52 



The consecutive processes of Mathematics led to the 

 consecutive processes of Logic ; but it was not until long 

 after mankind had attained to abstract ideas that they 

 attained to any clear notion of their connexion with one 

 another. The leading ideas of Mathematics became the 

 leading ideas of Logic. The " one " and the " many " 

 passed into the *' whole " and its '' parts " ; and thence 

 into the " Universal " and the " Particular." The fallacies 

 of Logic, such as the well known puzzle of Achilles and 

 the tortoise, partake of the nature of both Sciences. 

 And perhaps the conception of the Infinite and the 

 Infinitesimal, as well as of Negation, may have been 

 in early times transferred from Logic to Mathematics. 

 But the connexion of our ideas of number is probably 

 anterior to the connexion of any of our other ideas. 

 And as a matter of fact, geometry and arithmetic had 

 already made considerable progress when Aristotle in- 

 vented the Syllogism. 



General ideas there were, beside those of Mathema- 

 tics — true flashes of genius which saw that there must 

 be general laws to which the universe conforms, but 

 which saw them only by occasional glimpses, and 

 through the distortion of imperfect knowledge; and 

 although the only records of them now remaining are the 

 inadequate representations of later writers, yet we must 

 still remember that to the existence of such ideas is 

 due not only the conception but even the possibility of 

 Physical Science. But these general ideas were too 

 wide in their grasp, and in early days at least were 

 connected to their subjects of application by links too 

 shadowy, to be thoroughly apprehended by most minds ; 

 and so it came to pass that one form of such an idea 

 was taken as its only form, one application of it as the 

 idea itself ; and Philosophy, unable to maintain itself at 



