THE OTHER MODERN RIVALS.

Min. I consider the question, as to whether Euclid's system and numeration should be abandoned or retained, to be now set at rest: the subject of Parallels being disposed of, no minor points of difference can possibly justify the abandonment of our old friend in favour of any Modern Rival. Still it will be worth while to examine the other writers, whose works you have brought with you, as they may furnish some valuable suggestions for the improvement of Euclid's Manual.

Nie. The other writers are Chauvenet, Loomis, Morell, Reynolds, and Wright.

Min. There are a few matters, as to which we may consider them all at once. How do they define a straight Line?

Nie. All but Mr. Reynolds define it as the shortest distance between two points, or more accurately, to use the 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?