Understanding Euclid's Sixth Postulate: Exploring the Foundations of Geometry

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In summary, the conversation is about Euclid's sixth postulate and its mention in various sources. The speaker is unsure about the postulate's validity and asks for clarification. Another speaker mentions that it is a modern invention and provides a source for more information. The conversation ends with a request for insight on the postulate.
  • #1
chemistry1
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Hey,

Has anybody heard of Euclid's sixth postulate ? It says : Two lines do not contain a space.

I don't know why, but I'm only finding this postulate in my things, not anywhere else... Help ? Thanks
 
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  • #2
I'm not an expert on Euclidean geometry, but I've never heard of it.

Where exactly did you encounter this?
 
  • #4
It is purely a modern invention
http://en.wikipedia.org/wiki/Euclidean_geometry
'Some modern treatments add a sixth postulate, the rigidity of the triangle, which can be used as an alternative to superposition.'

p 5. Coxeter, H.S.M. (1961). Introduction to Geometry. New York: Wiley.
 
  • #5
Can someone give me insight on it ? Thank you.
 

What is Euclid's sixth postulate?

Euclid's sixth postulate, also known as the parallel postulate, is one of the five postulates that form the basis of Euclidean geometry. It states that if a line intersects two other lines forming two interior angles on the same side that sum to less than 180 degrees, then the two lines, if extended indefinitely, will intersect on that side on which the angles sum to less than 180 degrees.

Why is Euclid's sixth postulate important?

Euclid's sixth postulate plays a crucial role in the development of Euclidean geometry and is essential in proving many geometric theorems. It also distinguishes Euclidean geometry from other non-Euclidean geometries.

What are some implications of Euclid's sixth postulate?

Euclid's sixth postulate implies that parallel lines never intersect, and the sum of the interior angles of a triangle is always 180 degrees. It also implies that the shortest distance between two points is a straight line.

What is the history of Euclid's sixth postulate?

Euclid's sixth postulate was first stated by Euclid in his book "The Elements" around 300 BCE. It was later challenged by mathematicians, including Proclus and John Wallis, who attempted to prove it from the other four postulates. It was not until the 19th century that mathematicians like Johann Lambert and Carl Friedrich Gauss questioned the validity of the parallel postulate and developed non-Euclidean geometries.

What are some modern applications of Euclid's sixth postulate?

Euclid's sixth postulate is still used in modern geometry, architecture, and engineering to solve problems involving parallel lines and triangles. It is also essential in the study of non-Euclidean geometries, which have applications in fields such as physics and computer graphics.

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