176 DISCOURSE ON THE STUDY 



the nature of the inductions by which they are to 

 be arrived at. In their simplest or least general 

 stages (of which alone we speak at present) they 

 usually express some numerical relation between 

 two quantities dependent on each other, either as 

 collateral effects of a common cause, or as the 

 amount of its effect under given numerical circum- 

 stances or data. For example, the law of refrac- 

 tion before noticed ($ 22.) expresses, by a very 

 simple relation, the amount of angular deviation of a 

 ray of light from its course, when the angle at which 

 it is inclined to the refracting surface is known, 

 viz. that the sine of the angle which the incident 

 ray makes with a perpendicular to the surface is 

 always to that of the angle made by the refracted 

 ray with the same perpendicular, in a constant pro- 

 portion, so long as the refracting substance is the 

 same. To arrive inductively at laws of this kind, 

 where one quantity depends on or varies with another, 

 all that is required is a series of careful and exact 

 measures in every different state of the datum and 

 qucesitum. Here, however, the mathematical form 

 of the law being of the highest importance, the 

 greatest attention must be given to the extreme cases 

 as well as to all those points where the one quantity 

 changes rapidly with a small change of the other.* 

 The results must be set down in a table in which 

 the datum gradually increases in magnitude from 

 the lowest to the highest limit of which it is sus- 



* A very curious instance of the pursuit of a law completely 

 empirical into an extreme case is to be found in Newton's rule for 

 the dilatation of his coloured rings seen between glasses at great 

 obliquities. Optics, book ii. part i. obs. 7. 



