128 THE PRINCIPLES OF SCIENCE. 



sidering the principles of approximation we found that a 

 small portion of any curve will appear to be a straight 

 line. Whenever our modes of measurement are compara 

 tively rude, we must expect to be unable to detect the 

 curvature. Thus Kepler made meritorious attempts to 

 discover the law of refraction, and he slightly approxi 

 mated 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 very 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 propor 

 tional to the angle of elevation of the plane. This is 

 approximately the case when the angle is very small, and 

 it becomes true again when the angle is a right angle ; 

 but in reality the law is much more complicated, the 

 power required being proportional to the sine of the 

 angle. The early thermometer-makers were quite unaware 

 whether the expansion of mercury was exactly propor 

 tional 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 man 

 ner. Until some reason to the contrary is shown, we 

 should do well to look upon every law of simple propor 

 tion as only provisionally true. 



Nevertheless, there are many of the most important 

 laws of nature which are in the form of simple propor 

 tions. Wherever a uniform cause acts in independence 

 of its previous effects, we may expect this relation. Thus, 

 an accelerating force acts equally upon a moving and a 

 motionless body. Hence the velocity produced is always 

 in simple proportion to the force, and also to the duration 

 of its uniform action. As gravitating bodies never in 

 terfere with each other s gravity, this force is in direct 



