Page:Carroll - Euclid and His Modern Rivals.djvu/33

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Reasons for retaining it:—

needed to prove I. 8 ....................................................................................................................................................................................................................................................

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not included in new I. 8 ....................................................................................................................................................................................................................................................

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proves rigidity of Triangle ....................................................................................................................................................................................................................................................

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I. 7, 8 analogous to III. 23, 24 ....................................................................................................................................................................................................................................................

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bears on practical science ....................................................................................................................................................................................................................................................

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Omission of II. 8 suggested ....................................................................................................................................................................................................................................................

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Reason for retaining it, its use in Geometrical Conic Sections ....................................................................................................................................................................................................................................................

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Alterations suggested:—

New proofs for I. 5:—

'hypothetical construction' ....................................................................................................................................................................................................................................................

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superposition ....................................................................................................................................................................................................................................................

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treating sides as 'obliques' ....................................................................................................................................................................................................................................................

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treating sides„ as radii of a Circle ....................................................................................................................................................................................................................................................

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Inversion of order of I. 8, 24 rejected ....................................................................................................................................................................................................................................................

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Inversion of order of„ I. 18,19,20; do. ....................................................................................................................................................................................................................................................

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Fuller proof of I. 24; accepted ....................................................................................................................................................................................................................................................

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Algebraical proofs of II.; rejected ....................................................................................................................................................................................................................................................

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Additions suggested:— ....................................................................................................................................................................................................................................................

New Axiom; accepted ....................................................................................................................................................................................................................................................

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Two new Theorems; do. ....................................................................................................................................................................................................................................................

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§ 7. The summing-up.

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Euclid's farewell speech ....................................................................................................................................................................................................................................................

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