Page:Carroll - Euclid and His Modern Rivals.djvu/280

242 our time how the history of science teaches in the clearest language that the struggle against self-imposed restrictions has been of the most signal service in the advancement of knowledge.

The use of hypothetical constructions will not present itself often enough to produce any very great saving in the demonstrations; while the difficulty which they produce to many beginners, as shown by the experience to which I have already referred, is a fatal objection to them. Why should a beginner not assume that he can draw a circle through four given points if he finds it convenient?

Finally, I hold that Euclid, in his solution of the problems he requires, supplies matter which is simple and attractive to beginners, and which therefore adds practically nothing to their labours, while it has the advantage of rendering his treatise far more rigorous and convincing to them.

The objections against Euclid's order seem to me to spring mainly from an intrusion of natural history into the region of mathematics; I am not the first to print this remark, though it occurred to me independently. It is to the influence of the classificatory sciences that we probably owe this notion that it is desirable or essential in our geometrical course to have all the properties of triangles thrown together, then all the properties of rectangles, then perhaps all the properties of circles; and so on. Let me quote authority in favour of Euclid, far more impressive than any which on this point has been brought against him: "Euclid … fortunately for us, never dreamed of a geometry of triangles as distinguished from a geometry of circles, … but made one help out the other as he best could."

Euclid has been blamed for his adherence to the syllogistic method; but it is not necessary to say much on this point, because the reformers are not agreed concerning it: those who are against the syllogism may pair off with those who are for the syllogism. We are told in this connexion that, "the result