344 METHODOLOGY OF SCIENCE 



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the group B. Therefore A and C are equal. The same process of 

 reasoning can be carried out for any number of groups. 



Likewise it can be demonstrated that if A is poorer than B and B 

 poorer than C, then A is also poorer than C. For in the association 

 of B with A some members of B are left over by hypothesis, and 

 likewise some members of C are left over if one associates C with B. 

 Therefore in the association of C with A, not only those members are 

 left over which could not be associated with B, but also those mem- 

 bers of C which extend beyond B. This proposition can be extended 

 to any number of groups, and permits the arrangement of a number 

 of different groups in a simple series by beginning with the poorest 

 and choosing each following so that it is richer than the preceding 

 but poorer than the following. From the proposition just established, 

 it follows that every group is so arranged with reference to all other 

 groups that it is richer than all the preceding and poorer than all the 

 following. 1 



In this derivation of scientific proposition or laws of the simplest 

 kinds, the process of derivation and the nature of the result becomes 

 particularly clear. We arrive at such a proposition by performing 

 an operation and expressing the result of it. This expression enables 

 us to avoid the repetition of the operation in the future, since in 

 accordance with the law we can indicate the result immediately. 

 Thus an abbreviation and therefore a facilitation of the problem is 

 attained which is the more considerable the larger the number of 

 operations saved. 



If we have a number of equal groups, we know by the process of 

 association that all of the operations with reference to arrangement 

 which we can perform with one of them can be performed with all the 

 others. It is sufficient, therefore, to determine the properties of 

 arrangement of one of these groups in order to know forthwith the 

 properties of all the others. This is an extremely important pro- 

 position, which is continually employed for the most various purposes. 

 All speaking, writing, and reading rests upon the association of 

 thoughts with sounds and symbols, and by arranging the signs in 

 accordance with our thoughts we bring it to pass that our hearers 

 or readers think like thoughts in like order. In a similar fashion we 

 make use of various systems of formulae in the different sciences, 

 especially in the simpler sciences; and these formula? we correlate 

 with phenomena and use in place of the phenomena themselves, 

 and can therefore derive from them certain characteristics of phe- 

 nomena without being compelled to use the latter. The force of this 

 process appears very strikingly in astronomy where, by the use of 

 definite formula? associated with the different heavenly bodies, we 



1 Equal groups cannot be distinguished here, and therefore represent only a 

 group. 



