110 THE PRINCIPLES OF SCIENCE. 



Even when we have no means of accurately measuring 

 the variable quantities we may yet be convinced of their 

 connexion, if one always varies perceptibly at the same 

 time as the other. Fatigue increases with exertion ; 

 hunger with abstinence from food ; desire and degree of 

 utility decrease with the quantity of commodity con- 

 sumed. We know that the sun's heating power depends 

 upon his height in the sky ; that the temperature of the 

 air falls in ascending a mountain ; that the earth's crust 

 is found to be perceptibly warmer as w r e sink mines into 

 it; we infer the direction in which a sound comes from 

 the change of loudness as we approach or recede. The 

 facility with which we can time after time observe the 

 increase or decrease of one quantity with another suf- 

 ficiently shows the connexion, although we may be un- 

 able to assign any precise law of relation. The probability 

 in such cases depends upon frequent coincidence in time. 



Empirical Mathematical Laws. 



It is important to acquire a clear comprehension of the 

 part which is played in scientific investigation by em- 

 pirical formulae and laws. If we have a table containing 

 certain values of a variable and the corresponding values 

 of the variant, there are certain mathematical processes by 

 which we can infallibly discover a mathematical formula 

 yielding numbers in more or less exact agreement with 

 the table. We may generally assume that the quantities 

 will approximately conform to a law of the form 



y = A + B# + Cx 2 , 



in which x is the variable and y the variant. We can 

 then select from the table three values of y, and the cor- 

 responding values of x ; inserting them in the equation, 

 we obtain three equations by the solution of which we 

 gain the values of A, B, and C. It will be found as a 



