Page:Carroll - Euclid and His Modern Rivals.djvu/250

212 § 2. Euclid's Constructions.

I am told that you indulge too much in 'arbitrary restrictions.' Mr. Reynolds says (Pref. p. vi.) 'The arbitrary restrictions of Euclid involve him in various inconsistencies, and exclude his constructions from use. When, for instance, in order to mark off a length upon a straight Line, he requires us to describe five Circles, an equilateral Triangle, one straight line of limited, and two of unlimited length, he condemns his system to a divorce from practice at once and from sound reason.'

Euc. Mr. Reynolds has misunderstood me: I do not require all that construction in Prop. 3. To explain my meaning I must go back to Prop. 2, and I must ask your patience while I make a few general remarks on construction. The machinery I allow consists of a pencil, a ruler, and a pair of compasses to be used for drawing a Circle about a given centre and passing through a given point (that is what I mean by 'at any distance'), but not to be used for transferring distances from one part of a diagram to another until it has been shown that such transference can be effected by the machinery already allowed.

Min. But why not allow such transference without proving that possibility?

Euc. Because it would be introducing as a Postulate what is really a Problem. And I go on the general principle of never putting a Problem among my Postulates, nor a Theorem among my Axioms.