30 



ON THE MATHEMATICAL PROBABILITY 



probability of a common origin, when the words compared present a coincidence in more 

 than one of their meanings. For, if the concurrent meanings are apparently derived from 

 a single primitive, we have the case which has been long and generally recognized as 

 strengthening the evidence of family connection,— the case of parallelism in thought, which 

 comes next in point, of importance to parallelism in grammatical forms ; while, if the mean- 

 ings are radically distinct, the probability of each successive coincidence is diminished in a 

 geometrical ratio, the chance of a second being only i, — and of rn simultaneous coincidences, 

 only '„,. Therefore, although it is desirable to ascertain the primitive or radical signifi- 

 cation of words, before instituting a comparison, it is not absolutely essential, neither is it 

 even so important as it is often thought. 



But what can be said of synonymes X is it not probable that when the number of verbal 

 equivalents becomes large, the number of accidental resemblances will be proportionally 

 large! Here, if anywhere, is the stronghold of the believers in casual similitudes, but 

 even hen; their position may be easily assailed. 



Granting, as before, for the influence of known and unknown laws, that two given 

 languages represent the same ideas, and are also homonymous, the meanings being allotted, 

 to the several words at random, let us further suppose that each idea, is attached to m 

 synonymes in each language, the entire number of words being v. If the synonymes of a 

 given idea in one language are «, /S, y, — im we shall still have no probable grounds to ex- 

 pect that either of these words will represent the same idea in the other. For the chance 

 of the given idea being represented by any single word in the other language being -' 

 the chance of its being represented by some one of the m homonymous words is -, which 

 does not become a probability until m > '', a degree of synonymy that is wholly incredible. 



Finally, if the in words in the first language are not only synonymes of a, single 

 idea, but perfect equivalents that may be taken indiscriminately, each of the words 

 a, b, c, ...h/, being defined by the same set of ideas, a, /3, y, . . p., the chance ot a 

 coincidence between some one of the m ideas and either of the m homonyms in the 

 other language, would be — , which becomes a probability when m uM* But even m 

 that case, the chance of a second coincidence on the same word, would be only "V[ ' which 

 is less than a probability. 



That the change of form which the words of every living language are constantly under- 

 going, has no effect on the probability of accidental resemblance, is evident from the 

 fact that all our reasonings have been of the most general character, and they may be ap- 

 plied to any languages whatever, without regard to their family relationship, their present 



* There would still hardly bo a likelihood of more thnn one coincidence among all the homonyms. _For the 

 chance of such a, coincidence would be indicated bym x"'/, which docs not, become a probability until m (n) I 



