720 PROCEEDINGS OP SECTION J. 



mensurable denomination. Of wliat meaning is the number 23 .H. 

 for example ? What quantity is measured in thirty-sevenths ? No 

 very complex arithmetic expressions need be given to the pupil to 

 simplify ; in actual work the use of decimals obviates the need for using 

 such. In the ordinary case, in practice the numerator and denomi- 

 nator each consists of several factors to be multiplied and the denomi- 

 nator divided into the numerator. The result is taken out in decimals 

 to five significant figures, and the decimal portion converted, if 

 necessary, into aliquot parts of the quantity involved. 



The subject of proportion is one of great importance, and one 

 capable of improvement in its treatment. Originally it was taught 

 as the rule of three, with the dot symbols of proportionality. There 

 was no great power in this. A modern treatment is that known as 

 the unitary method, a method which I consider unsatisfactory because 

 illogical. One might cite many instances. The following will suffice : 

 The volume of a gas is directly proportional to its absolute temperature. 

 If the volume of a quantity of gas at the absolute temperature 350° c. 

 is 293 c.c, what is its volume at 372° c. ? According to the unitary 

 method we ask — if the volume is 293 c.c. at 350° c. what is its volume 

 at 1° c. ? ; from which is obtained its volume at 372° c. The result 

 is correct, but the proportionality does not hold down to 1° c, and there 

 is no meaning in the volume obtained at 1° c. The proportionality 

 between two quantities is best represented as a fraction. For example, 



372 

 The proportion between the above two temperatures is -qcj^' ^^^ ' ^^ 



proportion between x, the required volume, and the given volume is 

 -— . These fractions are identical in value, and x may be easily cal- 

 culated. Again, the subject of proportion should include not only 

 simple proportion, but also inverse proportion and proportionality to 

 any power. In all cases the proportionality would be expressed by the 



ct c 

 equality of two fractions of the type - = ,, and, three of the four quanti- 

 ties being given, the fourth may be calculated according to an obvious 

 and simple rule. As an example : The area of a circle of 8in. diameter 

 is 50 3 square inches, what is the area of a circle of 13in. diameter ? 

 The area of a circle being proportional to the square of the diameter, 



^^ ., .^ 169 X 50-3 X 169 ,_ 



we have the identity ^-p = ^-- .-, x ■= ^, = 133 square 



■^ 64 50-3 64 ^ 



inches. 



The nature of proportion is illustrated at a later stage by reference 

 to similar triangles, and more especially by its graphical representa- 

 tion the straight line plotted to cartesian co-ordinates. 



When the pupil passes from arithmetic to mathematics, generally, 

 he has to encounter a difficulty which requires time and patience for 

 its removal. In arithmetic he is at home with concrete cases, but now 

 he has to pass from the concrete to abstract, from the particular to the 



