608 THE AMERICAN JOURNAL OF SOCIOLOGY
ard, is limited; the magnitude sought is equally so. In the same way, geometry is the science of the measurement of extent. For example, a straight line is defined as the shortest distance between two points. This definition indicates a constant, neces- sary relationship between two points which serve as limits. There are, then, mathematical limits in the estimate of all natu- ral phenomena. As we are unable to form a correct idea of infinite space, or of infinite numerical quantities, without at the same time using finite measures and quantities, all observable phenomena are, from the mathematical point of view, condi- tioned and limited. All may be reduced to the calculation of probabilities. This amounts to saying that they have an average which constitutes their normal equilibrium, an equilibrium whose variations are limited in the sense that they decrease in propor- tion as they depart from this average. From the mathematical point of view, and as far as phenomena can be exactly calculated and measured, Fourier, Laplace, and Quetelet were right : all things, including social phenomena, can be reduced to averages.
When we apply the calculation of probabilities and the theory of averages to social facts, in order to reveal the social order, we must not, however, lose sight of the fact that these probabilities, these averages, can be applied only to homogeneous things, and not to heterogeneous phenomena. Now, the social structure is formed of unlike elements. In sociology, therefore, the theory of averages and of probabilities can be applied only to the cate- gories of similar facts. It is inapplicable to the organic struc- ture of either special societies or society as a whole ; that is to say, to combinations of heterogeneous elements in organized wholes. Except from the point of view of number and exten- sion, these organic combinations elude statistics and every mathe- matical theory.
Nevertheless, by co-ordinating the averages of each class of social phenomena in definite limits of space and time, we may obtain statistics summing up, at a fixed time and for a given territory, the elementary statics of a particular civilization. However, this statics, based upon statistics, will be insufficient to explain structure. Structure involves the higher idea of a
