different groups in a series by beginning with the poorest and choosing 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.[1]
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 arranging 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 important 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 accordance with our thoughts we cause our hearers or our readers to think the same thoughts in the same sequence. We manipulate many formulæ in a similar manner (especially in the simpler sciences), applying the results to phenomena, instead of dealing with the phenomena themselves; 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 forming new members of the series from old ones by adding a single member. By this process we at once obtain the different groups arranged
- ↑ Equal groups can not be distinguished here; and represent merely a single quantity.
