Euclid's Fifth Postulate

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Euclid

Euclid was a Mathematician from ancient Greece who lived around 300BC. He is sometimes referred to as the Father of Geometry, and there is no doubt that his work has probably influenced Mathematicians more than any other especially in the realm of geometry.

Axioms and Postulates

Euclid's Geometry is based on axioms and postulates. There is no clear distinction made between the words postulate and axiom in the world of Mathematics and at times both are used interchangeably.

An axiom or postulate is generally accepted to mean that what is stated is self evident or recognised as universally true and as such no proof is required. So an axiom is therefore a statement we are asked to accept as a truth above question.

Fallible Axioms

People often use axioms in their arguments because they contain a line of defence that places them above question. It doesn't matter how clever the argument as soon as here the words self evident and fact the alarm bells go off in my head. Let me explain why.

Deductive geometry which is the geometry of Euclid and which for a long time was accepted as accurately describing our spatial experience is based on Euclid's axioms. Following are the first five postulates published around 300BC:

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Proving Euclid's Parallel Postulate

Axiom #5 is also called Euclid's Parallel Postulate and it is on the basis of this postulate that squares were proved to exist. But for centuries after its publication Mathematicians all over the world attempted to prove the fifth postulate of Euclid but all to no avail.

A Jesuit Priest and Mathematician Girolamo Saccheri made it his life's work to prove Euclid's Parallel Postulate. Saccheri also used reductio ad absurdum or Proof by Contradiction to prove the Parallel Postulate. By making the assumption that the Parallel Postulate was false Saccheri then proceeded expecting a contradiction to pop up which would then allow him to conclude that the assumption that the postulate was false could not be true. Saccheri could not find such a contradiction so he had unwittingly discovered a completely new way of perceiving space completely separate from the space described by Euclid's Geometry and which we now call Hyperbolic Geometry. A space where something Euclid said was self evident was not so evident after all.

Do Squares Really Exist?

Slarty O'Brian a respected and much admired writer and philosopher in HubPages wrote a hub called "Nothing, what is it?" and in one of his comments he talks about the impossibility of a square circle and in this hub the possibility of the existence squares is questioned.

Without Euclid's Parallel Postulate upon which the existence of squares is based we are not in a position to categorically declare that squares actually exist! Q.E.D.

Non-Euclidean Geometry Part 1

Non-Euclidean Geometry Part 2

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Comments 14 comments

RealHousewife profile image

RealHousewife 5 years ago from St. Louis, MO

I am so glad you defined "postulate" now I've learned a new word and been confused in ways you and Slarty can not fathom:). Of course squares exist - here I am! Imagine that! Lol

You are too smart for me!


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Thank you RealHousewife for taking the time to visit and read this one. I am fascinated with Non-Euclidean geometry and thought the idea of writing about squares being an illusion would really annoy Slarty LOL! You are no illusion and neither are you a square.


Alastar Packer profile image

Alastar Packer 5 years ago from North Carolina

Geometries never been a fave or strong point here Spirit; but this was pretty interesting. Squares are nothing but lucid dreams anyway..LOL. Hope the goose is hanging high for you these days Irish.


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Thank you for the visit and the comment and I think my goose hangs well enough! LOL!

I was just watching a program on the History Channel about the Civil War and the enlisting of negroes in the Union to fight the Confederacy. It was a really interesting programme and I thought of you. Then I go to my computer and there you have made a comment on my hub! You have a good day Dixie and keep up the good work.


RealHousewife profile image

RealHousewife 5 years ago from St. Louis, MO

SW - I wish your little island was closer - the girls always need math help and my brain doesn't like to go there!

Do you notice that lots of writers don't like math? I seem to hear that a lot and wonder if that's the left brain/ right brain thing?


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Hello again RH, A man brought his parrot to the vet because it would not stop coughing. The vet simply said to the owner of the parrot: "Stop smoking, and then maybe your own cough will clear up. The parrot is imitating you."

The best advice I can give any parent is to be the person they want their children to be. If you accept a limiting belief then that is your gift to your children. Why would you expect your children to be mathematical geniuses if you openly accept that your brain doesn't like to go there?

Here is a great rap to help you rethink the beliefs you hold about yourself because we can change our beliefs at any time.

I hope you don't mind the lecture but I feel you can hear me. http://www.youtube.com/watch?v=y_7CtUzCA-k


RealHousewife profile image

RealHousewife 5 years ago from St. Louis, MO

I hear you SW! I don't tell my kids I don't understand it! lol They think I do! It takes me forever to go through their books to re-figure it out....but thank you...I totally know what you mean. I don't want to create ANY self fulfilled prophecies in my kids:)


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

It is not what we say that is important but our thoughts and feelings that count in that communication that goes on below the level of our conscious awareness. What we say accounts for only a small fraction of what we actually communicate to our children. This is the important point and one that few fully realise. How you think and feel about the subject is what you communicate regardless of what you say so change that and you will see a change in your children's attitude to the subject. Easier said than done though isn't it? That is what I do for a living ...teaching people how to change their own self limiting beliefs. Most of my hubs are about this very topic. Good night. Tomorrow is 11/11/11 and I will be 19 years married to my Mary.


RealHousewife profile image

RealHousewife 5 years ago from St. Louis, MO

Yes I agree with you - and I've told them in advance I have many flaws - I can accept that and they need to as well:) lol

Comgratulations on 19 years together! Best wishes to you both!


Alastar Packer profile image

Alastar Packer 5 years ago from North Carolina

HPs rampant with synchronicity these days Irish. As concerns what you saw on the History Channel, you'd like the 1989 movie 'Glory.'


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

Alastair, I love the movie Glory. You might write ahub about HPs synchronicity these days. That would be something!


Spirit Whisperer profile image

Spirit Whisperer 5 years ago from Isle of Man Author

RealHousewife I am actually saying the opposite ... the faults you say you have are not real but imagined and in accepting them you are telling your children to as well. Why would you this? I am saying that in order to change and be the very best that we can be we must not accept anything about ourselves that we perceive as negative.

Have a look at this video http://www.youtube.com/watch?v=SipVykiE6mY&feature...


farmvillefcu profile image

farmvillefcu 5 years ago

This is one of the most comprehensive reference regarding Euclid's Fifth Postulate. I could still recall during my master's class in Geometry which we are assigned to prove Euclid's Fifth Postulate. Thanks for the videos which was also very entertaining.

http://www.farmvillefcu.com/


vip 4 years ago

moody but not fukre

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