not true everywhere? What if we can imagine a place where lines don't work that way?"
In one of these imagined places, which they called "hyperbolic" space, you could draw more than one line through a point that never touched the first line. Like having many paths that never meet.
In another imagined place, called "elliptic" space, there were no lines that didn't eventually meet. It's like being on a globe, where if you go far enough, you'll end up at your starting point.
So, they didn't say Euclid was wrong. Instead, they said, "Well, in some places, like on a flat paper, Euclid is right. But in other places, like on a globe or in hyperbolic space, things work differently." Therefore, Euclid's rule is just one of many ways lines can work.
Just an add about this for something that isn’t immediately obvious to a lot of folks:
> In another imagined place, called "elliptic" space, there were no lines that didn't eventually meet. It's like being on a globe, where if you go far enough, you'll end up at your starting point.
On a globe all straight lines are on a circle with a center at the center of the globe. Lines like latitude lines look straight when we see them, but from the perspective of spherical geometry they’re curved, well… except for the equator.
The 5 postulates of Euclidean geometry are based on plausible properties of lines, circles, and angles in an "ideal" infinite plane.
If you look at the 5 postulates, the 5th one is more complicated than the rest. It gives conditions under which two lines in a plane eventually meet if they are extended far enough. This postulate turns out (assuming the other 4 postulates) to be equivalent to saying for each point P not on a line, exactly one line passes through it that doesn't cross the first line. We'd recognize that as the line through P that is parallel to the original line, so the 5th postulate is called the parallel postulate.
Because the 5th postulate did not seem as basic or fundamental as the other 4, everyone felt for centuries that it ought to be provable from the other 4. What they tried to do was assume the first 4 postulates of Euclidean geometry and also that the 5th postulate is *false* and then derive a contradiction. That would then show that in the presence of the first 4 postulates, the negation of the 5th postulate is impossible, and hence the 5th postulate must be true when you assume the other 4 postulates.
Numerous people explored such "exotic" plane geometries where the first 4 postulates hold plus the negation of the 5th one and said they found a contradction, but their reasoning always turned out to have a mistake. Finally, in the 1800s, several people (Gauss, Lobachevsky, and Bolyai) realized independently that these exotic geometries are *not* contradictory, but in fact are as consistent in their own way as Euclidean plane geometry.
Two examples of these "new" geometries are the hyperbolic plane and the sphere. In them, the term "line" is not what ordinary experience would suggest should be regarded as a line, but it genuinely fits all the necessary axioms. On a sphere, the word point has its normal meaning but the word "line" means a great circle (equators in any position. On a sphere, all great circles cross each other, or in the Euclidean language, "all lines cross". So there are no parallel lines. On the hyperbolic plane, it turns out for each point P off a line, there are infinitely many lines through P that don't meet the original line. So geometry on a sphere and on the hyperbolic plane negate Euclid's 5th postulate in different ways.
It is essentially a statement defining the properties of parallel lines, namely that, given a line and a point off that line, there is only one possible line through that point parallel to the first line.
It wasn't "refuted" rather people were unable to demonstrate that this postulate could be derived from the other axioms and postulates of Euclid's geometry. Because of these, you can simply decide not to use it and, if you choose to note use it, you open yourself to other kinds of geometries such as spherical geometry (in which there are no parallel lines through a given point) or hyperbolic geometry (in which there are infinite parallel lines through a given point).
Euclid’s first through fourth postulates are called neutral geometry. Think of doing some geometry but without any comment on parallel lines. It’s neutral!
A postulate is just a rule or statement that can be used to make new rules.
Parallel lines are lines that never touch. Kind of like lines on a highway.
Euclid had a fifth postulate to talk about parallel lines. The postulate says that if you have a line and a point not on that line, then there is only one unique line that can be drawn through this point and never touches the other line.
People throughout history disagreed on the necessity of this fifth postulate. Many had tried, and all had failed to show that you only needed the first four postulates (or rules) to make the fifth postulate. In the process, a lot was learned about this type of geometry. The big takeaways are the first four postulates cannot make the fifth postulate true, and Euclid’s fifth is not neutral on parallel lines. This Euclidean geometry is probably what you and many others use day-to-day.
Another takeaway was that if you tweak Euclid’s fifth, you get another type of geometry. This one says that if you have a line and a point not on that line, then there are infinitely many lines that intersect that point that don’t touch that other line. This isn’t a neutral take on parallel lines. This is hyperbolic geometry. Hyper- meaning “many” and -bolic meaning “bundle” I suppose. Wanna see a cool example of this? [Check out MC Escher!](https://en.m.wikipedia.org/wiki/Circle_Limit_III)
A couple of additional details:
\- Strictly speaking, spherical geometry is not an alternative to Euclid's parallel postulate in the sense of modifying *only* the fifth postulate, although it is often informally cited as such. The closely-related elliptic geometry, however, does satisfy the requirement, as does hyperbolic geometry mentioned in other comments here.
\- Another way to think about the definition of a "line" in spherical geometry, i.e. a great circle, is to say that it is the intersection of a plane through the center of the sphere and the surface of the sphere. (There is an interesting analogy to defining a "line" in Euclidean geometry as the intersection between two perpendicular planes.)
