Page:Carroll - Euclid and His Modern Rivals.djvu/279

Rh to go through the same matter in Euclid. So far as the half time is concerned this seems to be only an expression of belief as to the result of an untried experiment; it is based on the comparison of a few other books with Euclid, one of these being the Course of the late Professor Cape; as I have already stated, actual experience suggests a conclusion directly contrary to the present prediction. As to the number of propositions we readily admit that a reduction might be made, for it is obvious that we may in many cases either combine or separate according to our taste. But the difficulty of a subject does not vary directly as the number of propositions in which it is contained; a single proposition will in some cases require more time and attention than half a dozen others. I have no doubt that the mixture of easy propositions with the more difficult is a great encouragement to beginners in Euclid; and instead of diminishing the number of propositions I should prefer to see some increase: for example I should like to have Euclid i. 26 divided into two parts, and Euclid i. 28 into two parts.

Again, it has been said that Euclid is artificial, and that he "has sacrificed to a great extent simplicity and naturalness in his demonstrations;" it is a curious instance of the difference of opinion which we may find on the same subject, for, with a much wider experience than the writer whom I quote, I believe that Euclid maintains, and does not sacrifice, simplicity and naturalness in his system, assuming that we wish to have strictness above all things.

The exclusion of hypothetical constructions has been represented as a great defect in Euclid; and it has been said that this has made the confused order of his first book necessary. Confused order is rather a contradictory expression; but it may be presumed that the charge is intended to be one of confusion: I venture to deny the confusion. I admit that Euclid wished to make the subject depend on as few axioms and postulates as possible; and this I regard as one of his great merits; it has been shown by one of the most distinguished mathematicians of