QUANTITATIVE EVOLUTION OF KNOWLEDGE. 199 



in things and relations, simultaneously evolves by one pro 

 cess of culture the ideas of equality of things and equality 

 of relations ; which are the respective bases of exact con 

 crete reasoning and exact abstract reasoning Mathematics 

 and Logic. And once more, this idea of equality, in the 

 very process of being formed, necessarily gives origin to 

 two series of relations those of magnitude and those of 

 number: from which arise geometry and the calculus. Thus 

 the process throughout is one of perpetual subdivision and 

 perpetual intercommunication of the divisions. From the 

 very first there has been that consensus of different kinds of 

 knowledge, answering to the consensus of the intellectual 

 faculties, which, as already said, must exist among the sci 

 ences. 



Let us now go on to observe how, out of the notions of 

 equality and number, as arrived at in the manner described, 

 there gradually arose the elements of quantitative prevision. 



Equality, once having come to be definitely conceived, 

 was readily applicable to other phenomena than those of 

 magnitude. Being predicable of all things producing indis 

 tinguishable impressions, there natui ally grew up ideas of 

 equality in weights, sounds, colours, &c. ; and indeed it can 

 scarcely be doubted that the occasional experience of equal 

 weights, sounds, and colours, had a share in developing the 

 abstract conception of equality that the ideas of equality 

 in size, relations, forces, resistances, and sensible proper 

 ties in general, were evolved during the same period. 

 But however this may be, it is clear that as fast as the no 

 tion of equality gained definiteness, so fast did that lowest 

 kind of quantitative prevision which is achieved without 

 any instrumental aid, become possible. 



The ability to estimate, however roughly, the amount 

 of a foreseen result, implies the conception that it will be 

 equal to a certain imagined quantity ; and the correctness 

 of the estimate will manifestly depend upon the accuracy at 



