Page:Carroll - Euclid and His Modern Rivals.djvu/262

224 to remember. Algebraical proofs of course introduce the difficulty of 'incommensurables.'

Min. We will now take the new Propositions, &c. which have been suggested.

Here is an Axiom:—'Two lines cannot have a common segment.'

Euc. Good. I have tacitly assumed it, but it may as well be stated.

Min. Several new Theorems have been suggested, but only two of them seem to me worth mentioning. They are:—

'All right angles are equal.'

Euc. I have already approved of that (p. 219).

Min. The other is one that is popular with most of your Rivals:—

'Of all the Lines which can be drawn to a Line from a point without it, the perpendicular is least; and, of the rest, that which is nearer to the perpendicular is less than one more remote; and the lesser is nearer than the greater; and from the same point only two equal Lines can he drawn to the other Line, one on each side of the perpendicular.'

Euc. I like it on the whole, though so long an enunciation will be alarming to beginners. But it is strictly analogous to III. 7. Introduce it by all means in the revised edition of my Manual. It will be well, however, to lay it down as a general rule, that no Proposition shall be so interpolated, unless it be of such importance and value as to be thought worthy of being quoted as proved, in the same way in which candidates in examinations are now allowed to quote Propositions of mine.

Min. (with a fearful yawn) Well! I have no more to say.