Page:Carroll - Euclid and His Modern Rivals.djvu/21
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ARGUMENT OF DRAMA.
xvii
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Table III. Containing five Propositions, taken from Table II, which have been proposed as Axioms
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Euclid's Axiom;
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| (2) |
T. Simpson's Axiom;
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Clavius' Axiom„
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| (4) |
Playfair's Axiom„
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| (5) |
R. Simpson's Axiom.
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It will be shown (in Appendix III) that any Theorem of Table II is sufficient logical basis for all the rest
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§ 5. Playfair's Axiom.
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Is Euclid's 12th Axiom axiomatic?
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Need of test for meeting of finite Lines
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Considerations which make Euclid's Axiom more axiomatic
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Euclid's Axiom deducible from Playfair's
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Reasons for preferring Euclid's Axiom:—
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| (1) |
Playfair's does not show which way the Lines will meet;
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Playfair's asserts more than Euclid's, the additional matter being superfluous.
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Objection to Euclid's Axiom (that it is the converse of I. 17) untenable
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§ 6. Principle of Superposition.
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Used by Moderns in Euc. I. 5
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Used by Moderns„ in Euc.„ I. 24
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