The subjects of mean value and expectation, which have always played a central role in the theory of probabilities, have taken on additional importance in recent years, owing to the idea of dispersion, and its application to statistical series. For that reason they have been given a good deal of prominence. Per contra, geometrical probability, which is little more than a plaything, and the probability of causes, which rests on very shaky foundations, are treated briefly. Yet they should not be omitted entirely, for the former is related to statistical mechanics, and the latter gives the only answers we have to certain questions which recur insistently.

The most important part of the theory is that which deals with the distribution of errors of observation. The funda mental question here is what to do with the exponential law of Gauss. I have tried to make it as plausible as I could by basing it, on very broad assumptions, even though this adds somewhat to the length of the deduction. I have, however, given the principles of combining observations as far as possible independently of the Gaussian law. The study of errors in two dimensions, which formerly interested few but students of artillery practice, has taken on a new importance through its relation to statistical correlation.

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