122 SCIENCE AND METHOD. 



book that has been justly admired and has received 

 several awards of merit — Hilbert's " Grundlagen der 

 Geometric." Let us see how he begins. " Imagine 

 three systems of THINGS, which we will call points, 

 straight lines, and planes." What these " things " are 

 we do not know, and we do not need to know — it 

 would even be unfortunate that we should seek to 

 know ; all that we have the right to know about them 

 is that we should learn their axioms, this one, for 

 instance : " Two different points always determine 

 a straight line," which is followed by this comment- 

 ary : " Instead of determine we may say that the 

 straight line passes through these two points, or that 

 it joins these two points, or that the two points are 

 situated on the straight line." Thus " being situated 

 on a straight line" is simply defined as synonymous 

 with " determining a straight line." Here is a book 

 of which I think very highly, but which I should not 

 recommend to a schoolboy. For the matter of that 

 I might do it without fear ; he would not carry his 

 reading very tar. 



I have taken extreme examples, and no instructor 

 would dream of going so far. But, even though he 

 comes nowhere near such models, is he not still 

 exposed to the same danger? 



We are in a class of the fourth grade. The teacher 

 is dictating : " A circle is the position of the points 

 in a plane which are the same distance from an in- 

 terior point called the centre." The good pupil writes 

 this phrase in his copy-book and the bad pupil draws 

 faces, but neither of them understands. Then the 

 teacher takes the chalk and draws a circle on the 

 board. "Ah," think the pupils, "why didn't he say 



