Page:Carroll - Euclid and His Modern Rivals.djvu/21

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Table III. Containing five Propositions, taken from Table II, which have been proposed as Axioms ....................................................................................................................................................................................................................................................

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(1)

Euclid's Axiom;

(2)

T. Simpson's Axiom;

(3)

Clavius' mpsonAxiom„

(4)

Playfair's psonAxiom„

(5)

R. Simpson's Axiom.

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:—

(1)

Playfair's does not show which way the Lines will meet; ....................................................................................................................................................................................................................................................

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(2)

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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b