DIVISIONS OF MATHEMATICS, HOW BELATED. 137 



leal relations of entities, whatever be their special charac 

 ters. And what is the nature of the mental process by 

 which numbers are found capable of having their relations 

 expressed algebraically ? It is just the same. It is the for 

 mation of certain abstract conceptions of numerical funo- 

 tions which are the same whatever be the magnitudes of 

 the numbers. It is the invention of general symbols serv 

 ing to express the relations between numbers, as numbera 

 express the relations between things. And transcendental 

 analysis stands to algebra in the same position that algebra 

 stands in to arithmetic. 



To briefly illustrate their respective powers ; arithme 

 tic can express in one formula the value of a particular 

 tangent to a particular curve ; algebra can express in one 

 formula the values of all tangents to a particular curve ; 

 transcendental analysis can express in one formula the val 

 ues of all tangents to all curves. Just as arithmetic deals 

 with the common properties of lines, areas, bulks, forces, 

 periods ; so does algebra deal with the common properties 

 of the numbers which arithmetic presents ; so docs tran 

 scendental analysis deal with the common properties of the 

 equations exhibited by algebra. Thus, the generality of 

 the higher branches of the calculus, when compared with 

 the lower, is the same kind of generality as that of the lower 

 branches when compared with geometry or mechanics. 

 And on examination it will be found that the like relation 

 exists in the various other cases above given. 



Having shown that M. Comte s alleged law of progres 

 sion docs not hold among the several parts of the same 

 science, let us see how it agrees with the facts when applied 

 to separate sciences. &quot; Astronomy,&quot; says M. Comte, at the 

 opening of Book III., &quot; was a positive science, in its geo 

 metrical aspect, from the earliest days of the school of Alex- 

 andria ; but Physics, which we are now to consider, had no 

 positive character at all till Galileo made his great discov- 



