[FOM] Standard Language of Euclid

Vaughan Pratt pratt at cs.stanford.edu
Fri Oct 2 17:13:03 EDT 2009

Andrej Bauer wrote:
> Geometrically speaking, we're trying to see how to interpolate a
> parametric cubic polynomial curve through vertices of a convex
> quadrilateral. It's probably a judgement call as to whether that's "an
> increase in complexity", but it sure sounds "simple" to me.

The additional requirement that the norm of the curve also be a cubic is 
equivalent to the representability of a sixth-degree polynomial as the 
square of a cubic one.  This in turn is equivalent to requiring that its 
six roots form three double roots.

Book I of Euclid is all about iterated quadratics.  Cubics are beyond 
the "standard language of Euclid."

Vaughan Pratt

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