CHAP. V.] PRINCIPLES OF SYMBOLICAL REASONING. 69 



experience, but is derived, like the knowledge of the other laws 

 of the mind, from the clear manifestation of the general principle 

 in the particular instance. A single example of reasoning, in 

 which symbols are employed in obedience to laws founded upon 

 their interpretation, but without any sustained reference to that 

 interpretation, the chain of demonstration conducting us through 

 intermediate steps which are not interpretable, to a final result 

 which is interpretable, seems not only to establish the validity of 

 the particular application, but to make known to us the general 

 law manifested therein. No accumulation of instances can pro- 

 perly add weight to such evidence. It may furnish us with clearer 

 conceptions of that common element of truth upon which the ap- 

 plication of the principle depends, and so prepare the way for its 

 reception. It may, where the immediate force of the evidence is 

 not felt, serve as a verification, a posteriori, of the practical vali- 

 dity of the principle in question. But this does not affect the posi- 

 tion affirmed, viz., that the general principle must be seen in the 

 particular instance, seen to be general in application as well as 

 true in the special example. The employment of the uninterpre- 

 table symbol ^ - 1, in the intermediate processes of trigonometry, 

 furnishes an illustration of what has been said. I apprehend that 

 there is no mode of explaining that application which does not 

 covertly assume the very principle in question. But that prin- 

 ciple, though not, as I conceive, warranted by formal reasoning 

 based upon other grounds, seems to deserve a place among those 

 axiomatic truths which constitute, in some sense, the foundation 

 of the possibility of general knowledge, and which may properly 

 be regarded as expressions of the mind's own laws and consti- 

 tution. 



6. The following is the mode in which the principle above 

 stated will be applied in the present work. It has been seen, 

 that any system of propositions may be expressed by equations 

 involving symbols #, ?/, 2, which, whenever interpretation is pos- 

 sible, are subject to laws identical in form with the laws of a sys- 

 tem of quantitative symbols, susceptible only of the values and 

 1 (II. 15). But as the formal processes of reasoning depend only 

 upon the laws of the symbols, and not upon the nature of their 

 interpretation, we are permitted to treat the above symbols, 



