THE STUDY OF MATHEMATICS 63 



emerge as the residue of similarity amid such great 

 apparent diversity. In this way the abstract demon 

 strations should form but a small part of the instruction, 

 and should be given when, by familiarity with concrete 

 illustrations, they have come to be felt as *e natural 

 embodiment of visible fact. In this early stage proofs 

 should not be given with pedantic fullness ; definitely 

 fallacious methods, such as that of superposition, should 

 be rigidly excluded from the first, but where, without 

 such methods, the proof would be very difficult, the 

 result should be rendered acceptable by arguments and 

 illustrations which are explicitly contrasted with demon 

 strations. 



In the beginning of algebra, even the most intelligent 

 child finds, as a rule, very great difficulty. The use of 

 letters is a mystery, which seems to have no purpose 

 except mystification. It is almost impossible, at first, 

 not to think that every letter stands for some particular 

 number, if only the teacher would reveal what number it 

 stands for. The fact is, that in algebra the mind is first 

 taught to consider general truths, truths which are not 

 asserted to hold only of this or that particular thing, but 

 of any one of a whole group of things. It is in the power 

 of understanding and discovering such truths that the 

 mastery of the intellect over the whole world of things 

 actual and possible resides ; and ability to deal with the 

 general as such is one of the gifts that a mathematical 

 education should bestow. But how little, as a rule, is 

 the teacher of algebra able to explain the chasm which 

 divides it from arithmetic, and how little is the learner 

 assisted in his groping efforts at comprehension ! Usually 

 the method that has been adopted in arithmetic is con 

 tinued : rules are set forth, with no adequate explanation 

 of their grounds ; the pupil learns to use the rules blindly, 



