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On extensions of Pascal's theorem (Second paper) Paul Serret's theorem
Published online by Cambridge University Press: 20 January 2009
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The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.
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- Research Article
- Information
- Proceedings of the Edinburgh Mathematical Society , Volume 5 , Issue 3 , February 1938 , pp. 155 - 158
- Copyright
- Copyright © Edinburgh Mathematical Society 1938
References
page 155 note 1 Proceedings of the Edinburgh Mathematical Society (2), 5, pp. 55–62.CrossRefGoogle Scholar
page 155 note 2 See Encyk. d. math. Wiss. III 22A, pp. 836–837.Google Scholar
page 155 note 3 It was published in 1869 in a book bearing the misleading title Géométrie de Direction, which consisted almost entirely of deductions from the theorem. Serret also used the reciprocal theorem on sets of lines, and of its extension to three dimensions ; but at that date naturally did not mention the equally obvious extension to space of many dimensions. The mode of its publication explains and partly excuses the neglect of the theorem, but it is noteworthy that W. K. Clifford recognised its importance. (Collected Papers, XIV and XV).
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