310 PRESENTMKXT AND PROOF [X Cil'.OM ICTRV. 



than the bounds of common-sense, and you may have to reject 

 some of your results as not corresponding to reahty. The 

 super-abstracted paradoxes of Hegel and the 4th and ;;th 

 dimensions of some mathematicians are merely syml)olism 

 divorced from reality, and therefore from truth. Both logi- 

 cians and mathematicians need the touch of actualities to keep 

 them sane. I think I detect a tendency to hurry tow^ards and 

 exaggerate the symbolic side of Geometry — in the words of my 

 thesis, to sacrifice Presentment to Proof. 



2. Another im])ortant distinction is in our own nature. 

 There is the intuitive power which looks into the essence of 

 things and studies their character ; there is the rati'Ocinativc 

 power which loves to prove theorems and to arrange them in 

 order upon an irrefragable basis ; and there is the ingenuity 

 which delights in the solving of problems and conundrums. 

 These are not necessarily concurrent in the same pupil. To 

 me the first seems by far the most important ; the usual exami- 

 nation papers test only the second and third. The first corre- 

 sponds to Presentment ; the second, to Proof ; the third to 

 Puzzle : it is just a logical game, only it gets most marks in 

 the test, and therefore teachers are led to try to inspire the 

 faculty even into pupils who have no taste for conundrums. 

 Again I say. our text-books and examinations seem to negle':*: 

 Presentment for Proof and Puzzle. 



3. A third distinction is also psychological. It seems 

 obvious to say that the function of sense is to provide the raw 

 material for thought, to explore the universe and submit its 

 discoveries to the higher powers. But I notice a tendency in 

 text-books and in the University Syllabus to make the students' 

 drawing exercises retrospective ; they are carefully to measure 

 figures whose properties they have already proved, in order, 

 forsooth, to give them confidence that their reasoning has been 

 correct. Surely this is very topsy-turvy. .\n army might 

 as well employ its l)est scouts in consolidating the back trenches. 



Bearing these three distinctions in mind, I put forward a 

 few maxims, which 1 believe are consciously or instinctively 

 followed by all good teachers, but the theory of which seems 

 to require a little urging. 



(a) Present merit goes before Proof. — Presentment means 

 so to put facts before the senses as to make it easy to idealise 

 from them, and so to put inferences before the mind as to make 

 it easy to analyse, classify, and systematise them. We begin 

 well in this matter. We teach our little ones in Kindergarten 

 what they can grasp of the laws of form. But we do not 

 persevere. Just as there ought t(^ be a continuous gradation 

 of object-lessons leading up to some physical scien.ce to be 

 afterwards logically studied, so we need a continuous course 

 of drawing and other practical geometrical work to prepare 



