SOME USES OF MATHEMATICS 119 



Now these two facts may be expressed in the single 

 relation 



V oc AH (6) 



This is the law for the volume of a rectangular 

 parallelepiped. It is not yet in the form for numerical 

 calculation because we have not yet decided upon the 

 units hi which the three magnitudes, V, H y and A are 

 to be measured or expressed. 



We recognize that any magnitude, e.g. an amount of 

 money, is expressed by a number, a "numeric/' as we 

 say, and a unit. The numeric indicates how many 

 times the chosen unit is contained in the given magni- 

 tude. The greater the unit, the smaller, of course, the 

 corresponding numeric. Thus 



$3 = 300^, that is 3(1 dollar) =300(1 cent) 

 or 1 dollar, 300 _ 100 

 1 cent " ' 3 1 



In the expression of any given magnitude the numeric 

 is inversely as the unit. 



The numeric expressing the volume depends upon 

 the choice of unit for measuring volume. Similarly, 

 the product AH will depend upon the units in which 

 these two magnitudes are expressed. Consider for 

 example the concrete numerical problem of a corn- 

 crib of area 10 sq. ft. and height 4 ft. For this case 

 AH is obviously 40 cubic feet. But if we measured 

 the volume by pouring in corn we should find it to be 

 about 32 bushels. That is, in bushels the numeric 

 of the volume, V, is 32. By selecting the unit we may 

 make the numeric anything we please. It would, 

 however, be a convenience to choose the unit expressing 



