Euclid’s fifth axiom says that given a line $l$ and a point $P$ not on the line, there is

**exactly one line**parallel to $l$ passing through the point $P$. For centuries people thought that Euclid’s fifth axiom was “obvious”. But some mathematicians did not find it obvious.

Finally, Reimann and Lobachevsky, both modified the axiom and tried to derive a new geometry.

Reimann began with the axiom: Given a line $l$ and a point $P$ not on the line, there is

Similarly, if we take a hyperbola ($y=1/x$) and rotate it around the y-axis, then we obtain a surface where Lobachevsky’s geometry holds. Lobachevsky’s geometry contains the axiom: Given a line $l$ and a point $P$ not on the line, there is

There is a property of the surface (known as

**no line parallel**to $l$ passing through the point $P$. Reimann derived many geometrical theorems that are applicable on the surface of a sphere. For example, he showed that the sum of angles of a triangle is always greater than 180 degrees. Try drawing a triangle on a sphere and see why this has to be true.Similarly, if we take a hyperbola ($y=1/x$) and rotate it around the y-axis, then we obtain a surface where Lobachevsky’s geometry holds. Lobachevsky’s geometry contains the axiom: Given a line $l$ and a point $P$ not on the line, there is

**more than one line**parallel to $l$ passing through the point $P$. In this geometry, the sum of angles of a triangle is always less than $180$ degrees.There is a property of the surface (known as

**curvature**) that determines the geometry. Only surfaces with curvature zero follow the Euclidean geometry. Another example of a surface is that of a saddle (of a horse). Can you tell which geometry is applicable on this surface?