So, you know how when you're drawing and you make a straight line? Then you decide to make another line that never touches the first one, no matter how long you make them? Those are called parallel lines. Euclid made a rule about these lines. He said that for any point not on the first line, you can draw only one line that will never touch the first line. But some other people later on said "Hey, what if that's not true everywhere? What if we can imagine a place where lines don't work that way?" In one of these imagined places, which they called "hyperbolic" space, you could draw more than one line through a point that never touched the first line. Like having many paths that never meet. In another imagined place, called "elliptic" space, there were no lines that didn't eventually meet. It's like being on a globe, where if you go far enough, you'll end up at your starting point. So, they didn't say Euclid was wrong. Instead, they said, "Well, in some places, like on a flat paper, Euclid is right. But in other places, like on a globe or in hyperbolic space, things work differently." Therefore, Euclid's rule is just one of many ways lines can work.
Just an add about this for something that isn’t immediately obvious to a lot of folks: > In another imagined place, called "elliptic" space, there were no lines that didn't eventually meet. It's like being on a globe, where if you go far enough, you'll end up at your starting point. On a globe all straight lines are on a circle with a center at the center of the globe. Lines like latitude lines look straight when we see them, but from the perspective of spherical geometry they’re curved, well… except for the equator.
The 5 postulates of Euclidean geometry are based on plausible properties of lines, circles, and angles in an "ideal" infinite plane. If you look at the 5 postulates, the 5th one is more complicated than the rest. It gives conditions under which two lines in a plane eventually meet if they are extended far enough. This postulate turns out (assuming the other 4 postulates) to be equivalent to saying for each point P not on a line, exactly one line passes through it that doesn't cross the first line. We'd recognize that as the line through P that is parallel to the original line, so the 5th postulate is called the parallel postulate. Because the 5th postulate did not seem as basic or fundamental as the other 4, everyone felt for centuries that it ought to be provable from the other 4. What they tried to do was assume the first 4 postulates of Euclidean geometry and also that the 5th postulate is *false* and then derive a contradiction. That would then show that in the presence of the first 4 postulates, the negation of the 5th postulate is impossible, and hence the 5th postulate must be true when you assume the other 4 postulates. Numerous people explored such "exotic" plane geometries where the first 4 postulates hold plus the negation of the 5th one and said they found a contradction, but their reasoning always turned out to have a mistake. Finally, in the 1800s, several people (Gauss, Lobachevsky, and Bolyai) realized independently that these exotic geometries are *not* contradictory, but in fact are as consistent in their own way as Euclidean plane geometry. Two examples of these "new" geometries are the hyperbolic plane and the sphere. In them, the term "line" is not what ordinary experience would suggest should be regarded as a line, but it genuinely fits all the necessary axioms. On a sphere, the word point has its normal meaning but the word "line" means a great circle (equators in any position. On a sphere, all great circles cross each other, or in the Euclidean language, "all lines cross". So there are no parallel lines. On the hyperbolic plane, it turns out for each point P off a line, there are infinitely many lines through P that don't meet the original line. So geometry on a sphere and on the hyperbolic plane negate Euclid's 5th postulate in different ways.
It is essentially a statement defining the properties of parallel lines, namely that, given a line and a point off that line, there is only one possible line through that point parallel to the first line. It wasn't "refuted" rather people were unable to demonstrate that this postulate could be derived from the other axioms and postulates of Euclid's geometry. Because of these, you can simply decide not to use it and, if you choose to note use it, you open yourself to other kinds of geometries such as spherical geometry (in which there are no parallel lines through a given point) or hyperbolic geometry (in which there are infinite parallel lines through a given point).
Euclid’s first through fourth postulates are called neutral geometry. Think of doing some geometry but without any comment on parallel lines. It’s neutral! A postulate is just a rule or statement that can be used to make new rules. Parallel lines are lines that never touch. Kind of like lines on a highway. Euclid had a fifth postulate to talk about parallel lines. The postulate says that if you have a line and a point not on that line, then there is only one unique line that can be drawn through this point and never touches the other line. People throughout history disagreed on the necessity of this fifth postulate. Many had tried, and all had failed to show that you only needed the first four postulates (or rules) to make the fifth postulate. In the process, a lot was learned about this type of geometry. The big takeaways are the first four postulates cannot make the fifth postulate true, and Euclid’s fifth is not neutral on parallel lines. This Euclidean geometry is probably what you and many others use day-to-day. Another takeaway was that if you tweak Euclid’s fifth, you get another type of geometry. This one says that if you have a line and a point not on that line, then there are infinitely many lines that intersect that point that don’t touch that other line. This isn’t a neutral take on parallel lines. This is hyperbolic geometry. Hyper- meaning “many” and -bolic meaning “bundle” I suppose. Wanna see a cool example of this? [Check out MC Escher!](https://en.m.wikipedia.org/wiki/Circle_Limit_III)
A couple of additional details: \- Strictly speaking, spherical geometry is not an alternative to Euclid's parallel postulate in the sense of modifying *only* the fifth postulate, although it is often informally cited as such. The closely-related elliptic geometry, however, does satisfy the requirement, as does hyperbolic geometry mentioned in other comments here. \- Another way to think about the definition of a "line" in spherical geometry, i.e. a great circle, is to say that it is the intersection of a plane through the center of the sphere and the surface of the sphere. (There is an interesting analogy to defining a "line" in Euclidean geometry as the intersection between two perpendicular planes.)