OF NATURAL PHILOSOPHY. 177 



ceptible. It will depend then entirely on our habit 

 of treating mathematical subjects, how far we may 

 be able to include such a table in the distinct state- 

 ment of a mathematical law. The discovery of 

 such laws is often remarkably facilitated by the 

 contemplation of a class of phenomena to be noticed 

 further on, under the head of Collective Instances, 

 (see 194.) in which the nature of the mathe- 

 matical expression in which the law sought is com- 

 prehended, is pointed out by the figure of some 

 curve brought under inspection by a proper mode 

 of experimenting. 



(186.) After all, unless our induction embraces 

 a series of cases which absolutely include the 

 whole scale of variation of which the quantities 

 in question admit, the mathematical expression so 

 obtained cannot be depended upon as the true one, 

 and if the scale actually embraced be small, the 

 extension of laws so derived to extreme cases will 

 in all probability be exceedingly fallacious. For 

 example, air is an elastic fluid, and as such, if 

 enclosed in a confined space and squeezed, its bulk 

 diminishes : now, from a great number of trials made 

 in cases where the air has been compressed into a 

 half, a third, &c. even as far as a fiftieth of its bulk, 

 or less, it has been concluded that " the density of 

 air is proportional to the compressing force," or the 

 bulk it occupies inversely as that force ; and when 

 the air is rarefied by taking off part of its natural 

 pressure, the same is found to be the case, within 

 very extensive limits. Yet it is impossible that this 

 should be, strictly or mathematically speaking, the 

 true law ; for, if it were so, there could be no limit 



N 



