Page:Carroll - Euclid and His Modern Rivals.djvu/56

18 Min. A charming companion! And his name? Euc. Phantasms have no names—only numbers. You may call him 'Herr Niemand,' or, if you prefer it, 'Number one-hundred-and-twenty-three-million-four-hundred-and-fifty-six-thousand-seven-hundred-and-eighty-nine.'

Min. For constant use, I prefer 'Herr Niemand.' Let us now consider the question of the separation of Problems and Theorems.

§ 3. The combination, or separation, of Problems and Theorems.

Euc. I shall be glad to hear, first, the reasons given for separating them, and will then tell you my reasons for mixing them.

Min. I understand that the Committee of the Association for the Improvement of Geometrical Teaching, in their Report on the Syllabus of the Association, consider the separation as 'equivalent to the assertion of the principle that, while Problems are from their very nature dependent for the form, and even the possibility, of their solution on the arbitrary limitation of the instruments allowed to be used, Theorems, being truths involving no arbitrary element, ought to be exhibited in a form and sequence independent of such limitations.' They add however that 'it is probable that most teachers would prefer to introduce Problems, not as a separate section of