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We see now that a plane surface, usually abbreviated to the single word "plane," is a kind of surface. Perhaps the remainder of the statement is a definition of content, but, if so, some words are missing.
One interpretation often given is that if a plane surface contains two points, then it contains the line connecting the two points. If that were the meaning, then it would be just as well to make that the explicit definition or to make it a postulate. examples on online math tutors ; But that does not seem to be Euclid's intent. His proposition XI.7 has a detailed proof that the line joining two points on two parallel lines lies in the plane of the two parallel lines. No proof at all would be necessary if that line were by definition or by postulate contained in a plane that contained its ends.
Note that a plane surface may be infinite, but needn't be infinite. It can be a square, a circle, or any other plane figure. more examples on math tutors online.
We see now that a plane surface, usually abbreviated to the single word "plane," is a kind of surface. Perhaps the remainder of the statement is a definition of content, but, if so, some words are missing.
One interpretation often given is that if a plane surface contains two points, then it contains the line connecting the two points. If that were the meaning, then it would be just as well to make that the explicit definition or to make it a postulate. examples on online math tutors ; But that does not seem to be Euclid's intent. His proposition XI.7 has a detailed proof that the line joining two points on two parallel lines lies in the plane of the two parallel lines. No proof at all would be necessary if that line were by definition or by postulate contained in a plane that contained its ends.
Note that a plane surface may be infinite, but needn't be infinite. It can be a square, a circle, or any other plane figure. more examples on math tutors online.
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