The Art of the Infinite by Robert Kaplan & Ellen Kaplan

Author:Robert Kaplan & Ellen Kaplan
Language: eng
Format: epub
ISBN: 9780141937076
Publisher: Penguin Books Ltd
Published: 2004-11-14T16:00:00+00:00

x2+y2 = r2

or

y =

The final modification displaces the circle’s center from (0,0) to some other point (h,k) on the plane:

Now the circle’s equation is

(x − h)2 + (y − k)2 = r2

so

(y − k)2 = r2 − (x − h)2

which gives us

or

This is the algebraic form of a circle. It is the form latent in all the circles drawn in sand, on paper, or on your thigh. Without sweating any details we see that if x, r, h, and k lie in some square root extension field F., y will lie in at worst the next link from it.

Now we can take on the intersection of a circle and a line that both arise from some Fi. We hope that what points they have in common are in Fi. too, or in a square root extension link from it.

Because the equation for a circle is more complicated than that of a line, the tactics for doing this will be more intricate than they were when we looked at the intersection of two lines—but the strategy remains exactly the same: to show that whatever happens, no more than arithmetic combinations and square rooting will be involved.

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