1. According to Prof. Skeat (_Etymological Dictionary_) the earliest known meaning is that of _furious_ action, as in a charge of cavalry. The etymology, he considers, is connected with the Teutonic word _rand_ (brim), and implies the furious and irregular action of a river full to the brim. 2. See the problem paper of Jan. 18, 1854, in the Cambridge Mathematical Tripos. 3. As, according to Mr H. Godfray, the majority of the candidates _did_ assume when the problem was once proposed in an examination. See the _Educational Times_ (Reprint, Vol. VII. p. 99.) 4. Vide p. 68. 5. It would of course be more complete to take _ten_ alternatives of direction, and thus to omit none of the digits; but this is much more troublesome in practice than to confine ourselves to eight. 6. Any more than we picture the shape of an equiangular spiral _at the centre_. CHAPTER VI.[1] _THE SUBJECTIVE SIDE OF PROBABILITY. MEASUREMENT OF BELIEF._ 1. Originally written in somewhat of a spirit of protest against what seemed to me the prevalent disposition to follow De Morgan in taking too subjective a view of the science. In reading it through now I cannot find any single sentence to which I could take distinct objection, though I must admit that if I were writing it entirely afresh I should endeavour to express myself with less emphasis, and I have made alterations in that direction. The reader who wishes to see a view not substantially very different from mine, but expressed with a somewhat opposite emphasis, can refer to Mr F. Y. Edgeworth's article on "The Philosophy of Chance" (_Mind_, Vol. IX.) 1. Having now obtained a clear conception of a certain kind of series, the next enquiry is, What is to be done with this series? How is it to be employed as a means of making inferences? The general step that we are now about to take might be described as one from the objective to the subjective, from the things themselves to the state of our minds in contemplating them. The reader should observe that a substitution has, in a great number of cases, already been made as a first stage towards bringing the things into a shape fit for calculation. This substitution, as described in former chapters, is, in a measure, a process of _idealization_. The series we actually meet with are apt to show a changeable type, and the individuals of them will sometimes transgress their licensed irregularity. Hence they have to be pruned a little into shape, as natural objects almost always have before they are capable of being accurately reasoned about. The form in which the series emerges is that of a series with a fixed type. This imaginary or ideal series is the basis of our calculation.