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Note on the Parallel-Postulate

Published online by Cambridge University Press:  03 November 2016

A. A. Krishnaswami Ayyangar
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Extract

Some teachers may not have noticed that there is a real connection between the parallel postulate and another and more fundamental property of space which it is also more natural to assume, viz. that stated by John Wallis in the seventeenth century:

Type
Research Article
Information
The Mathematical Gazette , Volume 12 , Issue 172 , October 1924 , pp. 191 - 194
Copyright
Copyright © Mathematical Association 1924

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References

page 191 note * In a footnote on p. 167, vol. xii., Mathematical Gazette, July 1924, Prof. 31. J. M. Hill remarks that either his proof or Prof. Nunn’s or some equivalent proof should find a place in every textbook of elementary geometry in place of Euclid’s Postulate of Parallels. I have been therefore tempted to sketch here an equivalent proof, which, I believe, will be at least as commendable as the others. On p. 413, vol. xi., Prof. Hill mentions that he has received a full statement of Prof. Nunn’s Arguments which, so far as I know, has not been published anywhere, in extenso and I do not know therefore how far I may have been anticipated by him.

page 192 note * Vide ii. (3) infra where we set off AM=BE in AB and in order that M may fall between A and B, AB should be greater than BE and this will be secured if AB ≥ BC, BCbeing obviously greater than BE.