XXH.] QUANTITATIVE INDUCTION. 487 



hunger with abstinence from food; desire and degree of 

 utility decrease with the quantity of commodity con- 

 sumed. We kno,w that the sun's heating power depends 

 upon his height ^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 we 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 connection, 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 mathematical processes by which 

 we can infallibly discover a mathematical formula yield- 

 ing 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 + BZ + Cx\ 



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 

 general rule that the formula thus obtained yields the 

 other numbers of the table to a considerable degree of 

 approximation. 



In many cases even the second power of the variable 

 will be unnecessary; Regnault found that the results 

 of his elaborate inquiry into the latent heat of steam at 

 different pressures were represented with sufficient ac- 

 curacy by the empirical formula 



A, = 606*5 + 0-305 t, 

 in which X is the total heat of the steam, and t the tern- 



