102 COSMIC FHILOSOPHY. [ft. ii. 



siniplification may be tliereljy achieved, it is usual to throw 

 any two forms of an equation into a proportion — a procedure 

 in whicli the equality of the relations is specifically asserted." 

 To cite Mr. Spencer's simple illustration : if we take any 

 equation, 2 xy=y^, and, dividing it by y, obtain a second 

 equation, 2 x=y, the legitimacy of our proceeding is at once 

 rendered apparent when the two equations are thrown together 

 in a proportion, in which it is asserted that the ratio of 2 ccy 

 to y^ is equal to the ratio of 2 a; to y. Or, if any doubt still 

 remain as to the correctness of this, we resort to the familiar 

 device of multiplying extremes and means, and obtain the 

 identical proposition 2 xy'^ = 2 ocy^, in which the identity of the 

 two terms is immediately cognized, because the states of con- 

 sciousness which they evoke are indistinguishable from one 

 another. 



Thus the complicated quantitative reasoning by which an 

 astronomer determines the distance of a heavenly body con- 

 sists in the long-continued compounding of immediate cogni- 

 tions of the equality or inequality of two or more given 

 relations or groups of relations of position and magnitude. 



Before proceeding to unfold all that is implied by this 

 conclusion, let us consider another concrete example of a 

 somewhat different kind. When a certain horned animal, of 

 slender figure, with cloven hoofs, and a hairy integument, is 

 presented to the inspection of a naturalist, he at once re- 

 cognizes it as a girafie ; and, if required further to describe 

 it, he observes that, as having four stomachs and chewing 

 the cud, it belongs to the sub-order of ruminants ; as having 

 its toes firmly united in a solid hoof, it belongs to the order 

 of ungulata; as having mammary glands and suckling its 

 young, it belongs to the class of mammals ; and, as having 

 an internal bony skeleton, it belongs to the sub-kingdom of 

 vertebrates. What, now, is the mental act which is repeated 

 at each stage of this description ? It is " a cognition of the 

 fact that the relation between particular attributes in this 



