DIVISIONS OF MATHEMATICS, HOW RELATED. 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 func- 

 tions which are the same whatever be the magnitudes of 

 the numbers. It is the invention of general symbols serv- 

 mg to express the relations between numbers, as number^ 

 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 a^^ tangents to a/? 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 does 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 does not hold among the several parts of the same 

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

 to separate sciences. " Astronomy," says M. Comte, at the 

 opening of Book III., " 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- 



