INTRODUCTION TO SOCIOLOGY 607
divided by twelve that we shall see it rise again the eleventh day. If the periodical return of this phenomenon has been observed one thousand times, the probability will be represented by -frnhz-- It is apparent, therefore, that after centuries of repeated performance the probability must be equivalent to a certainty and that the contrary event would be considered miraculous.
The difficulty increases and becomes enormous when we have to deal with complex facts, and especially with social facts. By themselves, the seven classes of social phenomena, considered as distinct groups without subdivisions, give rise to 127 combina- tions of classes. It is apparent, therefore, that the relationships which may be produced by the phenomena included in these classes .are so nearly innumerable that they cannot be exactly calculated, although we can say that the possible variations of these relations are not unlimited. Nevertheless, from the mathematical point of view, whatever may be the character of the observations, the results observed do not fall indiscrimi- nately on the two sides of an average, but in determined order, which is that of the scale of possibilities. The nature of this scale is determined by the number of the observations. The more numerous the latter, the more nearly they approach the average, and the relatively fewer variations or errors will they contain. If, for example, I measure the height of a mountain, and if my first estimates vary between nine hundred and eleven hundred meters, the more frequently I repeat my measurements the larger will be the number of the estimates that border upon the average of one thousand meters and the less numerous will be those which deviate from it.
In arithmetic as well as in geometry the field of observation and of variation is limited. One can form an idea of a magni- tude only by comparing it with another known, limited magni- tude, unity. Numbers are, then, the relationships between this known unit and a magnitude to be known. " Every number,' said Newton, "is the relationship of a magnitude to unity." Arithmetic is, then, the science of the measurement of magni- tudes. Measure means limit ; unity, which serves as the stand-
