2 3 o POPULAR SCIENCE MONTHLY 



different groups in a series by beginning with the poorest and choos- 

 ing each succeeding one so that it is richer than its predecessor though 

 poorer than its successor. Through a proposition that has been already 

 proved (p. 229) it follows that each group is thus also arranged with 

 reference to all other groups in such a way that it is richer than all 

 its predecessors as well as poorer than all its successors. 2 



In developing most simple propositions or laws, this method of 

 their discovery and the nature of the results become most clear to 

 us. We achieve such a proposition by carrying out an operation and 

 giving expression to its results. This expression enables us thereafter 

 to save ourselves the trouble of repeating the operation. We are able 

 to give the result immediately in accordance with the law. Thus we 

 shorten and facilitate the procedure more or less according to the 

 number of operations avoided. 



Given any number of equal groups, we recognize that by arrang- 

 ing them with relation to one another as above, we are able to carry out 

 upon all of them each and every operation involving arrangement that 

 we are able to carry out upon one of them. It is therefore sufficient 

 to determine the characteristics of arrangement of any one of these 

 groups in order to know those of all the others. This is a most im- 

 portant proposition which is constantly applied for manifold purposes. 

 Thus talking, writing and reading are founded upon the coordinating of 

 thoughts to sounds and signs; and by arranging the signs in accord- 

 ance with our thoughts we cause our hearers or our readers to think the 

 same thoughts in the same sequence. We manipulate many formulae 

 in a similar manner (especially in the simpler sciences), applying the 

 results to phenomena, instead of dealing with the phenomena them- 

 selves; and we are able to deduce some properties of the latter without 

 being compelled to work with the phenomena themselves. The force of 

 this procedure is most striking in astronomy, where, by manipulating 

 certain formulas which have been applied to certain celestial bodies, we 

 are able to predict their future positions with a great degree of accuracy. 



From the science of order we pass to the science of numbers or 

 arithmetic by the systematic development of an operation that has 

 just been indicated. We are able to arrange any given number of 

 quantities in such a manner that the richer always succeeds the poorer. 

 The system obtained in this fashion is, however, quite accidental as 

 regards the number and richness of its members. Obviously we can 

 only obtain an orderly structure of all possible groups by starting 

 with a group having but one member, i. e., a simple thing, and form- 

 ing new members of the series from old ones by adding a single mem- 

 ber. By this process we at once obtain the different groups arranged 



2 Equal groups can not be distinguished here ; and represent merely a single 

 quantity. 



