Geometry Without Axioms; Or the First Book of Euclid's Elements: With Alterations and Familiar Notes; And an Intercalary Book in Which the Straight Line and Plane Are Derived from Properties of the Sphere; Being an Attempt to Get Rid of Axioms and Postula

Excerpt from Geometry Without Axioms; Or the First Book of Euclid's Elements: With Alterations and Familiar Notes; And an Intercalary Book in Which the Straight Line and Plane Are Derived From Properties of the Sphere; Being an Attempt to Get Rid of Axioms and Postulates; And Particularly to Establish the Theory of Parallel Lines W

XXVIII D in the First Book, where the conclusion rests entirely on the impossibility of a certain line ceasing to cut a series of other lines during a continuous motion, in a way incapable of being supplied by any succession of insulated positions as like wise in part of Prop. XII Cor. 8 of the Intercalary Book, and else where. On the whole therefore it may be an interesting question in what place a geometer would be warranted in saying, I could have proved these preliminaries, but it would have been necessary to disturb the order which directs that lines be treated of first and solids afterwards, and to introduce motion; for which reason it was considered better that they should be adopted without proof.'

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