4<J8 THE PRINCIPLES OF SCIENCE. [CHAP. 



proceed upon the same principles. If possible we must 

 detect the exact laws in action, and then trust to them as 

 a guide when we have no experience. If not, an empirical 

 formula of the same character as those employed in inter- 

 polation is our only resource. But to extend our inference 

 tar beyond the limits of experience is exceedingly unsafe. 

 Our knowledge is at the best only approximate, and 

 takes no account of small tendencies. Now it usually 

 happens that tendencies small within our limits of ob- 

 servation become perceptible or great under extreme 

 circumstances. When the variable in our empirical 

 formula is small, we are justified in overlooking the higher 

 powers, and taking only two or three lower powers. But 

 as the variable increases, the higher powers gain in impor- 

 tance, and in time yield the principal part of the value of 

 the function. 



This is no mere theoretical inference. Excepting the 

 few primary laws of nature, such as the law of gravitj', 

 of the conservation of energy, &c., there is hardly any 

 natural law which we can trust in circumstances widely 

 different from those with which we are practically ac- 

 quainted. From the expansion or contraction, fusion or 

 vaporisation of substances by heat at the surface of the 

 earth, we can form a most imperfect notion of what would 

 happen near the centre of the earth, where the pressure 

 almost infinitely exceeds anything possible in our experi- 

 ments. The physics of the earth give us a feeble, and pro- 

 bably a misleading, notion of a body like the sun, in 

 which an inconceivably high temperature is united with an 

 inconceivably high pressure. If there are in the realms of 

 space nebulae consisting of incandescent and unoxidised 

 vapours of metals and other elements, so highly heated 

 perhaps that chemical composition is out of the question, 

 we are hardly able to treat them as subjects of scientific 

 inference. Hence arises the great importance of experi- 

 ments in which we investigate the properties of substances 

 under extreme circumstances of cold or heat, density or 

 rarity, intense electric excitation, &c. This insecurity 

 in extending our inferences arises from the approximate 

 character of our measurements. Had we the power of 

 appreciating infinitely small quantities, we should by 

 the principle of continuity discover some trace of every 



