`Circular Definitions`
` `
`Subject:      RE: Reply to "Do Points Have Area?"`
`Author:       Jesse Yoder < jesse@flowresearch.com>`
`Date:         Mon, 2 Feb 1998 09:01:27 -0500`
` `
`Bill - On the face of it, I agree. I think it is very difficult to come`
`up with a noncircular definition of a circle. Notice, however, that the`
`definition you quote is not viciously circular, since it does not`
`contain the word 'circle' - instead, it contains the word 'circular.' On`
`the other hand, as I have argued elsewhere (in a post to Candice), the`
`Euclidean definition of the circle as a set of points equidistant from a`
`fixed point is BANKRUPT, since it allows for the possibility that four`
`points form a circle.`
` `
`In any system, some concepts or terms must be accepted as undefined. Not`
`every concept can be defined in terms of previously defined concepts.`
`Garry (Mr K K G Yau) has urged me in the past to accept 'point' as`
`undefined. I am unwilling to do this, because the difference between`
`points and Points (which are points with area) is fundamental to the`
`Circular Geometry I have developed. `
` `
`On the other hand, it may be necessary to accept certain operations as`
`fundamental and undefined. In particular, it may be necessary to accept`
`as undefined the concept of circular motion (as in a Circle is generated`
`by moving a Point in a circular motion). Likewise, if we are to suffer`
`with the concept of a straight line (which I have it appears been forced`
`to accept against my wishes, since I need the concept of radius and`
`diameter), perhaps the concept of "straight" should also be undefined,`
`as in a straight Line is generated by moving a Point in a single uniform`
`direction--the path so generated is a straight Line. (Note that a curve`
`is generated by moving a Point in a nonuniform direction). `
` `
`Possibly the idea of circular motion could be specified without using`
`the concept of circle or circularity, e.g., by moving a Point in such a`
`way as to maintain a constant distance from a fixed point until the`
`Point intersects with itself--the path so generated is a Circle. If you`
`accept this, then this is a noncircular definition of the circle.`
` `
`Jesse`
` `

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