Page:Carroll - Euclid and His Modern Rivals.djvu/286

248 statement. He may be trained to suppression: but the true way to abbreviation is from the full length. Mr. Wilson does not use the phrases of reasoning consistently. He tells the student that a corollary is 'a geometrical truth easily deducible from a theorem': and then, to the theorem that only one perpendicular can be drawn to a straight line, he gives as a corollary that the external angle of a triangle is greater than the internal opposite. This is not a corollary from the theorem, but a matter taken for granted in proving it.

Leaving this, with a recommendation to the author to strengthen his armour by the study of logic, we pass on to the system. There is in it one great point, which brings down all the rest if it fall, and may perhaps—but we must see Part II. before we decide—support the rest if it stand. That point is the treatment of the angle, which amounts to this, that certain notions about direction, taken as self-evident, are permitted to make all about angles, parallels and all, immediate consequences. The notion of continuous change, and consequences derived from it, enters without even an express assumption: 'continually' is enough.

Mr. Wilson would not have ventured expressly to postulate that when a magnitude changes continuously, all magnitudes which change with it also change continuously. He knows that when a point moves on a line, an angle may undergo a sudden change of two right angles. He trusts to the beginner's perception of truth in the case before him: the whole truth would make that beginner feel that he is on a foundation of general principles made safe for him by selection, and only safe because the exceptions are not likely to occur to his mind. On this we write, as Newton wrote on another matter, ''Falsa! Falsa! Non est Geometria!''

What 'direction' is we are not told, except that 'straight lines which meet have different directions.' Is a direction a magnitude? Is one direction greater than another? We should suppose so; for an angle, a magnitude, a thing which is to