PROCEEDINGS OF SECTION J. 721 



general. A single chapter on numerical substitution wMcli prefaces 

 most textbooks on algebra is treating this difficulty very lightly. The 

 average textbook on algebra assumes that this difficulty is really over- 

 come by studying the first chapter, and launches out immediately 

 into the ocean of symbols. From this moment the average boy is 

 lost. He henceforth juggles, and often dexterously, with symbols, 

 but the realities behind the symbols are veiled from his eyes. The 

 late Professor Clifford said that " Algebra which cannot be translated 

 into good English and sound common sense is bad algebra." How 

 many boys can translate their algebra into good English ? Beyond 

 time-honored problems on the market price of eggs, or the ages of 

 Tom, Dick, and Harry, there is no interpretation of the symbols in 

 terms of things. 



Along with algebra a boy generally takes up the study of Euclid. 

 I have referred already to the unsuitability of Euclid, and will state 

 the reasons. Euclid is a system of geometry. Given a knowledge 

 of geometrical facts, Euclid shows that they are all deducible from a 

 minimum of premises. These premises are arbitrary. The effort is 

 purely philosophical ; but the point to recognise is that the geometrical 

 facts are antecedent to and independent of any philosophical system. 

 No deductive process gives any boy a clearer perception of the fact that 

 the join of two points on the circumference of a circle lies within the 

 circle, than he acquires from his sense of sight. The boy in the first 

 place should acquire his knowledge of geometry. His inductions may 

 be classified, but there is no importance attaching to the fact that they 

 are all capable of deduction from certain arbitrary assumptions. There 

 is no more certain theorem in education than that the concrete must 

 precede the abstract, and yet it is disregarded when a boy has to pass 

 through the portals of metaphysics to learn that the angles at the base 

 of an isosceles triangle are equal. 



Another serious defect in the teaching of mathematics, to my 

 mind, is its specialisation. The various branches are kept separate 

 instead of combined into a harmonious whole. I will outline what I 

 consider to be a sound, useful, and intelligent course of mathematics 

 within the range of the average schoolboy. To meet the difficulty 

 which the student experiences in passing from the particular to the 

 general, every symbol used refers to something definite and apparent 

 to the pupil. This is accomplished by basing the work largely on 

 mensuration. In the experimental study of mathematics it should 

 be regarded as the first stage of physics. The pupil should make 

 actual measurements himself as far as possible, draw to scale, measure 

 and calculate results. The common geometrical surfaces are thus 

 dealt with. After working out calculations in particular cases of the 

 same type, he passes to the general case, using symbols instead of 

 letters. He thus obtains the formula suiting the general case. For 

 example, he draws and calculates the area of various equilateral tri- 

 angles. He then expresses the area of an equilateral triangle as a 

 function of its side, and from the formula deduces the side as a fraction 

 of the area, from which he can construct an equilateral triangle of 

 z2 



