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To the pure geometer the radius of curvature is an incidental characteristic - like the grin of the Cheshire cat. To the physicist it is an indispensable characteristic. It would be going too far to say that to the physicist the cat is merely incidental to the grin. Physics is concerned with interrelatedness such as the interrelatedness of cats and grins. In this case the "cat without a grin" and the "grin without a cat" are equally set aside as purely mathematical phantasies.
The Expanding Universe..
Loci: Convergence
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Van Schooten's Ruler Constructions
Third solution to Problem II
Another way: Assume, as before, C is a point away from AB, and from B through C is drawn an indefinitely long line, and find on it CD equal to CB, and join it to A to form AD. Then in DA, assume DE equals DC and DF equals CB. [Here, van Schooten gives five illustrations, showing each of the following five cases: 1) DF = DA, 2) DF < DA, 3) DV > DA but DE < DA, 4) DE = DA, and 5) DE > DA. We illustrate case # 2)] Make VE, FC intersecting at G, then DGH cutting FB in H.
If now F falls on the point A, then the lines FB and AB coincide, and then H bisects it.
But if the point F falls beyond or within A, I say that if the line is drawn through the points C and H, it will bisect the line AB at I.
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