Book contents
- Frontmatter
- PROLOGUE
- Contents
- Act I
- Act II
- Act III
- ACT IV
- Appendices
- I Extract from Mr. Todhunter's essay on ‘Elementary Geometry,’ included in ‘The Conflict of Studies, &c.’
- II Extract from Mr. De Morgan's review of Mr. Wilson's Geometry, in the ‘Athenæum’ for July 18, 1868
- III The enunciations of Tables I—IV, stated in full: with references to the various writers who have assumed or proved them
- IV The various methods of treating Parallels, adopted by Euclid and his Modern Rivals
- V Proof that, if any one proposition of Table II be granted as an Axiom, the rest can be deduced from it
V - Proof that, if any one proposition of Table II be granted as an Axiom, the rest can be deduced from it
Published online by Cambridge University Press: 29 August 2010
- Frontmatter
- PROLOGUE
- Contents
- Act I
- Act II
- Act III
- ACT IV
- Appendices
- I Extract from Mr. Todhunter's essay on ‘Elementary Geometry,’ included in ‘The Conflict of Studies, &c.’
- II Extract from Mr. De Morgan's review of Mr. Wilson's Geometry, in the ‘Athenæum’ for July 18, 1868
- III The enunciations of Tables I—IV, stated in full: with references to the various writers who have assumed or proved them
- IV The various methods of treating Parallels, adopted by Euclid and his Modern Rivals
- V Proof that, if any one proposition of Table II be granted as an Axiom, the rest can be deduced from it
Summary
“…and so we make it quite a merry-go-rounder.” I was obliged to consider a little before I understood what Mr. Peggotty meant by this figure, expressive of a complete circle of intelligence.
It is to be proved that, if any one of the propositions of Table II be granted, the rest can be proved.
Euclid I, 1 to 28, is assumed as proved.
It is assumed that, where two propositions are contranominals, so that each can be proved from the other, it is not necessary to include both in the series of proofs.
It is assumed as axiomatic that either of two finite magnitudes of the same kind may be multiplied so as to exceed the other.
The following Lemmas are assumed as proved. They may easily be proved by the help of Euc. I. 4, 16, and 26.
LEMMA 1.
If, in two quadrilateral figures, three sides and the included angles of the one be respectively equal to three sides and the included angles of the other, the figures are equal in all respects.
[Analogous to Euc. I. 4.
LEMMA 2.
If, in two quadrilateral figures, three angles and the adjacent sides of the one be respectively equal to three angles and the adjacent sides of the other, the figures are equal in all respects.
[Analogous to Euc. I. 26 (a).
LEMMA 3.
It is not possible to draw two perpendiculars to a Line from the same point.
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- Euclid and His Modern Rivals , pp. 275 - 284Publisher: Cambridge University PressPrint publication year: 2009First published in: 1879