SOME USES OF MATHEMATICS 121 



about 600 B.C., by their ability to express and reason 

 with abstract ideas. If one catches up with them he 

 is only about 400 years behind the times in point of 

 view, for Greek thought dominated such scientific 

 spirit as was shown in the medieval ages, until Galileo 

 in the 17th century connected theory and practice 

 through experimentation. To catch up with the 

 present-clay trend is not, however, as hard a task as 

 these dates would seem to imply, for, if we accept the 

 inheritance, we are heirs to the successes, not the 

 failures and trials, of our scientific forebears. 



Returning to our concrete problem we may now make 

 a general statement. If a law l is expressed as Z oc XY, 

 then Z = XY if the units (a;), (y), and (z) are so chosen 

 that they are related as (z) = (x)(y). In the case of a 

 rectangular area, if the sides are measured in feet 

 (x) = 1 ft. and (y) = 1 ft., hence (z) = (1 ft.)(l ft.) or lit: 2 

 More generally, if the unit of length which we adopt 

 is symbolized by (L) the unit of area will be (L) 2 and 

 the unit of volume will be (L) 3 e.g. 1 ft. 3 



Starting with a chosen unit of length we derive, as 

 a consequence of the law of area, a unit of area which 

 is expressed in terms of our fundamental unit. Simi- 

 larly, from the law of volume we derive a unit of 

 volume which is expressed in terms of the unit of 

 length. 



In the sense of being derived from some other unit, 

 the unit of length is not itself a derived unit. It is a 

 fundamental unit, as we have implied above. Such 



1 More generally if Z&XY then Z = kXY, where k is a factor of 

 proportionality. This is the defining equation of Z in terms of 

 X and y, and k depends upon the choice of units. 



