The proof in 'What is mathematics' reads: 'there exists a transformation of the plane into itself which has the following properties:a)the given circle is fixed under the transformation b)any straight line is carried into a straight line c)the center of the circle is carried into some other point.
The mere exsitence of such a transformation shows the impossibility of constructing with the straightedge alone the center of the given circle. For, whatever the construction maight be, it would consist in drawing a certain number of straight lines and finding their interections with one another and with the given circle. Now if the whole figure.... is subjected to the transformation above..... it will yeild as result a point othter than the center of the given circle. Hence such construction is impossible.'
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