440 THE PRINCIPLES OF SCIENCE. [CHAP. 



In induction we aim at establishing a general law, and 

 if we deal with quantities that law must really be expressed 

 more or less obviously in the form of an equation, or 

 equations. We treat as before of conditions, and of what 

 happens under those conditions. But the conditions will 

 now vary, not in quality, but quantity, and the effect will 

 also vary in quantity, so that the result of quantitative in- 

 duction is always to arrive at some mathematical expression 

 involving the quantity of each condition, and expressing 

 the quantity of the result. In other words, we wish to 

 know what function the effect is of its conditions. We 

 shall find that it is one thing to obtain the numerical 

 results, and quite another thing to detect the law obeyed 

 by those results, the latter being an operation of an inverse 

 and tentative character. 



The Variable and the Variant. 



Almost every series of quantitative experiments is 

 directed to obtain the relation between the different 

 values of one quantity which is varied at will, and an- 

 other quantity which is caused thereby to vary. We 

 may conveniently distinguish these as respectively the 

 variable and the variant. When we are examining the 

 effect of heat in expanding bodies, heat, or one of its 

 dimensions, temperature, is the variable, length the 

 variant. If we compress a body to observe how much 

 it is thereby heated, pressure, or it may be the dimensions 

 of the body, forms the variable, heat the variant. In 

 the thermo-electric pile we make heat the variable and 

 measure electricity as the variant. That one of the two 

 measured quantities which is an antecedent condition of 

 the other will be the variable. 



It is always convenient to have the variable entirely 

 under our command. Experiments may indeed be made 

 with accuracy, provided we can exactly measure the vari- 

 able at the moment when the quantity of the effect is 

 determined. But if we have to trust to the action of 

 some capricious force, there may be great difficulty in 

 making exact measurements, and those results may not 

 be disposed over the whole range of quantity in a con- 

 venient manner. It is one prime object of the experi- 



