Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Replies: 62   Last Post: Feb 22, 2014 12:11 AM

 Search Thread: Advanced Search

 Messages: [ Previous | Next ]
 David Hartley Posts: 463 Registered: 12/13/04
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 17, 2014 6:20 PM
 Plain Text Reply

In message <0c7fbbac-f296-456f-bf77-68290a164ba3@googlegroups.com>, Ken
Quirici <kquirici@yahoo.com> writes
>> Consider a system consisting of a single straight line and the points
>>
>> thereon. It satisfies all Euclid's explicit postulates (and Pasch's
>>
>> axiom) but circles have only two points in their circumferences and
>>
>> circles with a common radius do not intersect. There are no equilateral
>>
>> triangles.
>>
>>

>
>there's problem in that his definitions include (at least in Heath's
>translation) surface - that which has length and breadth only.
>
>Does this vitiate your statement?
>

No, the axioms don't imply that a surface exists.

Of course the problem of non-standard models like the one I've suggested
is trivial to fix. Just add axioms to say that there is at least one
line and a point not on that line, (and a plane which does not contain
the line if you want 3-D). I assume, from the discussions I've seen on
the web, there are models which satisfy this but still have circles
which don't intersect even though they "should". Further Googling would
probably turn up an example but for now it's more fun trying to find one
on my own.

....
>Thanks. I'm hoping Moise's book on Elementary Geometry from an Advanced
>Standpoint will provide a sufficiently rigorous approach. Have you read
>it? If so, is it sufficiently rigorous?

No I haven't read it. I've never studied Euclid's axioms before.

--
David Hartley

Date Subject Author
2/14/14 ken quirici
2/14/14 ross.finlayson@gmail.com
2/15/14 thenewcalculus@gmail.com
2/15/14 bert
2/15/14 thenewcalculus@gmail.com
2/16/14 ross.finlayson@gmail.com
2/16/14 thenewcalculus@gmail.com
2/15/14 thenewcalculus@gmail.com
2/15/14 Brian Q. Hutchings
2/15/14 ken quirici
2/15/14 FredJeffries@gmail.com
2/15/14 ken quirici
2/15/14 thenewcalculus@gmail.com
2/16/14 ken quirici
2/16/14 ken quirici
2/16/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 YBM
2/18/14 thenewcalculus@gmail.com
2/17/14 David Hartley
2/17/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 David Hartley
2/17/14 ken quirici
2/19/14 David Hartley
2/20/14 thenewcalculus@gmail.com
2/17/14 ross.finlayson@gmail.com
2/18/14 thenewcalculus@gmail.com
2/17/14 ken quirici
2/17/14 thenewcalculus@gmail.com
2/17/14 Brian Q. Hutchings
2/17/14 thenewcalculus@gmail.com
2/17/14 Ken.Pledger@vuw.ac.nz
2/17/14 thenewcalculus@gmail.com
2/18/14 David Bernier
2/18/14 thenewcalculus@gmail.com
2/18/14 Ken.Pledger@vuw.ac.nz
2/18/14 thenewcalculus@gmail.com
2/18/14 thenewcalculus@gmail.com
2/18/14 ken quirici
2/18/14 thenewcalculus@gmail.com
2/18/14 Wizard-Of-Oz
2/18/14 thenewcalculus@gmail.com
2/18/14 Wizard-Of-Oz
2/19/14 thenewcalculus@gmail.com
2/19/14 Wizard-Of-Oz
2/19/14 Brian Q. Hutchings
2/18/14 thenewcalculus@gmail.com
2/18/14 thenewcalculus@gmail.com
2/18/14 ken quirici
2/18/14 thenewcalculus@gmail.com
2/18/14 Brian Q. Hutchings
2/18/14 thenewcalculus@gmail.com
2/18/14 Brian Q. Hutchings
2/20/14 Brian Q. Hutchings
2/22/14 thenewcalculus@gmail.com

© The Math Forum at NCTM 1994-2018. All Rights Reserved.