Page:Carroll - Euclid and His Modern Rivals.djvu/197

Rh words of Mr. Chauvenet, 'a Line of which every portion is the shortest Line between the points limiting that portion.

Min. We discussed that Definition in M. Legendre's book. How does Mr. Reynolds define it?

Nie. Not at all.

Min. Very cautious. What of angles?

Nie. Some of them allow larger limits than Euclid does. Mr. Wright talks about 'angles of continuation' and 'angles of rotation.'

Min. Good for Trigonometry: not so suitable to early Geometry. How do they define Parallels?

Nie. As in Euclid, all of them.

Min. And which Proposition of Tab. II. do they assume?

Nie. Playfair's, or else its equivalent, 'only one Line can be drawn, parallel to a given Line, through a given point outside it.'

Min. Now let us take them one by one.