History of Hyperbolic Geometry

Lobachevsky-Bolyai-Gauss Geometry

Hyperbolic geometry is a non-Euclidean geometry is also known as Lobachevsky-Bolyai-Gauss geometry, due to its cofounders. This geometry satisfies all of Euclid's postulates except for the fifth postulate, the parallel postulate. Euclid's Fifth Postulate Reads: "If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." This postulate is modified for Hyperbolic Geometry to read: "For any infinite straight line L and any point P not on it, there are many other infinitely extending straight lines that pass through P and which do not intersect L.

Euclids Fifth Postulate

Since the postulate itself can be difficult to understand just from reading it, interact with the display below to understand what Euclid's 5th Postulate means

Drag the sliders to change the angles n and m

Make one of the angles less than 90 degrees

Do the lines intersect on that side?

Select all that apply

A
Yes
B
No

Check my answer (3)

Next We Have The Modified Parallel Postulate for Hyperbolic Geometry

Try constructing these lines in the applet below

After constructing the four lines (PQ, PR, PS, PT)

Did any of these lines intersect the given line L?

Select all that apply

A
Yes
B
No

Check my answer (3)

Try constructing additional lines

How many lines do you think you could construct that go through point P, but do not intersect with line L? Why?