124 THE PRINCIPLES OF SCIENCE. [CHAP. 



are no infallible and general rules for its accomplishment ; 

 it must be done by trial, by guesswork, or by remembering 

 the results of differentiation, and using them as a guide. 



Coming more nearly to our own immediate subject, 

 exactly the same difficulty exists in determining the law 

 which certain things obey. Given a general mathematical 

 expression, we can infallibly ascertain its value for any 

 required value of the variable. But I am not aware that 

 mathematicians have ever attempted to lay down the rules 

 of a process by which, having given certain numbers, one 

 might discover a rational or precise formula from which 

 they proceed. The reader may test his power of detecting 

 a law, by contemplation of its results, if he, not being a 

 mathematician, will attempt to point out the law obeyed 

 by the following numbers : 



-, * i, -- , - 5 -, - 691 , 7 -> &Z > * 386? , c tc 



6 30 ' 42 30 66 2730 6 510 798 



These numbers are sometimes in low terms, but un- 

 expectedly spring up to high terms ; in absolute magnitude 

 they are very variable. They seem to set all regularity 

 and method at defiance, and it is hardly to be supposed 

 that anyone could, from contemplation of the numbers, 

 have detected the relations between them. Yet they are 

 derived from the most regular and symmetrical laws of 

 relation, and are of the highest importance in mathematical 

 analysis, being known as the numbers of Bernoulli. 



Compare again the difficulty of decyphering with that 

 of cyphering. Anyone can invent a secret language, and 

 with a little steady labour can translate the longest letter 

 into the character. But to decypher the letter, having no 

 key to the signs adopted, is a wholly different matter. 

 As the possible modes of secret writing are infinite in 

 number and exceedingly various in kind, there is no direct 

 mode of discovery whatever. Repeated trial, guided more 

 or less by knowledge of the customary form of cypher, and 

 resting entirely on the principles of probability and logical 

 induction, is the only resource. A peculiar tact or skill is 

 requisite for the process, and a few men, such as Wallis or 

 Wheatstone, have attained great success. 



Induction is the decyphering of the hidden meaning of 

 natural phenomena. Given events which happen in certain. 



