136 



SCIENCE. 



[N. S. Vol. III. No. 56. 



horse ' does not differ so essentially as our 

 authors think from the ' fixed unit ' conception 

 of this number against which they protest so 

 strenuously. And this fictitious operation is no 

 more the essence of multiplication and division 

 than it is of counting. Multiplication of integers 

 is abbreviated addition. The product ' three 

 times two' is the sum of three two's not, 

 happily, the measure in terms of a primary un- 

 defined unit of something whose measure in 

 terms of a secondary undefined unit is three, 

 when the measure of the secondary unit itself 

 in terms of this primary unit is two. 



On the other hand, measuring in the ordinary 

 sense — the process which leads to the represen- 

 tation of continuous magnitudes as lines or sur- 

 faces, in terms of some unit of measure — deserves 

 all the prominence which our authors would give 

 it in arithmetic. We do not mean measuring in 

 the exact mathematical sense, of course, but the 

 rough measuring of common life, in which the 

 magnitude measured and the unit are always 

 assumed to be commensurable. 



Compared with counting, or even addition and 

 multiplication, an operation which involves the 

 use of an arbitrary unit, and the comparison of 

 magnitudes by its aid, is artificial. But this 

 metrical use of number is of immense practical 

 importance and of great interest to any child 

 mature enough to understand it. No doubt a 

 child may use a twelve-inch rule to advantage 

 when practicing multiplication and division of 

 integers. Certainly such an aid is almost indis- 

 pensable in learning fractions. Without it the 

 fraction is more than likely to be a mere symbol 

 to him, without exact meaning of any kind. 

 ' Two-thirds ' has a reality for the child who 

 can interpret it as the measure of a line two 

 inches long in terms of a unit three inches long, 

 which it quite lacks for him who can only repeat 

 that it is ' two times the third part of unity.' 

 Mathematicians now define the fraction as the 

 symbolic result of a division which cannot be 

 actually effected, but that definition will not 

 serve the purposes of elementary instruction. It 

 is as certain that the fraction had a metrical or- 

 igin as it is that the integer had not, and in learn- 

 ing fractions, as in learning integers, the child 

 cannot do better than follow the experience of 

 the race. 



Our authors must, therefore, be credited with 

 doing the cause of rational instruction in arith- 

 metic a real service by laying the stress they do 

 on this proper metrical use of number. Their 

 chapters on the practical teaching of arithmetic, 

 moreover, though unduly prolix, contain many 

 excellent suggestions. It is a pity that a book 

 in the main so sound in respect to practice 

 should be wrong on fundamental points of 

 theory. One can but regret that its authors 

 did not take pains before writing it to read 

 what mathematicians of the present century 

 have had to say on the questions with which 

 they meant to deal. Their conception of num- 

 ber might have been modified by the considera- 

 tions which have led mathematicians to ' arith- 

 metise ' the higher analysis itself by replacing 

 the original metrical definition of the irrational 

 number by a purely arithmetical one. At all 

 events their notions of certain mathematical 

 concepts would not have been so crude ; they 

 would not have made such a use of mathematical 

 terms as this: "Quantity, the unity measured, 

 whether a ' collection of objects ' or a physical 

 whole, is continuous, an undefined how much; 

 number as measuring value is discrete, how 

 many. ' ' 



H. B. Fine. 



Princeton, December 31, 1895. 



Experimental Farms. Reports for 1894. Printed 



by order of Parliament. Ottawa, 1895. 422 



pp. 8°. 



The direct application of scientific methods of 

 investigation to practical questions has, perhaps, 

 in no field found greater extension during the 

 last decade on this continent than in agriculture. 



The establishment of the experiment stations 

 in connection with agricultural colleges in all 

 our States by the Hatch Act of 1887 has revo- 

 lutionized the possibilities of agricultural pur- 

 suits, and what this act did for the United 

 States, Canada did the same year in perhaps a 

 more efiicient if not as extensive manner for its 

 people. This greater efliciency we would at- 

 tribute to the fact that the direction of the five 

 experimental farms located in different parts of 

 the country is concentrated in one director and 

 one staff, thereby producing that unity of pur- 

 pose which insures success. 



