PROGRESS OF GEOMETRY. 149 



classes of relations ; and by so multiplying the experiences 

 of them, or making them relatively frequent, facilitate their 

 generalization. Moreover, various classes of phenomena 

 presented by society itself, as for instance those which 

 political economy formulates, become relatively frequent, and 

 therefore recognizable, in advanced social states ; while in 

 less advanced ones they are either too rarely displayed to 

 have their relations perceived, or, as in the least advanced 

 ones, are not displayed at all. 



That, where no other circumstances interfere, the order in 

 which different uniformities are established varies as their 

 complexity, is manifest. The geometry of straight lines was 

 understood before the geometry of curved lines ; the proper 

 ties of the circle before the properties of the ellipse, parabola, 

 and hyperbola ; and the equations of curves of single cur 

 vature were ascertained before those of curves of double 

 curvature. Plane trigonometry comes in order of time and 

 simplicity before spherical trigonometry ; and the mensura 

 tion of plane surfaces and solids before the mensuration of 

 curved surfaces and solids. Similarly with mechanics : the 

 laws of simple motion were generalized before those of com 

 pound motion ; and those of rectilinear motion before those 

 of curvilinear motion. The properties of equal-armed levers 

 or scales, were understood before those of levers with un 

 equal arms ; and the law of the inclined plane was formulated 

 earlier than that of the screw, which involves it. In chemis 

 try, the progress has been from the simple inorganic com 

 pounds to the more involved or organic compounds. And 

 where, as in the higher sciences, the conditions of the explo 

 ration are more complicated, we still may clearly trace 

 relative complexity as determining the order of discovery 

 where other things are equal. 



The progression from concrete relations to abstract ones, 

 and from the less abstract to the more abstract, is equally 

 obvious. Numeration, which in its primary form concerned 



