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

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 ken quirici Posts: 336 Registered: 1/29/05
Re: Euclid's Elements Book 1 Proposition 1 - something Euclid missed?
Posted: Feb 17, 2014 9:06 PM

On Monday, February 17, 2014 6:20:09 PM UTC-5, David Hartley wrote:
>
> 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.
>
>

Righto. They don't. The Heath translation (and I think most of the others) divide the axioms into
Postulates and Common Notions and none of them relate to the plane.

>
> 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

OK. I'll just have to check it out (literally) when it arrives from ILL.

Thanks & Regards,

Ken

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