xx. i.] QUANTITATIVE INDUCTION. 501 



divided into several variable strata which by their uncon- 

 nected changes frustrate the exact calculations of astro- 

 nomers. Human life may be subject at different ages to 

 a succession of different influences incapable of reduction 

 under any one law. The results observed may in fact be 

 aggregates of an immense number of separate results each 

 governed by its own separate laws, so that the subjects 

 may be complicated beyond the possibility of complete 

 resolution by empirical methods. This is certainly true 

 of the mathematical functions which must some time or 

 other be introduced into the science of political economy. 



Simple Proportional Variation. 



When we first treat numerical results in any novel kind 

 of investigation, our impression will probably be that one 

 quantity varies in simple proportion to another, so as to 

 obey the law y = mx + n. We must learn to distinguish 

 carefully between the cases where this proportionality is 

 really, and where it is only apparently true. In con- 

 sidering the principles of approximation we found that a 

 small portion of any curve will appear to be a straight line. 

 When our modes of measurement are comparatively rude, 

 we must expect to be unable to detect the curvature. 

 Kepler made meritorious attempts to discover the law of 

 retraction, and he approximated to it when he observed 

 that the angles of incidence and refraction if small bear 

 a constant ratio to each other. Angles when small are 

 nearly as their sines, so that he reached an approximate 

 result of the true law. Cardan assumed, probably as a 

 mere guess, that the force required to sustain a body on 

 an inclined plane was simply proportional to the angle of 

 elevation of the plane. This is approximately the casr 

 when the angle is small, but in reality the law is inucL 

 more complicated, the power required being proportional 

 to the sine of the angle. The early thermometer-makers 

 were unaware whether the expansion of mercury was 

 proportional or not to the heat communicated to it, and 

 it is only in the present century that we have learnt it 

 to be not so. We now know that even gases obey the 

 law of uniform expansion by heat only in an approximate 



