xxviii.] ANALOGY. 613 



of the solid, liquid, and gaseous states remove the basis 

 of the speculation. 



From these and many other instances which might be 

 adduced, we learn that analogical reasoning leads us to 

 the conception of many things which, so far as we can 

 ascertain, do not exist. In this way great perplexities 

 have arisen in the use of language and mathematical 

 symbols. All language depends upon analogy; for we 

 join and arrange words so that they may represent the 

 corresponding junctions or arrangements of things and 

 their equalities. But in the use of language we are 

 obviously capable of forming many combinations of words 

 to which no corresponding meaning apparently exists. 

 The same difficulty arises in the use of mathematical 

 signs, and mathematicians have needlessly puzzled them- 

 selves about the square root of a negative quantity, which 

 is, in many applications of algebraic calculation, simply a 

 sign without any analogous meaning, there being a failure 

 of analogy. 



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