Page:Carroll - Euclid and His Modern Rivals.djvu/105

.] Min. We had better have that first.

Nie. Very well. 'The distance of one point from another is the shortest path from one to the other.'

Min. Might we not say 'is the length of the right Line joining them?'

Nie. Yes, that is the same thing.

Min. And similarly we may modify the Definition you gave just now.

Nie. Certainly. 'The distance of a point from a right Line is the length of the perpendicular let fall upon it from the given point.'

Min. What is your next step?

P. 33. ''Ded. G''. 'If points be taken along one of the arms of an angle farther and farther from the vertex, their distances from the other arm will at length be greater than any given straight line.'

In proving this we assume as an Axiom that the lesser of two magnitudes of the same kind can be multiplied so as to exceed the greater.

Min. I accept the Axiom and the proof.

P. 34. Ax. 'If one right Line be drawn in the same Plane as another, it cannot first recede from and then approach to the other, neither can it first approach to and then recede from the other on the same side of it.'

Min. Here, then, you assume, as axiomatic, one of the Propositions of Table II. After this, you ought to have