Page:Carroll - Euclid and His Modern Rivals.djvu/195

. § 3.] that a man could have all those three maladies, and survive? And yet the thing is possible!

Let me now read you a statement (at p. 112) about incommensurables:—

'When one of the magnitudes can be represented only by an interminable decimal, while the other is a finite whole number, or finite decimal, no finite common submultiple can exist; for, though a unit be selected in the last place of the whole number or finite decimal, yet the decimal represented by all the figures which follow the corresponding place in the interminable decimal, being less than that unit in that place and unknown in quantity, cannot be a common measure of the two magnitudes, and is only a remainder.'

Now can you lay your hand upon your heart and declare, on the word of an honest man, that you understand this sentence—beginning at the words 'yet the decimal'?

Nie. (vehemently) I cannot!

Min. Of the two reasons which are mentioned, to explain why it 'cannot be a common measure of the two magnitudes,' does the first—that it is 'less than that unit in that place'—carry conviction to your mind? And does the second—that it is 'unknown in quantity' ripen that conviction into certainty?

Nie. (wildly) Not in the least!

Min. Well, I will not 'slay the slain' any longer. You may consider Dr. Willock's book as rejected. And I think we may say that the whole theory of 'direction' has collapsed under our examination.

Nie. I greatly fear so.