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Chapter 10: Problem 32

State the assumption that Euclid made about parallel lines that was altered in both hyperbolic and elliptic geometry.

Short Answer

Expert verified
Euclid's postulation states that exactly one parallel line can be drawn through a point not on a given line. Hyperbolic geometry modifies this assumption by allowing multiple parallel lines to pass through the point not on the given line. However, elliptic geometry rejects the concept of parallel lines entirely, stating that any two lines in elliptic geometry intersect.

Step by step solution

01

Identify Euclid's Assumption

Euclid's concept related to parallel lines, most commonly known as 'Euclid's Fifth Postulate', or the 'Parallel Postulate', implies that through a point not on a given straight line, exactly one line can be drawn that is parallel to the given line.
02

Explain Hyperbolic Geometry Modification

In Hyperbolic geometry, the similar postulate, also known as the Hyperbolic Parallel Postulate, implies that through a point not on a given line, there pass at least two distinct lines that do not intersect the given line. This modification allows infinite parallel lines through the given point.
03

Explain Elliptic Geometry Modification

In Elliptic geometry, the modification is quite drastic, as it rejects Euclid's postulate completely. There are no parallel lines at all. Any two lines will always intersect each other, thus eliminating the concept of 'parallelism' present in Euclidean and Hyperbolic Geometry.

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