Page:Carroll - Euclid and His Modern Rivals.djvu/53

. II. § 2.] dealing with them which differs much from mine—not with a view of substituting the new Manual for mine, but in order to make such changes in my proofs as may be thought desirable. There are other matters again, where changes have been suggested, which you ought to consider, but on general grounds, not by examining particular writers.

Let me enumerate what I conceive should be the subjects of your enquiry, arranged in order of importance.


 * (1) The combination, or separation, of Problems and theorems.
 * (2) The treatment of Pairs of Lines, especially Parallels, for which various new methods have been suggested. These may be classified as involving—
 * Infinite series: suggested by.
 * Angles made with transversals:.
 * Equidistance:.
 * Revolving Lines:.
 * Direction:.
 * The substitution of 'Playfair's Axiom' for my Axiom 12.

If your decision, on these two crucial questions, be given in my favour, we may take it as settled, I think, that my Manual ought to be retained as a whole: how far it should be modified to suit modern requirements will be matter for further consideration.


 * (3) The principle of superposition.
 * (4) The use of diagonals in Book II.