Postulates that Euclid neglected to mention?

  • Thread starter LeonhardEuler
  • Start date
  • Tags
    Postulates
In summary, Euclid did not mention several postulates in his presentation of geometry, including the ability to draw a line from any point to any other point, the extension of a straight line, the drawing of a circle with any center and radius, the equality of all right angles, and the condition for two lines to eventually intersect. These postulates were likely already widely accepted and known in Euclid's time. They are fundamental in constructing geometric figures and proving theorems in Euclidean geometry. While considered to be true by many, some argue that they are merely assumptions. Other postulates, such as the parallel postulate, were also not mentioned by Euclid and have sparked much debate and the development of non-Euclidean geometries
  • #1
LeonhardEuler
Gold Member
860
1
Out of curiosity, I was recently reading through Euclid's Elements online (http://aleph0.clarku.edu/~djoyce/java/elements/elements.html" )

My question is, is this the general opinion of all mathematicians? Also, has anyone tried to fix this problem, and how many more postulates would they need? I'm really surprised because I thought for a long time that only those 5 postulates were needed to axiomize Euclidean geometry.
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
It's generally accepted. Hilbert pretty much fixed it. See Foundations of Geometry, D. Hilbert (trans. L. Unger), Open Court Publ.
 
  • #3
Thanks a lot, I'll do that!
 

1. What are the postulates that Euclid did not mention?

The postulates that Euclid did not mention are the following:

  • A line can be drawn from any point to any other point.
  • A finite straight line can be extended continuously in a straight line.
  • A circle can be drawn with any center and radius.
  • All right angles are equal to each other.
  • If a straight line intersects two other straight lines, and the interior angles on the same side of the transversal are less than two right angles, then the two straight lines will eventually intersect on that side.

2. Why did Euclid neglect to mention these postulates?

It is believed that Euclid did not mention these postulates because they were already widely accepted and known among mathematicians of his time. He focused on presenting a rigorous and logical system of geometry based on a small number of basic postulates.

3. How do these postulates impact Euclidean geometry?

The postulates that Euclid neglected to mention have a significant impact on Euclidean geometry. They provide the foundation for constructing geometric figures and proving theorems. Without these postulates, many fundamental concepts and theorems in Euclidean geometry would not be possible.

4. Are these postulates considered to be true or just assumptions?

These postulates are considered to be true since they have been accepted and used by mathematicians for centuries. However, some mathematicians argue that they are just assumptions and cannot be proven to be true. The validity of these postulates is still a topic of debate in the mathematical community.

5. Are there any other postulates that Euclid did not mention?

Yes, there are other postulates that Euclid did not mention, such as the parallel postulate. This postulate states that if a line intersects two other lines and the interior angles on the same side of the transversal are less than two right angles, then those two lines will never intersect. This postulate has been a subject of much interest and debate in the history of mathematics and has led to the development of non-Euclidean geometries.

Similar threads

A Non-Euclidean geometry help
    • General Math
Replies
3
Views
1K
Is Euclid's Fourth Postulate Redundant?
    • General Math
Replies
4
Views
2K
Why do we need postulate 4 in Euclid's element (P14)?
    • General Math
Replies
2
Views
2K
Is Euclid's Postulate 5 really a postulate of parallels ?
    • Differential Geometry
Replies
18
Views
9K
Euclid's elements book 3 proposition 20
    • General Math
Replies
4
Views
2K
A How did Saccheri prove Euclid's Fifth Postulate?
    • General Math
Replies
6
Views
2K
Euclid's elements proposition 15 book 3
    • General Math
Replies
2
Views
1K
I The world from a bat's perspective (sonic images)
    • Classical Physics
Replies
21
Views
1K
Why is there a C? - Exploring the Singularities of Fixed Speed
    • Special and General Relativity
Replies
16
Views
3K
Why Study Curved Space? Exploring the Assumptions and Realities of the Universe
    • Astronomy and Astrophysics
Replies
24
Views
5K

Hot Threads

Recent Insights

Back
Top