Page:Carroll - Euclid and His Modern Rivals.djvu/178

140 straight lines.' Is it fair to give this without any limitation? What if the given lines were parallel?

P. 84. 'If A, B, C … as conditions involve D as a result, and the failure of C involves a failure of D; then A, B, D … as conditions involve C as a result.' If not-C proves not-D, then D proves C. A and B are irrelevant and obscure the statement. I observe, in passing, the subtle distinction which you suggest between the failure of C and 'a failure of D.' D is a habitual bankrupt, who has often passed through the court, and is well used to failures: but, when C fails, his collapse is final, and 'leaves not a wrack behind'!

P. 90. 'Given a curve, to ascertain whether it is an arc of a circle or not.' What does 'given a curve' mean? If it means a line drawn with ink on paper, we may safely say at once 'it is not a circle.'

P. 96. Def. 15. 'When one of the points in which a secant cuts a circle is made to move up to, and ultimately coincide with, the other, the ultimate position of the secant is called the tangent at that point.' (The idea of the position of a Line being itself a Line is queer enough: I suppose you would say 'the ultimate position of Whittington was the Lord Mayor of London.' But this is by the way: of course you mean 'the secant in its ultimate position.') Now let us take three points on a circle, the middle one fixed, the others movable; and through the middle one let us draw two secants, each passing through one of the other points; and then let us make the other points 'move up to, and ultimately coincide with,' the middle one. We have no ground for saying that these